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:1901.00330 · Physics Letters B

Entanglement of purification and projection operator in conformal field theories

Wu-zhong Guo

Published Sep 12, 2019• 13 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1016/j.physletb.2019.134934

Related papers

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

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

Towards entanglement of purification for conformal field theories

Hayato Hirai, Kotaro Tamaoka, Tsuyoshi Yokoya

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|${}_3$| 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.

arXiv:1803.10539•Progress of Theoretical and Experimental Physics•May 7, 2018•61 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

The connection between holographic entanglement and complexity of purification

Mahdis Ghodrati, Xiao-Mei Kuang, Bin Wang +2 more

Abstract In this work we study how entanglement of purification (EoP) and the new quantity of “complexity of purification” are related to each other using theEP=EWconjecture. First, we consider two strips in the same side of a boundary and study the relationships between the entanglement of purification of this mixed state and the parameters of the system such as dimension, temperature, length of the strips and the distance between them. Next, using the same setup, we introduce two definitions for the complexity of mixed states, complexity of purification (CoP) and the interval volume (VI). We study their connections to other parameters similar to the EoP case. Then, we extend our study to more general examples of BTZ black holes solution in massive gravity, charged black holes and multipartite systems. Finally, we give various interpretations of our results using resource theories such as LOCC and also bit thread picture.

arXiv:1902.02475•Journal of High Energy Physics•Sep 1, 2019•54 citations

Alternative to purification in conformal field theory

Xin Jiang, Peng Wang, Houwen Wu +1 more

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.

arXiv:2406.09033•Physical review. D/Physical review. D.•Jan 23, 2025•9 citations
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.
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 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 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 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 In this work we study how entanglement of purification (EoP) and the new quantity of “complexity of purification” are related to each other using theEP=EWconjecture. First, we consider two strips in the same side of a boundary and study the relationships between the entanglement of purification of this mixed state and the parameters of the system such as dimension, temperature, length of the strips and the distance between them. Next, using the same setup, we introduce two definitions for the complexity of mixed states, complexity of purification (CoP) and the interval volume (VI). We study their connections to other parameters similar to the EoP case. Then, we extend our study to more general examples of BTZ black holes solution in massive gravity, charged black holes and multipartite systems. Finally, we give various interpretations of our results using resource theories such as LOCC and also bit thread picture.
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.