Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1902.02369 · Physical Review Letters

Entanglement of Purification in Many Body Systems and Symmetry Breaking

Arpan Bhattacharyya, A. Jahn, Tadashi Takayanagi, Koji Umemoto

Published May 22, 2019• 47 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.122.201601

Related papers

Entanglement of purification in free scalar field theories

Arpan Bhattacharyya, Tadashi Takayanagi, Koji Umemoto

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.

arXiv:1802.09545•Journal of High Energy Physics•Apr 1, 2018•55 citations

Holographic Study of Entanglement of Purification for Excited States

Mohammad Sahraei, Mohammad Javad Vasli, M. Reza Mohammadi Mozaffar +1 more

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.

•May 27, 2021•4 citations

Bit threads and holographic entanglement of purification

Dong-Hui Du, Chong-Bin Chen, Fu-Wen Shu

Abstract The entanglement of purification (EoP), which measures the classical correlations and entanglement of a given mixed state, has been conjectured to be dual to the area of the minimal cross section of the entanglement wedge in holography. Using the surface-state correspondence, we propose a “bit thread” formulation of the EoP. With this formulation, proofs of some known properties of the EoP are performed. Moreover, we show that the quantum advantage of dense code (QAoDC), which reflects the increase in the rate of classical information transmission through quantum channel due to entanglement, also admits a flow interpretation. In this picture, we can prove the monogamy relation of QAoDC with the EoP for tripartite states. We also derive a new lower bound for S(AB) in terms of QAoDC, which is tighter than the one given by the Araki-Lieb inequality.

arXiv:1904.06871•Journal of High Energy Physics•Aug 1, 2019•45 citations

Entanglement of purification and disentanglement in CFTs

Wu-zhong Guo

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.

arXiv:1904.12124•Journal of High Energy Physics•Sep 1, 2019•14 citations

Long Distance Entanglement of Purification and Reflected Entropy in Conformal Field Theory

Hugo A. Camargo, Lucas Hackl, Michał P. Heller +2 more

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.

arXiv:2102.00013•Physical Review Letters•Sep 29, 2021•24 citations

Rényi entanglement of purification and half Rényi reflected entropy in free scalar theory

Liangyu Chen

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 ≥ $$ \frac{1}{2}{S}_R^{(n)} $$ 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.

arXiv:2501.10944•Journal of High Energy Physics•Jun 5, 2025•0 citations
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 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.
Abstract The entanglement of purification (EoP), which measures the classical correlations and entanglement of a given mixed state, has been conjectured to be dual to the area of the minimal cross section of the entanglement wedge in holography. Using the surface-state correspondence, we propose a “bit thread” formulation of the EoP. With this formulation, proofs of some known properties of the EoP are performed. Moreover, we show that the quantum advantage of dense code (QAoDC), which reflects the increase in the rate of classical information transmission through quantum channel due to entanglement, also admits a flow interpretation. In this picture, we can prove the monogamy relation of QAoDC with the EoP for tripartite states. We also derive a new lower bound for S(AB) in terms of QAoDC, which is tighter than the one given by the Araki-Lieb inequality.
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.
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.
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.