PRLPRLTAOPhysical Review LettersPhys. Rev. Lett.0031-90071079-7114American Physical Society10.1103/PhysRevLett.122.141601LETTERSElementary Particles and FieldsEntanglement Wedge Cross Section from the Dual Density MatrixTamaokaKotaro^{*}Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan

k-tamaoka@het.phys.sci.osaka-u.ac.jp

9April201912April2019122141416014October2018Published by the American Physical Society2019authorsPublished by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP^{3}.

We define a new information theoretic quantity called odd entanglement entropy (OEE) which enables us to compute the entanglement wedge cross section in holographic conformal field theories (CFTs). The entanglement wedge cross section has been introduced as a minimal cross section of the entanglement wedge, a natural generalization of the Ryu-Takayanagi surface. By using the replica trick, we explicitly compute the OEE for two-dimensional holographic CFT (three-dimensional anti–de Sitter space and planar Bañados-Teitelboim-Zanelli black hole) and see agreement with the entanglement wedge cross section. We conjecture this relation will hold in general dimensions.

Introduction and summary.—

The entanglement entropy (EE) quantifies the quantum entanglement between two subsystems for a given pure state. It is defined by the von Neumann entropy of a reduced density matrix ρA on a subsystem A, S(ρA)=-TrHAρAlogρA. If one considers states in the conformal field theories (CFTs) with the gravity dual [1], the EE tells us the area of the minimal surface anchored on the boundary of asymptotically anti–de Sitter space (AdS) [2,3]. This strongly suggests that the bulk gravity is encoded into the structure of quantum entanglement in the boundary. Since the EE cannot tell us all the structure of the entanglement, the minimal surface also cannot tell us the whole structure of geometry in the same manner. Therefore, finding the generalization on both sides is quite important in order to decode the profound connection between the entanglement and the geometry [4–6].

Recently, the entanglement wedge cross section EW, a generalization of the minimal surface, has been introduced [7,8]. The definition of the minimal surface and the EW will be reviewed in the next section. The cross section has been conjectured to be dual to the entanglement of purification (EOP) [9], which is a correlation measure for mixed states (for recent progress, refer to Refs. [10–15]). The EOP is also a generalization of the EE and has many nice properties consistent with the entanglement wedge cross section. However, computing the EOP is a really hard task because we need to find the minimized value from all possible purifications.

Can we then extract the entanglement wedge cross section directly from a given mixed state in QFT? In this Letter, we answer yes to this question and demonstrate it explicitly, however, from another (rather “odd”) generalization of the EE.

We first summarize the main result of the present Letter. Let ρA1A2 be a mixed state acting on bipartite Hilbert space H=HA1⊗HA2. Then we define a quantity So(no)(ρA1A2)≡11-no[TrH(ρA1A2TA2)no-1],where TA2 is the partial transposition [16] with respect to the subsystem A2. Namely, we will consider the Tsallis entropy [17] for the partially transposed ρA1A2. We are especially interested in the limit no→1, So(ρA1A2)≡limno→1So(no)(ρA1A2),where no is analytic continuation of an odd integer. (We should keep in mind that it is not enough to fix a unique analytic continuation as like other quantities computed from the replica trick.) Since the odd integer analytic continuation is crucial in the later discussion, we will call So as “odd entanglement entropy” or OEE in short. Loosely speaking, the OEE is the von Neumann entropy with respect to ρA1A2TA2; however, ρA1A2TA2 potentially contains negative eigenvalues. We will be more precise on that point further on. In particular, we will demonstrate the following three facts: First, So(ρA1A2) reduces to the EE S(ρA1) if ρA1A2 is a pure state. Second, So(ρA1A2) reduces to the von Neumann entropy S(ρA1A2) if ρA1A2 is a product state. Third, if one considers two-dimensional holographic CFT, direct calculation indeed agrees with EW(ρA1A2)≡So(ρA1A2)-S(ρA1A2)=EW(ρA1A2).In particular, we consider the subregion of the vacuum state and the thermal state. We conjecture this relation will hold even for the higher dimensional cases. From our viewpoint, the EW is similar to the coherent information [18,19], which can take even a negative value. We conclude with a discussion on this point.

Entanglement wedge cross section.—

In this section, we briefly review the holographic prescription of the EE and definition of the entanglement wedge cross section. For the rigorous definition, refer to Ref. [7]. Throughout this Letter, we assume a static geometry in the bulk and take a conventional time slice M.

To this end, we first recall the holographic EE for the static geometries [2]. Let us consider the subregion A in ∂M. We can imagine a series of co-dimension 2 surfaces ΓA which satisfy ∂ΓA=∂A and are homologous to A. Then, a minimal area one ΓAmin is called the minimal surface. The holographic EE is given by S(ρA)=(Area ofΓAmin)4GN,where GN is the Newtonian constant. Here ρA is a reduced density matrix on the subregion A. The ρA is supposed to be dual to a bulk subregion called the entanglement wedge [20–22]. We define the (time slice of) entanglement wedge as a bulk region surrounded by ΓAmin and A. (On the definition of the entanglement wedge, we should include the domain of dependence of the region surrounded by ΓAmin and A. Since we will focus on static geometries, it is enough to consider its time slice.) Note that we can start even from disconnected subregions in the boundary. For later use, we further divide A=A1∪A2≡A1A2. See Fig. 1 for a nontrivial example.

110.1103/PhysRevLett.122.141601.f1

Left: An example of the entanglement wedge cross section ΣA1A2min. (a blue dotted line). The vertical direction corresponds to the radial one of Poincare AdS3. The horizontal line coincides with a time slice of CFT2. Blue curved lines show the minimal surfaces ΓAmin(A=A1A2) for S(ρA1A2) where the ρA1A2 is a state acting on bipartite Hilbert space HA1⊗HA2 (associated with geometrical subregions A1 and A2) and is supposed to be dual to the entanglement wedge. A blue shaded region represents a time slice of the entanglement wedge. Right: If A1 and A2 are sufficiently distant, we have no connected entanglement wedge and EW(ρA1A2)=0.

Next, we define the entanglement wedge cross section. Let us regard the boundary of entanglement wedge A∪ΓAmin as a new boundary of the bulk geometry. Then, we can find a new minimal area surface ΣA1A2min, which separates A1 and A2. An important point is that we let the ΣA1A2min end not only on A but also on the ΓAmin. Since the ΣA1A2min can be regarded as a minimal cross section of the entanglement wedge, we define the entanglement wedge cross section as EW(ρA1A2)=(Area ofΣA1A2min)4GN.As an example, see again Fig. 1.

Definition and properties of <inline-formula><mml:math display="inline"><mml:msub><mml:mi>S</mml:mi><mml:mi>o</mml:mi></mml:msub></mml:math></inline-formula> & <inline-formula><mml:math display="inline"><mml:msub><mml:mi mathvariant="script">E</mml:mi><mml:mi>W</mml:mi></mml:msub></mml:math></inline-formula>.—

Partial transposition and a generalized EE: We first introduce the partial transposition that is relevant to the definition of Eq. (1). Let ρAB be a state acting on Hilbert space H=HA⊗HB and let |ei(A,B)⟩s(i=1,2,…,dimHA,B) be a complete set thereof. Using this basis, we can expand a given density matrix, ρAB=∑ik∑jℓ⟨ei(A)ej(B)|ρAB|ek(A)eℓ(B)⟩|ei(A)ej(B)⟩⟨ek(A)eℓ(B)|.We define the partial transposition of the ρAB with respect to HA,B as ⟨ei(A)ej(B)|ρABTA|ek(A)eℓ(B)⟩=⟨ek(A)ej(B)|ρAB|ei(A)eℓ(B)⟩,⟨ei(A)ej(B)|ρABTB|ek(A)eℓ(B)⟩=⟨ei(A)eℓ(B)|ρAB|ek(A)ej(B)⟩.Note that the partial transposition does not change its normalization TrHρABTA=TrHρABTB=TrHρAB=1, whereas it changes the eigenvalues. Since the partial transposition is not a completely positive map, the ρABTB can include negative eigenvalues. This negative property is actually a sign of the quantum entanglement [16] and utilized to, for example, the negativity [23]. See also a recent argument on the entanglement wedge cross section and the negativity [24].

The nth power of the ρABTB depends on the parity of n: TrH(ρABTB)n={∑λi>0|λi|n-∑λj<0|λj|n(n:odd),∑λi>0|λi|n+∑λj<0|λj|n(n:even),where λis are the eigenvalues of the ρABTB. This argument is completely the same as the negativity using the replica trick [25,26]. The main difference in the present Letter is that we are just choosing the odd integer. Therefore, OEE can be formally written as So(ρAB)=-∑λi>0|λi|log|λi|+∑λj<0|λj|log|λj|.

Pure states: Let |ΨAB⟩ be a pure state in bipartite Hilbert space H=HA⊗HB. Using the Schmidt decomposition, we can write the |ΨAB⟩ as a simple form, |ΨAB⟩=∑n=1Npn|nA⟩|nB⟩,where 0≤pn≤1, ∑npn=1. The N can be taken as min(dimHA,dimHB). One can show that the corresponding density matrix ρAB=|ΨAB⟩⟨ΨAB| and its partial transposition ρABTB have the eigenvalues, Spec(ρAB)={1,0,…,0},Spec(ρABTB)={p1,…,pN,+p1p2,-p1p2,⋯,+pN-1pN,-pN-1pN}.Here each ±pipj(i≠j) in Eq. (13) appears just once, respectively. In particular, from the definition of Eq. (10), these contributions completely cancel out. Thus, one can conclude that EW(ρAB)=So(ρAB)=S(ρA)(for pure states).

Product states: Let ρA1B1⊗σA2B2 be a product state with respect to the bipartition HA1B1⊗HB2A2. Then the So is additive, So(ρA1B1⊗σA2B2)=So(ρA1B1)+So(σA2B2),so is EW. Here we took the partial transposition with respect to B1 and B2. In particular, if τAB=τA′⊗τB′′,we have TrHτABn=TrH(τABTB)n. This fact immediately leads So(τAB)=S(τAB). Thus, we also obtain EW(τAB)=0. Note that all properties discussed the above are consistent with the entanglement wedge cross section.

Vacuum state in holographic <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>CFT</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>.—

In this section, we compute the So and the EW for mixed states in CFT on R2. We divide the total Hilbert space of CFT into HA⊗HAc, where the corresponding subregion A and its complement Ac are not necessarily to be connected. Then we can prepare a mixed state ρA1A2≡TrHAc|0⟩⟨0|, where |0⟩ is the vacuum state in CFT. Here we further divided the remaining subspace HA into two pieces, HA1 and HA2. We will focus on the holographic CFT2.

Two disjoint intervals: First, we consider disjoint interval A1=[u1,v1],A2=[u2,v2] on a time slice τ=0. In order to compute the So and the EW, we can apply the replica trick as usual [25,26]. In particular, one can write the nth power of the density matrix and its partial transposition in terms of the correlation functions for a cyclic orbifold theory CFTn/Zn, TrHA(ρA1A2)n=⟨σn(u1)σ¯n(v1)σn(u2)σ¯n(v2)⟩CFTn/Zn,TrHA(ρA1A2TA2)n=⟨σn(u1)σ¯n(v1)σ¯n(u2)σn(v2)⟩CFTn/Zn,where σn(σ¯n) is the (anti-)twist operator with scaling dimension hσn=h¯σn=(c/24)[n-(1/n)]. In terms of the original n-fold geometry, these operators map the nth replica sheet to n±1th ones. For later use, we also introduce σn2 and σ¯n2 which map the nth replica sheet to n±2th ones. The scaling dimension of the σn2 depends on the parity of n[25,26], hσn2=h¯σn2={c24(n-1n)(n:odd),c12(n2-2n)(n:even).Since we are interested in the odd integer case, this coincides with hσn. Hereafter, we will omit the suffix of the correlation function, CFTn/Zn, for brevity. Since Eq. (17) is studied in Ref. [27], we focus on the latter one.

Let us expand Eq. (18) into the conformal blocks in the t channel, where F(c,hσn,hp,x) and F¯(c,h¯σn,h¯p,x¯) are the Virasoro conformal blocks and bp’s are the OPE coefficients. We defined the cross ratio, x=(u1-v1)(u2-v2)(u1-u2)(v1-v2),and impose x=x¯, since we are interested in the time slice τ=0. The dominant contribution at the large-c limit will come from a conformal family with the lowest scaling dimension in the channel [27]. This approximation should be valid only for some specific region xc<x<1. We do not specify the lower bound xc, but just expect xc∼12. In this channel, the dominant one is universally σn2 (and σ¯n2) due to the twist number conservation, ⟨σn(u1)σ¯n(v1)σ¯n(u2)σn(v2)⟩/(|u1-v2||v1-u2|)-(c/6)[n-(1/n)]∼bσn2F(c,hσn,hσn2,1-x)F¯(c,h¯σn,h¯σn2,1-x¯).Next we would like to specify the analytic form of the above conformal blocks. This contribution of the conformal block consists only of light operators in the heavy-light limit [28]. In this case, these analytic forms are known in the literature [28–31]. In our situation, the block for σn2 has a simple form, logF(c,hσno,hσno2,1-x)=-hσnolog(1+x1-x),where we assumed analytic continuation of odd integer n≡no and the light limit c≫1 with fixed hi/c,hp/c≪1. Here we took the normalization in Ref. [29]. Therefore, we have obtained So(ρA1A2)=S(ρA1A2)+c6log[1+x1-x]+const,where S(ρA1A2)=c3log|u1-v2|ε+c3log|v1-u2|ε.Here we introduced UV cutoff ε. The constant terms do not depend on the position. For a while, we just assume the contribution from bσn2 is negligible at the large-c limit. This assumption will be justified when we consider the pure state limit discussed next. In the same way, we can compute the s-channel limit x→0. In this case, the dominant contribution will be the vacuum block as like the EE. Hence, we obtain So(ρA1A2)=c3log|u1-u2|ε+c3log|v1-v2|ε=S(ρA1A2).Therefore, we have confirmed EW(ρA1A2)={14GNlog[1+x1-x](tchannel,x∼1),0(schannel,x∼0)in the two disjoint interval case. Here we used the relation between the central charge and the three-dimensional Newtonian constant c=(3/2GN)[32]. Equation (28) precisely matches the minimal entanglement wedge cross section for AdS3 (see Fig. 1). Extension to the multi-interval cases is straightforward.

Pure state limit: Let us consider the single interval limit u2→v1 and v2→u1. This corresponds to the pure state limit for the initial mixed state. In this case, our calculation reduces to a two-point function of the twist operators. Hence, we can get the usual EE with single interval A=[u1,v1] in this limit. This is generic statement for any CFT2, but let us see this behavior from Eq. (24). If one takes the distance |u2-v1| and |v2-u1| to the cutoff scale ε, the second term of the right-hand side of Eq. (24) reduces to the length of the geodesics anchored on the boundary points u1 and v1. Moreover, this argument guarantees the constant terms from bσp2 is irrelevant at the large-c limit because of the position independence.

Thermal state in holographic <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>CFT</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>.—

In this section, we consider the thermal state in holographic CFT2, which is genuinely mixed state and is dual to the static (planar) Bañados-Teitelboim-Zanelli (BTZ) black hole [33]. Namely, we will consider the CFT2 on cylinder Sβ1×R, with single interval on the time slice, A=[-ℓ/2,ℓ/2]. The Ac denotes its complement.

To compute TrH(ρAAcTAc)no by using the replica trick, one needs to take care about the location of the branch cut, which cannot be realized as the naive conformal map from the plane z (previous results in section) to the cylinder w=σ+iτ. The correct prescription [34] is given by Tr(ρAAcTAc)no=⟨σno(-L/2)σ¯no2(-ℓ/2)σno2(ℓ/2)σ¯no(L/2)⟩β,where we introduced a finite but large cutoff L so that the conformal map can work. Thus, our “complement” Ac is now [-L/2,-ℓ/2]∪[ℓ/2,L/2], although the true time slice is the infinite line. After taking the limit no→1, we let L→∞[34]. Here the suffix of correlation function β denotes the inverse temperature. Then the corresponding Tr(ρAAc)no should be Tr(ρAAc)no=⟨σno(-L/2)σ¯no(L/2)⟩β.By using the conformal map z=e2πw/β, one can write the above correlation function as Tr(ρAAcTAc)no=(2πβ)8hσno⟨σno(e-(πL/β))σ¯no2(e-(πℓ/β))σno2(e(πℓ/β))σ¯no(e(πL/β))⟩,Tr(ρAAc)no=(2πβ)4hσno⟨σno(e-(πL/β))σ¯no(e(πL/β))⟩.Then one can expand Eq. (31) by using the conformal blocks. The dominant contribution can be again approximated by the single conformal block contribution which depends on the value of the cross ratio. Here the cross ratio is x=e-(2π/β)ℓ for sufficiently large L.

First, we consider the t-channel (x→1) limit, ℓ≪β. Then the dominant contribution from the channel is the vacuum block; hence, Eq. (31) reduces to the product of two point functions. After simple calculation, we obtain EW=c3logβπε(sinhπℓβ)+const(x∼1),where we introduced the UV cutoff ε form the dimensional analysis. The constant term comes from the normalization of two-point functions. This precisely matches the EW for the planar BTZ black hole (see Fig. 2).

210.1103/PhysRevLett.122.141601.f2

Calculation of EW for the static planar BTZ black hole (BH). The inverse temperature β is determined by the radius of the horizon. If the subsystem A is sufficiently small ℓ≪β, the EW computes the geodesics anchored on the boundary of A (black curve), which agrees with Eq. (33). For ℓ≫β, the EW does the disconnected surfaces (dotted vertical lines), which is consistent with Eq. (34).

Next, we consider the s-channel (x→0) limit, ℓ≫β. The dominant contribution in the channel is now the twist operator σn(σ¯n) because of the twist number conservation. Then we have obtained EW=c3logβπε+const(x∼0),where the constant terms come from the normalization of two-point functions and the OPE coefficients. This again agrees with the EW; however, it is important to note that this result is exact at the leading order of small x expansion. There is the position- dependent deviation of order O(x1).

Discussion.—

The EW can be negative. For example, one can find the Werner state in a 2-qubit system can have negative EW; thus, the EW is farther from the entanglement measure than the EOP. The EW(ρAB) is rather similar to the coherent information I(A⟩B)≡S(ρB)-S(ρAB)[18,19], or equivalently, the conditional entropy with the minus sign S(A|B)≡-I(A⟩B). Remarkably, these quantities can have both positive and negative values. The conditional entropy has already been discussed in the context of the differential entropy from which one can draw the bulk convex surfaces [35,36]. In particular, these were defined together with its orientation (with ± sign) [37,38]. For the differential entropy, one needs infinite series of density matrices associated with each infinitesimal subregion. On the other hand, our present result has been derived from a single density matrix ρAB dual to the entanglement wedge. This is a crucial difference compared with the differential entropy. It is very interesting to study the operational interpretation of So as like the differential entropy [39]. Further studies on generic properties of the So are also important.

From the viewpoint of the EW in the present Letter, the inequality [7,8]EW(ρAB)≥I(A:B)/2 is not always true and can impose new constraints on states dual to the classical geometries. Here we introduced the mutual information I(A:B)=S(ρA)+S(ρB)-S(ρAB). Understanding when such constraints can be satisfied is a very interesting future direction. It might be understood as the specific nature of the “holographic states” such as the absolutely maximally entangled (AME) states [40]. Derivation of the EW using the on-shell gravity action [41,42] would test our conjecture in general dimensions. Another obvious extension is to study the time-dependent setup on both sides. We would like to report on these issues in the near future.

We are grateful to Hayato Hirai, Norihiro Iizuka, Sotaro Sugishita, Tadashi Takayanagi, Satoshi Yamaguchi, and Tsuyoshi Yokoya for useful discussion and interesting conversation. The author thanks Hayato Hirai, Norihiro Iizuka, Tadashi Takayanagi, and Satoshi Yamaguchi for useful comments on the draft.

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