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arXiv:1709.07424 · Journal of High Energy Physics

Entanglement of purification: from spin chains to holography

Phuc Nguyen, Trithep Devakul, Matthew G. Halbasch, Michael P. Zaletel, Brian G. Swingle

Published Jan 1, 2018• 238 citations
arXivHTMLPDF

Abstract

Purification is a powerful technique in quantum physics whereby a mixed quantum state is extended to a pure state on a larger system. This process is not unique, and in systems composed of many degrees of freedom, one natural purification is the one with minimal entanglement. Here we study the entropy of the minimally entangled purification, called the entanglement of purification, in three model systems: an Ising spin chain, conformal field theories holographically dual to Einstein gravity, and random stabilizer tensor networks. We conjecture values for the entanglement of purification in all these models, and we support our conjectures with a variety of numerical and analytical results. We find that such minimally entangled purifications have a number of applications, from enhancing entanglement-based tensor network methods for describing mixed states to elucidating novel aspects of the emergence of geometry from entanglement in the AdS/CFT correspondence.

DOI
10.1007/jhep01(2018)098

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Purification is a powerful technique in quantum physics whereby a mixed quantum state is extended to a pure state on a larger system. This process is not unique, and in systems composed of many degrees of freedom, one natural purification is the one with minimal entanglement. Here we study the entropy of the minimally entangled purification, called the entanglement of purification, in three model systems: an Ising spin chain, conformal field theories holographically dual to Einstein gravity, and random stabilizer tensor networks. We conjecture values for the entanglement of purification in all these models, and we support our conjectures with a variety of numerical and analytical results. We find that such minimally entangled purifications have a number of applications, from enhancing entanglement-based tensor network methods for describing mixed states to elucidating novel aspects of the emergence of geometry from entanglement in the AdS/CFT correspondence.
We argue that the entanglement of purification for 2D holographic conformal field theories (CFT) can be obtained from conformal blocks with internal twist operators. First, we explain our formula from the viewpoint of a tensor network model of holography. Then, we apply it to bipartite mixed states dual to a subregion of AdS|3{}_33​| and the static Bañados–Teitelboim–Zanelli black hole geometries. The formula in CFT agrees with the entanglement wedge cross section in the bulk, which has been recently conjectured to be equivalent to the entanglement of purification.
A bstract We introduce a new information-theoretic measure of multipartite correlations Δ P , by generalizing the entanglement of purification to multipartite states. We provide proofs of its various properties, focusing on several entropic inequalities, in generic quantum systems. In particular, it turns out that the multipartite entanglement of purification gives an upper bound on multipartite mutual information, which is a generalization of quantum mutual information in the spirit of relative entropy. After that, motivated by a tensor network description of the AdS/CFT correspondence, we conjecture a holographic dual of multipartite entanglement of purification Δ W , as a sum of minimal areas of codimension-2 surfaces which divide the entanglement wedge into multi-pieces. We prove that this geometrical quantity satisfies all properties we proved for the multipartite entanglement of purification. These agreements strongly support the Δ P = Δ W conjecture. We also show that the multipartite entanglement of purification is larger than multipartite squashed entanglement, which is a promising measure of multipartite quantum entanglement. We discuss potential saturation of multipartite squashed entanglement onto multipartite mutual information in holographic CFTs and its applications.
In conformal field theories, in contrast to some auxiliary states into the bipartite mixed state ρ A B as the usual purifications do, we show a pure entangled state ψ A B can be constructed by the undetectable regions. In this pure state ψ A B , the von Neumann entropy S vN ( A : B ) naturally captures quantum entanglement between A and B . We verify that S vN ( A : B ) is equal to the entanglement wedge cross section E W in anti–de Sitter spacetime, which is conjectured to be the holographic dual of the entanglement of purification.
We explore a conformal field theoretic interpretation of the holographic entanglement of purification, which is defined as the minimal area of the entanglement wedge cross section. We argue that, in AdS_{3}/CFT_{2}, the holographic entanglement of purification agrees with the entanglement entropy for a purified state, obtained from a special Weyl transformation, called path-integral optimizations. By definition, this special purified state has minimal path-integral complexity. We confirm this claim in several examples.
Quantifying entanglement properties of mixed states in quantum field theory via entanglement of purification and reflected entropy is a new and challenging subject. In this work, we study both quantities for two spherical subregions far away from each other in the vacuum of a conformal field theory in any number of dimensions. Using lattice techniques, we find an elementary proof that the decay of both the entanglement of purification and reflected entropy is enhanced with respect to the mutual information behavior by a logarithm of the distance between the subregions. In the case of the Ising spin chain at criticality and the related free fermion conformal field theory, we compute also the overall coefficients numerically for the both quantities of interest.
Many of the tools and ideas of quantum information theory have been employed to shed light on the holographic principle.The entanglement entropy in the AdS/CFT correspondence tells us that information of the bulk geometries is encoded in the boundary correlations.This thesis is aimed to deepen our understanding of this direction by studying various correlation measures in holography.We first derive a generalization of the holographic entanglement of purification conjecture for multipartite states.It turns out that this proposal is consistently supported in arbitrary n-partite states.We next study the entanglement of purification in quantum many body systems by numerical methods.That reveals a remarkable non-monotonicity of the entanglement of purification, attributed to the different sensitivities to quantum and classical correlations.We finally discuss the holographic duals of two classes of correlation measures: optimized correlation measures and axiomatic entanglement measures.The results suggest that classical correlation also plays an important role in encoding the geometrical information.i