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arXiv:1802.09545 · Journal of High Energy Physics

Entanglement of purification in free scalar field theories

Arpan Bhattacharyya, Tadashi Takayanagi, Koji Umemoto

Published Apr 1, 2018• 55 citations
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Abstract

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.

DOI
10.1007/jhep04(2018)132

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Informational properties of holographic Lifshitz field theory *

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Abstract In this paper, we explore the properties of holographic entanglement entropy (HEE), mutual information (MI) and entanglement of purification (EoP) in holographic Lifshitz theory. These informational quantities exhibit some universal properties of holographic dual field theory. For most configuration parameters and temperatures, these informational quantities change monotonically with the Lifshitz dynamical critical exponent z. However, we also observe some non-monotonic behaviors for these informational quantities in some specific spaces of configuration parameters and temperatures. A particularly interesting phenomenon is that a dome-shaped diagram emerges in the behavior of MI vs z, and correspondingly a trapezoid-shaped profile appears in that of EoP vs z. This means that for some specific configuration parameters and temperatures, the system measured in terms of MI and EoP is entangled only in a certain intermediate range of z.

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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.
We study the entanglement of purification (EOP), a measure of total correlation between two subsystems A and B, for free scalar field theory on a lattice and the transverse-field Ising model by numerical methods. In both of these models, we find that the EOP becomes a nonmonotonic function of the distance between A and B when the total number of lattice sites is small. When it is large, the EOP becomes monotonic and shows a plateaulike behavior. Moreover, we also show that the original reflection symmetry which exchanges A and B can get broken in optimally purified systems. We provide an interpretation of our results in terms of the interplay between classical and quantum correlations.
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.
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 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.
Abstract In this paper, we explore the properties of holographic entanglement entropy (HEE), mutual information (MI) and entanglement of purification (EoP) in holographic Lifshitz theory. These informational quantities exhibit some universal properties of holographic dual field theory. For most configuration parameters and temperatures, these informational quantities change monotonically with the Lifshitz dynamical critical exponent z. However, we also observe some non-monotonic behaviors for these informational quantities in some specific spaces of configuration parameters and temperatures. A particularly interesting phenomenon is that a dome-shaped diagram emerges in the behavior of MI vs z, and correspondingly a trapezoid-shaped profile appears in that of EoP vs z. This means that for some specific configuration parameters and temperatures, the system measured in terms of MI and EoP is entangled only in a certain intermediate range of z.