Quantum and Classical Correlations Inside the Entanglement Wedge

TL;DR

This paper explores the relationship between quantum and classical correlations within the entanglement wedge in holography, introducing a new geometrical measure called the entanglement wedge mutual information that captures more than just quantum entanglement.

Contribution

It demonstrates that the entanglement wedge cross section can exceed quantum entanglement measures and introduces the entanglement wedge mutual information as a novel dual to the Q-correlation.

Findings

EWCS can be larger than entanglement of formation.

EWMI satisfies properties of Q-correlation and superadditivity.

Both EWCS and EWMI capture classical correlations in holography.

Abstract

We show that the entanglement wedge cross section (EWCS) can become larger than the quantum entanglement measures such as the entanglement of formation in the AdS/CFT correspondence. We then discuss a series of holographic duals to the optimized correlation measures, finding a novel geometrical measure of correlation, the \textit{entanglement wedge mutual information} (EWMI), as the dual of the $Q$-correlation. We prove that the EWMI satisfies the properties of the $Q$-correlation as well as the strong superadditivity, and that it can become larger than the entanglement measures. These results imply that both of the EWCS and the EWMI capture more than quantum entanglement in the entanglement wedge, which enlightens a potential role of classical correlations in holography.

Figures

Equations

$$S_A=\underset{{\gamma }_A}{\min }\frac{\mathrm{Area}({\gamma }_A)}{4G_N},$$

$$E_W(A:B):=$$

$$=\underset{{\gamma }_{\mathcal{A}}}{\min }\frac{\mathrm{Area}({\gamma }_{\mathcal{A}})}{4G_N},$$

$$E_P(A:B):=\underset{{|\psi ⟩}_{A{A}^{\prime }B{B}^{\prime }}}{\min }{S}_{A{A}^{\prime }}=\frac{1}{2}\underset{{|\psi ⟩}_{A{A}^{\prime }B{B}^{\prime }}}{\min }I(A{A}^{\prime }:B{B}^{\prime }),$$

$$S_A+S_{AB}\ge S_B,$$

$$d{s}^2=\frac{d{z}^2-d{t}^2+d{x}^2}{z^2}.$$

$$I(A:B)$$

$$=\min \{\frac{2c}{3}\log \frac{2a}{ϵ},\frac{2c}{3}\log \frac{\sqrt{a/b}(b-a)}{ϵ}\},$$

$$p_{\mathrm{MI}}^{\ast }\equiv \frac{a_{\mathrm{MI}}^{\ast }(b)}{b}=3-2\sqrt{2}.$$

$$E_W$$

$$=\min \{\frac{c}{3}\log \frac{2a}{ϵ},\frac{c}{3}\log [\frac{b^2-a^2}{bϵ}]\}.$$

$$p_{\mathrm{EW}}^{\ast }\equiv \frac{a_{\mathrm{EW}}^{\ast }(b)}{b}=\sqrt{2}-1.$$

$$d{s}^2$$

$$f(z)$$

$$l_{\mathrm{MI}}^{\ast }(z_H)$$

$$l_{\mathrm{EW}}^{\ast }(z_H)$$

$${\rho }_{AB}=|\psi ⟩{⟨\psi |}_{A{B}_L}\otimes {\rho }_{B_R},$$

$$E_W(A:B_1{B}_2)=E_W(A{B}_1:B_2)+E_W(A{B}_2:B_1).$$

$${|\psi ⟩}_{\mathcal{A}{\mathcal{B}}_1{\mathcal{B}}_2}={|{\varphi }^1⟩}_{\mathcal{A}{\mathcal{B}}_1}\otimes {|{\varphi }^2⟩}_{\mathcal{A}{\mathcal{B}}_2}.$$

$$E_{sq}(A:B)$$

$$=\frac{1}{2}I(A:B)-\frac{1}{2}\underset{{\rho }_{ABE}}{\max }{I}_3(A,B,E),$$

$$E_{sq}(A:B)=\frac{1}{2}I(A:B).$$

$$E_I(A:B)$$

$$:=\frac{1}{2}\underset{{\rho }_{AB{A}^{\prime }{B}^{\prime }}}{\min }(I(A{A}^{\prime }:B{B}^{\prime })-I(A^{\prime }:B^{\prime }))$$

$$=\frac{1}{2}I(A:B)$$

$$+\frac{1}{2}\underset{{\rho }_{AB{A}^{\prime }{B}^{\prime }}}{\min }(I(A{A}^{\prime }:B{B}^{\prime })-I(A:B)-I(A^{\prime }:B^{\prime })),$$

$$E_I(A:B)=\frac{1}{2}I(A:B).$$

$$E_Q(A:B)$$

$$\equiv \underset{{\rho }_{ABE}}{\min }{f}^Q(A,B,E).$$

$$E_R(A:B)$$

$$\equiv \underset{{\rho }_{ABE}}{\min }{f}^R(A,B,E).$$

$$E_Q(A:B)$$

$$=\frac{1}{2}\underset{{|\psi ⟩}_{A{A}^{\prime }B{B}^{\prime }}}{\min }(S_A+S_B+\frac{S_{A{A}^{\prime }}+S_{B{B}^{\prime }}-S_{B{A}^{\prime }}-S_{A{B}^{\prime }}}{2})$$

$$\equiv \underset{{|\psi ⟩}_{A{A}^{\prime }B{B}^{\prime }}}{\min }{f}^Q(A,A^{\prime },B,B^{\prime }).$$

$$E_R(A:B)$$

$$=\frac{1}{2}\underset{{|\psi ⟩}_{A{A}^{\prime }B{B}^{\prime }}}{\min }(S_{AB}+S_{A{A}^{\prime }}+S_{B{B}^{\prime }}-S_{A^{\prime }}-S_{B^{\prime }})$$

$$\equiv \underset{{|\psi ⟩}_{A{A}^{\prime }B{B}^{\prime }}}{\min }{f}^R(A,A^{\prime },B,B^{\prime }).$$

$$\frac{1}{2}I\le E_Q,E_R\le E_P.$$

$$E_R(A:B)$$

Full text

I Introduction

Quantum entanglement has provided a key tool to study various aspects of modern physics from condensed matter theory to the black hole evaporation. In the AdS/CFT correspondence (Maldacena99:TheLargeNlimit, ; GubserKlebanovPolyakov98:GaugeTheoryCorrelators, ; Witten98:Anti-deSitter, ), quantum entanglement also plays a central role in the investigation of how the bulk geometrical data are encoded in the boundary field theory (Maldacena03:Eternalblackhole, ; VanRaamsdonk10:Buildingup, ; MaldacenaSusskind13:Coolhorizons, ; Takayanagi18:HolographicSpacetimes, ; VanRaamsdonk18:Buildingup, ). The Ryu-Takayanagi formula (RyuTakayanagi06:Holographic, ; LewkowyczMaldacena13:GeneralizedGravitational, ) (or the Hubeny-Rangamani-Takayanagi formula (HubenyRangamaniTakayanagi07:ACovariantholographic, ; DongLewkowyczRangamani16:DerivingCovariant, ) for covariant cases) tells us that the von Neumann entropy associated with a spacial subregion $A$ in CFTs $S_{A}\equiv S(\rho_{A})=-{\rm Tr}\rho_{A}\log\rho_{A}$ is equivalent to the area of codimension-2 minimal surface $\gamma_{A}$ which is anchored on the entangling surface $\partial A$ and homologous to $A$,

[TABLE]

at the leading order of the large $N$ limit. The von Neumann entropy $S_{A}$ is commonly called the entanglement entropy (EE) because this quantifies an amount of quantum entanglement between $A$ and its complement $A^{c}$ when the total state is pure (DonaldHorodeckiRudolph01:TheUniqueness, ). For mixed states, however, the von Neumann entropy no longer deserves to be a measure of correlation, and thus we need to find another geometrical way to measure correlations.

A generalization of the Ryu-Takayanagi surface, the entanglement wedge cross section (EWCS), was introduced in (UmemotoTakayanagi18:Entanglementofpurificationthroughholographicduality, ; NguyenDevakulHalbaschZaletelSwingle17:Enanglementofpurification, ) as the minimal cross section of the entanglement wedge (Wall12:Maximinsurfaces, ; CzechKarczmarekNogueiraVanRaamsdonk12:TheGravityDual, ; HeadrickHubenyLawrenceRangamani14:Causalityandholographic, ). This is a geometrical measure of correlations between the boundary subsystems connected by the entanglement wedge which are usually in mixed states. Thus the EWCS in boundary theories is expected to be dual to some correlation measure which is a generalization of EE for mixed states.

The EWCS was originally conjectured to be the dual of the entanglement of purification (EOP) (TerhalHorodeckiLeungDiVincenzo02:TheEntanglementofPurification, ), based on agreements of their various information-theoretic properties (UmemotoTakayanagi18:Entanglementofpurificationthroughholographicduality, ; NguyenDevakulHalbaschZaletelSwingle17:Enanglementofpurification, ) as well as compatibility with the tensor network description of AdS/CFT (Swingle09:EntanglementRenormalization, ; MiyajiTakayanagi15:SurfaceStateCorrespondence, ). The proposal has passed further consistency checks in the multipartite generalization (UmemotoZhou18:Entanglement, ) and in the conditional generalization (BaoHalpern17:HolographicInequalities, ; BaoHalpern18:Conditional, ). Refer to (HiraiTamaokaYokoya18:TowardsEntanglementofPurification, ; EspindolaGuijosaPedraza18:EntanglementwedgereconstructionandEntanglementofPurification, ; BaoChatwin-DaviesRemmem18:EntanglementofPurificationandMultiboundaryWormhole, ; YangZhangLi18:HolographicEntanglementofPurificationforthermofielddouble, ; CaputaMiyajiTakayanagiUmemoto18:HolographicEntanglementofPurificationFromConformalFieldTheories, ; LiuLingNiuWu19:EntanglementofPurificationinHolographicSystems, ; Guo19:EntanglementofPurificationandProjectiveMeasurement, ; BhattacharyyaJahnTakayanagiUmemoto19:EntanglementofPurificationinManybody, ; GhodratiKuangWangZhangZhou19:TheconnectionbetweenholographicentanglementandComplexityofPurification, ; Prudenziati19:AgeodesicWittendiagramdescriptionofholographicentanglement, ; BaoPeningtonSorceWall19:HolographidTensorNetworksinFullAdSCFT, ; BabaeiVelniMohammadiMozaffarVahidinia19:SomeAspectsofEntanglementwedgecrosssection, ; Ota19:CommentsonholographicentanglementsincutoffAdS, ; JokelaPonni19:Notesonentanglementwedgecrosssections, ; Guo19:EntanglementofPurificationanddisentanglement, ; KusukiTamaoka19:DynamicsofEntanglementWedge, ) for recent progress.

Surprisingly, several correlation measures other than EOP have been shown to be essentially equal to the EWCS with appropriate coefficients, including the logarithmic negativity (Kudler-FlamRyu18:EntanglementNegativityandMinimalEntanglementWedge, ; Kudler-FlamMacCormackRyu19:HolographicentanglementcontourBitthreadsandtheEntanglementTsunami, ; Kudler-FlamNozakiRyuTan19:QuantumvsClassical, ; KusukiKudler-FalmRyu19:DerivationofHolographicNegativity, ), the odd entropy (Tamaoka18:EntanglementWedgeCrossSection, ), and the reflected entropy (DuttaFaulkner19:Acanonicalpurification, ). With the monogamy of holographic mutual information (HaydenHeadrickMaloney11:HolographicMutualInformation, ) in mind, which strongly suggests that quantum entanglement dominates holographic correlations, we may speculate that some axiomatic measure of quantum entanglement (see e.g. (HHHH:QuantumEntanglement, )) would also be equivalent to the EWCS in holographic CFTs.

In this paper, however, we present a no-go theorem in this direction: the EWCS is not dual of various entanglement measures. Furthermore, we show that the EWCS can be strictly larger than various entanglement measures at the leading order $O(N^{2})$. It is particularly shown in a holographic configuration near to the saturation of the Araki-Lieb inequality (HubenyMaxfieldRangamaniTonni13:Holographicentanglementplateaux, ; Headrick14:Generalpropertiesof, ). We also point out that the EWCS is also larger than another type of quantum correlation, the quantum discord (HendersonVedral01:ClassicalQuantumandTotal, ; OllivierZurek01:QuantumDiscord, ). It implies that the EWCS captures more than quantum entanglement in the entanglement wedge, and it must be sensitive to classical correlations as well.

Next, we introduce a series of holographic duals for the optimized correlation measures, which are akin to the EOP. This class includes two entanglement measures; the squashed entanglement (ChristandlWinter04:SquashedEntanglement, ) and the conditional entanglement of mutual information (CEMI) (YangHorodeckiWang08:AnAdditive, ), and three total correlation measures; the EOP, the $Q$-correlation, and the $R$-correlation (LevinSmith19:OptimizedMeasures, ). We show that the CEMI reduces to half of the holographic mutual information and the $R$-correlation does to the EWCS, when they are optimized over the geometrical extensions. These two duals thus do not lead to new geometrical object in the bulk.

However, we find that the holographic dual of the $Q$-correlation provides us with a new bulk measure of correlation inside the entanglement wedge, which we call the entanglement wedge mutual information (EWMI). This quantity appropriately satisfies all of the properties of the $Q$-correlation, as well as the strong superadditivity like the EWCS. Furthermore, we show that the EWMI can also strictly become larger than the various quantum correlation measures in the same holographic configurations. It again implies that classical correlations are included in holographic correlations and they are geometrically encoded in the entanglement wedge.

This paper is organized as follows: In section 2, we review the basic notion of the EWCS and information-theoretic correlation measures. In section 3, we show that the EWCS is strictly larger than various measures of quantum correlation in a holographic configuration near to the saturation of the Araki-Lieb inequality. In section 4, we argue holographic duals of the optimized correlation measures, introduce the EWMI, and discuss the aspects of the EWMI. In section 5, we discuss some future problems. In the appendix, we prove new inequalities of the multipartite EOP and the multipartite EWCS, complementing the work of (UmemotoZhou18:Entanglement, ).

Note: We became aware that an independent work (LevinSmith19:Toappear, ) which partially overlaps with the present paper will appear soon.

II Preliminaries

II.1 Entanglement Wedge Cross Section

In the present paper we deal with static spacetime for simplicity (a generalization to non-static spacetime is straightforward using the HRT-formula (HubenyRangamaniTakayanagi07:ACovariantholographic, ; DongLewkowyczRangamani16:DerivingCovariant, ) instead of the RT-formula). The boundary subsystems are denoted by $A$ and $B$ and the entanglement wedge of $AB\equiv A\cup B$ (on a canonical time slice) is denoted by $\mathcal{M}_{AB}$ (Wall12:Maximinsurfaces, ; CzechKarczmarekNogueiraVanRaamsdonk12:TheGravityDual, ; HeadrickHubenyLawrenceRangamani14:Causalityandholographic, ). Given an entanglement wedge $\mathcal{M}_{AB}$, we may define the minimal cross section as follows (UmemotoTakayanagi18:Entanglementofpurificationthroughholographicduality, ; NguyenDevakulHalbaschZaletelSwingle17:Enanglementofpurification, ):

Suppose the boundary of $\mathcal{M}_{AB}$ is divided into two “subsystems” $\mathcal{A}$ and $\mathcal{B}$ i.e. $\partial\mathcal{M}_{AB}=\mathcal{A}\cup\mathcal{B}$ under the condition $\mathcal{A}=A\cup A^{\prime}$, $\mathcal{B}=B\cup B^{\prime}$. We include the asymptotic AdS boundary and (if it exists) black hole horizon in the boundary of $\mathcal{M}_{AB}$. The EWCS of $\mathcal{M}_{AB}$, $E_{W}(A:B)$, is defined as the minimum of the holographic entanglement entropy $S_{\mathcal{A}}$ optimized over all possible partitions (Fig.1)

[TABLE]

where $\gamma_{\mathcal{A}}$ is the RT-surface of $\mathcal{A}$. It gives a generalization of (1) for mixed states in the sense that $\gamma_{\mathcal{A}}$ reduces to the usual RT-surface when $\rho_{AB}$ is a pure state. The EWCS always satisfies the inequalities $\frac{1}{2}I(A:B)\leq E_{W}(A:B)\leq\min\{S_{A},S_{B}\}$, where $I(A:B)\coloneqq S_{A}+S_{B}-S_{AB}$ is the mutual information. The above definition can be generalized to $n$-partite subsystems (UmemotoZhou18:Entanglement, ). Remarkably, the EWCS can be regarded as a generalization of the area of a wormhole horizon in the canonical purification (DuttaFaulkner19:Acanonicalpurification, ).

II.2 Information-theoretic Correlation Measures

The EWCS was originally conjectured to be dual to the entanglement of purification (EOP) at the leading order $O(N^{2})$. The EOP is defined for a bipartite state $\rho_{AB}$ by (TerhalHorodeckiLeungDiVincenzo02:TheEntanglementofPurification, )

[TABLE]

where the minimization is performed over all possible purifications. The information-theoretic properties of EOP (TerhalHorodeckiLeungDiVincenzo02:TheEntanglementofPurification, ; BagchiPati15:MonogamyPolygamy, ) are proven for the EWCS geometrically, including the multipartite cases (UmemotoZhou18:Entanglement, ). Moreover, the surface/state correspondence of the tensor network description (MiyajiTakayanagi15:SurfaceStateCorrespondence, ) allows us to find a heuristic derivation of $E_{W}=E_{P}$ (UmemotoTakayanagi18:Entanglementofpurificationthroughholographicduality, ).

The EOP, the mutual information, the $Q$-correlation and $R$-correlation (LevinSmith19:OptimizedMeasures, ) (which will be defined in the section IV) are monotonically non-increasing under local operations (LO), but may increase by classical communication (CC). We call such non-negative quantities on $\rho_{AB}$ as (bipartite) total correlation measures. On the other hand, entanglement measures are defined by monotonicity under LOCC. There is a class of entanglement measures which satisfy additional axioms such as asymptotic continuity, which we call (bipartite) *axiomatic entanglement measures *(see e.g. (HHHH:QuantumEntanglement, )). There are various choices of additional axioms one can impose. In what follows we make a somewhat minimal requirement motivated by the uniqueness theorem (DonaldHorodeckiRudolph01:TheUniqueness, ): They coincide with EE for pure states. This may be regarded as a normalization condition for different measures. Such a class includes, for instance, the distillable entanglement $E_{D}$ (BennettDiVincenzoSmolinWootters96:MixedstateEntanglement, ; Rains99:Rigoroustreatmentofdistillable, ), the squashed entanglement $E_{sq}$ (Tucci02:Entanglement, ; ChristandlWinter04:SquashedEntanglement, ), the conditional entanglement of mutual information $E_{I}$ (YangHorodeckiWang08:AnAdditive, ), the relative entropy of entanglement $E_{RE}$ (VedralPlenioRippinKnikght97:QuantifyingEntanglement, ), the entanglement cost $E_{C}$ (BennettDiVincenzoSmolinWootters96:MixedstateEntanglement, ; HaydenHorodeckiTerhal01:TheAsymptoticEntanglementCost, ), and the entanglement of formation $E_{F}$ (BennettDiVincenzoSmolinWootters96:MixedstateEntanglement, ). There is another measure of quantum correlation, called the quantum discord $D$ (HendersonVedral01:ClassicalQuantumandTotal, ; OllivierZurek01:QuantumDiscord, ). It captures wider types of quantum correlation than quantum entanglement, and coincides with EE for pure states.

III EWCS is Not Dual of Axiomatic Entanglement Measures

First of all, we can use the generic upper bounds $E_{D},E_{sq},E_{I},E_{RE},D\leq I$ to exclude $E_{D},\ E_{sq},\ E_{I},\ E_{RE}$ and $D$ as a dual candidate of $E_{W}$, since $E_{W}(A:B)>I(A:B)$ can be observed near to the $O(1)$ phase transition of $I(A:B)$ (UmemotoTakayanagi18:Entanglementofpurificationthroughholographicduality, ). It already gives us intuition that the entanglement (or quantum correlation) measures are usually less than the EWCS in holographic CFTs. In this way, however, we cannot exclude $E_{C}$ and $E_{F}$ since they may exceed $I(A:B)$ (they can be greater than $I(A:B)/2$ LiLuo07:Totalversusquantum ). In order to do that, we consider another particular holographic setup as follows.

III.1 The EWCS in the Araki-Lieb Transition

One of outstanding characteristics of holographic CFTs is the fact that the Araki-Lieb inequality,

[TABLE]

can be saturated at the leading order $O(N^{2})$ in some particular configurations (HubenyMaxfieldRangamaniTonni13:Holographicentanglementplateaux, ; Headrick14:Generalpropertiesof, ). It is typically realized by a subsystem $A$ completely surrounded by sufficiently large $B$ (Fig.2)

. Though the following discussion is valid for the more generic setups, we focus on a configuration in Poincaré ${\rm AdS}_{3}$ with the metric

[TABLE]

Suppose the subsystems $A$ and $B$ are given by $A=[-a,a],\ B=[-b,-a]\cup[a,b]\equiv B_{1}\cup B_{2}$ for $0<a<b$ w.l.o.g. We also define the relative size of subsystems by $p\equiv\frac{a}{b}$ for $p\in(0,1)$. The mutual information $I(A:B)$ exhibits a phase transition due to that of $S_{B}=S_{B_{1}B_{2}}$ depending on the relative size $p$. The connected phase $I(B_{1}:B_{2})>0$ is preferred if $p$ is small, and the disconnected phase $I(B_{1}:B_{2})=0$ is if it is large. Thus $I(A:B)$ can be computed as

[TABLE]

where $c$ is the central charge of holographic $2$d CFTs and $\epsilon$ is the UV cutoff. It is divergent since we are taking the adjacent limit. The phase transition point of $I(A:B)$ can be read off as

[TABLE]

The Araki-Lieb inequality is saturated for $0<p<p_{{\rm MI}}^{*}$ but not for $p_{{\rm MI}}^{*}<p<1$.

The EWCS also exhibits a phase transition depending on the relative size of $A$ (Fig.3). The formula for the EWCS in Poincaré ${\rm AdS}_{3}$ is given in (UmemotoTakayanagi18:Entanglementofpurificationthroughholographicduality, ) by

[TABLE]

The phase transition of EWCS therefore happens at

[TABLE]

The phase transition points of the mutual information and the EWCS do not match, and the strict inequality $p_{{\rm MI}}^{*}<p_{{\rm EW}}^{*}$ holds (Fig.4).

This means that the EWCS saturates its upper bound $E_{W}(A:B)=S_{A}$ while the Araki-Lieb inequality is not saturated for $p\in(p_{{\rm MI}}^{*},p_{{\rm EW}}^{*})$. This observation provides us a crucial benchmark: A correlation measure $E$ can not be dual to the EWCS if $E(A:B)=S_{A}$ automatically implies the saturation of the Araki-Lieb inequality $S_{A}+S_{AB}=S_{B}$.

The Araki-Lieb inequality is also holographically saturated in the global BTZ black hole

[TABLE]

with the inverse temperature $\beta=2\pi z_{H}$ and the periodic boundary condition $x\simeq x+2\pi$. We choose $A=[-l/2,l/2]$ for $l\in(0,\pi)$ and $B$ as the reminder. It exhibits the Araki-Lieb saturation when the size of $A$ is small enough. We find the phase transition points (see e.g. (HubenyMaxfieldRangamaniTonni13:Holographicentanglementplateaux, ; UmemotoTakayanagi18:Entanglementofpurificationthroughholographicduality, ; NguyenDevakulHalbaschZaletelSwingle17:Enanglementofpurification, ))

[TABLE]

This leads to $l_{{\rm MI}}^{*}(z_{H})<I_{{\rm EW}}^{*}(z_{H})$ for any $z_{H}>0$, which confirms the above conclusion.

III.2 The Axiomatic Entanglement Measures and the Araki-Lieb Saturation

We can now cite the following fact: the entanglement of formation saturates its upper bound $E_{F}(A:B)\leq S_{A}$ if and only if the Araki-Lieb inequality is saturated (XiLuWangLi12:NecessaryandSufficient, ). This immediately means that $E_{F}$ is not dual of $E_{W}$ from the above observation. Furthermore, this statement can be generalized to other measure $E$ which satisfies (i) the monotonicity $E(A:B_{1}B_{2})\geq E(A:B_{1})$, (ii) $E(A:B)=S_{A}=S_{B}$ for pure states, and (iii) $E\leq E_{F}$. Indeed, the saturation $E(A:B)=S_{A}$ leads to $E_{F}(A:B)=S_{A}$ from (iii), which is equivalent to the Araki-Lieb saturation. The opposite is shown by using the unique structure (up to isometries) of states which saturate the Araki-Lieb inequality (ZhangWu11:Onconjectures, )

[TABLE]

where the Hilbert space of $B$ is decomposed into $\mathcal{H}_{B}=\mathcal{H}_{B_{L}}\otimes\mathcal{H}_{B_{R}}$. This form with (i) and (ii) leads to the saturation by $S_{A}\geq E(A:B_{L}B_{R})\geq E(A:B_{L})=S_{A}$. This class of correlation measures especially includes the entanglement cost $E_{C}$, and the other entanglement measures mentioned at the beginning of this section as well. Thus we can also exclude $E_{C}$ and the other measures as a dual candidate of $E_{W}$.

In addition, it also means that this class of entanglement measures must be strictly less than $E_{W}$ for the states of $p\in(p_{{\rm MI}}^{*},p_{{\rm EW}}^{*})$. Since there seems to be no reason to believe that these states are singular among the holographic states, we argue that the EWCS is generically larger than these entanglement measures in holographic CFTs (unless the Araki-Lieb inequality is saturated).

III.3 Interpretation from the Holographic Entanglement of Purification

By contrast, the EOP evades the above criteria since (iii) does not hold, and still deserves consideration as a possible dual of the EWCS. In addition, there exists a class of states for which $E_{P}(A:B)=S_{A}$, but the Araki-Lieb inequality is not saturated (CheistandlWinter05:Uncertainty, ). Similarly, the logarithmic negativity, the odd entropy, and the reflected entropy do not satisfy (iii), and we expect that they also coincide with $S_{A}$ with an appropriate coefficient for some mixed states without the Araki-Lieb saturation.

Furthermore, we can understand the behavior of the EWCS through the Araki-Lieb transition based on the surface/state correspondence. First, we note a remarkable equality which holds after the phase transition $p>p_{{\rm EW}}^{*}$,

[TABLE]

This equation can be explained as follows: the two configurations $p>p_{{\rm EW}}^{*}$ and $p<p_{{\rm EW}}^{*}$ are equivalent to whether the correlation** $I(\mathcal{B}_{1}:\mathcal{B}_{2})$** vanishes or not. For $p>p_{{\rm EW}}^{*}$, we see $I(\mathcal{B}_{1}:\mathcal{B}_{2})=0$, and it immediately leads to the unique form (up to isometries on $\mathcal{H}_{\mathcal{A}}$) of any purification (Uhlmann76:Thetransitionprobabilityeltit, )

[TABLE]

This form of optimal purification, common to each of the three EWCSs, clearly establishes the equality (16).

On the other hand, if $p<p_{{\rm EW}}^{*}$, remaining correlation $I(\mathcal{B}_{1}:\mathcal{B}_{2})>0$ will drastically change the structure of purifications from (17). In this case, the optimal purification would be simply given by the standard purification (TerhalHorodeckiLeungDiVincenzo02:TheEntanglementofPurification, ) i.e. setting $A^{\prime}$ as empty. In this sense, the phase transition point $p_{{\rm EW}}^{*}$ is thus understood as a point at which the standard purification switches with the decoupled purification (17) as the optimal purification.

IV Holographic Duals of the Optimized Correlation Measures

We observed that the EWCS can not be the dual of any axiomatic entanglement measures. Then a natural question is as follows: Is there any axiomatic entanglement measure which deserves a geometrical dual?

IV.1 Holographic Dual of the Optimized Entanglement Measures

Here we discuss two possible candidates: the squashed entanglement $E_{sq}$ (ChristandlWinter04:SquashedEntanglement, ) and the conditional entanglement of mutual information $E_{I}$ (YangHorodeckiWang08:AnAdditive, ). Their definitions are reminiscent of the EOP (4). The squashed entanglement is defined as

[TABLE]

where $\rho_{ABE}$ is an extension such that ${\rm Tr}_{E}\rho_{ABE}=\rho_{AB}$, and $I_{3}(A,B,C)=S_{A}+S_{B}+S_{C}-S_{AB}-S_{BC}-S_{CA}+S_{ABC}$ is the tripartite information.

We now impose a crucial assumption to find a possible geometrical dual of $E_{sq}$: Performing the minimization over a class of extensions which have classical geometrical duals, is sufficient to achieve the minimum. It implies that the monogamy of mutual information $I_{3}(A,B,E)\leq 0$ (HaydenHeadrickMaloney11:HolographicMutualInformation, ) must hold for the extensions $\rho_{ABE}$. A holographic dual of the squashed entanglement is then given by half of the holographic mutual information (HaydenHeadrickMaloney11:HolographicMutualInformation, ),

[TABLE]

This is achieved by a trivial extension $E=\emptyset$. This relation implies that the holographic mutual information should satisfy the properties of the squashed entanglement, such as the monogamy relation $E_{sq}(A:BC)\geq E_{sq}(A:B)+E_{sq}(A:C)$ (KoashiWinter04:Monogamy, ) which is generically considered as a characteristic of quantum entanglement. The holographic mutual information indeed satisfies the monogamy relation as mentioned above. It is worth noting that the saturation of $E_{sq}\leq\frac{1}{2}I$ occurs if $\rho_{AB}$ saturates Araki-Lieb inequality, but this is not the only possibility (CheistandlWinter05:Uncertainty, ).

The relation (20), or the monogamy property of the mutual information, suggests a striking conclusion: the mutual information captures only quantum entanglement in holography, even though it is usually a total correlation measure (HaydenHeadrickMaloney11:HolographicMutualInformation, ).

We give support for this argument by elaborating on the conditional entanglement of mutual information $E_{I}$ (YangHorodeckiWang08:AnAdditive, ). It is defined by

[TABLE]

where $\rho_{ABA^{\prime}B^{\prime}}$ is again any extension of $\rho_{AB}$. It is an additive measure of quantum entanglement (YangHorodeckiWang08:AnAdditive, ). Suppose the monogamy of mutual information for some geometric extensions $\rho_{AA^{\prime}BB^{\prime}}$ is enough to find the minimum. Then we find $I(AA^{\prime}:BB^{\prime})-I(A:B)-I(A^{\prime}:B^{\prime})\geq I(A:B^{\prime})+I(B:A^{\prime})\geq 0$, which leads to the holographic dual of the CEMI as half of the holographic mutual information (with a trivial extension $A^{\prime}B^{\prime}=\emptyset$)

[TABLE]

It again implies that the holographic mutual information only captures quantum entanglement. This is in contrast to the EWCS which still captures classical correlations in holography. Indeed, it was pointed out in (BhattacharyyaJahnTakayanagiUmemoto19:EntanglementofPurificationinManybody, ) that the EOP could be more sensitive to classical correlations than the mutual information.

These proposals about $E_{sq}$ and $E_{I}$ are obviously consistent with the Araki-Lieb transition discussed above, since the holographic dual of $E_{sq}$ and $E_{I}$ would be the holographic mutual information itself.

We emphasize the fact that two differently defined measures of entanglement reduce to the same quantity $\frac{1}{2}I$ in holography. To our knowledge, there seems to be no obstruction to speculate that the other entanglement measures such as $E_{C}$ and $E_{F}$ also coincide with $\frac{1}{2}I$. We leave investigating their holographic duals as an interesting future work.

IV.2 Holographic Duals of the Optimized Total Correlation Measures

All of the correlation measures $E_{P},\ E_{sq},\ E_{I}$ are defined as the minimum of a linear combination of von Neumann entropies over all possible purifications or extensions. This class of correlation measures is called the optimized correlation measures (LevinSmith19:OptimizedMeasures, ). There are two other such measures, the $Q$-correlation and the $R$-correlation, introduced in (LevinSmith19:OptimizedMeasures, )

[TABLE]

The symmetry between $A$ and $B$ becomes obvious in the equivalent expression in terms of purifications ($E\equiv A^{\prime}$),

[TABLE]

The $Q$-correlation and the $R$-correlation are non-increasing under local operations, but not necessarily under LOCC. They satisfy the inequality (LevinSmith19:OptimizedMeasures, )

[TABLE]

We note a close relationship between the $R$-correlation and the CEMI, which is clear from the following expression of $E_{R}$

[TABLE]

It is similar to the CEMI (21), though the minimization of the CEMI is performed over all possible extensions.

The Holographic Counterparts

Here we investigate holographic duals of the $Q$-correlation and the $R$-correlation. The definition of the holographic dual candidate of $E_{Q}$ is stated as follows (we focus on static geometries):

Given an entanglement wedge $\mathcal{M}_{AB}$, divide its boundary into $\partial\mathcal{M}_{AB}=\mathcal{A}\cup\mathcal{B}$ so that $\mathcal{A}=A\cup A^{\prime}$ and $\mathcal{B}=B\cup B^{\prime}$. Then minimize the combination of holographic entanglement entropy $f^{Q}(A,A^{\prime},B,B^{\prime})$ over all possible partitions. We define the minimum as the entanglement wedge mutual information (EWMI), denoted by $E_{M}$

[TABLE]

An example of the EWMI is depicted in the Fig.5. It may be regarded as the half of the mutual information between $A$ (or $B$) and the “subsystem” $M$ assigned to the codimention-2 cross section of $S_{AA^{\prime}}\ (=S_{BB^{\prime}})$.

Generically, the EWMI requires us to consider many complicated configurations of $A^{\prime}$ and $B^{\prime}$ in order to minimize $f^{Q}$. For some simple cases, however, such as the two disjoint intervals in $\rm{AdS}_{3}/\rm{CFT}_{2}$ or the (symmetric) Araki-Lieb saturating configurations, there is an intuitive way to compute $E_{M}$ owing to the symmetry of setup: Minimize (half of) the mutual information $\max\{I(A:M),I(B:M)\}$ over all possible choices of the cross sections,

[TABLE]

where $M$ corresponds to the cross section of some partition $A^{\prime}\cup B^{\prime}$. This form also clarifies a useful relation

[TABLE]

for at least one of the optimal cross sections $M^{*}$. Note that the optimal purification for $E_{M}$ is not necessarily unique, nor does it necessarily agree with that of $E_{W}$ (Fig.5). We will see both concrete examples in the below discussion of the Araki-Lieb transition of $E_{M}$. There is another suggestive form of $E_{M}$ for these cases,

[TABLE]

where we have used $I(A^{\prime}:B^{\prime})=0$ which holds for the ancillary subsystems on the RT-surface. At least one of $I(A:B^{\prime})$ and $I(B:A^{\prime})$ must vanish at this point because the whole system is homologically trivial. Moreover, the balancing condition (36) is equivalent to the condition $I(A:B^{*})=I(B:A^{*})$. Thus we can conclude that both of $I(A:B^{\prime})$ and $I(B:A^{\prime})$ should vanish for the balanced optimal partition. As a result, we reach a formula

[TABLE]

where $A_{b}^{*}$ is the balanced optimal partition. We may define the deviation from the EWCS due to the balancing term $S_{BA^{\prime}}$ as

[TABLE]

We will check this formula (38) by direct computation in the Araki-Lieb transition.

A caveat is that neither the formula (35) nor (38) is necessarily valid for any configurations, and there possibly exists other types of optimal configurations of $A^{\prime}$ and $B^{\prime}$ for more complicated subsystems. Indeed, for example, if we set $|B_{1}|>|B_{2}|$ in the Araki-Lieb saturating configuration, $E_{M}$ can be realized by an optimal configuration neither of $I(A:M^{*})$ nor $I(B:M^{*})$, but of a combination $I(A:M^{*}_{A})+I(B:M^{*}_{B})$ where $M^{*}_{A}\cup M^{*}_{B}=M$. Such a configuration is not preferred for the disjoint two intervals (LevinSmith19:Toappear, ) nor for the symmetric Araki-Lieb configuration. This example indicates that we need to replace $\max\{I(A:M),I(B:M)\}$ in (35) with $\max_{M_{A}\cup M_{B}=M}\{I(A:M_{A})+I(B:M_{B})\}$ in general. We leave proving or disproving it for generic configurations as an important future work.

The EWMI satisfies the properties of $E_{Q}$. For example, it can not be greater than the EWCS,

[TABLE]

which must hold to be consistent with $E_{W}=E_{P}$ from (32). One can prove this inequality by drawing a picture, but an easier way is to use the von Neumann entropy to represent the corresponding geometrical areas. Suppose the optimal partition of $E_{W}$ is given by $A_{W}^{*}$ and $B_{W}^{*}$. Then we can show $E_{W}=S_{AA_{W}^{*}}\geq\frac{1}{2}(S_{A}+S_{B}+S_{AA_{W}^{*}}-S_{BA_{W}^{*}})\geq E_{M}$, where we have used strong subadditivity.

Similarly, $E_{M}$ can not be less than half of holographic mutual information,

[TABLE]

It is clear from (38) as $S_{AA_{b}^{*}}\geq E_{W}(A:B)\geq\frac{1}{2}I(A:B)$. These properties also guarantee that $E_{M}(A:B)=S_{A}=S_{B}$ for pure states, and that $E_{M}$ vanishes if and only if $I(A:B)=0$ (with $A^{*}=\gamma_{A}$ and $B^{*}=\gamma_{B}$). It also shows the extensivity $E_{M}(A_{1}:B)\geq E_{M}(A_{2}:B)$ when $A_{1}\supset A_{2}$ (Fig.6). The additivity $E_{M}(\rho_{A_{1}B_{1}}\otimes\sigma_{A_{2}B_{2}})=E_{M}(\rho_{A_{1}B_{1}})+E_{M}(\sigma_{A_{2}B_{2}})$ is also clear because the decoupled state corresponds to disjoint geometries. All of these consistent properties tempt us to propose the relation (at the leading order $O(N^{2})$)

[TABLE]

In pure ${\rm AdS}_{3}$, the $E_{M}$ for two disjoint intervals has a simple expression (Fig.7).

In such cases, the optimal partition coincides with that of $E_{W}$, as it is obvious from the conformal symmetry. Thus $E_{M}$ becomes just the average of $\frac{1}{2}I$ and $E_{W}$ by (38),

[TABLE]

From this expression, we can easily confirm all of the properties of $E_{M}$ mentioned above. This expression is not necessarily true in generic setups such as three or more multipartite intervals or black hole geometry.

Surprisingly, the EWMI also satisfies the strong superadditivity

[TABLE]

which can be proven geometrically (Fig.8). It is similar to the proof of the strong superadditivity of $E_{W}$ (UmemotoTakayanagi18:Entanglementofpurificationthroughholographicduality, ).

The relation (44) is not a generic property of $E_{Q}$. Thus we may regard it as a characteristic of holographic correlations, as with the holographic entropy cone (HaydenHeadrickMaloney11:HolographicMutualInformation, ; BaoNezamiOoguriStoicaSullyWalter15:TheHolographicEntropyCone, ; RotaWeinberg18:Newconstraintsforholographicentropyfrommaximin, ; HubenyRangamaniRota18:TheHolographicentropyArrangement, ; Hernandez19:HolographicentropyconeforFive, ).

The dual of $E_{R}$ is defined in the same manner, replacing $f^{Q}$ with $f^{R}$ in the above procedure. However, it turns out that this definition is equivalent to that of the EWCS. It stems from the fact we implicitly used in the definition of $E_{M}$ (and $E_{W}$) that it is sufficient to consider the ancillary systems $A^{\prime}$ and $B^{\prime}$ located only on the RT-surface $S_{AB}$ for minimization. For such subsystems we find $I(A^{\prime}:B^{\prime})=S_{A^{\prime}}+S_{B^{\prime}}-S_{AB}=0$, resulting in $E_{R}=E_{P}=E_{W}$ from (33). We also state it as a holographic proposal

[TABLE]

The additivity of the EWCS is consistent with that of the $R$-correlation (LevinSmith19:OptimizedMeasures, ). The relation between $E_{R}$ and $E_{I}$ then gives an interesting perspective on the geometrical extensions: if only pure geometries are available, the correlations can reduce to $E_{W}$ at most. If mixed geometries are also allowed, then the inaction gives a further reduction to $\frac{1}{2}I$.

IV.3 The EWMI in the Araki-Lieb Transition

Let us study $E_{M}$ in the Araki-Lieb transition discussed in section III.1 in detail. First, we remark that $E_{M}=S_{A}$ should hold for $p<p_{{\rm MI}}^{*}$ from the inequality $\frac{1}{2}I\leq E_{M}\leq E_{W}$, while it also can be checked by direct computation. For $p>p_{{\rm MI}}^{*}$, the situation is more complicated than $E_{W}$ due to the four configurations of $S_{AA^{\prime}}-S_{BA^{\prime}}$. For simplicity, we fix the size $b$ to unit size in the setup and always deal with the relative size $p$ as the parameter.

The two phases of $S_{AA^{\prime}}$ and the two phases of $S_{BA^{\prime}}$ are depicted in Fig.9.

The minimal configuration depends not only on the parameter $p$ but also on the size of $A^{\prime}$, parameterized by $q\in(0,1)$. We can easily find out the minimal configurations in the extremal cases: in the small $A^{\prime}$ limit ($q\to 0$), the phase (A1) for $S_{AA^{\prime}}$ and the phase (B1) for $S_{BA^{\prime}}$ are preferred (recall $I(B_{1}:B_{2})=0$ for $p>p_{{\rm MI}}^{*}$). Similarly, we have the phase (A2) for $S_{AA^{\prime}}$ and the phase (B2) for $S_{BA^{\prime}}$ in the large $A^{\prime}$ limit ($q\to 1$). Therefore, as we increase $q$ from [math] to $1$, we will see a phase transitions of $S_{AA^{\prime}}-S_{BA^{\prime}}$ for the fixed $p>p_{{\rm MI}}^{*}$ in either path

[TABLE]

Note that $S_{BA^{\prime}}$ is always in (B2) regardless to $q$ for $p<p_{{\rm MI}}^{*}$.

It is not hard to show that increasing $q$ may decrease $S_{AA^{\prime}}-S_{BA^{\prime}}$ only in the phase (A2, B1). In the phase (A1, B1), changing $q$ has no effect at all, and in the phase (A1, B2) and (A2, B2), increasing $q$ does increase $S_{AA^{\prime}}-S_{BA^{\prime}}$. Therefore, a nontrivial optimal partition for $E_{M}$ is observed only when the phase transition follows the path (II).

With this in mind, we find the phase transition points $q^{*}$ of $S_{AA^{\prime}}$ and $S_{BA^{\prime}}$ as a function of $p$

[TABLE]

[TABLE]

These are plotted in Fig. 10.

The region $q<q_{AA^{\prime}}^{*}$ corresponds to the phase (A1) for $S_{AA^{\prime}}$, and the region $q<q_{BA^{\prime}}^{*}$ to the phase (B1) for $S_{BA^{\prime}}$. The phase transition of $E_{M}$ happens at the crossing point $p_{{\rm EM}}^{*}$ at which $q_{AA^{\prime}}^{*}(p_{{\rm EM}}^{*})=q_{BA^{\prime}}^{*}(p_{{\rm EM}}^{*})$ holds,

[TABLE]

The ancillary system $A^{\prime}$ of any size $q\leq q_{BA^{\prime}}^{*}$ achieves the minimum $E_{M}=S_{A}$ for $p<p_{{\rm EM}}^{*}$. For $p>p_{{\rm EM}}^{*}$, the minimum of $S_{AA^{\prime}}-S_{BA^{\prime}}$ is obtained at $q=q_{BA^{\prime}}^{*}(p)$.

Therefore, we have found $E_{M}=S_{A}$ for $p\leq p_{{\rm EM}}^{*}$, and $E_{M}=\frac{1}{2}(S_{A}+S_{B}+S_{AA^{*}}-S_{BA^{*}})=\frac{1}{2}(S_{A}-S_{A^{*}}+S_{AA^{*}})$ for $p>p_{{\rm EM}}^{*}$. In the latter case, the size of $A^{*}$ is given by $q_{BA^{\prime}}^{*}(p)$, and $S_{AA^{*}}$ is in the phase (A2) and $S_{BA^{*}}$ is at the phase transition point (B1)$=$(B2). We may compute $S_{A^{*}}$ as $S_{A^{*}}=\frac{1}{2}(S_{A}+S_{AB}-S_{B})$ from the equality condition (B1)=(B2).

After all, we obtain $E_{M}$ in the Araki-Lieb transition as

[TABLE]

where $S_{AA^{*}}\equiv S_{AA^{*}}(p,q_{BA^{\prime}}^{*}(p))$ denotes a contribution from the geodesics between $\partial A$ and $\partial A^{*}$,

[TABLE]

This result (51) confirms the shortcut formula (38). Note that $\frac{1}{2}I(A:B)=S_{A}$ for $p<p_{{\rm MI}}^{*}$ and that the balanced optimal partition for $p\in(p_{{\rm MI}}^{*},p_{{\rm EM}}^{*})$ is given by $A^{\prime}$ of the size $q=q_{BA^{\prime}}^{*}(p)$, not the trivial partition (though it is also optimal). The balancing condition $I(A:M^{*})=I(B:M^{*})$ generically corresponds to the condition (B1)$=$(B2). The deviation (39) is given by

[TABLE]

The plots of $E_{M}$ and $D_{b}$ are given in Fig.11 and Fig.12.

In particular, the strict inequality $p_{{\rm MI}}^{*}<p_{{\rm EM}}^{*}$ indicates that $E_{M}$ must be strictly greater than the axiomatic entanglement measures for $p\in(p_{{\rm MI}}^{*},p_{{\rm EM}}^{*})$, based on the same logic as the EWCS. One can also confirm that $E_{M}$ exhibits the same kind of phase transition in the global BTZ black hole.

V Discussion

We have introduced a series of possible holographic duals to the optimized correlation measures. The crucial assumption for the equivalence was that the geometrical extensions are enough to achieve their minimum in holographic CFTs. They demonstrate many properties which are completely consistent with the original information-theoretic measures.

We showed that the EWCS and the EWMI can be larger than the wide class of entanglement measures, while the holographic mutual information is not necessarily. This implies that the EWCS and the EWMI should be more sensitive to classical correlations than the holographic mutual information. Note that both of the EWCS and the EWMI satisfy the strong superaddivity, which is a weaker property of quantum entanglement than the monogamy relation (since the latter induces the former). In addition, the EOP or the reflected entropy is supposed to be more sensitive to classical correlations than the mutual information (BhattacharyyaJahnTakayanagiUmemoto19:EntanglementofPurificationinManybody, ; KusukiTamaoka19:DynamicsofEntanglementWedge, ). It will be interesting future work to investigate a role of classical correlation in holographic CFTs.

There is a caveat that all of our discussions are restricted to the leading order $O(N^{2})$. In particular, the Araki-Lieb transition relies on the property of the holographic entanglement entropy at this order. If one includes quantum corrections from bulk entanglement entropy at $O(N^{0})$ (FaulknerLewkowyczMaldacena13:QuantumcorrectiontoHolographic, ; BarrellaDongHartnollMartin13:HolographicEntanglementBeyondClassical, ), the rigorous relation will be violated. For instance, the structure of state (15) is not robust against small correction to the exact saturation (HaydenJozsaPetzWinter04:StructureofStates, ). The saturation of the EWCS or the EWMI should also be found only in the large $N$ limit. We expect, however, that our conclusion itself still survives: the EWCS and the EWMI with some appropriate quantum corrections will still capture classical correlations.

The bit thread formalism (FreedmanHeadrick17:BitThreads, ) has been cooperative with these holographic optimized correlation measures. The bit threads for the bipartite EWCS was discussed in (AgonDeBouerPedraza18:GeometricAspectsofHolographicBitThreads, ; DuChenShu19:BitthreadsandholographicEntanglementOfPurification:2019emy, ; BaoChatwinPollack19:TowardsaBitThreads, ; HarperHeadrick19:BitthrreadsandHolographicEntanglementofPurification, ) and generalized to the multipartite EWCS (HarperHeadrick19:BitthrreadsandHolographicEntanglementofPurification, ). It will be interesting to seek a bit thread formalism for $E_{M}$ as well. Also, a multipartite generalization of $E_{M}$ would provide us a new tool to probe a specific aspect of the holographic correlations.

Acknowledgments

We are grateful to Arpan Bhattacharyya, Souvik Dutta, Jonathan Harper, Masamichi Miyaji, Yoshifumi Nakata, Tadashi Takayanagi, and Yang Zhou for valuable comments on a draft of the present paper. We thank Jesse C. Cresswell and Massimiliano Rota for a lot of stimulating discussions which essentially motivated this work. We are grateful to "Quantum Information and String Theory 2019" at the Yukawa Institute for Theoretical Physics in Kyoto University where this work was initiated. We also thank “Gravity - New perspectives from strings and higher dimensions 2019” at the Centro de Ciencias de Benasque Pedro Pascual, where this work was completed. K.U. is supported by Grant-in-Aid for JSPS Fellows No.18J22888.

VI Appendix

VI.1 The Multipartite Generalization

In this appendix, we complement some missing pieces in the previous study (UmemotoZhou18:Entanglement, ) of the multipartite generalization of the mutual information, the EOP, and the squashed entanglement as well as their holographic duals. The mutual information $I(A:B)$ has various multipartite generalizations. One of them is called the total correlation defined by

[TABLE]

where $S(\rho||\sigma)={\rm Tr}\rho(\log\rho-\log\sigma)$ is the relative entropy. There is another generalization called the dual total correlation

[TABLE]

where $S(A|B)=S_{AB}-S_{B}$ is the conditional entropy and $\overset{i}{\check{\cdots}}$ denotes the exclusion of $A_{i}$. The $T_{n}$ and $D_{n}$ are monotonically non-increasing under strict local operations, vanish if and only if the state is totally decoupled, and $T_{n}=D_{n}=\sum_{i=1}^{n}S_{A_{i}}$ if the state is pure.

A multipartite generalization of the EOP (UmemotoZhou18:Entanglement, ; BaoHalpern18:Conditional, ) and the squashed entanglement (YangHorodeckiHorodeckiHorodeckiOppenheimSong09:Squashed, ; AvisHaydenSavov08:DistributedCompressionandMultipartySquashedEntanglement, ) are given as follows:

[TABLE]

where $T_{n}(A_{1}:\cdots:A_{n}|E)=I(A_{1}:A_{2}|E)+I(A_{1}A_{2}:A_{3}|E)+\cdots+I(A_{1}\cdots A_{n-1}:A_{n}|E)$. The multipartite EOP is monotonically non-increasing under strict local operations. The holographic dual of the multipartite EOP was proposed as the multipartite EWCS (UmemotoZhou18:Entanglement, ).

We can generalize the discussion of the holographic dual of the bipartite squashed entanglement as follows. The multipartite squashed entanglement can be written as

[TABLE]

where we define $Q_{n}(A;E)\coloneqq I(A_{1}\cdots A_{n}:E)-\sum_{i=1}^{n}I(A_{i}:E).$ This can be both positive and negative in generic quantum system. In holography, however, the monogamy of mutual information implies $Q_{n}\geq 0$. For $n=2$, it reproduces the non-positivity of tripartite information $Q_{2}(AB;E)=I(AB:E)-I(A:E)-I(B:E)=-I_{3}(A:B:E)\geq 0$. It again results in a conjecture that holographic multipartite squashed entanglement is equivalent to half of the total correlation,

[TABLE]

For the latter convenience, we introduce two non-negative quantities for $n\geq 3$:

[TABLE]

They are normalized so that $X_{n}=Y_{n}=\sum_{i=1}^{n}S_{A_{i}}$ holds for pure states. They are positive semi-definite as it is clear from the following expressions,

[TABLE]

where $D_{n}(A_{1}:\cdots:A_{n}|E)=I(A_{1}:A_{2}\cdots A_{n}|E)+I(A_{2}:A_{3}\cdots A_{n}|A_{1}E)+\cdots+I(A_{n-1}:A_{n}|A_{1}\cdots A_{n-2}E)$. $X_{n}$ is monotonically non-increasing under strict local operations, while the $Y_{n}$ is not necessarily. Both $X_{n}$ and $Y_{n}$ are not faithful i.e. there exists a state which is not decoupled $\rho_{A}\neq\rho_{A_{1}}\otimes\cdots\otimes\rho_{A_{n}}$ but $X_{n}=0$ or $Y_{n}=0$. Thus we do not consider each of them as a good correlation measure. Note a balance equation,

[TABLE]

For holographic states, the monogamy of mutual information leads to a generic ordering

[TABLE]

Indeed, $T_{n}\leq D_{n}$ follows from the monogamy of mutual information,

[TABLE]

Then $X_{n}\leq T_{n}$ and $D_{n}\leq Y_{n}$ are obvious by their definition.

Now we present some lower bounds on the multipartite EOP, which generalizes and complements the inequalities proven in (UmemotoZhou18:Entanglement, ). The multipartite EOP is bounded from below by half of any multipartite correlation measure $\Theta$ which satisfies (i) $\Theta=\sum_{i=1}^{n}S_{A_{i}}$ for pure $n$-partite states, and (ii) is non-increasing under strict local operations,

[TABLE]

It is obvious from the definition (59) following the same logic as in (UmemotoZhou18:Entanglement, ). Here $T_{n}$, $D_{n}$, and $X_{n}$ satisfy both conditions, but $Y_{n}$ does not satisfy (ii). Thus, we get three inequalities for generic multipartite states

[TABLE]

The lower bounds by $T_{n}$ and $X_{3}$ were proven in (UmemotoZhou18:Entanglement, ), and the above inequality gives $n$-partite generalization for $X_{n}$. On the other hand, the bound by $D_{n}$ is totally new. Interestingly, $D_{n}$ gives a stricter lower bound on $E_{P}$ than $T_{n}$ in holography by the ordering (67). One can check that $E_{W}\geq\frac{1}{2}D_{n}$ always holds, while $E_{W}\geq\frac{1}{2}Y_{n}$ is not true in general.