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Jorge Pullin: Okay. So our speaker today is, I'm on Lagosite, who will speak about entanglement on holography and spin networks.

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Simon L: Thank you for the opportunity to speak here.

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Simon L: So today, I would like to give a sort of a big picture overview of some aspects of calculating entanglement entropy.

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Simon L: particularly in gauge theories, and that is gauge views and spin networks, and in the end I would like to apply this to a holographic reconstruction

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Simon L: in the context of reconstructing intertwiners from boundaries. Links in spin networks

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Simon L: at

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Simon L: this is partly based on work with Daniela Oditi and Diogenia called Francesca.

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Simon L: So to begin this talk, I would like to give a

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Simon L: sort of general overview of algebraic definitions and methods in understanding different contributions of

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Simon L: entropy that can it go into entropy calculations in particular, choices of sub algebra, and how they affect what you're going to calculate.

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Simon L: Then I would like to specify the discussion on spin networks, the types of algebra that you have there.

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Simon L: I would like to into exemplify how you can

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Simon L: see some parts of entanglement in spin networks as being inherent to the graph structure itself.

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Simon L: and then also talk about the an entropy of a generic sub region in a generic state.

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Simon L: and how to calculate this using techniques from random tensor networks.

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Simon L: Then in the end I would like to apply the results from that section to see whether there's a sense of a notion of

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Simon L: holographic reconstruction for the fixed spin spin network case, and then also hopefully discuss how the how that gets generalized to the

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Simon L: a non-fixed spin case.

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Simon L: which is more recent work.

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Simon L: So to begin I would like to record the very basic notion of entanglement that we all know.

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Simon L: which is that if you have a tensor product structure in a in a Hilbert space, then anything that is not a tensor product, state or separable state will be known as a tangled

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Simon L: and attachment. Entropy is known as a measure that tells you how much a State deviates from such a separable state.

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Simon L: This is all fine if you have a tensor product structure. But if your subsystems are and do not admit such a decomposition.

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Jerzy Lewandowski: Should I see anything?

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Simon L: Yes.

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Jerzy Lewandowski: Except for the title, Boundary conditions, Algebra.

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Simon L: Yes, this is still fine. This is still fine. Yes, don't worry.

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Jerzy Lewandowski: Oh, okay. Yes.

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Simon L: This is just a preamble

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Simon L: to that section.

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Simon L: So if this is, if you do no longer have a tensor product, search of Hilbert spaces. You rather need a notion of subsystems in terms of algebra.

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Simon L: and the point of view that we will take here is that we describe a system in terms of sub in terms of algebras with units.

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Simon L: So we will see those algebra as being the accessible operations on a given system

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Simon L: in particular. If you have a subsystem, then it will be described in terms of a sub algebra with unit.

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Simon L: So the operations that you can perform on a subsystem

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Simon L: for entanglement. Then we will need a notion of complementary subsystems which, in this algebraic context, just means that you have an algebra Al and an algebra AR,

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Simon L: with a letter of which is known as the as the commenting of the first.

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Simon L: and of course vice versa.

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Simon L: If you have that the algebra generated by these 2 algebra sub algebra's can be written as a tensor product. Then this setting gives you back the original naive, simple Hibbert Space definition.

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Simon L: and that you represent it. And then you have a tensor product structure on Hilbert spaces, and you can see complementary subsystems as just tensor product factors.

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Simon L: However, in general those 2 complimentary subsystems will have a non-trivial intersection

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Simon L: of operators which commute with everything, and so which are known as a center.

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Simon L: This in particular also means that the full algebra, that it cannot been written as a tensor product.

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Simon L: What this means in terms of representations is that we can diagonalize all of the operators in the center.

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Simon L: And so we can write the full representation as being of as consisting of block diagonal matrices.

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Simon L: Each of the sectors in the decomposition, for example, on this layer level of filbert spaces

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Simon L: will then have a tensor product decomposition.

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Simon L: but will be a direct sum over all of the labels, all of the eigenvalues of the central operators.

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Simon L: The the further furthermore, this is important because density of matrices in this subsystem, so the data that consists just of the 2 regions

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Simon L: must commute with the center, and so they also feature, direct some decomposition, a block. Diagonal decomposition.

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Simon L: which makes us makes it possible to interpret them as classical mixtures of states which live in each of the blocks separately.

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Simon L: Then

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Simon L: there's a certain type of subsystem which is very important in the case of lattice, gauge theories and gauge theories more general or field theories even.

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Simon L: which is that of a finite region. For example, a spatial one, on which you impose some boundary conditions.

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Simon L: Generically speaking, you can implement these boundary conditions by choosing a certain sub algebra of boundary operators. So bound operators which are supported on the boundary of the region.

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Simon L: For example, in the case of a scalar field theory, you have on this corner

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Simon L: operators given by the fields and their contributor.

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Simon L: and if you remove from the algebra

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Simon L: the momentum operators, you only have the commuting field operators left.

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Simon L: and then, because they all commute. They form a center, and you can diagonalize them and give them fixed values. This is what you would call Dirichlet boundary conditions.

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Simon L: So overall you can connect choices of boundary sub algebra with choices of boundary conditions on the system.

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Simon L: In turn. Then.

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Simon L: as we just see

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Simon L: that the this implies that the Hilbert space has a direct Sunday composition.

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Simon L: and that that similarly density matrices do as well.

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Simon L: Now for lattice gauge degrees. There's so there's an additional complication in that there is a preferred gauge, invariant subalgebra of observables, of operations

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Simon L: which is specified by having a center given by the Gauss operators.

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Simon L: This gauge, and whether one calculates the entropy on the gauge, invariant algebra, or the full algebra will be a point of discussion

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Simon L: later on.

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Simon L: Now there are different choices, for what what can mean by a boundary in this case? In the lattice? One can, for example, mean that there is a given boundary, and there are some links piercing out of it.

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Simon L: And in that case one will have about and a corner algebra generated by operators on those radio links

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Simon L: on the other end one can mean a strict boundary.

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Simon L: and then only tangential links will be in the corners of algebra

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Simon L: hint.

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Simon L: depending on what you choose for them for what you mean by a boundary, you will have different algebras, and therefore also different

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Simon L: decompositions of the density matrix

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Simon L: in particular. If you choose the first type. So radio links.

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Simon L: and you choose the boundary sub algebra generated only by the electric fields.

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Simon L: Then you have a center which is generated by the Casimirs

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Simon L: of the of each of the links.

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Simon L: This is what one would call electric boundary conditions in the latter sketch theory case.

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Simon L: If one works with a non Abelian lettuce gauge theory.

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Simon L: Then there is the further requirement on gauge invariant states

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Simon L: that they are constant across the representation that is given by each, by the by, the boundary, Casimir.

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Simon L: What this means in turn for the density matrices in each block

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Simon L: is that they further decompose into a data invariant piece which I've denoted here with a bar

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Simon L: and a piece which is a flat state. So in completely invariant state

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Simon L: on the representation space itself, the finite dimensional one that's labeled by the boundary spin values. S.

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Simon L: So in mind you in this case here the like, the sector label E, here is the set of Casimir values for the entire boundary of the graph.

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Hal Haggard: Simon, can we briefly interrupt? There's a question.

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Simon L: Yes, I see. Lauren has a question. Yes.

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Laurent Freidel: Hi, Simon, just maybe a little bit of precision, this this about this magnetic boundary condition. So if I put it randomly, I would always kind of cut it in the middle of a link, whereas magnetic cut it relies being extremely precise where I cut.

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Simon L: Yes.

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Laurent Freidel: Tangentially in the app, so any little perturbation will destroy that

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Laurent Freidel: that magnetic boundary condition. So in that sense, I would imagine that the first one is the one that survives the continuum limit, but the second one.

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Simon L: That's a very good question, I think, in the typical. Let us gate theory literature. People would also consider the second type in that they just have sharp boundaries

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Simon L: and also boundary holonomies.

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Simon L: For example, in the

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Simon L: in the paper

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Simon L: up here

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Simon L: you will find precisely that they discuss the led the second of the 2.

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Laurent Freidel: Yeah, I know Cassini usually. But what I'm saying is this one is doesn't have any any.

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Laurent Freidel: It doesn't survive the continuum, maybe because any little perturbation of your cut will destroy that boundary condition, whereas the other one is kind of stable, so.

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Simon L: Move it.

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Laurent Freidel: At.

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Simon L: I I I understand what you mean. Yes, yes.

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Laurent Freidel: Sort of ladies artifacts that.

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Laurent Freidel: And that's that's why. Usually we use the electric one. Because.

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Simon L: Yes, yes, I understand. Yes.

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Laurent Freidel: No, you could. Yeah, you you could use the Teta term to rotate one into the other. But anyway, you you're good.

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Laurent Freidel: Thanks.

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Simon L: Now. With this decomposition of the density matrix, we can plug this into the for Neumann entropy directly

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Simon L: and get 3 different terms in the in the

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Simon L: Get 3 different terms.

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Simon L: The first of these we will call the distillable entanglement, and is given as sort of an average in the classical distribute in a classical mixture distribution

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Simon L: of the gauge, invariant data of the State.

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Simon L: The second piece is a Shannon entropy of the mixture weights themselves.

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Simon L: and the last piece is only present for non Abelian theories, and is associated to edge mode contributions directly.

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Simon L: Of course, we can do this as well, not just with for Norman entropy, but also with rainy entropy

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Simon L: and other types of entropies. If you wish.

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Simon L: Now, with these 3 pieces we can combine them in several ways to have different interpretations.

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Simon L: The first is the this syllable entanglement itself.

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Simon L: This is known to not have contributions from edge modes.

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Simon L: but instead has an operationally an operational interpretation

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Simon L: in terms of the maximum number of Epr pairs, you can extract from the State

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Simon L: by transferring them using an algorithm doing, using a protot distillation protocol into an external cubic reservoir.

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Simon L: There's another content combination which is known as the gauge invariant entropy which further includes the Shannon entropy.

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Simon L: This

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Simon L: you can also see as just all of the terms minus the non obedient piece.

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Simon L: and is the entropy calculated when you consider only the gauge, invariant degrees of freedom, the gauge, invariant algebra.

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Simon L: Lastly, if you use the full expression which is known as the full entropy.

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Simon L: this will include all echoes. It will include the Kabat contact term.

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Simon L: and is basically just what you would naively calculate.

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Simon L: What should become clear from this is that any prescription, for when you calculate an entropy.

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Simon L: must also come attached with the note of what it's actually calculating.

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Simon L: The typical methods which cut, which are the replica trick via path integrals or the extended Hibs based procedure of Donelli Wall and Fridle, calculate the full entropy.

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Simon L: whereas, if you use the replica trick and also fix gauge, fix some boundary conditions on the entangling surface.

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Simon L: It will give you instead the gauge, invariant entropy.

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Simon L: So in principle you have 3 types, and there are different, and there are ways to calculate either of them.

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Simon L: and whether you prefer one over the other just depends on what question you want to ask.

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Simon L: Now let us specify the discussion to spin networks. I don't think I need to go over the details too much in this, in this kind of audience

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Simon L: what we will need is the sit with network vertex, Hilbert space of a single disconnected vertex.

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Simon L: and the one of the of the whole graph

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Simon L: in particular. What we will need is that the single vertex hip space decomposes into sectors labeled by the spins.

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Simon L: and each of these sectors decomposes again into a tensor product of link spaces and at an intertwiner space

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Simon L: for the graph boundary. We choose the standard choice here of just having a bunch of one valid vertices which are connected through boundary links to the rest of the graph

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Simon L: for the corn algebra. Then we have holonomy, flux, algebras on all of the boundary links, and we choose electric boundary conditions, which means we only keep the fluxes inside the algebra.

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Simon L: Then the corner center

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Simon L: is generated again by the by the casmers on each of the links.

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Simon L: and we will have a super a super selection sector decomposition of the Hilbert Space.

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Simon L: Now there's already some entanglement which is included directly in spin network states on a given graph.

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Simon L: This is a

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Simon L: this is most easily exemplified on a single link which we can decompose into semi-links on the 2 vertices that it starts and ends on

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Simon L: by taking the full, and by taking the space of a link and writing it as the space of the the tool semi links, but applying a projector on it.

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Simon L: We can then realize these link states the glute link states as gauge invariant states on 2 semi-links. The gauge invariance in question is just write and write translations onto the 2 elements of the 2 arguments of the wave function

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Simon L: on the level of the basis states. Then we can realize this sort of gluing operation of the 2 sides

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Simon L: by taking the basis states on each of the vertices and vertices.

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Simon L: and and applying to them, projecting them onto a maximally entangled link state such as this one here.

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Simon L: What this means is that we can glue the at least for fixed spins.

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Simon L: Vertex! Hilbert spaces together to form a glute

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Simon L: spin network here. But space on a given graph.

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Simon L: using a projection operator

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Simon L: for fixed spins. This works all the way, and this goes all the way through.

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Simon L: and we can write a spin. The spin network basis functions on a given graph

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Simon L: as projected, entangled a pair of states.

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Simon L: This is a type of tensor network state which is very common, and which simply consists of a tensor product state of some vertex states

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Simon L: which are then projected onto maximillent pair states.

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Simon L: this is, not so surprising. Many different types of bases in different contexts have this, and it's very useful in condensed metaphysics.

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Simon L: Because they have a very clean entanglement structure.

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Simon L: This can be seen, for example, that the products that, of course, has no entanglement in terms of the vertex factorization.

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Simon L: but there is entanglement which is introduced on the links

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Simon L: by the projection here.

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Simon L: so we can extend this to a larger class of states, and specify a kind of corner of the Hilbert Space, in which we have a similar structure.

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Simon L: So we would like to have a set of spin network states which still have this peps like structure in which you can talk of

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Simon L: entanglement coming from gluing.

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Simon L: And in order to get this, we take the fixed spin procedure.

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Simon L: but we weigh all of the maximally entangled link states in the sectors

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Simon L: by some arbitrary coefficients and superpose them.

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Simon L: Then we can. We can do the same procedure. We can take arbitrary spin network vertic states with no restriction on the spin labels

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Simon L: and project them onto these new, modified, maximally entangled pair states.

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Simon L: this will give what I would give us what I would call a spin tensor network state.

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Simon L: They are. They also feature very clean entanglement structure.

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Simon L: and we can probably expect something like an area law in general from them as well.

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Simon L: What's maybe important to mention is that the taking the full spin network, vertex civil spaces, and applying this sort of state to them

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Simon L: is only a proper projection. If all of the coefficients are equal to one.

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Simon L: The resulting States, if they are not.

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Simon L: still have a similar interpretation in terms of spin network states. But

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Simon L: it's just that the operation itself is no longer a projection.

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Simon L: Now, before we get to calculating the

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Simon L: reduce the entropy or entanglement entropy between 2 sub regions, I would like to give some definitions, some terminology

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Simon L: of different types of entanglement we can have in spin networks by choosing different types of subsystems.

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Simon L: The first type that would be relevant is that of link entanglement. For that we just choose a subsystem given by a single link, just as we had before.

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Simon L: and then we can split the link into semi links in order to calculate the entanglement between the 2 sides

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Simon L: for intertwine entanglement. It's similar. We can choose 2 vertices and a sub algebra subsystems, which was the operators on intertwining degrees of freedom.

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Simon L: and then, by reducing the system to those into the into that subsystem.

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Simon L: We can then check the entanglement between the 2 intertwiners

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Simon L: intertwina spaces.

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Simon L: Those 2 will be the most important for us, but just for completing, as I also mentioned, that there's a possibility to consider entanglement between boundaries and meetings, as well.

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Laurent Freidel: So sorry. Could you go back and explain a little bit more? Your notation? What do you call E, and ge is the Casimir and E. What is it?

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Simon L: G, in this case.

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Laurent Freidel: So ge. Is the Casimir, the opposite.

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Simon L: Yes.

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Laurent Freidel: Yeah, I'm not sure. And what do you call E.

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Simon L: He is the is just any link.

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Simon L: any link in the graph.

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Laurent Freidel: Oh, it's the edge. Okay.

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Simon L: Yes, it's the edge. Yes.

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Laurent Freidel: Okay.

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Laurent Freidel: and what you call a peps is simply an arbitrary proposition of

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Laurent Freidel: of edge.

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Laurent Freidel: Of representation associated with the edge.

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Simon L: Yes. So the so the so the way the link states here. Yes, those are just arbitrary super positions.

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Laurent Freidel: Are you going to find a way to select the the coefficient.

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Laurent Freidel: Or Gj.

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Simon L: There are some conditions which come from requiring holographic reconstruction. But I'm not aware that you can get anything particularly explicit out of it. No.

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Simon L: I think they need to be sort of

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Simon L: peaked around some, but around some

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Simon L: spin configuration, in order to support.

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Simon L: to support holographic reconstruction.

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Laurent Freidel: Okay. But what about? Okay, you're not going to discuss

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Simon L: I'm not going to get into that today.

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Laurent Freidel: Well. The modular Hamiltonian, for instance, which is the the central aspect of.

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Simon L: I'm not going to get into that today. No.

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Simon L: now.

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Simon L: moving on we will consider

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Simon L: and entanglement between 2 sub regions of a spin network.

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Simon L: We will consider the sub regions to be split in in into by A

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Simon L: by a set of links along a set of links inside the graph.

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Simon L: and we will. This induces a new center that comes along the entangling surface, which again is given by Cassimir's on the edges of the links.

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Simon L: This again induces a decomposition of the density matrix

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Simon L: where now the sum is over, spin values on the. For example, if we reduce to the left-hand side.

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Simon L: then it will be given by the boundary, but by the spins on the boundary edges and along the entangling edges. Here

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Simon L: and again, importantly, we have this contribution here, which is flat and

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Simon L: does, which comes from considering gauge invariant states

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Simon L: that we can again

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Simon L: without even knowing anything about the State, makes some statements about the entropy entropy overall

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Simon L: the in particular, the non Abelian piece will localize on edges that come from the button that sit on the boundary.

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Simon L: and if we

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Simon L: for example, reduce to fixed the fixed spin set case

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Simon L: where the classical mixture weight is a conical delta.

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Simon L: then we have no Shannon piece.

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Simon L: and we have a contribution. From the non Abelian piece.

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Simon L: which is precisely just the logarithms of the dimension

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Simon L: counted over all of the boundary links.

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Simon L: This is actually the leading order, contribution,

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Simon L: in what's and in what's known as the area law behavior.

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Simon L: And it's also clear from comparison with the K with the single link case before

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Simon L: that. This accounts for all link entanglement between the 2 regions.

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Simon L: Yes, Carlos.

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Carlo Rovelli: just. I'm confused. That what is a claim here? That

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Carlo Rovelli: this particular kind of entropy as the structure, or that the

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Carlo Rovelli: There can be entanglement with entropy between the 2 regions, which has nothing to do with this particular entanglement.

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Simon L: The State. The statement is in, for example, in the fixed spin case.

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Simon L: There the entropy consists of a contribution from the distillable part.

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Simon L: and this non Abelian piece G,

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Simon L: which has the form of something which sits on the boundary.

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Carlo Rovelli: Okay, okay? So the following entropy can still have a bulk, bulk.

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Carlo Rovelli: product, features and.

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Simon L: Yeah, through the week.

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Simon L: Yes, in fact, in fact, as as I was, as I would say, with the last line on the slide here.

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Simon L: This kind of raises the suspicion that the bulk bike

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Simon L: contribution is precisely only coming from intertwina entanglement.

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Simon L: and that this corresponds directly to the distillable piece of the entropy.

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Carlo Rovelli: Right? So the reason I'm asking this is because this is a sort of non holographic part of it

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Carlo Rovelli: for Norman. Would you agree with that?

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Simon L: It depends on what you mean by holographic, but definitely.

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Carlo Rovelli: But that's that's what. Okay? Yeah.

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Simon L: It's the only part that that quite that is like the leading order.

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Simon L: area law term.

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Carlo Rovelli: No, I'm I'm lost. Is that.

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Simon L: The other way around. Yes, you spoke about.

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Carlo Rovelli: That I intend to.

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Simon L: Why not entertainment? Yes.

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Simon L: that's the non-allographic piece. Yes.

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Carlo Rovelli: Is a nonographic piece.

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Simon L: Yes, but this here would be the area law piece.

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Carlo Rovelli: Right. But somehow

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Carlo Rovelli: the reason I'm

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Carlo Rovelli: posing this question and I least get confused is that the the idea of holography?

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Carlo Rovelli: Is often

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Carlo Rovelli: associated to the idea to the sort of hypothesis

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Carlo Rovelli: that the only possible entanglement between

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Carlo Rovelli: the 2 sides is of this area kind, and that's not the case here. Right.

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Simon L: Oh, I mean it. It is. It is. It is true that it's not in general the case, but we will find later on that only when the intertwine entanglement is very small, do you actually get to reconstruct stuff

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Simon L: so holographic reconstruction really does correspond in a way to asking that the intertwine and intertwine entanglement is relatively small.

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Carlo Rovelli: Right. But physically, why does it have to be small.

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Simon L: We will get to it later. But essentially, it's just a quantum information reason.

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Simon L: It's just that the boundary that the boundary space from which you reconstruct the bulk. Data

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Simon L: is, relatively speaking.

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Carlo Rovelli: And why should you

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Carlo Rovelli: reconstructed bulk data from the boundary.

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Simon L: Isn't that what holography is about.

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Carlo Rovelli: That's exactly the point. So the question.

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Simon L: I mean, we will get to this later on. I will give a concrete notion of what I mean by photography, and then I think it will

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Simon L: be easier to discuss.

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Carlo Rovelli: Okay, okay, so, but let's postpone this to the to the the end. Thank you.

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Simon L: Yes.

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FAU: Hello, I I want some clarification of Number 13, equation number 13. So you say, area law. But but this is not area, right. It's logarithmic of area.

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Simon L: Oh, okay, it's area law in the sense of the graph area.

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Simon L: So the number of boundary links.

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FAU: Yeah, so it's a different sense of area.

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Simon L: Well, it's it's a different sense. Yes, in that we are working with an abstract graph. And there is a priori not really

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FAU: Not really, if you'll measure the area, because.

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Simon L: It's not geometric area. No, definitely, not. But this is precisely the kind of sense that you would see. People claim area law behavior in tensor network stories.

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FAU: Thank you.

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Simon L: Okay.

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Simon L: Yes, moving on.

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Simon L: Now there is. Now let's look at the written. Let's look at a case of

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Simon L: more typical spin network state. Let's first look at the fixed spin cases, but then also expand to the spin tensor network case.

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Simon L: So what we would like to what we could do now is, we could calculate the the entanglement entropy between 2 fixed regions

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Simon L: for an, for a typical, for a random state.

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Simon L: So how we will do this is, we will look at spin tensor networks where we fix the graph pattern we fix the States on which we project, but we choose the vertex States randomly from the distribution.

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Simon L: The way we do this is, we first fix a reference state, and then we rotate those States with some randomly chosen unitary on the vertex civil spaces.

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Simon L: The Unitarians themselves will be chosen by the hard measure on the, on the unitary group.

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Simon L: So just already giving you the caveat.

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Simon L: This requires that the Hilbert space in question are finite, dimensional. You implement this by introducing cutoffs.

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Simon L: Now.

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Simon L: if you do all of this, and you choose your cutoffs, so that the representation labels, the spins are relatively high.

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Simon L: Then, by using results from the literature in random tensor networks, we know that the distribution localizes on its average, so fluctuations can be neglected, and we can rewrite the, for example, second Revenue entropy, which is relatively easy to calculate

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Simon L: in terms of a quotient of some numbers. Do you want it to be 0?

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Simon L: Those numbers can be calculated using a mapping onto a set of random icing models. Okay, not random, but

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Simon L: they will become random. So the idea is that we first use the replica trick to rewrite the product in the numerator of the Andrani entropy.

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Simon L: and on 2 copies on 2 replica copies and an additional swap operator.

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Simon L: Then by linearity, we can pull in the unitary average

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Simon L: and calculate the average squared density matrix under the unitary average.

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Simon L: and express that as some swap operators on vertices

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Simon L: of the graph

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Simon L: with what is.

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Jerzy Lewandowski: What is replica? Trick.

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Simon L: Of course.

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Jerzy Lewandowski: I know that this is a famous term, but maybe I can learn now.

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Simon L: okay, I wish I had written this down now. But the basic idea is that

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Simon L: if you have a trace over a product of A and B.

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Simon L: Then you can rewrite this on the say on 2 copies of the same Hilbert Space

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Simon L: as a trace of a tensor B

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Simon L: times a swap operator.

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Simon L: So something which just swaps 2 copies of the Hilbert Space.

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Simon L: I would try. I would try it out on paper. Just to

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Simon L: just to get the idea across. But that's the idea. There you can

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Simon L: products on multiple copies of hybrid spaces, using these swap operators.

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Jerzy Lewandowski: Day. Thank you.

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Simon L: Yeah. So the the tensor product here can then be expanded into a sum.

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Simon L: And in the sum we will have a different number of swap operators

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Simon L: on on some different sets of vertices that we can label the sets of where the swap operators live by some icing spins that point up or down.

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Simon L: This way we can convert

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Simon L: the the whole expansion of the object c. One and Z. 0

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Simon L: into partition sums of an icing model

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Simon L: in the icing model. The couplings are determined by

349
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Simon L: by the spins on the given graph. So here we're still in the fixed spin setting

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Simon L: and the temperature of the icing model is given by the mean spin across the graph

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Simon L: in the high spin regime, then.

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Simon L: which corresponds to the low temperature regime of the icing model.

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Simon L: We can then, just take the ground state of the IC model as the main contribution. And this will be coming from something which is like something like a domain work close to the entangling surface.

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Simon L: What we then get is that for fixed spins.

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Simon L: the entropy of a entanglement entropy. Okay, there are any 2 entropy of the reduced state

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Simon L: has 2 main contributions.

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Simon L: the first coming from the edge piece, which is the area law piece.

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Simon L: and another piece coming from the reduced Intertwina state. So this is the true bulk entropy.

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Laurent Freidel: Could could you? Oh, you go!

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Carlo Rovelli: This is for those particular States that you get averaging

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Carlo Rovelli: sort of the sort of locally right. This is not a general statement about

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Carlo Rovelli: average states of the full hipaa space. Is that correct? So the class of state? I want to say.

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Simon L: Just said.

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Carlo Rovelli: If you you're integrating over this use, but it's use a local sort of screen by spin, so to say.

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Simon L: But there are several caveats in additions to that.

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Carlo Rovelli: So the fool

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Carlo Rovelli: unitary, sir, that would be completely destroyed, because you have all sort of.

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Simon L: Yes and no. Yes, in that a prior you could expect so. But no, because when you actually do it, you find the same kind of result.

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Carlo Rovelli: Well, I'm not sure.

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Simon L: I mean the the calculation is out there in particular. You can do it without choosing this particular class of states, and you with doing a different kind of

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Simon L: argument. Eugenio Bianchi has a paper on this in which he just looks at a generic subsystem state. And basically it's the same result.

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Simon L: It's just.

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Eugenio Bianchi: You get some volume low, right? You get some volume.

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Carlo Rovelli: Really

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Carlo Rovelli: you don't tell Gary. Hello! That's the point

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Carlo Rovelli: in Eugenio's. Calculation is very clear. If you go that, the second term would be much larger in a sense.

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Simon L: Yes, yes, so the the the calculation here does not make any statement about the sat relative size of the terms.

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Carlo Rovelli: Okay, so

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Carlo Rovelli: there's nothing

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Carlo Rovelli: Ariello generic

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Carlo Rovelli: so far. Right.

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Simon L: In principle, no.

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Carlo Rovelli: Okay.

384
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Carlo Rovelli: It's a.

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Simon L: For instance, it's.

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Carlo Rovelli: I mean somehow to squeeze in a a subtle, graphic apology somewhere. You get to some holographic conclusion.

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Carlo Rovelli: That's what I would insist if you don't sort of associately bring brings in somehow graphic hypothesis. You don't get an autographic.

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Carlo Rovelli: a

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Carlo Rovelli: definitely.

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Simon L: I did not. No, okay, good

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Simon L: Lahore.

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Laurent Freidel: Yeah, it was just a question about what is your notation? Zen? One slash 0. What does that mean?

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Simon L: This is just that it's an or it's sort of it refers to what and to what into z one or z 0.

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Laurent Freidel: Where z is the swapping or no swapping.

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Simon L: Z is the is these quantities that.

396
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Laurent Freidel: Appear, when you, when you.

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Simon L: The tracing and the unitary average.

398
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Simon L: Yes. So in this sort of setting.

399
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Simon L: what you can extract is that you achieve maximal entropy for the reduced state.

400
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Simon L: If you have, for example, small intertwana dimensions.

401
00:36:51.790 --> 00:36:55.799
Simon L: So because the boundary in this case is the only

402
00:36:56.300 --> 00:37:01.560
Simon L: so the the maximum entropy you. The maximum bound on the entropy you can have in this case

403
00:37:01.760 --> 00:37:03.779
Simon L: is already

404
00:37:05.610 --> 00:37:07.630
Simon L: saturated. Wait a second.

405
00:37:12.200 --> 00:37:16.849
Simon L: Okay. I had an argument in my, but this is not the correct one for this case.

406
00:37:20.100 --> 00:37:26.010
Simon L: okay, I will. I will keep this this remark here, for later on, where it's actually relevant. Actually.

407
00:37:27.880 --> 00:37:32.289
Simon L: now. If we extend the calculation now to

408
00:37:32.830 --> 00:37:35.260
Simon L: the spin tensor networks of before.

409
00:37:35.550 --> 00:37:40.079
Simon L: So where we do not have a restriction on spins anymore

410
00:37:40.801 --> 00:37:50.449
Simon L: the calculation is still doable, but it requires that the partition sums of the icons now have an additional sum over spin labels throughout the graph.

411
00:37:50.640 --> 00:37:53.969
Simon L: one for each copy in the replica. So for

412
00:37:54.020 --> 00:37:56.919
Simon L: the second venue entropy, there is now 2 sums.

413
00:37:57.910 --> 00:38:05.199
Simon L: There's some additional factors in the sum which depend only on the dimensionalities of the representations

414
00:38:05.520 --> 00:38:10.660
Simon L: which now at the point on of the classical mixture distribution.

415
00:38:11.850 --> 00:38:20.189
Simon L: Then there's also, furthermore, some restrictions on the combinations of

416
00:38:20.260 --> 00:38:21.949
Simon L: the spin sectors

417
00:38:22.200 --> 00:38:24.510
Simon L: and the Isaac configurations.

418
00:38:24.780 --> 00:38:30.719
Simon L: So some configuration, some combinations, are forbidden just by orthogonality of some spaces.

419
00:38:31.480 --> 00:38:40.020
Simon L: and this leads to the inclusion of a of an additional Boolean factor in the definition of the of the icing models. Partition sums.

420
00:38:41.580 --> 00:38:49.580
Simon L: What's important is that in this sitting there is no longer a universal notion of temperature like there was in the fixed spin case.

421
00:38:49.930 --> 00:39:00.620
Simon L: The reason is simply that before we had a mean spin across all of the across the whole graph. But this is no longer the case. When we have multiple different assignments.

422
00:39:01.630 --> 00:39:10.130
Simon L: we can still, however, cut and go ahead and calculate stuff by performing a cumulant expansion of this expression here.

423
00:39:11.828 --> 00:39:24.379
Simon L: In the distribution given by the by 2 p. And 2 copies of the classical mixture. It's 2 copies here because we're calculating the second Renyi entropy. If it were K copies, then it would be

424
00:39:24.550 --> 00:39:26.429
Simon L: care copies of the distribution.

425
00:39:27.360 --> 00:39:36.250
Simon L: and this overall allows you to still find and to still calculate, approximately speaking, what the entropy is.

426
00:39:37.320 --> 00:39:40.889
Simon L: Now with this technology in mind.

427
00:39:41.300 --> 00:39:48.200
Simon L: I would like to turn now to the notion of holographic reconstruction that I'm interested in.

428
00:39:48.690 --> 00:39:51.190
Simon L: and that this has been applied to in the past.

429
00:39:51.990 --> 00:40:03.620
Simon L: So for the simpler case we will just work with fixed spins. So holographic reconstruction of intertwina data from the boundary links.

430
00:40:04.520 --> 00:40:11.760
Simon L: The way that we that this is supposed to work is that given a state of the full graph. Again fixed spins.

431
00:40:12.300 --> 00:40:14.169
Simon L: We induce a mapping

432
00:40:14.270 --> 00:40:17.209
Simon L: from the Hilbert space of intertwiners

433
00:40:17.240 --> 00:40:20.379
Simon L: to the Hilbert space of boundaries and meetings.

434
00:40:20.770 --> 00:40:23.019
Simon L: and require that this is an isometry.

435
00:40:23.520 --> 00:40:27.639
Simon L: The map itself is just given by taking a partial scalar product

436
00:40:27.810 --> 00:40:39.210
Simon L: with the intertwina state. So this is a state of the full graph. This is the intertwinus, and if we contract all of the indices here, what's left is a state that sits only on the boundary links.

437
00:40:39.670 --> 00:40:44.530
Carlo Rovelli: But you're restricting of all the States side those for which this is possible.

438
00:40:45.938 --> 00:40:47.880
Simon L: What do you mean? Restricting?

439
00:40:48.460 --> 00:40:50.530
Simon L: I mean at this point it's just a definition.

440
00:40:52.731 --> 00:40:57.950
Carlo Rovelli: Isometry. Oh, sorry isometric doesn't. Isometric means only that the.

441
00:40:59.710 --> 00:41:02.090
Simon L: Isometric as a map between Edward spaces.

442
00:41:02.780 --> 00:41:07.059
Carlo Rovelli: Okay? So it's does not mean that they are.

443
00:41:07.250 --> 00:41:08.709
Carlo Rovelli: This is something virtual.

444
00:41:10.240 --> 00:41:11.305
Simon L: Invertible

445
00:41:13.980 --> 00:41:15.560
Simon L: depends on dimensions.

446
00:41:15.790 --> 00:41:16.840
Simon L: I mean

447
00:41:18.200 --> 00:41:25.569
Simon L: in principle, you would like for it to be invertible. But that's not a requirement, a priority.

448
00:41:27.260 --> 00:41:32.540
Simon L: Of course, what this means is at least that the dimension here is bigger than the one here.

449
00:41:34.320 --> 00:41:36.250
Simon L: otherwise it cannot be isometric.

450
00:41:38.230 --> 00:41:41.099
Carlo Rovelli: So this, this is crucial. So let me. This is your

451
00:41:48.570 --> 00:41:50.989
Carlo Rovelli: You're going to the bulk to the boundary right.

452
00:41:50.990 --> 00:41:51.640
Simon L: Yes.

453
00:41:52.660 --> 00:41:56.469
Simon L: we're turning. We're mapping a bulk state to a boundary state.

454
00:41:56.790 --> 00:42:00.959
Carlo Rovelli: So you you start from a much larger state to a much smaller state. That's right.

455
00:42:00.960 --> 00:42:02.500
Simon L: In principle.

456
00:42:02.930 --> 00:42:06.619
Simon L: If you, the scaling between the 2 is

457
00:42:06.910 --> 00:42:12.879
Simon L: very different, and the bulk Intertwina stays will be very large. Yes.

458
00:42:13.620 --> 00:42:22.469
Simon L: so what you have to do is either restrict to sets of spin labels where the intertwana spaces are small.

459
00:42:22.980 --> 00:42:27.520
Simon L: or you have to restrict to what's called code subspaces

460
00:42:27.820 --> 00:42:32.270
Simon L: in which you can actually do an isometric reconstruction.

461
00:42:34.920 --> 00:42:40.919
Simon L: So I get what you mean in principle. The intertwined spaces are pretty large, and the boundary spaces are pretty small.

462
00:42:41.080 --> 00:42:41.680
Carlo Rovelli: Yeah.

463
00:42:41.780 --> 00:42:44.770
Carlo Rovelli: So why, this is like strong, ordinary

464
00:42:45.390 --> 00:42:49.410
Carlo Rovelli: shrinking of the spaces be network that one usually consider.

465
00:42:51.600 --> 00:42:52.430
Simon L: So

466
00:42:53.000 --> 00:42:56.069
Simon L: so in principle, what the what this refers to is

467
00:42:57.570 --> 00:42:59.130
Simon L: for certain.

468
00:42:59.670 --> 00:43:04.789
Simon L: So this is a definition. It doesn't refer to how generic this is right.

469
00:43:05.440 --> 00:43:08.520
Carlo Rovelli: That's what I'm saying. That was my question. It's not generic at all.

470
00:43:08.520 --> 00:43:09.869
Simon L: No, absolutely not.

471
00:43:10.380 --> 00:43:11.080
Carlo Rovelli: Okay.

472
00:43:11.550 --> 00:43:13.040
Simon L: It's absolutely not generic.

473
00:43:13.770 --> 00:43:17.910
Carlo Rovelli: I somehow want to avoid the impression that there is anything natural in olography.

474
00:43:18.300 --> 00:43:21.419
Simon L: I mean, I'm not claiming anything of this vote right?

475
00:43:21.740 --> 00:43:23.500
Simon L: And making a definition.

476
00:43:23.800 --> 00:43:27.750
Carlo Rovelli: Alright. So this is a this is a definition that picks up a very

477
00:43:28.760 --> 00:43:31.679
Carlo Rovelli: small set of cases in all the possibilities.

478
00:43:32.000 --> 00:43:33.260
Carlo Rovelli: An old spinaker. But.

479
00:43:33.260 --> 00:43:36.010
Simon L: Isn't that what you would call non generic.

480
00:43:37.770 --> 00:43:39.799
Carlo Rovelli: Right? So it's highly non generic.

481
00:43:39.800 --> 00:43:43.550
Simon L: Yes, so it captures what you want to. What you want to say.

482
00:43:46.150 --> 00:43:52.250
Carlo Rovelli: No, because the the holography thesis is that there's something generic about being allographic.

483
00:43:53.380 --> 00:43:54.939
Simon L: Where that's a hypothesis.

484
00:43:55.540 --> 00:43:56.970
Carlo Rovelli: Okay. Good.

485
00:43:56.970 --> 00:43:57.550
Simon L: Brett.

486
00:43:58.010 --> 00:43:58.740
Simon L: good.

487
00:43:59.120 --> 00:44:00.100
Simon L: Okay.

488
00:44:00.100 --> 00:44:02.448
Hal Haggard: Simone had something to add also.

489
00:44:02.840 --> 00:44:05.299
Simone SPEZIALE: Your gamma allows loops.

490
00:44:08.640 --> 00:44:10.379
Simon L: in principle. Yes.

491
00:44:10.980 --> 00:44:16.319
Simone SPEZIALE: Right. So there's really an infinite amount of information that is being thrown away with this projection right?

492
00:44:17.538 --> 00:44:20.890
Simon L: In principle. Yeah, I I think so.

493
00:44:20.890 --> 00:44:22.465
Simone SPEZIALE: In practice, not just in.

494
00:44:22.780 --> 00:44:24.769
Simon L: In principle and in practice. Yes.

495
00:44:25.330 --> 00:44:26.450
Simone SPEZIALE: Thank you.

496
00:44:27.010 --> 00:44:35.530
Simon L: Yes, I mean, we would get late to to to to like an interpretation of what this kind of photography lets you recover, and

497
00:44:35.980 --> 00:44:39.049
Simon L: I think on some level, it's quite restrictive.

498
00:44:39.760 --> 00:44:40.790
Simone SPEZIALE: Okay. Thank you.

499
00:44:43.420 --> 00:44:44.065
Simon L: So

500
00:44:44.920 --> 00:44:45.666
Laurent Freidel: Maybe another.

501
00:44:46.040 --> 00:44:46.720
Simon L: Yes.

502
00:44:47.580 --> 00:44:48.229
Laurent Freidel: And I'm

503
00:44:48.370 --> 00:45:08.680
Laurent Freidel: I mean here, maybe to to add to that discussion at some point. I mean, we we want. I think that's maybe where where you're going I don't know, but you know, at some point you want the State side to be a solution of the constraints. Right? So I mean in general, which means the admin constraints and the default constraints, whatever that means in.

504
00:45:08.730 --> 00:45:24.070
Laurent Freidel: you know. And of course, so you know, here I'm assuming. So of course we don't have that that expression, but that that restricts right? So there's a restriction coming coming from the fact that site cannot be an arbitrary state, as you know. Is that what you.

505
00:45:24.070 --> 00:45:29.930
Simon L: Yes, yes, that is true, but it's but it's due to the restriction of the definition.

506
00:45:31.390 --> 00:45:32.810
Simon L: Yeah, more general definition. But yeah.

507
00:45:32.810 --> 00:45:45.330
Laurent Freidel: This business. P. Network in the bark is very, very big, but the the subsets of States which satisfies the constraints like invariance of the support I mean translation, etcetera, is much smaller, right?

508
00:45:45.470 --> 00:45:46.000
Laurent Freidel: I feel.

509
00:45:46.000 --> 00:45:54.269
Simone SPEZIALE: We all agreed, since many years. That is not a subset, because Wilton and constraint is not generating compact orbits. So it's not.

510
00:45:54.270 --> 00:45:56.620
Laurent Freidel: Yeah. But psi, psi is

511
00:45:57.090 --> 00:45:58.769
Simone SPEZIALE: There will have to be some distribution.

512
00:45:58.770 --> 00:46:01.589
Laurent Freidel: The dual sigh, is the dual state here.

513
00:46:01.590 --> 00:46:02.720
Simone SPEZIALE: Right, right, so.

514
00:46:02.720 --> 00:46:08.279
Laurent Freidel: That that projection map should be done with, you know, with something that satisfies the constraints, which, of course.

515
00:46:08.280 --> 00:46:14.869
Simone SPEZIALE: Well, it's not an element, is is not a subset of the space. I mean. Of course, you could try to construct it by looking at the duo.

516
00:46:15.770 --> 00:46:19.540
Laurent Freidel: Sure put put a dual in his age, and then you have the same

517
00:46:21.230 --> 00:46:27.750
Simone SPEZIALE: Well, but then we are talking about much, much more general quantities than what he's been talking about, which were L. 2 States.

518
00:46:28.850 --> 00:46:30.019
Laurent Freidel: I don't know. I mean.

519
00:46:30.020 --> 00:46:30.279
Simon L: 'kay

520
00:46:30.540 --> 00:46:31.630
Laurent Freidel: Comments, but.

521
00:46:31.630 --> 00:46:32.180
Simone SPEZIALE: Okay.

522
00:46:32.730 --> 00:46:33.420
Laurent Freidel: I mean so.

523
00:46:33.420 --> 00:46:41.709
Simon L: Perhaps perhaps the the second definition I'm about to give a bit later in terms of algebra will give a bit more food for discussion later on.

524
00:46:42.760 --> 00:46:43.435
Simon L: So

525
00:46:44.430 --> 00:46:49.930
Simon L: in order to check how generic this is, and we would see that it's not so generic.

526
00:46:51.550 --> 00:46:58.350
Simon L: we first want to transfer this state in this this condition of isometry

527
00:46:58.610 --> 00:47:02.560
Simon L: into a statement that we can quantify using some numbers

528
00:47:02.810 --> 00:47:12.000
Simon L: in particular, we can turn the isometry question into a maximum entropy question of the in the reduced intertwin state.

529
00:47:12.820 --> 00:47:16.650
Laurent Freidel: Could you record the definition of being isometric? I don't think you write it. There.

530
00:47:16.891 --> 00:47:18.580
Simon L: The one on the left here, basically.

531
00:47:18.580 --> 00:47:23.778
Laurent Freidel: Yeah, that's what you mean. Okay, is proportion to the identity on on.

532
00:47:24.620 --> 00:47:25.330
Laurent Freidel: Yeah, that's.

533
00:47:25.330 --> 00:47:30.640
Simon L: The prefactor comes from it being being induced from a state. Basically.

534
00:47:32.390 --> 00:47:47.850
Simon L: Now, we can again, verify this this condition on reduced into intertwana states by doing a similar mapping to an icing model calculation. So, using the same set of fixed spin, random tensor network

535
00:47:47.870 --> 00:47:52.330
Simon L: states, we can calculate the entropy in that setting.

536
00:47:52.450 --> 00:47:58.096
Simon L: and in which case the IC model acquires some additional bike.

537
00:47:59.727 --> 00:48:01.940
Simon L: external magnetic fields

538
00:48:01.960 --> 00:48:06.390
Simon L: which are which tell it where the grounds, and which modify the ground, state a little bit.

539
00:48:07.380 --> 00:48:15.119
Simon L: What we find from this is, that if the intertwiners in the given sector, that we that we consider

540
00:48:15.250 --> 00:48:16.479
Simon L: are smaller.

541
00:48:16.890 --> 00:48:26.769
Simon L: So there's a lot of inhomogeneity in the spins, and the dimension of of the intertwined spaces are very small, like one dimensions, 2 dimensions, 3 dimensions. Perhaps.

542
00:48:27.300 --> 00:48:33.290
Simon L: then, it's relatively generic to find holographic and reconstruction behavior.

543
00:48:34.210 --> 00:48:42.239
Simon L: But of course, what this means is. If the intertwina spaces are small. Then there's not really a lot of stuff to reconstruct. In the first place.

544
00:48:43.730 --> 00:48:45.747
Simon L: Now I will.

545
00:48:46.590 --> 00:48:52.779
Simon L: continue this to the generous spin tensor network case where we have spin super positions.

546
00:48:53.650 --> 00:48:58.385
Simon L: in which case we will, we can no longer identify the center.

547
00:48:58.840 --> 00:49:00.670
Simon L: the super selection sectors.

548
00:49:01.310 --> 00:49:05.040
Simon L: and here we no longer have a Hilbert space decomposition

549
00:49:05.380 --> 00:49:12.580
Simon L: between a bike system and a boundary system. So what we can do instead is replace it by a mapping between algebras.

550
00:49:13.320 --> 00:49:18.526
Simon L: The algebra here in question. Would be chosen to be this

551
00:49:19.000 --> 00:49:22.350
Simon L: to be the operators acting on.

552
00:49:22.440 --> 00:49:23.750
Simon L: intertwine us.

553
00:49:24.080 --> 00:49:27.999
Simon L: and the fixed label here is just the boundary spins.

554
00:49:28.910 --> 00:49:32.250
Simon L: so we will still sum over bulk, spins

555
00:49:32.700 --> 00:49:35.100
Simon L: off the intertwine into the spaces

556
00:49:35.190 --> 00:49:38.449
Simon L: and the intertwinos here are really the ones of the full graph.

557
00:49:39.560 --> 00:49:42.300
Simon L: Similarly, on the boundary end of the story.

558
00:49:42.370 --> 00:49:45.879
Simon L: we take all of the but all of the boundaries and meeting spaces.

559
00:49:46.570 --> 00:49:47.803
Simon L: and we

560
00:49:48.480 --> 00:49:55.739
Simon L: and we sum over all of we take a direct sum over all of the operator spaces for all of the boundary spin values.

561
00:49:57.320 --> 00:49:59.820
Simon L: Then, with this choice of

562
00:49:59.940 --> 00:50:02.419
Simon L: bikes and boundary subsystems.

563
00:50:02.540 --> 00:50:08.180
Simon L: We can verify that they are, in fact, complimentary in the sense of the beginning of the talk.

564
00:50:08.960 --> 00:50:13.859
Simon L: and the center is given by the boundary spin, and by the boundary spins.

565
00:50:15.170 --> 00:50:20.229
Simon L: Further, there is there is a notion of extension and partial trace maps.

566
00:50:20.370 --> 00:50:23.680
Simon L: So here is an extension map

567
00:50:23.900 --> 00:50:26.200
Simon L: share the eyes for injection.

568
00:50:26.660 --> 00:50:35.150
Simon L: So this takes an intertwine operator and extends it to the into the boundary links by simply doing it sector-wise.

569
00:50:36.050 --> 00:50:43.490
Simon L: Similarly, on the partial trace site, we take a general operator on both boundary and bulk.

570
00:50:43.520 --> 00:50:48.790
Simon L: and reproduce an operator on the boundary. The output system

571
00:50:48.950 --> 00:50:53.089
Simon L: from it, by taking the partial trace in each sector where we remove the bulk.

572
00:50:54.340 --> 00:50:57.599
Simon L: we can use these partial trays and extension maps

573
00:50:57.720 --> 00:51:00.900
Simon L: to define what's known as a Choi mapping.

574
00:51:00.920 --> 00:51:10.629
Simon L: But in this context. I would like to call it the transport super operator transport, because it transports an operator from the bulk to an operator on the boundary

575
00:51:10.780 --> 00:51:12.530
Simon L: from input to output

576
00:51:12.820 --> 00:51:16.610
Simon L: and super operator because it maps operators to operators.

577
00:51:17.990 --> 00:51:23.240
Simon L: So in particular, what this does is extends the operator to the full system

578
00:51:23.330 --> 00:51:29.940
Simon L: takes a product with something written with a partial partial transpose of the full graph state.

579
00:51:30.210 --> 00:51:35.340
Simon L: and then trace this out so that we get an operator on the boundary system.

580
00:51:36.540 --> 00:51:42.090
Simon L: This is a complicated definition, but for a trivial center in the algebra that we chose.

581
00:51:42.200 --> 00:51:47.249
Simon L: and for pure States this actually reduces to the Hilbert Space formulation. Before

582
00:51:48.550 --> 00:51:54.300
Simon L: now. The more generally kind of definition of what I mean by a holographic state, then.

583
00:51:54.320 --> 00:52:00.418
Simon L: for given fixed choice of sub algebra, in which you can choose the sets of

584
00:52:00.900 --> 00:52:03.019
Simon L: spins you include.

585
00:52:03.390 --> 00:52:06.670
Simon L: If you want, you can choose that and manipulate that by hand

586
00:52:08.710 --> 00:52:17.810
Simon L: in that setting. In in that setting I call a State holographic. If the induced transport super operator is asymmetric in the Hilbert Schmidt sense.

587
00:52:18.240 --> 00:52:27.329
Simon L: So once again, Hibbert Schmidt, we we choose because we are on. We suppose that the hip that the spaces and operator algebra here are finite, dimensional.

588
00:52:27.420 --> 00:52:29.899
Simon L: So we once again need to impose some cutoffs.

589
00:52:31.230 --> 00:52:37.259
Simon L: With this definition in mind, we can once again transfer all of this into an entropy, calculation.

590
00:52:38.530 --> 00:52:46.390
Simon L: and the expansion calculations are a bit difficult because of this whole cumulant expansion thing that I've mentioned before.

591
00:52:46.580 --> 00:52:52.669
Simon L: but we can look for some necessary criteria for the for the State to be holographic.

592
00:52:52.750 --> 00:52:59.539
Simon L: The first is that the total dimension of all of the boundaries and meetings

593
00:52:59.560 --> 00:53:04.409
Simon L: which is, in a sense which is equivalent to asking. The total area of the state

594
00:53:04.460 --> 00:53:10.640
Simon L: of the system needs to be constant across all of the sectors you include in the super in the

595
00:53:10.860 --> 00:53:12.360
Simon L: in the full algebra.

596
00:53:12.910 --> 00:53:19.600
Simon L: So you're sort of looking at. An algebra which has a lax boundary condition where the total area is fixed.

597
00:53:20.640 --> 00:53:30.900
Simon L: This introduces, then, a notion of scale to the system which you can use to rewrite the icing model partition functions, and then you can take again a low temperature limit.

598
00:53:31.070 --> 00:53:36.379
Simon L: The low temperature limit in this case means you look at fixed large total area.

599
00:53:38.380 --> 00:53:40.410
Simon L: So if you do this.

600
00:53:40.420 --> 00:53:48.389
Simon L: if you reduce, if you restrict to this kind of setting, then the isometry condition just reduces to one that goes per boundary sector.

601
00:53:48.960 --> 00:53:51.299
Simon L: So in each of the boundary sectors.

602
00:53:51.460 --> 00:53:53.000
Simon L: So each of the

603
00:53:53.200 --> 00:53:57.829
Simon L: sort of local. So what we fix is the total area right.

604
00:53:57.930 --> 00:54:00.690
Simon L: But the local area densities

605
00:54:01.010 --> 00:54:03.500
Simon L: are still not not fixed.

606
00:54:03.830 --> 00:54:12.860
Simon L: And so each of these local area density still labels a super selection sector, and in each of those sectors we require, again, isometry to be there.

607
00:54:14.500 --> 00:54:22.309
Simon L: So with that in mind, those are the necessary conditions we know about. We know about this by boundary reconstruction

608
00:54:22.680 --> 00:54:28.299
Simon L: and the conditions for it we can reduce to the fixed spin case again.

609
00:54:29.880 --> 00:54:32.630
Simon L: Now for some interpretation of this

610
00:54:33.950 --> 00:54:42.219
Simon L: what we are doing here is that we are reconstructing intertwiner degrees of freedom of the public

611
00:54:42.340 --> 00:54:46.420
Simon L: from pure gauged mode, edge modes, degrees of freedom on the boundary.

612
00:54:47.240 --> 00:54:51.680
Simon L: and the constraint of fixing the full total area

613
00:54:51.770 --> 00:55:00.269
Simon L: very easily understood, as that this is not actually part of the edge modes. This is not part, and the and this is not part of the

614
00:55:00.370 --> 00:55:11.269
Simon L: sort of soft degrees of freedom, but rather this is part of the global gauge group which fits together with whatever spin defects you have in the bulk.

615
00:55:12.680 --> 00:55:16.320
Simon L: So you can take many different conclusions from this.

616
00:55:16.560 --> 00:55:43.570
Simon L: 2 that I'd like to highlight. Here is the first that we have a preferred corner of states here which do seem to be close to holography. We have identified something here which is which is specific enough so that we can do bulk reconstruction. That's one possible conclusion. The other conclusion is that somehow, because we need to restrict the intertwana degrees of freedom, so much.

617
00:55:43.990 --> 00:55:48.049
Simon L: those are difficult to reconstruct from those edge modes alone.

618
00:55:48.800 --> 00:56:03.899
Simon L: So, in a sense, what we can reconstruct from edge modes is anything but the intertwining degrees of freedom. And that's those are very, very simple. So with regards to the earlier question about what if I have loops in the bike? What if there's some curvature?

619
00:56:04.960 --> 00:56:05.855
Simon L: Well,

620
00:56:06.780 --> 00:56:10.850
Simon L: I'd imagine that it's difficult to encode this in the edge modes alone.

621
00:56:10.850 --> 00:56:21.790
Simone SPEZIALE: You knew it was coming right. So this preferred the corner of the Baltimore space. It's safer to assume for the time being that it excludes loops.

622
00:56:23.010 --> 00:56:25.359
Simon L: I don't know if there's

623
00:56:26.220 --> 00:56:27.919
Simon L: Eugenio's already there.

624
00:56:28.720 --> 00:56:33.809
Simon L: Perfect the answer. So I would say I'm not sure.

625
00:56:33.900 --> 00:56:37.320
Simon L: but more prudently, probably not.

626
00:56:38.470 --> 00:56:39.520
Simone SPEZIALE: Let's see if your journeys will.

627
00:56:39.520 --> 00:56:39.890
Simon L: Daniel.

628
00:56:39.890 --> 00:56:41.019
Simone SPEZIALE: Prudent, or bring.

629
00:56:41.355 --> 00:56:49.070
Eugenio Bianchi: I just wanted to ask, have you checked the volume? We are seeing. The dimensions of Delbert. Spaces are either one or drop.

630
00:56:49.080 --> 00:56:50.460
Eugenio Bianchi: maybe 2.

631
00:56:51.150 --> 00:56:51.680
Simon L: But I checked.

632
00:56:51.680 --> 00:56:53.239
Eugenio Bianchi: The volume of the interior.

633
00:56:53.900 --> 00:56:54.370
Simon L: Oh!

634
00:56:54.370 --> 00:56:57.150
Eugenio Bianchi: Volume of the interior hits the area. Is it 0.

635
00:56:57.150 --> 00:57:07.079
Simon L: It's it's not a sharp criterion, right? So it's sort of the dimensions need to be small relative to the total scale that you fix for the system.

636
00:57:07.420 --> 00:57:08.460
Simon L: and

637
00:57:08.600 --> 00:57:20.679
Simon L: the the larger the Hibots, the larger the dimensions, the worse it gets. It's sort of an approximate isometry. In the first place, because the entropy is never quite maximal. It's only close to being maximum.

638
00:57:21.460 --> 00:57:22.419
Simon L: So it's such a.

639
00:57:22.420 --> 00:57:27.080
Eugenio Bianchi: Really, can you get something closer to 3 dimensional exuding space

640
00:57:27.160 --> 00:57:31.140
Eugenio Bianchi: where that the area goes like length, square and the volume like the

641
00:57:31.230 --> 00:57:32.690
Eugenio Bianchi: let's you.

642
00:57:33.250 --> 00:57:38.499
Eugenio Bianchi: or is it that it's just a crumpled surface? It has an area that 0 volume is.

643
00:57:38.710 --> 00:57:43.539
Simon L: I feel like it's close. It's probably close to a crumpled area of sorts.

644
00:57:43.700 --> 00:57:52.159
Simon L: or perhaps or perhaps one interpretation that I like to at least have in my head with this

645
00:57:52.180 --> 00:57:53.220
Simon L: is that

646
00:57:53.380 --> 00:58:03.739
Simon L: if we have something which is entirely reconstructed, but from the a small degrees, gauge group, degrees of freedom alone, then it's probably because there's somehow flatteness inside.

647
00:58:04.570 --> 00:58:06.380
Simon L: And with that in mind

648
00:58:06.430 --> 00:58:10.619
Simon L: it may not be that the volume is necessarily so constrained.

649
00:58:10.730 --> 00:58:13.669
Simon L: but it will still have some.

650
00:58:13.930 --> 00:58:15.900
Simon L: It will still be very rigid.

651
00:58:18.130 --> 00:58:19.110
Eugenio Bianchi: Okay. Thank you.

652
00:58:19.230 --> 00:58:25.910
Simone SPEZIALE: I related to a genius question, but what fixes the valency of the interior nodes.

653
00:58:26.798 --> 00:58:29.879
Simon L: In this case I just fixed them arbitrarily to the.

654
00:58:30.430 --> 00:58:52.130
Simone SPEZIALE: Right genius question cannot even be asked. I guess right, because the whether you're if they're for Valent, or 5, 5 or 6 volunt, the properties of the volume will depend. So you have to first make the sharp, and then you will be able to. It can be asked. In order to answer it. I guess one needs also to have control over these right.

655
00:58:52.360 --> 00:59:15.270
Simon L: Yes. So in the case that I've been presenting here, you have fixed a graph, and I just studied a fixed valency. But there's nothing that changes in the formalism. If you just let the balance see vary, so you can study both the 3D gravity case where everything is trivalent, but you could also do something where everything is for balance. You could mix everything you could have arbitrarily high balances.

656
00:59:15.270 --> 00:59:21.489
Simone SPEZIALE: Just to make sure. I understand. You mean that your autographic reconstruction would be a fixed bulk graph.

657
00:59:21.740 --> 00:59:22.390
Simon L: Yes.

658
00:59:22.840 --> 00:59:31.830
Simon L: so all of this is very limited in that you need to have a fixed graph structure. Otherwise you cannot apply the random tensor network methods. As far as we are aware.

659
00:59:32.180 --> 00:59:37.800
Simone SPEZIALE: So even in this small corner we are really far from holography. I would say right.

660
00:59:38.650 --> 00:59:40.159
Simon L: In principle, but.

661
00:59:40.160 --> 00:59:45.189
Simone SPEZIALE: Because you're already assigning a huge amount of information about the bulk by specifying the graph.

662
00:59:45.430 --> 00:59:47.923
Simon L: Yes, but in the same way it's sort of

663
00:59:48.600 --> 00:59:51.960
Simon L: It's sort of washed out once you perform the average.

664
00:59:52.660 --> 00:59:57.479
Simon L: so I would not be surprised if you can choose different sorts of graph structures

665
00:59:57.720 --> 01:00:00.400
Simon L: and most of the results.

666
01:00:00.400 --> 01:00:00.720
Simone SPEZIALE: But yeah.

667
01:00:00.720 --> 01:00:01.430
Simon L: Sort of.

668
01:00:02.170 --> 01:00:06.539
Simone SPEZIALE: Th. There's you mean washed out when you module by defamorphisms.

669
01:00:07.542 --> 01:00:11.869
Simon L: No, when you perform the group averaging. But I guess that's not really.

670
01:00:12.010 --> 01:00:13.980
Simon L: It's just about like I mean.

671
01:00:13.980 --> 01:00:21.850
Simone SPEZIALE: Existed, you should be able to, from your boundary it alone to reconstruct the actual graph inside that.

672
01:00:21.850 --> 01:00:27.309
Simon L: So I find that to be difficult, at least from given from the data that we use here.

673
01:00:27.310 --> 01:00:28.769
Simone SPEZIALE: Agree, though so, if.

674
01:00:28.770 --> 01:00:38.640
Simon L: If you don't augment the boundary with some extra degrees of freedom which allow you to reconstruct more stuff, I would find it difficult to reconstruct anything more than.

675
01:00:39.450 --> 01:00:40.080
Simone SPEZIALE: Right.

676
01:00:40.080 --> 01:00:40.539
Simon L: They were funny.

677
01:00:40.540 --> 01:00:41.540
Simone SPEZIALE: Difficult to even.

678
01:00:41.540 --> 01:00:43.460
Simon L: Instruct a single bulk, spin.

679
01:00:43.600 --> 01:00:44.999
Simone SPEZIALE: Right? Yeah, I agree.

680
01:00:45.620 --> 01:00:46.200
Simon L: So.

681
01:00:46.200 --> 01:00:52.809
Carlo Rovelli: Simon. This. This obviously answers the sort of question that I was raising when you were giving definitions. There is.

682
01:00:52.810 --> 01:00:53.610
Simon L: Yes, sir.

683
01:00:53.610 --> 01:00:54.710
Carlo Rovelli: Then there are different.

684
01:00:55.330 --> 01:00:59.060
Carlo Rovelli: different a priori assumptions people have in their hand, in their head.

685
01:00:59.170 --> 01:01:02.150
Carlo Rovelli: But I think you your your what you present is very clear.

686
01:01:02.440 --> 01:01:10.319
Simon L: Yes, that's that's what I meant. The the whole, the whole step up from giving the Hibbert Space definition to the algebra one.

687
01:01:10.610 --> 01:01:14.180
Simon L: I think, having that in mind makes it more clear what I mean.

688
01:01:15.470 --> 01:01:23.130
Laurent Freidel: But you have. Can you commit now on the role of the constraints which here it's? You know what your your discussion is when you don't talk about any

689
01:01:23.330 --> 01:01:24.809
Laurent Freidel: any constraints

690
01:01:24.960 --> 01:01:26.489
Laurent Freidel: right in the bulk.

691
01:01:26.530 --> 01:01:27.799
Laurent Freidel: so we haven't studied that.

692
01:01:27.800 --> 01:01:28.299
Simon L: And all the.

693
01:01:28.790 --> 01:01:31.729
Laurent Freidel: And let's say, the translation, constraint.

694
01:01:31.730 --> 01:01:34.004
Simon L: Haven't studied this at all so far.

695
01:01:34.470 --> 01:01:35.020
Simon L: So.

696
01:01:35.330 --> 01:01:36.252
Laurent Freidel: Hey? We'd happen.

697
01:01:36.560 --> 01:01:46.709
Carlo Rovelli: If I may. Lauren Lauren. What you're saying is that there's a there's an interesting hypothesis here which is that preferred corner of the open space. Photography might be precisely

698
01:01:46.850 --> 01:01:50.049
Carlo Rovelli: where the constraints project, the

699
01:01:50.819 --> 01:02:04.130
Carlo Rovelli: the the kinematical illustration 2. That's the that's another formulation of the holographic hypothesis. I I I would say, or for which you might have intuitions, or or or or

700
01:02:04.230 --> 01:02:06.659
Carlo Rovelli: or indirect arguments, and and.

701
01:02:06.660 --> 01:02:16.840
Laurent Freidel: Yeah, I was. I was maybe making a statement in between. It's clear that here he's working on some kinematic a little bit space. So a lot of a lot of these States that you count as difference are completely redundant.

702
01:02:16.860 --> 01:02:45.269
Laurent Freidel: So once you impose the constraints, the question, there's a drastic as reduction of what our admissible bug states. Whether thatuction it's drastic, right? It's like a, you know, because there's there's whether that drastic reduction means that you project onto the edge mode. That's that's another extra purchases. But we have to appreciate that. What we're talking about here is is vastly over counting in the box, right? Because we we are not positioning by the action of the gauge group.

703
01:02:45.280 --> 01:02:47.229
Laurent Freidel: The by, the yeah. So

704
01:02:48.120 --> 01:02:51.290
Laurent Freidel: there's there's a lot of space in between. Yeah.

705
01:02:51.310 --> 01:02:54.000
Laurent Freidel: you know, counting speed network states as valid states.

706
01:02:54.000 --> 01:02:54.600
Simon L: Smoke.

707
01:02:54.890 --> 01:02:56.679
Laurent Freidel: And looking at the physical ones.

708
01:02:56.680 --> 01:02:58.380
Simon L: I have 2 more slides.

709
01:02:58.540 --> 01:03:00.639
Hal Haggard: I was. Gonna say exactly that, Simon, wedding

710
01:03:01.150 --> 01:03:02.810
Hal Haggard: up, and then that'll.

711
01:03:02.810 --> 01:03:04.070
Simon L: Yeah, then we can discuss more.

712
01:03:04.070 --> 01:03:05.760
Hal Haggard: Can continue discussion.

713
01:03:05.760 --> 01:03:12.940
Simon L: Yes, so to add on the on the fire of the holography. And what is the right notion story?

714
01:03:13.518 --> 01:03:24.779
Simon L: I would like to add a further complication in that. It's also possible to just talk about entanglement between boundary regions. And this is an entirely separate question.

715
01:03:25.440 --> 01:03:36.560
Simon L: One way you can do this is, you can take one fixed graph state and you fix some intertwining data. For example, if it's fixed spin, then you just fix one state.

716
01:03:36.780 --> 01:03:45.029
Simon L: But if it's multiple sectors, then you fix one for each separate state and you sort of condition the boundary state on the spike data.

717
01:03:45.730 --> 01:03:48.500
Simon L: So you now have an induced boundary state

718
01:03:48.610 --> 01:03:50.749
Simon L: which only sits on the semi links.

719
01:03:51.330 --> 01:04:01.109
Simon L: And this is now in a factorizing boundary head space. So this is very easy from the point of view of trying to figure out what entanglement means.

720
01:04:01.560 --> 01:04:07.409
Simon L: There is no necessary necessity for an area law or something like that because there's

721
01:04:07.490 --> 01:04:13.540
Simon L: no center, there's no center which would give you the kinds of contributions we saw earlier.

722
01:04:14.410 --> 01:04:15.360
Simon L: So

723
01:04:15.920 --> 01:04:27.260
Simon L: this is more similar, perhaps, to what people do in Ads. Cft, where they say that the entanglement between different regions of the Cft, which sits entirely on the boundary

724
01:04:27.650 --> 01:04:32.940
Simon L: is what gives rise to bike connectivity and other properties of the bike space time.

725
01:04:33.420 --> 01:04:40.220
Simon L: So maybe, in a sense, the actual geometry from entanglement that we should be looking for

726
01:04:40.260 --> 01:04:50.669
Simon L: is actually of this kind, and is something that also would need to be studied in order to better understand what kind of stuff we can reconstruct.

727
01:04:51.020 --> 01:04:56.399
Simon L: But it's more difficult to compute this sort of thing in general, so

728
01:04:57.203 --> 01:05:03.020
Simon L: so far we don't know that much about it, especially not in the case of spin, tensor networks

729
01:05:03.320 --> 01:05:04.110
Simon L: and

730
01:05:04.220 --> 01:05:15.740
Simon L: observables would also need to be covered, calculated in this kind of context, in order to have to have an idea of what the criteria really mean in terms of, say, geometry.

731
01:05:16.820 --> 01:05:17.500
Simon L: So

732
01:05:18.550 --> 01:05:21.209
Simon L: let me just wrap up with some take-home messages.

733
01:05:21.756 --> 01:05:23.740
Simon L: The first is that

734
01:05:24.060 --> 01:05:31.520
Simon L: whatever entropy you calculate, given a density. Matrix depends on what algebra the density matrix sits in

735
01:05:31.990 --> 01:05:44.039
Simon L: what algebra you choose to work with the gauge invariant one, the kinematical one which kinds of gates invariance, for example, just gals, or all the way to diffu invariance.

736
01:05:44.120 --> 01:05:54.340
Simon L: Those different types mean different things in particular. If you ask a quantum information purist. They might tell you that only the distalable entanglement is really one that they are interested in

737
01:05:55.290 --> 01:05:56.859
Simon L: depending on who you ask.

738
01:05:57.700 --> 01:06:03.439
Simon L: Different types of algebra can also be. That's the second point assigned to regions.

739
01:06:03.650 --> 01:06:07.989
Simon L: So you have different choices in that, and if have different choice of boundary conditions.

740
01:06:08.330 --> 01:06:10.290
Simon L: it needs to be kept in mind.

741
01:06:11.150 --> 01:06:15.230
Simon L: Furthermore, spin network states have a super selection

742
01:06:15.300 --> 01:06:17.960
Simon L: by the boundary values of the spins.

743
01:06:18.620 --> 01:06:26.389
Simon L: That's very, and that's very important for getting the edge mode contribution, this a boundary term in the entanglement, entropy.

744
01:06:27.700 --> 01:06:34.739
Simon L: and in particular they are universal, in that, if your state is gauge invariant, it will have this contribution.

745
01:06:36.420 --> 01:06:46.580
Simon L: Furthermore, the last point is that if you have, if you have a spin network state, and you impose some restrictions on what kind of State. It is

746
01:06:46.750 --> 01:07:01.520
Simon L: then, on average, the typical State will support this kind of holographic reconstruction for match modes. So, of course, in the last point I was, and being a bit sparse with the details. But no, I think the details should be clear as well.

747
01:07:04.830 --> 01:07:07.679
Simon L: Okay, there was an empty slide there. Okay.

748
01:07:09.600 --> 01:07:10.560
Simon L: thank you.

749
01:07:12.481 --> 01:07:16.900
Hal Haggard: Thank you, Simon. Thank you for the nice overview. Carlo.

750
01:07:19.290 --> 01:07:21.734
Carlo Rovelli: Only me says nobody else would definitely be.

751
01:07:22.350 --> 01:07:22.970
Hal Haggard: Yeah, so.

752
01:07:22.970 --> 01:07:30.060
Carlo Rovelli: Anyway. Okay, so first of all, that was very clear. I think a follow. I'm not sure I got the last

753
01:07:30.090 --> 01:07:35.380
Carlo Rovelli: of your comments, but that's maybe just number 5. But let me let me make 2.

754
01:07:35.380 --> 01:07:38.420
Simon L: Just the stuff from before, but more not more short.

755
01:07:38.960 --> 01:07:48.139
Carlo Rovelli: Let me make to so I I think I understood you. I think it's very nice, very clean. I think I got it. So it's very good, very useful, I would say.

756
01:07:48.420 --> 01:07:49.930
Carlo Rovelli: I want to make 2

757
01:07:50.080 --> 01:07:52.105
Carlo Rovelli: to side comments.

758
01:07:53.410 --> 01:08:01.720
Carlo Rovelli: one may be related to your last, the the very last end of your very vague end of your your talk.

759
01:08:03.670 --> 01:08:05.370
Carlo Rovelli: It might be useful

760
01:08:06.500 --> 01:08:09.399
Carlo Rovelli: to think about the spin network state.

761
01:08:09.910 --> 01:08:11.870
Carlo Rovelli: the super space you're talking about.

762
01:08:13.137 --> 01:08:14.629
Carlo Rovelli: Not does the

763
01:08:14.730 --> 01:08:17.310
Carlo Rovelli: the same of a sort of a cushy surface?

764
01:08:19.582 --> 01:08:23.120
Carlo Rovelli: But rather as a boundary of 4 dimensional region.

765
01:08:24.380 --> 01:08:26.924
Carlo Rovelli: namely, not the analog

766
01:08:27.660 --> 01:08:28.990
Carlo Rovelli: of the

767
01:08:29.210 --> 01:08:32.550
Carlo Rovelli: sort of help, the spaces of no relativistic quantum mechanics. So.

768
01:08:32.840 --> 01:08:40.540
Carlo Rovelli: but the analog of the sort of tensor product of the in state and the outstate.

769
01:08:42.290 --> 01:08:45.790
Carlo Rovelli: The what I call the boundary state in my book.

770
01:08:46.490 --> 01:08:49.460
Simon L: Like a boundary state for spin foot models.

771
01:08:49.850 --> 01:08:58.325
Carlo Rovelli: Yes, exactly, which is not the sense in your sense. It's not the sort of in physical terms. It's not the 2 dimensional bandwidth, the 3 dimensional band, the band of

772
01:09:00.488 --> 01:09:02.379
Carlo Rovelli: Because if it is so

773
01:09:03.517 --> 01:09:11.509
Carlo Rovelli: if you think in these terms, which is just definitions perfectly possible. And I think that's clean way of viewing, of viewing the

774
01:09:12.790 --> 01:09:19.059
Carlo Rovelli: then the dynamics is given by just a brown, the state just a linear function of this state. Right?

775
01:09:19.189 --> 01:09:20.069
Carlo Rovelli: That's it.

776
01:09:20.670 --> 01:09:23.100
Simon L: I couldn't. I couldn't acoustically hear you. Sorry.

777
01:09:23.109 --> 01:09:23.759
Carlo Rovelli: Oh, sorry!

778
01:09:23.760 --> 01:09:25.700
Simon L: By by given by a brown State.

779
01:09:25.700 --> 01:09:29.600
Carlo Rovelli: If you do so, then the dynamics is just Abra on this state.

780
01:09:29.609 --> 01:09:31.579
Simon L: A brow on the state. Okay? Yes, yes, yes.

781
01:09:31.580 --> 01:09:38.829
Carlo Rovelli: Linear function, this state. So you don't have to protect anything down to to a subspace. You just have to write right.

782
01:09:39.130 --> 01:09:42.214
Carlo Rovelli: which, in the no relativistic case would be

783
01:09:43.130 --> 01:09:49.290
Carlo Rovelli: just transform is transforming sort of psi final U evolution psi initial

784
01:09:49.670 --> 01:09:53.070
Carlo Rovelli: into a bra on the

785
01:09:53.543 --> 01:09:58.306
Carlo Rovelli: acting on some initial tense or sci vinyl dagger may start.

786
01:09:59.710 --> 01:10:05.819
Carlo Rovelli: You think of these terms, then? This what you're sort of. Yeah, what you're sort of hinting in these slides

787
01:10:06.290 --> 01:10:08.740
Carlo Rovelli: is that it is in this sense

788
01:10:08.750 --> 01:10:10.540
Carlo Rovelli: that the

789
01:10:11.390 --> 01:10:14.360
Carlo Rovelli: boundary through this place, in in my sense.

790
01:10:15.048 --> 01:10:18.441
Carlo Rovelli: We should give different names because Bandar, hipaa space no has.

791
01:10:18.750 --> 01:10:20.480
Simon L: Foundry versus Corner.

792
01:10:20.690 --> 01:10:24.049
Carlo Rovelli: Yeah. Boundaries. This is called alright. Thank you. The boundary hit. The space is

793
01:10:24.530 --> 01:10:26.880
Carlo Rovelli: include something about the book.

794
01:10:26.960 --> 01:10:34.205
Carlo Rovelli: because in the classical case it always. It's clearly does right, because if you give that down the boundary you have.

795
01:10:35.067 --> 01:10:41.989
Carlo Rovelli: if you think in terms of Hamit, Hamilton, Hamilton, Jacobi logic, Hamilton. Function, logic, that sort of

796
01:10:42.120 --> 01:10:45.731
Carlo Rovelli: define something in between. So that was my, it's very big. It's not something.

797
01:10:45.990 --> 01:10:48.980
Simon L: When you say bark. In this case you mean the 4.

798
01:10:48.980 --> 01:10:49.410
Carlo Rovelli: To Mr.

799
01:10:49.410 --> 01:10:50.602
Simon L: Facetime. Bug, okay.

800
01:10:51.940 --> 01:10:52.910
Simon L: okay, so.

801
01:10:53.070 --> 01:10:59.069
Carlo Rovelli: Maybe we should determine, we should think, in terms of 4 dimensional geometry, not 3 dimensional geometry. When we think of the bulk, because that's.

802
01:10:59.070 --> 01:10:59.800
Simon L: But do you.

803
01:11:00.190 --> 01:11:02.050
Carlo Rovelli: After all, that's yeah.

804
01:11:02.340 --> 01:11:03.892
Carlo Rovelli: That's one comment.

805
01:11:05.220 --> 01:11:11.520
Carlo Rovelli: the second comment. I'm just repeating something which I know it's controversial. But I am.

806
01:11:11.660 --> 01:11:16.070
Carlo Rovelli: I I think it's correct. And so it's a it's a

807
01:11:16.754 --> 01:11:29.860
Carlo Rovelli: I think it's important to remember the there is a way of interpreting the now 3 dimensional here to space for spin networks as something which doesn't know at all about the the dynamics and Newtonian constraint.

808
01:11:30.490 --> 01:11:34.338
Carlo Rovelli: but knows already fully about the

809
01:11:36.550 --> 01:11:40.400
Carlo Rovelli: The diphomorphic constraint, the three-dimensional dermorphic constraint.

810
01:11:40.700 --> 01:11:46.060
Carlo Rovelli: because at even given a graph, this is a truncation.

811
01:11:46.320 --> 01:11:47.730
Carlo Rovelli: is not the full theory.

812
01:11:48.240 --> 01:11:52.660
Carlo Rovelli: and as such it can be seen

813
01:11:52.870 --> 01:11:55.719
Carlo Rovelli: a sequentization of a truncation

814
01:11:56.020 --> 01:11:57.826
Carlo Rovelli: off the

815
01:11:58.730 --> 01:11:59.330
Simon L: I mean.

816
01:11:59.550 --> 01:12:03.649
Carlo Rovelli: Of of of of 3 dimensional geometries, not metrics. So

817
01:12:03.930 --> 01:12:09.080
Carlo Rovelli: this is the way that Eugenio has been thinking for for a long time. So you can take this piece of.

818
01:12:09.320 --> 01:12:10.429
Simon L: And see it as a top of.

819
01:12:10.736 --> 01:12:14.409
Carlo Rovelli: Up to exactly up to different offices that they just think to.

820
01:12:14.410 --> 01:12:16.840
Simon L: Eventually up to December.

821
01:12:17.160 --> 01:12:23.570
Carlo Rovelli: And in this space you just look for a for a subset. That's a truncation.

822
01:12:24.097 --> 01:12:30.050
Carlo Rovelli: You quantize that. Then you get this thing here. So that's a logic of Regent. That's regular logic of the.

823
01:12:30.050 --> 01:12:31.799
Simon L: But easy. Yes.

824
01:12:32.430 --> 01:12:37.160
Carlo Rovelli: I'm not saying that that's the only way of doing things or thinking things so one can think differently and do something differently.

825
01:12:37.160 --> 01:12:38.669
Simon L: Yes, I think it's okay.

826
01:12:39.540 --> 01:12:45.389
Carlo Rovelli: So what is missing from this mathematics to the physics from this perspective?

827
01:12:45.740 --> 01:12:47.260
Carlo Rovelli: It's just a dynamics

828
01:12:47.610 --> 01:12:51.840
Carlo Rovelli: I just wanted to throw there. This 2 comments are not directly related to your talk, but your talk.

829
01:12:51.840 --> 01:12:52.740
Simon L: Was this.

830
01:12:52.740 --> 01:13:06.670
Carlo Rovelli: It gives a. It gives sort of the mathematics of these things and the relation between book and boundary in a way. And I think that this 2 different perspective may help using all this going toward the physics.

831
01:13:07.040 --> 01:13:11.267
Simon L: Okay? So just so, I understand your first comment correctly.

832
01:13:11.880 --> 01:13:22.510
Simon L: you're saying that looking at the results of the failure to reconstruct a generic spin network evidence base from the edge modes alone.

833
01:13:22.520 --> 01:13:31.889
Simon L: We should rather see the kind of holography that might be more generic, as the kind that reconstructs for the data

834
01:13:31.920 --> 01:13:36.110
Simon L: from a boundary through which is code dimension. One.

835
01:13:37.350 --> 01:13:38.200
Simon L: Okay.

836
01:13:38.530 --> 01:13:39.979
Carlo Rovelli: Yes, and in in doing this case.

837
01:13:39.980 --> 01:13:52.649
Simon L: Would that? But would that not be almost too generic in the sense that if, for example, we were in classical physics then, and if I don't have a sort of timeline boundary and just have a 7 of sorts.

838
01:13:52.690 --> 01:14:00.460
Simon L: And I have initial data and final data, then the 4 of the data is sort of specified just by having a unique solution to the Cauchy problem.

839
01:14:01.380 --> 01:14:04.519
Carlo Rovelli: It is but the the

840
01:14:04.970 --> 01:14:12.520
Carlo Rovelli: for me. I I I've always hoped not in terms of Tamil boundary, but the 2 space like surface that meet at corner.

841
01:14:12.520 --> 01:14:13.200
Simon L: So the.

842
01:14:13.200 --> 01:14:19.669
Carlo Rovelli: Of the corner. I've always it might think I've always neglected the role of the corner and the corner. The role of corner cannot be neglected.

843
01:14:19.830 --> 01:14:22.560
Carlo Rovelli: Obviously so, the corner is crucial here.

844
01:14:23.316 --> 01:14:23.873
Carlo Rovelli: And

845
01:14:24.570 --> 01:14:31.070
Carlo Rovelli: somehow, the the the generatistic evolution is multi. Finger devolution is local, is all global.

846
01:14:31.070 --> 01:14:31.840
Simon L: Yes.

847
01:14:31.840 --> 01:14:34.770
Carlo Rovelli: So given a cushy surface, you can evolve just a bump.

848
01:14:35.170 --> 01:14:35.910
Simon L: Yes.

849
01:14:35.910 --> 01:14:43.200
Carlo Rovelli: That you forget all the rest. You just look at the bump. You have a 4 dimensional thing, and you have a sort of initial and final piece of the cushy surfaces.

850
01:14:43.200 --> 01:14:45.759
Simon L: Yes, but there's no time involved. Yeah.

851
01:14:45.760 --> 01:14:52.972
Carlo Rovelli: Right, but of course there's a corner whether to meet, and I've been neglecting all my overall. My book is sort of neglected corner, and there all this.

852
01:14:53.350 --> 01:14:57.419
Carlo Rovelli: looking at the corner, is crucial, and that's why I find it's it's.

853
01:14:57.420 --> 01:14:59.110
Simon L: Thank you. Okay.

854
01:14:59.110 --> 01:14:59.520
Carlo Rovelli: Meredith.

855
01:14:59.520 --> 01:15:00.719
Simon L: Okay, I see what you mean.

856
01:15:00.920 --> 01:15:03.508
Carlo Rovelli: Honest as as in in this.

857
01:15:04.140 --> 01:15:11.680
Carlo Rovelli: What I'm suggesting is that now remember that what we have to do we have to do the dynamics. But there's a way of simple way of doing the dynamics.

858
01:15:12.466 --> 01:15:13.739
Carlo Rovelli: Which is

859
01:15:13.750 --> 01:15:14.989
Carlo Rovelli: the way just it.

860
01:15:16.920 --> 01:15:18.740
Simon L: Okay. Thank you.

861
01:15:19.590 --> 01:15:21.080
Simon L: The home. Yeah.

862
01:15:21.080 --> 01:15:24.238
Laurent Freidel: Well, maybe Jersey first, since he hasn't asked.

863
01:15:24.790 --> 01:15:25.690
Simon L: Okay.

864
01:15:25.940 --> 01:15:27.020
Simon L: Jesse.

865
01:15:29.027 --> 01:15:39.759
Jerzy Lewandowski: Yes, so thank thank you. So I have a very general and stupid question, so why do we actually need

866
01:15:39.780 --> 01:15:50.600
Jerzy Lewandowski: the edge modes. It seems that all the Hilbert space is defined by the Hilbert spaces of intertwiners that are

867
01:15:50.620 --> 01:15:52.040
Jerzy Lewandowski: additionally

868
01:15:52.500 --> 01:15:58.319
Jerzy Lewandowski: reduced by so so, so, so, so I would say that edge

869
01:15:58.330 --> 01:16:09.339
Jerzy Lewandowski: edges don't introduce extra degrees of freedom. They introduce extra, and they reduce the degrees of freedom. So so why do we need to count edges, as also.

870
01:16:09.340 --> 01:16:10.279
Simon L: Let's say.

871
01:16:10.280 --> 01:16:12.280
Jerzy Lewandowski: You think, to the Hebrew Space degree.

872
01:16:12.280 --> 01:16:13.000
Simon L: It's

873
01:16:13.830 --> 01:16:18.168
Simon L: So as far as the hipaa space of speed networks is concerned.

874
01:16:19.030 --> 01:16:27.420
Simon L: the hologies live on there so I wouldn't argue for removing them. But, as far as the entropy is concerned, to raise a valid question

875
01:16:27.550 --> 01:16:39.859
Simon L: in principle, the data that is there, once you have fixed the graph is just the data of this Hibbert space here, and the data on the internal edges is fixed. It's fixed to be the maximally entangled state.

876
01:16:40.770 --> 01:16:43.500
Simon L: for example in the States that I considered here.

877
01:16:43.800 --> 01:16:53.360
Simon L: And then you can actually just say, Okay, I don't want to count them. They don't. They are just constrained. They are just fixed already, so they are not. There are no degrees of freedom.

878
01:16:53.870 --> 01:16:55.310
Simon L: You could say that.

879
01:16:55.970 --> 01:16:58.620
Simon L: And then you count a certain type of entropy.

880
01:16:59.290 --> 01:17:00.780
Simon L: Then you count

881
01:17:00.820 --> 01:17:03.469
Simon L: basically just the distillable entanglement.

882
01:17:04.640 --> 01:17:11.390
Simon L: But if you count also the contribution coming from the edges, then you're calculating a different type of entropy.

883
01:17:13.010 --> 01:17:15.550
Simon L: So in the end. It's sort of you can do both.

884
01:17:15.880 --> 01:17:20.790
Simon L: and whether you want one or the other depends on what you want to do with it, I would say.

885
01:17:21.080 --> 01:17:28.959
Simon L: because the I don't know, having having just the distillable entanglement has an operational interpretation.

886
01:17:29.270 --> 01:17:39.819
Simon L: But in many ways there are reasons to say that the edge MoD contribution is what you get. Naturally, if you just, if you just calculate it, via the replicatric.

887
01:17:43.030 --> 01:17:44.270
Jerzy Lewandowski: Okay, thinking.

888
01:17:44.270 --> 01:17:46.619
Simon L: So if you dislike edge modes, you can remove them.

889
01:17:47.098 --> 01:17:49.600
Jerzy Lewandowski: Yeah, this is what I what I understood. Thanks.

890
01:17:50.370 --> 01:17:51.550
Simon L: Okay. Simon.

891
01:17:52.130 --> 01:17:53.750
Laurent Freidel: Sorry I had a question before.

892
01:17:53.750 --> 01:17:55.770
Simon L: Yes, yes, Laurent go first.

893
01:17:56.047 --> 01:18:01.322
Laurent Freidel: Yeah, I wanna go back to this. Maybe this comment about the Defamorphism cause. I think it's very

894
01:18:01.750 --> 01:18:03.549
Laurent Freidel: appreciate that

895
01:18:03.640 --> 01:18:11.809
Laurent Freidel: in the traditional picture of Lqg, we think that you know, deformism doesn't create new entanglement across the spin. But

896
01:18:11.950 --> 01:18:13.580
Laurent Freidel: but in fact, that's

897
01:18:14.420 --> 01:18:29.480
Laurent Freidel: you know, I would say that that is not correct. and so there's this work that we did with with Daniel. And and I think it's an important work for for what you're discussing. And then we, of course, we're going to continue. Is that this is what we did with that nearly there are. And then

898
01:18:29.851 --> 01:18:54.310
Laurent Freidel: where we kind of looked at the at the, you know, covariance, phase, space, action of the defamism on the face, base, variable. And then you what you can prove there is that this defamism can be written as some kind of translation constraint. So there's there's a part of the deformism which act as a notion of local translation on the spin, and of course, a local translation, in fact, you know, changes the spin.

899
01:18:54.370 --> 01:19:05.420
Laurent Freidel: and therefore it means that you know there is a part of the of the constraints. And hopefully, we'll discuss more with Sima about that. But it's very important to appreciate that the defam

900
01:19:05.420 --> 01:19:27.569
Laurent Freidel: action as a quantum operator, not as a kinematical things on the embedded graph, but as a as a quantum operator acting on the facepace. Viable changes. You know the relationship between the flux and autonomy, etc, and that relationship can be quantized and will and does, in fact, lead to a a very interesting extra measure of entanglement.

901
01:19:27.570 --> 01:19:37.440
Laurent Freidel: because what it means is that now you you you know, anytime you cut, or if you look at the edges, then this edges needs to carry now a representation of some kind of local point of group.

902
01:19:37.440 --> 01:19:58.179
Laurent Freidel: which is what it should be like even in a local space. You know, you don't have it just local, Lawrence. You also have some kind of form of local point of it. So implementing that local point cafe entanglement is going to be absolutely crucial in understanding what is the spin network entanglement? And and I think it's very important for the field that we move

903
01:19:59.421 --> 01:20:04.889
Laurent Freidel: towards understanding this this translation, and then, before even we, we understand

904
01:20:04.970 --> 01:20:07.389
Laurent Freidel: the the, the constraints and thing of it.

905
01:20:09.930 --> 01:20:10.610
Simon L: Thank you.

906
01:20:11.910 --> 01:20:13.250
Simon L: Simon.

907
01:20:14.460 --> 01:20:18.090
Simone SPEZIALE: Yeah, I just had a quick taking a question. I was wondering if these

908
01:20:18.600 --> 01:20:23.770
Simone SPEZIALE: projector to the peps, as you call them. I can obtain it by doing

909
01:20:24.799 --> 01:20:30.289
Simone SPEZIALE: you know, an integral implementing the area matching constraint

910
01:20:30.350 --> 01:20:33.583
Simone SPEZIALE: in the spinorial representation.

911
01:20:34.230 --> 01:20:38.356
Simon L: I think so. Yes, in principle. Yes, because all it's doing really.

912
01:20:39.680 --> 01:20:41.290
Simon L: wait, this is the wrong one.

913
01:20:41.310 --> 01:20:47.849
Simon L: Yeah, this one here, all this is doing like the one on the top. Here is just taking the

914
01:20:48.361 --> 01:21:00.280
Simon L: that say in Su, 2 data, the Su 2 representation on the left, and the 2 representations on the right and sort of matching them together to like have the same action.

915
01:21:00.340 --> 01:21:05.519
Simon L: So in. So that includes the area matching constraint. I think it's basically equivalent to that.

916
01:21:05.770 --> 01:21:14.809
Simone SPEZIALE: Okay, good. And another similar question very quick. If you remove the party team map you have in 7,

917
01:21:14.900 --> 01:21:18.469
Simone SPEZIALE: you lose the maximal entanglement, or you still have it.

918
01:21:18.470 --> 01:21:21.190
Simon L: No, those are related by Unitarian transformation.

919
01:21:22.310 --> 01:21:23.727
Simone SPEZIALE: So you don't lose it. You said.

920
01:21:24.200 --> 01:21:25.970
Simon L: You don't lose maximum entanglement.

921
01:21:25.970 --> 01:21:26.320
Simone SPEZIALE: It's.

922
01:21:26.320 --> 01:21:31.139
Simon L: Really that you can use any. It just leads to a change in parity.

923
01:21:31.140 --> 01:21:32.523
Simone SPEZIALE: Right, yeah, yeah, so.

924
01:21:32.870 --> 01:21:35.540
Simon L: Entanglement of the State itself doesn't care about that.

925
01:21:35.540 --> 01:21:40.474
Simone SPEZIALE: Okay, right? Exactly. That's what I was expecting would have sounded weird. They just wanted to confirm. Thank you very much.

926
01:21:41.220 --> 01:21:41.930
Simon L: Welcome!

927
01:21:44.490 --> 01:21:47.119
Hal Haggard: Any further questions or comments.

928
01:21:49.240 --> 01:21:55.530
Carlo Rovelli: maybe there's nothing else a technical question can can you say, can can you flash again your definition of the

929
01:21:56.420 --> 01:21:59.669
Carlo Rovelli: of the States? Demographic states.

930
01:22:00.865 --> 01:22:02.800
Simon L: For the here!

931
01:22:02.920 --> 01:22:03.490
Simon L: Yep.

932
01:22:03.490 --> 01:22:06.135
Carlo Rovelli: The state, the one, the the one with the map.

933
01:22:06.850 --> 01:22:12.659
Simon L: Yeah, I would. I would use the general one, if you don't mind. But we can also take the one here.

934
01:22:12.660 --> 01:22:14.100
Carlo Rovelli: This one, this one.

935
01:22:14.100 --> 01:22:14.415
Simon L: Yes.

936
01:22:15.840 --> 01:22:18.549
Carlo Rovelli: Because that's where you sort of use starting from

937
01:22:18.850 --> 01:22:19.540
Simon L: Yes.

938
01:22:22.330 --> 01:22:24.030
Carlo Rovelli: So metric the

939
01:22:28.660 --> 01:22:30.310
Carlo Rovelli: the reason you

940
01:22:33.160 --> 01:22:34.873
Carlo Rovelli: right? Why?

941
01:22:35.830 --> 01:22:38.700
Carlo Rovelli: just trying to formulate question. Right?

942
01:22:43.860 --> 01:22:46.250
Carlo Rovelli: what? What is exactly? What is the

943
01:22:46.390 --> 01:22:50.839
Carlo Rovelli: what is the action on on the elastic quality on the left hand side?

944
01:22:50.930 --> 01:22:51.520
Carlo Rovelli: Yes.

945
01:22:52.260 --> 01:22:56.699
Carlo Rovelli: What is? What is the mathematics here? Exactly, I understand, said what is

946
01:22:57.360 --> 01:22:58.590
Carlo Rovelli: seal site?

947
01:22:58.990 --> 01:23:03.690
Simon L: So this doesn't just doesn't have the index. Psi, but it's the same thing.

948
01:23:06.000 --> 01:23:14.029
Simon L: So it's just a linear operator from intertwina to boundary spin. And then the adjoint going from boundary spin to intertwine.

949
01:23:16.400 --> 01:23:18.039
Carlo Rovelli: Right you don't use

950
01:23:19.926 --> 01:23:21.320
Carlo Rovelli: you you!

951
01:23:21.480 --> 01:23:24.370
Carlo Rovelli: Why don't you write it in the same notation as the

952
01:23:24.650 --> 01:23:27.229
Carlo Rovelli: what are just trying to figure out.

953
01:23:27.460 --> 01:23:29.189
Simon L: And the same notation as spot.

954
01:23:30.270 --> 01:23:34.859
Simon L: I'll be. Ca, I just forgot to write the index lowercase. Sorry, that's really it.

955
01:23:40.430 --> 01:23:42.313
Carlo Rovelli: No, no, this understand? But

956
01:23:45.253 --> 01:23:54.140
Carlo Rovelli: phi! With a with index. I it's Bra. Is that what you're thinking?

957
01:23:55.120 --> 01:23:56.279
Carlo Rovelli: It's a little.

958
01:23:57.260 --> 01:23:59.799
Simon L: It's a it's, it's well, it's

959
01:24:01.460 --> 01:24:07.510
Simon L: I mean, it's a. It's a linear map from from the intertwina space to the.

960
01:24:07.510 --> 01:24:09.680
Carlo Rovelli: Oh, I see what you're saying. Okay.

961
01:24:11.190 --> 01:24:13.759
Carlo Rovelli: okay, okay, okay, fine. And the

962
01:24:16.140 --> 01:24:17.650
Carlo Rovelli: right is that

963
01:24:17.670 --> 01:24:19.090
Carlo Rovelli: is that to?

964
01:24:19.860 --> 01:24:23.080
Carlo Rovelli: Is that the only way of capturing the idea of this

965
01:24:23.240 --> 01:24:28.550
Carlo Rovelli: allography, both to bulk or or this can be done differently. I guess that's what I want to do.

966
01:24:29.660 --> 01:24:30.870
Simon L: Apologies.

967
01:24:30.870 --> 01:24:33.420
Carlo Rovelli: End of the detail of this definition, the

968
01:24:34.210 --> 01:24:36.100
Carlo Rovelli: the world conclusion.

969
01:24:37.190 --> 01:24:40.630
Carlo Rovelli: Is there something else that one could require? Instead of that? Besides.

970
01:24:40.630 --> 01:24:42.380
Simon L: Instead of isometry, you mean.

971
01:24:45.590 --> 01:24:48.960
Simon L: I mean, what kind of photographic reconstruction, or

972
01:24:49.330 --> 01:24:51.100
Simon L: what? What do you have in mind?

973
01:24:52.220 --> 01:24:53.379
Carlo Rovelli: No, I'm just

974
01:24:54.460 --> 01:24:56.150
Carlo Rovelli: wondering whether

975
01:24:57.720 --> 01:24:58.660
Carlo Rovelli: so.

976
01:24:59.170 --> 01:25:01.329
Simon L: So this is this is basically

977
01:25:01.350 --> 01:25:13.980
Simon L: so maybe a bit of historical context for this the same the same exact definition is what was used in the context of tensor network ads. Cft.

978
01:25:15.410 --> 01:25:23.609
Simon L: In that they have some bike Hilbert spaces which they reconstruct through a tensor network state from some boundary links.

979
01:25:25.280 --> 01:25:29.919
Simon L: And the isometry question is, really it's it's it's used in that.

980
01:25:30.120 --> 01:25:39.820
Simon L: If you have the boundaries. If you have a boundary state, and you know it's coming from the bike, and you have some operators acting on the bike. Then you can reconstruct it from all operators on the boundary.

981
01:25:40.700 --> 01:25:45.588
Carlo Rovelli: Okay, okay? So that maybe that's the answer, the answer. What I wanted to know whether it's just

982
01:25:46.810 --> 01:25:53.760
Carlo Rovelli: your way of capturing as a general idea, or it was deriving from

983
01:25:53.940 --> 01:25:54.800
Carlo Rovelli: from a

984
01:25:55.230 --> 01:25:57.780
Carlo Rovelli: specifically analogous formulation.

985
01:25:58.020 --> 01:25:58.480
Carlo Rovelli: Oh, it's

986
01:25:58.710 --> 01:26:05.049
Simon L: Latter, but I do think it's sort of a very concrete implementation of a general idea.

987
01:26:05.270 --> 01:26:05.690
Carlo Rovelli: Okay.

988
01:26:05.690 --> 01:26:12.940
Simon L: If you if you want, you could make this more generally by replacing the naive boundary, Hibbert Space by something bigger.

989
01:26:13.520 --> 01:26:19.970
Simon L: Yeah, by placing some additional degrees of freedom on on the boundary vertices or something.

990
01:26:20.230 --> 01:26:24.870
Simon L: Then I could imagine you can just be more. You can just reconstruct more stuff.

991
01:26:25.470 --> 01:26:26.140
Carlo Rovelli: Okay.

992
01:26:26.370 --> 01:26:27.759
Carlo Rovelli: Okay. Thank you.

993
01:26:28.000 --> 01:26:32.409
Laurent Freidel: I mean, this is just the definition of Alo Yoshida, etc. You know.

994
01:26:32.410 --> 01:26:33.390
Simon L: Exactly.

995
01:26:33.390 --> 01:26:34.500
Laurent Freidel: That's how it is.

996
01:26:34.570 --> 01:26:36.580
Laurent Freidel: They kind of formalize.

997
01:26:36.840 --> 01:26:40.330
Carlo Rovelli: Okay, okay, so there's a history behind this definition. Okay.

998
01:26:40.330 --> 01:26:45.439
Laurent Freidel: Yeah, yeah, that's that's kind of the quantum information definition of holographic intensive network.

999
01:26:45.490 --> 01:26:48.220
Laurent Freidel: I do have a question, though, about 20

1000
01:26:48.694 --> 01:26:52.950
Laurent Freidel: because on the right hand side. You write the the identity on I but

1001
01:26:52.970 --> 01:27:03.239
Laurent Freidel: you mean I is a code subspace, right? It it cannot be the entire sets of, so I would. I would imagine that 5 5 dagger in general is a projection right.

1002
01:27:03.584 --> 01:27:04.960
Simon L: Yes, yes, I mean.

1003
01:27:05.349 --> 01:27:21.330
Laurent Freidel: Projection. So maybe for people we're not familiar. That's maybe the confusion here is that you know in in this, in this photographic map, what what you're imagining is that of course, you're not looking at the space of all possible

1004
01:27:21.410 --> 01:27:23.119
Laurent Freidel: bulk intertwiner. But you're looking.

1005
01:27:23.120 --> 01:27:24.469
Simon L: Yes, that's true. Yes.

1006
01:27:24.470 --> 01:27:34.230
Laurent Freidel: Call a good subspace, so maybe it should be called IC, or right, which is a subspace of intertwiner for which you have isometry right.

1007
01:27:34.230 --> 01:27:45.310
Simon L: Yes, I've been cavalier about including that part here, but you're completely right in principle. You can only reconstruct a small subspace of the full intertwining space, and that's the only requirement you have really.

1008
01:27:45.760 --> 01:27:52.909
Laurent Freidel: And and the idea about the constraints when I keep saying about the constraints differ, and a Miltonian is that that space

1009
01:27:52.930 --> 01:28:00.010
Laurent Freidel: is maybe supposed to be the the image of the projector onto the physical little bit space. Right? So that's.

1010
01:28:00.010 --> 01:28:00.560
Simon L: Hmm.

1011
01:28:00.690 --> 01:28:06.950
Laurent Freidel: So it's clear that there's a code subspacing gravity. And okay, that's right. That's the one thing which is clear.

1012
01:28:07.740 --> 01:28:12.650
Laurent Freidel: Yeah, whether that could subspace is reconstructable from the edge. That's the central question.

1013
01:28:12.950 --> 01:28:13.340
Simon L: Yes.

1014
01:28:13.340 --> 01:28:18.320
Carlo Rovelli: Yeah, agree? That's a that's a super interesting question. And it's a it's a

1015
01:28:18.580 --> 01:28:27.939
Carlo Rovelli: and now it's in this language is, there's a nice sort of technical way of formulate. Anyone could still turn it around in one way or the other is not

1016
01:28:28.300 --> 01:28:33.349
Carlo Rovelli: unique. But that's and I think that puts clear where the hypothesis is.

1017
01:28:33.882 --> 01:28:38.310
Carlo Rovelli: This is this is the this is in this language.

1018
01:28:38.682 --> 01:28:44.120
Carlo Rovelli: One way of formulating the holographic hypothesis. A lot. A lot follows from that right

1019
01:28:44.630 --> 01:28:48.009
Carlo Rovelli: black. What is black pollen to be, for instance.

1020
01:28:59.820 --> 01:29:00.440
Simon L: Hey!

1021
01:29:00.440 --> 01:29:05.009
Hal Haggard: Good. Shall we thank our speaker one more time? Are there any last comments.

1022
01:29:05.200 --> 01:29:06.830
Simon L: From the people at Western.

1023
01:29:08.700 --> 01:29:11.149
Hal Haggard: I think they they just had the video on.

1024
01:29:13.890 --> 01:29:14.600
Simon L: Yes.

1025
01:29:14.750 --> 01:29:16.330
Western: Sorry. Just a small one, just.

1026
01:29:16.330 --> 01:29:16.980
Simon L: Yes.

1027
01:29:17.190 --> 01:29:18.430
Western: Partner. Discussion.

1028
01:29:20.940 --> 01:29:29.460
Western: if I make correctly, for instance, indograph is, I think, most of the box reconstruction only concern the tongue. Language right.

1029
01:29:29.990 --> 01:29:38.100
Western: So not the bulk. Do you have an intuition of whether something similar is happening here, and whether your restriction on the Internet space

1030
01:29:39.150 --> 01:29:41.309
Western: somehow sending us that.

1031
01:29:41.530 --> 01:29:42.520
Simon L: So

1032
01:29:42.920 --> 01:29:50.060
Simon L: there are hints towards that. For example, from the Ad. Cft discussion. They also find that it's

1033
01:29:50.190 --> 01:29:51.830
Simon L: possible to

1034
01:29:52.370 --> 01:29:59.490
Simon L: to reconstruct bike operators only in some sort of causal wedge

1035
01:29:59.580 --> 01:30:01.989
Simon L: which is included in the entanglement, which

1036
01:30:02.140 --> 01:30:07.259
Simon L: but it's difficult to say that really, because even on that level.

1037
01:30:07.780 --> 01:30:11.510
Simon L: it's really only a given single time. Slice.

1038
01:30:12.010 --> 01:30:18.269
Simon L: and those co entanglement and causes wedges are usually constructed from having a full spacetime picture.

1039
01:30:18.670 --> 01:30:24.810
Simon L: so the interpretation of that stuff is a little bit. I would take it with a grain of salt.

1040
01:30:25.410 --> 01:30:26.320
Simon L: and

1041
01:30:26.540 --> 01:30:35.280
Simon L: on the side I don't, I think, in the formulation I give on this slide here. What you could do to test this sort of thing is

1042
01:30:35.550 --> 01:30:39.219
Simon L: you sort of reduce the algebra of inputs

1043
01:30:39.430 --> 01:30:41.359
Simon L: and the algebra of outputs

1044
01:30:41.380 --> 01:30:44.759
Simon L: to be sub regions of the of the bulk.

1045
01:30:45.590 --> 01:30:49.789
Simon L: So, for example, for the output regions, you just say, you see. But take a boundary segment.

1046
01:30:50.340 --> 01:30:54.969
Simon L: and then you take some other bike region of intertwiners.

1047
01:30:55.090 --> 01:31:01.190
Simon L: And then you ask, how do I have to modify these regions so that I get reconstruction?

1048
01:31:02.530 --> 01:31:15.370
Simon L: So that's something I could imagine you can do, but that is hampered a little bit by the question of What do you do when the state in question here, when the row is a mixed state.

1049
01:31:16.130 --> 01:31:20.149
Simon L: and it seems like there. The and the analysis with this sort of

1050
01:31:20.980 --> 01:31:24.359
Simon L: with this sort of stuff is quite difficult.

1051
01:31:25.080 --> 01:31:28.190
Simon L: So far I've not been able to

1052
01:31:28.500 --> 01:31:36.820
Simon L: make it work so well. It seems like the the criteria with like maximum entropy, only seems to work so well when you have a pure state.

1053
01:31:38.590 --> 01:31:40.059
Western: I see. Thanks.

1054
01:31:43.310 --> 01:31:46.700
Hal Haggard: Alright with that, let's thank Simon again. Thank you so much, Simon.

1055
01:31:48.090 --> 01:31:48.939
Simon L: You're welcome.

1056
01:31:49.500 --> 01:31:50.910
Simon L: Thank you for inviting me.