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arXiv:2406.09033 · Physical review. D/Physical review. D.

Alternative to purification in conformal field theory

Xin Jiang, Peng Wang, Houwen Wu, Haitang Yang

Published Jan 23, 2025• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevd.111.l021902

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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 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.
Abstract We study the entanglement of purification (EoP) of subsystem A and B in conformal field theories (CFTs) stressing on its relation to unitary operations of disentanglement, if the auxiliary subsystem A~ \tilde{A} A~ A ˜ adjoins A and A~B~ \tilde{A}\tilde{B} A~B~ A ˜ B ˜ is the complement of AB. We estimate the amount of the disentanglement by using the inequality of Von Neumann entropy as well as the surface/state correspondence. Denote the state that produces the EoP by |ψ〉 M . We calculate the variance of entanglement entropy of A A~ \tilde{A} A~ A ˜ in the state ∣ψ(δ)⟩:=eiδHA~B~∣ψ⟩M \left|\psi \left(\delta \right)\right\rangle := {e}^{i\delta H}\tilde{A}\tilde{B}{\left|\psi \right\rangle}_M ∣ψ(δ)⟩:=eiδHA~B~∣ψ⟩M​ ψ δ ≔ e iδH A ˜ B ˜ ψ M . We find a constraint on the state ∣ψ⟩M[KAA~,M,OA~]=0 {\left|\psi \right\rangle}_M\left[{K}_{A\tilde{A},M},{O}_{\tilde{A}}\right]=0 ∣ψ⟩M​[KAA~,M​,OA~​]=0 ψ M K A A ˜ , M O A ˜ = 0 , where KAA~,M {K}_{A\tilde{A},M} KAA~,M​ K A A ˜ , M is the modular Hamiltonian of A A~ \tilde{A} A~ A ˜ in the state |ψ〉 M , OA~∈R(A~) {O}_{\tilde{A}}\in \mathcal{R}\left(\tilde{\mathrm{A}}\right) OA~​∈R(A~) O A ˜ ∈ ℛ A ˜ is an arbitrary operator. We also study three different states that can be seen as disentangled states. Two of them can produce the holographic EoP result in some limit. But we show that none of they could be a candidate of the state |ψ〉 M , since the distance between these three states and |ψ〉 M is very large.
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.
A bstract In AdS/CFT we consider a class of bulk geometric quantities inside the entanglement wedge called reflected minimal surfaces. The areas of these surfaces are dual to the entanglement entropy associated to a canonical purification (the GNS state) that we dub the reflected entropy . From the bulk point of view, we show that half the area of the reflected minimal surface gives a reinterpretation of the notion of the entanglement wedge cross-section. We prove some general properties of the reflected entropy and introduce a novel replica trick in CFTs for studying it. The duality is established using a recently introduced approach to holographic modular flow. We also consider an explicit holographic construction of the canonical purification, introduced by Engelhardt and Wall; the reflected minimal surfaces are simply RT surfaces in this new spacetime. We contrast our results with the entanglement of purification conjecture, and finally comment on the continuum limit where we find a relation to the split property: the reflected entropy computes the von Neumann entropy of a canonical splitting type-I factor introduced by Doplicher and Longo.
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.
A recent proposal by Ryu and Takayanagi for a holographic interpretation of entanglement entropy in conformal field theories dual to supergravity on anti-de Sitter space is generalized to include entanglement entropy of black holes living on the boundary of anti-de Sitter space. The generalized proposal is verified in boundary dimensions d=2 and d=4 for both the uv-divergent and uv-finite terms. In dimension d=4 an expansion of entanglement entropy in terms of size L of the subsystem outside the black hole is considered. A new term in the entropy of dual strongly coupled conformal-field theory, which universally grows as L(2)lnL and is proportional to the value of the obstruction tensor at the black hole horizon, is predicted.