Quantum entanglement provides us a key to understanding how gravity emerges from field theories. This is manifest in the anti–de Sitter space (AdS)/conformal field theory (CFT) correspondence (or holography) [1]. In AdS/CFT, the entanglement entropy, the unique measure of quantum entanglement for pure states, is computed as a minimal area in AdS [2,3].

For mixed states, entanglement entropy can measure neither quantum entanglement nor correlations between two subsystems. Recently, it was conjectured that a quantity called entanglement of purification [4], which is a good measure of total correlations, has a simple geometric interpretation in AdS/CFT, called the holographic entanglement of purification [5,6]. This quantity is a natural extension of entanglement entropy for mixed states. An extension to multipartite correlations is also given in Ref. [7].

The entanglement of purification (EOP), written as EP(ρAB), is a measure of correlations between two subsystems A and B for a mixed state ρAB and is defined as follows [4]. Consider a purification of ρAB, given by |Ψ⟩AA˜BB˜, by enlarging the Hilbert space as HA⊗HB→HA⊗HB⊗HA˜⊗HB˜ such that ρAB=TrA˜B˜[|Ψ⟩AA˜BB˜⟨Ψ|AA˜BB˜].(1)Among infinitely many different choices of the purifications |Ψ⟩AA˜BB˜, the EOP is defined by minimizing the entanglement entropy SAA˜=-Tr[ρAA˜logρAA˜] as EP(ρAB)=min|Ψ⟩AA˜BB˜[SAA˜].(2)

The holographic entanglement of purification (HEOP) EW(ρAB) is defined by the area of the minimal cross section of entanglement wedge, denoted by ΣABmin: EW(ρAB)=A(ΣABmin)4GN,(3)and the equality EW(ρAB)=EP(ρAB) was conjectured in Refs. [5,6] based on quantum information theoretic properties. In this Letter, we focus on static backgrounds and always take a canonical time slice in the AdS.

The entanglement wedge is the region inside the AdS which is dual to the density matrix ρAB [8–10]. Therefore, the HEOP [Eq. (3)] is quite fundamental to understanding how the geometry in gravity corresponds to density matrices in CFTs. It also has an important meaning in the bit threads interpretation of AdS/CFT [11–14].

Nevertheless, direct comparisons between the HEOP and EOP have not been done so far, mainly because the minimization procedure in Eq. (2) is very hard in quantum field theories. Refer to Refs. [6,15,16] for numerical lattice calculations of EOP in free field theories. Also in Refs. [17–19], connections between EW(ρAB) and quantities other than the EOP have been proposed. For other recent progresses on the HEOP, refer to Refs. [20–32]

In this Letter, we would like to present a direct comparison between the HEOP EW(ρAB) and the entanglement entropy SAA˜ for a special class of purification |Ψ⟩AA˜BB˜, obtained by the path-integral optimization [33–35]. We will focus on several examples in AdS3/CFT2 and will find both quantities always agree with each other in the regime of validity of our computations.

Consider a two-dimensional (2D) CFT on a Euclidean flat space R2, which is described by the complex coordinate (w,w¯)=(ξ+iτ,ξ-iτ) with the metric ds2=dwdw¯=dτ2+dξ2.(4)The theory is invariant under the Weyl transformation of the metric: ds2=e2ϕ(τ,ξ)(dτ2+dξ2),(5)where ϕ(τ,ξ) is any function.

The path-integral optimization [33] is a special choice of Weyl transformation which (i) preserves the quantum state |Ψ⟩ at a particular time τ=τ0 and (ii) minimizes the path-integral complexity CL[ϕ] (defined below).

Since the wave functional of |Ψ⟩ can be computed by the Euclidean path integral on the lower half plane -∞<τ≤τ0, the first condition (i) is given by the boundary condition e2ϕ(τ0,ξ)=1.(6)The path-integral complexity is defined by the Liouville action CL[ϕ]=c24π∫-∞τ0dτ∫-∞∞dξ((∂τϕ)2+(∂ξϕ)2+e2ϕε2),(7)where ε is the UV cutoff or equally lattice spacing and c is the central charge. The reason why we identify CL[ϕ] with the complexity in path integrations is because the path-integral measure is proportional to eCL[ϕ] due to the conformal anomaly [36]. This quantity CL[ϕ] provides a field theoretic counterpart of holographic complexity [37] as explained in Ref. [38].

In this argument, we consider the discretization of the path integral such that each cell has the area ε2. The original flat metric [Eq. (4)] corresponds to a square lattice with lattice spacing Δξ=Δτ=ε. The optimization which changes the metric such that e2ϕ≤1 means coarse-graining lattice sites such that Δξ=Δτ=εe-ϕ.

The minimization of CL[ϕ] can be found by solving the Liouville equation (∂τ2+∂ξ2)ϕ=e2ϕ/ε2. Choosing τ0=-ε using the time translational symmetry, we find the solution which satisfies the condition [Eq. (6)] e2ϕ=ε2τ2.(8)This describes a hyperbolic space H2. Note that in Ref. [33] we had e2ϕ=1/z2 because we defined z=-ε with the rescaling eϕ→εeϕ. This is the simplest example of the path-integral optimization. Even though we optimized a pure state |Ψ⟩, it is straightforward to extend the formulation such that we optimize a given mixed state density matrix.

Consider the case where the subsystem AB(≡A∪B) is given by a single interval in a vacuum. We parametrize the subsystem A and B as follows: A=[a,p],B=[p,b],(-∞<a<p<b<∞),(9)where they share the point P given by y=p. The reduced density matrix ρAB is given by the Euclidean path integral over a complex plane with a slit along AB, whose coordinate is denoted by y. Next we consider a purification |Ψ⟩AA˜BB˜ of ρAB by introducing the subsystem A˜ and B˜ as in Fig. 1 such that they share the point Q parametrized by y=q.

The original definition of EOP is the minimum of SAA˜ against any purifications [Eq. (2)]. Here we would like to restrict ourselves to a class of purification which is realized by the Weyl transformation [Eq. (5)] and focus on the one given by the path-integral optimization (i.e., the one which minimizes CL[ϕ]). The purification condition [Eq. (1)] is equivalent to the requirement [Eq. (6)].

To find the optimization, let us map the y space into an upper half plane (w plane) by w=y-ab-y.(10)The points P and Q are mapped into wP=-(p-a/b-p) and wQ=i(q-a/q-b). Note that such a conformal map transforms a solution of the Liouville equation in one coordinate into that in another coordinate. In the above, we mapped into the w plane as we know very well solutions of the Liouville equation in the setup of w coordinate.

Since the path-integral optimization on the upper half plane is given by the hyperbolic space [Eq. (8)], the final optimized metric reads ds2=ε2τ2dwdw¯=ε2τ2(b-a)24|b-y|3|y-a|dydy¯≡e2ϕ˜dydy¯.(11)The coordinate τ takes values in the range -∞<τ<-δ1, where δ1(>0) is an appropriate cutoff so that we maintain the condition Eq. (1). However, a straightforward analysis shows the detail of this regularization is not important in our calculations. In this setup, we have e2ϕ˜P=1, which follows from the condition Eq. (4), and we find e2ϕ˜Q=[ε2(b-a)2/4(q-a)2(q-b)2], where ϕ˜P and ϕ˜Q are the value of ϕ˜ at P and Q.

By identifying the purification |Ψ⟩AA˜BB˜ in Eq. (1) with the path integration on the upper half plane in the y coordinate with the optimized metric [Eq. (11)], the entanglement entropy SAA˜=SBB˜ is found to be SAA˜=c3log(q-pε)+c6ϕ˜P+c6ϕ˜Q=c6log[(b-a)(q-p)22ε(q-a)(q-b)],(12)where in the first line we performed a scale transformation of the standard formula of Ref. [39]. Remember that the entanglement entropy can be found from the two point function of twist operators in the replica method and this transforms in a standard way under the Weyl transformation. Then we minimize SAA˜ with respect to q, leading to SAA˜min=c6log[2(p-a)(b-p)ε(b-a)],(13)at  q=2ab-(a+b)pa+b-2p.(14)

The holographic EOP is computed from the length of the entanglement wedge cross section ΣABmin in the hyperbolic space ds2=(dx2+dz2)/z2, which is a time slice of Poincare AdS3. The geodesic curve ΣABmin is a part of a circle which connects P: (x,z)=(p,ε) and Q: (x,z)=(q,ε), as depicted in Fig. 2. The minimal length condition requires that ΣABmin is perpendicular to the circle [x-(a+b/2)]2+z2=[(b-a)2/4] at their intersection point (x,z)=(x0,z0). This fixes the value of q as Eq. (14) and we obtain x0=[pq-ab/p+q-(a+b)] and z0=[(b-a)2/4]-[x0-(a+b/2)]2. Its length A(ΣABmin) is computed as q-p2∫εz0dzz(q-p)24-z2=log[2(p-a)(p-b)ε(b-a)].Thus the holographic EOP Eq. (3) perfectly matches with Eq. (13).

It is straightforward to extend the above agreement between EW(ρAB) and SAA˜ to the finite temperature case by performing a conformal transformation y=e(2π/β)ζ. Similarly, we can extend it to the finite size case.

As the second class of examples, we consider the case where the subsystems A and B are mutually disconnected intervals on an infinite line. Though we focus on the vacuum state in a two-dimensional CFT below, it is straightforward to extend to the finite temperature setup by the conformal map y=e(2π/β)ζ as before.

The reduced density matrix ρAB is described by a Euclidean path integral on the complex plane (y plane) with two slits along A and B as in the upper left picture of Fig. 3. Since ρAB depends only on the cross ratio η owing to the conformal invariance, without losing generality, we can fully parametrize the subsystem A and B as A=[1,1/x],B=[-1/x,-1],(0<x<1).(15)The cross ratio is given by η=(1-x)2/4x. We can choose the subsystem A˜ and B˜ which purify ρAB such that AA˜=[0,L] and take the limit L→∞, owing to the reflection symmetry. We confirmed numerically that this reflection symmetry is not spontaneously broken.

The y plane with two slits is mapped into a cylinder, which is described by the coordinate w=ξ+iτ, as w=i∫0ydy˜1(1-y˜2)(1-x2y˜2),(16)as in the upper right of Fig. 3.

Finally, we perform the path-integral optimization on a cylinder as in the lower picture of Fig. 3. We obtain the optimized metric by solving the Liouville equation: ds2=e2ϕdwdw¯=e2ϕ˜dydy¯,(17)where e2ϕ=π24K(x)2ε2cos2(πτ2K(x)),e2ϕ˜=e2ϕ1|(1-y2)(1-x2y2)|2.(18)We introduced the elliptic function K(x): K(x)=∫01dy˜1(1-y˜2)(1-x2y˜2).(19)

The coordinates τ takes values in the range -K(x)+δ2≤τ≤K(x)-δ2,(20)and ξ is periodically identified: ξ∼ξ+2K(1-x2). Here δ2 is an infinitesimally small regularization parameter fixed by the condition Eq. (1). Again, the detail of δ2 is not important in our calculations below. The boundary points of AA˜, y=0 and y=L(→∞), are mapped into the antipodal two points in the middle: (ξ,τ)=(0,0) and (ξ,τ)=(K(1-x2),0).

Now the entanglement entropy SAA˜min after the optimization can be computed as follows: SAA˜min=c3logLε+c6ϕ˜|y=L+c6ϕ˜|y=0=c3logLε+c6log[πε2L2xK(x)]+c6log[πε2K(x)]=-c6logx-c3log[2K(x)π].(21)Note that before the optimization, SAA˜ was UV divergent. The optimization procedure performs the Weyl transformation such that it squeezes the metric or equally reduces the number of lattice sites. Accordingly, SAA˜ gets smaller until it becomes finite. As we will argue later, our field theoretic description of this procedure is valid within a certain range.

Since K(x)≃(π/2)+(π/8)x2+… for x≪1, we have SAA˜min=-c6logx+O(x2).(22)

We can compare this with the holographic EOP [5,6]: EW(ρAB)=-c6logx,(0<x<3-22).(23)For 3-22<x<1 we have EW(ρAB)=0 as the entanglement wedge gets disconnected.

It is also useful to note that the holographic mutual information (HMI) I(A:B)=SA+SB-SAB behaves 12I(A:B)=c6logz=c6log(1-x)24x(0<x<3-22),(24)while I(A:B)=0 for 3-22<x<1. We plotted these quantities in Fig. 4.

We notice that SAA˜min almost coincides with EW(ρAB) for 0<x<3-22 in the plot. However, one may worry that they deviate from each other at the order O(x2). Also, SAA˜min continuously decreases and becomes incorrectly negative as x gets larger. We would like to argue that our Weyl invariance, Eq. (17), breaks down except for x≪1. This is because the length of minimal cross section of the optimized cylinder, given by Lcyl=[πεK(1-x2)/K(x)], is much greater than the cutoff scale ε only if x≪1. If the size of the manifold, on which we path integrate, becomes the same order of the lattice spacing ε, we cannot trust field theoretic properties of path integrals, such as the Weyl invariance. Therefore, the result, Eq. (21), is trustable only for x≪1 and in this range the equality SAA˜min=EW(ρAB) is confirmed. Indeed, in the single interval case, where we found the perfect matching Eq. (13), the Weyl transformation is trustable for any values of a, b and p because we always have Lcyl=∞.

It is also useful to note that at the holographic phase transition point x=3-22 (or equally η=1), the minimum cross section of the cylinder becomes Lcyl=2πε. This indeed agrees with the point where a counterpart of confinement or deconfinement transition is expected in the path-integral optimization for holographic CFTs as argued in Ref. [33], though we cannot fully trust this argument because Lcyl gets as small as the lattice spacing ε.

In this Letter, for a given mixed state ρAB, we introduced a minimized entanglement entropy SAA˜min as follows. First we define a special purified state |Ψmin⟩ABC for ρAB by minimizing the path-integral complexity CL [Eq. (7)]. Then SAA˜min is given by taking the minimum of SAA˜ for the purified state |Ψmin⟩ABC among all possible decompositions of the ancilla space for purification C=A˜∪B˜.

We analytically computed this quantity SAA˜min in several setups in two-dimensional CFTs. Remarkably, we found that it matches with the area of the entanglement wedge cross section EW(ρAB) in AdS, called the holographic entanglement of purification.

It is not obvious at present if our special purification is identical to the one which minimizes SAA˜ as in the original definition. Eq. (2). However, we expect they are at least very close to each other because the former minimizes the path-integral complexity and thus efficiently compresses the sizes of A˜ and B˜. In this sense, our result can be regarded as the first quantitative evidence for the holographic EOP conjecture [5,6].

In the original argument [5], the equivalence between EW(ρAB) and EP(ρAB) was explained by assuming the surface-state duality [40], based on the conjectured relation between tensor networks [41,42] and AdS/CFT [43]. In this argument, the minimization is taken only for quantum states with classical gravity duals. We would like to leave for future works the precise meaning of the minimum in the EOP [Eq. (2)] which correctly matches with EW(ρAB). Higher dimensional generalizations are also an intriguing future problem.