Entangle This: Tensor Networks and Gravity


  Monday 8 Tuesday 9   Wednesday 10


The continuous multi-scale entanglement renormalization ansatz (cMERA)


Ryu-Takayanagi Formula from Quantum Error Correction



AdS from Optimization of CFT Path-Integrals and Continuous Tensor Networks

11:00 Coffee Coffee & Posters 11:00


Entanglement and thermodynamics after a quantum quench in integrable systems


Van Raamsdonk



Bit threads in space and time

12:00 Coffee
12:30 Lunch Lunch 12:30

Colloquium: van Raamsdonk

Gravity and Entanglement



Approximate Quantum Markov Chains with applications to tensor networks and holography


Quantum source-channel codes as a way to capture the holographic correspondence.

13:30 End of Workshop
15:30 Coffee & Posters Coffee & Posters    




Scrambling the spectral

form factor



Remarks on entanglement phases and holography


Many-body systems: Entanglement spectra and symmetries at the boundary






G. Vidal: The continuous multi-scale entanglement renormalization ansatz (cMERA)


The first half of the talk will introduce the cMERA, as proposed by Haegeman, Osborne, Verschelde and Verstratete in 2011 [1], as an extension to quantum field theories (QFTs) in the continuum of the MERA tensor network for lattice systems. The second half of the talk will review recent results [2] that show how a cMERA optimized to approximate the ground state of a conformal field theory (CFT) retains all of its spacetime symmetries, although these symmetries are realized quasi-locally. In particular, the conformal data of the original CFT can be extracted from the optimized cMERA.
[1] J. Haegeman, T. J. Osborne, H. Verschelde, F. Verstraete, Entanglement renormalization for quantum fields, Phys. Rev. Lett, 110, 100402 (2013), arXiv:1102.5524
[2] Q. Hu, G. Vidal, Spacetime symmetries and conformal data in the continuous multi-scale entanglement renormalization ansatz, arXiv:1703.04798


F Brandao: Approximate Quantum Markov chains with applications to tensor networks and holography

We prove an upper bound on the conditional mutual information of Gibbs states of one- dimensional short-range quantum Hamiltonians at finite temperature.


F Verstraete: Remarks on entanglement phases and holography

We show that anyon condensation is in one-to-one correspondence to the behavior of the virtual entanglement state at the boundary (i.e., the entanglement spectrum) under those symmetries, which encompasses both symmetry breaking and symmetry protected (SPT) order, and we use this to characterize all anyon condensations for abelian double models through the structure of their entanglement spectrum.


D. Harlow: I explain how an analogue of the Ryu-Takayanagi formula is a general property of quantum error correcting codes, and use this to illuminating some puzzling issues about the formula in AdS/CFT.


M. Headrick: Bit threads in space and time

 I will briefly review the bit-thread picture of static holographic entanglement developed by myself and Michael Freedman. I will then explain how to extend this picture to time-dependent states.


F. Pastawski:  Quantum source-channel codes as a way to capture the holographic correspondence.

While originally motivated by quantum computation, quantum error correction (QEC) is currently providing valuable insights into many-body quantum physics such as topological phases of matter. Furthermore, mounting evidence originating from holography research (AdS/CFT), indicates that QEC should also be pertinent for conformal field theories. With this motivation in mind, we introduce quantum source-channel codes, which combine features of lossy-compression and approximate quantum error correction, both of which are predicted in holography. Approximate quantum error-correcting codes are codes only guarantee the approximate recovery of information, but this guarantee is provided uniformly for all code states. Here, we propose a complementary relaxation, wherein the code-subspace assumption is replaced by a weighted prior distribution; states drawn from this distribution can be approximately recovered on average. Through a recent construction for approximate recovery maps, we derive guarantees on its erasure decoding performance from calculations of an entropic quantity called conditional mutual information. As an example of possible prior distribution, we consider Gibbs states of the transverse field Ising model at criticality and provide evidence that they show non-trivial protection from local erasure. This gives rise to the first concrete interpretation of a bona fide conformal field theory as a quantum error correcting code. While, the motivation for this interpretation is to elucidate the information structure of holography; We argue that quantum source-channel codes are of independent interest beyond this scope.


J. Molina: Scrambling the spectral form factor

Quantum speed limits set an upper bound to the rate at which a quantum system can evolve and as such can be used to analyze the scrambling of information. To this end, we consider the survival probability of a thermofield double state under unitary time-evolution which is related to the analytic continuation of the partition function. We will comment on a tensor network which decribes  the time evolution of the thermofield double state.  An exponential lower bound to the survival probability with a rate governed by the inverse of the energy fluctuations of the initial state is provided. Further, we elucidate universal features of the non-exponential behavior at short and long times of evolution that follow from the analytic properties of the survival probability and its Fourier transform, both for systems with a continuous and a discrete energy spectrum. We find the spectral form factor in a number of illustrative models, notably we obtain the exact answer in the Gaussian unitary ensemble for any N with excellent agreement with recent numerical studies. We also discuss the relationship of our findings to models of black hole information loss, such as the Sachdev-Ye-Kitaev model dual to AdS2 as well as higher-dimensional versions of AdS/CFT.


N. Schuch:  Many-body systems: Entanglement spectra and symmetries at the boundary

We show that long-range order in tensor network models for symmetry protected phases is accompanied by a degeneracy in the so-called transfer operator of the system.


T. Takayanagi: AdS from Optimization of CFT Path-Integrals and Continuous Tensor Networks


We introduce a new optimization procedure for Euclidean path integrals which compute wave functionals in CFTs. We optimize the background metric in
the space on which the path integration is performed. Equivalently this is interpreted as a position dependent UV cut-off. For two dimensional CFT vacua, we find the optimized metric is given by that of a hyperbolic space and we interpret this as a continuous limit of the conjectured relation between tensor networks and AdS/CFT. We also show that when applied to reduced density matrices, it reproduces entanglement wedges and holographic entanglement entropy. We suggest that our optimization prescription is analogous to the estimation of computational complexity.


P. Calabrese: Entanglement and thermodynamics after a quantum quench in integrable systems

Entanglement and entropy are key concepts standing at the foundations of quantum and statistical mechanics. In the last decade the study of quantum quenches revealed that these two concepts are intricately intertwined. Although the unitary time evolution ensuing from a pure initial state maintains the system globally at zero entropy, at long time after the quench local properties are captured by an appropriate statistical ensemble with non zero thermodynamic entropy, which can be interpreted as the entanglement accumulated during the dynamics. Therefore, understanding the post-quench entanglement evolution unveils how thermodynamics emerges in isolated quantum systems. An exact computation of the entanglement dynamics has been provided only for non-interacting systems, and it was believed to be unfeasible for genuinely interacting models. In this talk I show that the standard quasiparticle picture of the entanglement evolution, complemented with integrability-based knowledge of the asymptotic state, leads to a complete analytical understanding of the entanglement dynamics in the space-time scaling limit

M. Van Raamsdonk: IFT Colloquium on "Gravity and Entanglement"


The AdS/CFT correspondence from string theory provides a quantum theory of gravity in which spacetime and gravitational physics emerge from an
ordinary non-gravitational quantum system with many degrees of freedom. In this talk, I will explain how quantum entanglement between these degrees
of freedom is crucial for the emergence of a classical spacetime, and describe progress in understanding how spacetime dynamics (gravitation)
arises from the physics of quantum entanglement.


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