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arXiv:1904.12124 · Journal of High Energy Physics

Entanglement of purification and disentanglement in CFTs

Wu-zhong Guo

Published Sep 1, 2019• 14 citations
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Abstract

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 $$ \tilde{A} $$ A ˜ adjoins A and $$ \tilde{A}\tilde{B} $$ 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 $$ \tilde{A} $$ A ˜ in the state $$ \left|\psi \left(\delta \right)\right\rangle := {e}^{i\delta H}\tilde{A}\tilde{B}{\left|\psi \right\rangle}_M $$ ψ δ ≔ e iδH A ˜ B ˜ ψ M . We find a constraint on the state $$ {\left|\psi \right\rangle}_M\left[{K}_{A\tilde{A},M},{O}_{\tilde{A}}\right]=0 $$ ψ M K A A ˜ , M O A ˜ = 0 , where $$ {K}_{A\tilde{A},M} $$ K A A ˜ , M is the modular Hamiltonian of A $$ \tilde{A} $$ A ˜ in the state |ψ〉 M , $$ {O}_{\tilde{A}}\in \mathcal{R}\left(\tilde{\mathrm{A}}\right) $$ 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.

DOI
10.1007/jhep09(2019)080

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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.
We investigate entanglement of purification (EoP) in conformal field theory. By using Reeh-Schlieder theorem, we construct a set of the purification states for ρAB, where ρAB is reduced density matrix for subregion AB of a global state ρ. The set can be approximated by acting all the unitary observables, located in the complement of subregion AB, on the global state ρ, as long as the global state ρ is cyclic for every local algebra, e.g., the vacuum state. Combining with the gravity explanation of unitary operations in the context of the so-called surface/state correspondence, we give an explanation of holographic EoP formula. We also explore the projection operator with the conformal basis in conformal field theory. In some limit we may produce the holographic EoP results by using the projection operator. Finally, we discuss the similarity and difference between the projection operator and unitary operations for calculating EoP.
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.
A bstract We compute the entanglement of purification (EoP) in a 2d free scalar field theory with various masses. This quantity measures correlations between two subsystems and is reduced to the entanglement entropy when the total system is pure. We obtain explicit numerical values by assuming minimal gaussian wave functionals for the purified states. We find that when the distance between the subsystems is large, the EoP behaves like the mutual information. However, when the distance is small, the EoP shows a characteristic behavior which qualitatively agrees with the conjectured holographic computation and which is different from that of the mutual information. We also study behaviors of mutual information in purified spaces and violations of monogamy/strong superadditivity.
We evaluate the entanglement of purification (EoP), which measures correlations between two subsystems, for excited states dual to asymptotically AdS spacetimes using holography. In this framework EoP is given by the minimal cross sectional area of an entanglement wedge. We carry out a perturbative analysis for calculating EoP between the vacuum and other states for a symmetric configuration consisting of two disjoint strips and obtain analytical results in the specific regimes of the parameter space. In particular, when the states described by purely gravitational excitations in the bulk we find that the leading correction to EoP is negative and the correlation between the subregions decreases. We also study other types of excitations upon adding the extra matter fields including current and scalar condensate. For the latter class of excitations, we find that the variation of EoP is negative, when the dimension of the scalar operator respects the unitarity bound. Finally, we discuss how these results are consistent with the behavior of other correlation measures including the holographic mutual information.
A bstract In the AdS/CFT context, the entanglement of purification (EoP, denoted as E P ) of CFT is conjectured to be dual to the entanglement wedge cross section (EWCS) in bulk. However, another quantity called reflected entropy S R is also supposed to be dual to two times the EWCS. A natural question is whether they are the same in holographic CFTs even though they are different in general. Previous studies have shown E P ≥ 12SR(n) \frac{1}{2}{S}_R^{(n)} 21​SR(n)​ 1 2 S R n , n ≥ 2 for random tensor networks. In this paper, we study this inequality beyond n ≥ 2, and we focus on the range 0 < n < 2. However, the calculations of EoP are notoriously difficult in general. Thus, our calculations mainly focus on the free scalar theory which is close to the holographic CFTs. We generalized the previous strategy for EoP in [1] to the Rényi case. And we have also presented two methods for Rényi reflected entropy, one is using correlators, the other one is Gaussian wavefunction ansatz. Our calculations show that the inequality still holds for 0 < n < 2, and it may give us some insights into the equivalence of EoP and half reflected entropy in holographic CFTs. As byproducts of our research, we have also demonstrated the positivity of the Rényi Markov gap and the monotonicity of the Rényi reflected entropy in the free scalar theory.
Abstract Understanding quantum entanglement in interacting higher-dimensional conformal field theories is a challenging task, as direct analytical calculations are often impossible to perform. With holographic entanglement entropy, calculations of entanglement entropy turn into a problem of finding extremal surfaces in a curved spacetime, which we tackle with a numerical finite-element approach. In this paper, we compute the entanglement entropy between two half-spaces resulting from a local quench, triggered by a local operator insertion in a CFT 3 . We find that the growth of entanglement entropy at early time agrees with the prediction from the first law, as long as the conformal dimension Δ of the local operator is small. Within the limited time region that we can probe numerically, we observe deviations from the first law and a transition to sub-linear growth at later time. In particular, the time dependence at large Δ shows qualitative differences to the simple logarithmic time dependence familiar from the CFT 2 case. We hope that our work will motivate further studies, both numerical and analytical, on entanglement entropy in higher dimensions.