First, we introduce quantities which provide analogues of the inner product of two density matrices. One of the best quantities is the fidelity defined by (2.1)

The fidelity is symmetric under an exchange of and , and takes values in the range (2.2)

Moreover, it satisfies (2.3) (2.4)

Therefore, we can employ the fidelity to distinguish two quantum states.

There are many other measures which satisfy the basic properties in Eqs. (2.2), (2.3), and (2.4)—they are listed in Appendix D. One of them is the affinity [33]: (2.5)

This quantity has upper and lower bounds in terms of the fidelity: (2.6)

For the actual computations, taking a square root of a given density matrix is not always tractable. This motivates us to consider the quantity : (2.7)

This quantity is called geometric mean fidelity; it was introduced in Ref. [34] (see also Refs. [35, 36]) and satisfies the basic properties in Eqs. (2.2), (2.3), and (2.4). it was employed to study non-equilibrium dynamics of quantum systems in Ref. [37]. We might be able to think that this quantity is analogous to the second Rényi entropy, while the fidelity is analogous to von Neumann entropy. Indeed, the total power of and is two in the former, and one in the latter.

It is also useful to evaluate these quantities when the states are pure, expressed as and . From the definitions, we obtain (2.8) (2.9) (2.10)

Now we move on to distance measures between two quantum states and . First of all, the Bures distance is defined from the fidelity as (2.11)

It is obvious that this quantity is symmetric and takes values in the range (2.12)

In addition, it satisfies (2.13)

There are several other important distance measures: the trace distance [38], relative entropy distance , Hellinger distance , and geometric mean fidelity distance , given respectively by (2.14) (2.15) (2.16) (2.17)

Three of them, namely , , and , satisfy the basic properties in Eqs. (2.12) and (2.13). On the other hand, the relative entropy distance is not symmetric and takes the values , though Eq. (2.13) holds. Refer to Refs. [39, 40] for computations in integrable 2d CFTs, and to Ref. [41] for an application to locally excited states (see also Ref. [42]).

It is useful to note the following relations between these distances: (2.18) (2.19)

Furthermore, we can introduce the so-called information metric when the density matrix is parameterized by continuous variables , denoted by . For the Bures distance, this metric is defined as (2.20) where the are infinitesimally small and denotes the higher powers of . This metric is called the Bures metric. In the same way, we can define another metric from the relative entropy distance , called the quantum Fisher metric . It is also possible to define the metrics and for the distance measures and , respectively.

The quantum version of the Cramér–Rao theorem [43] (see also the textbook Ref. [32]) tells us that when we try to estimate the value of from physical measurements, the errors of the estimated value are bounded by the inverse of the Bures metric : (2.21)

In particular, when , the uncertainty becomes divergent and we cannot estimate the value of at all. This is simply because the density matrix does not depend on and we cannot distinguish density matrices for various values of .

More precisely, the quantum Cramér–Rao theorem is stated as follows. A physical measurement is described by the positive operator-valued measure such that , where corresponds to each value of the measurement. Tr denotes the probability that the measured value is given by . We would like to estimate the value of from the measured value following an arbitrarily chosen function . We introduce an error in this process as (2.22)

To be exact, we actually consider copies of the system and take the asymptotic limit (2.23)

The quantum Cramér–Rao theorem [43] argues that the lower bound is given by the inverse of the Bures metric: (2.24)

For pure states and , the distance measures look like (2.25) (2.26)

We omit the relative entropy distance because becomes divergent when .

Consider locally excited states in a 2d CFT. We simply find (2.27)

This leads to the Bures metric (2.28) and the Hellinger metric (2.29)

Interestingly, the information metric is proportional to the two-dimensional hyperbolic space . This looks like a time slice of the gravity dual, i.e. the Poincaré AdS of Eq. (1.3). This coincidence is very natural because the distinguishability between two excitations should increase when the corresponding bulk points are geometrically separated. This was already noted essentially in Ref. [44]. However, this result is universal for any 2d CFT as the computation only involves two-point functions. This implies that the study of the information metric of the reduced density matrix has more opportunities to explore deep mechanisms of AdS/CFT, which is the main motivation of this paper.

Distinguishability measures for the reduced density matrices of Eq. (1.1) crucially depend on the nature of CFTs such as multi-point correlation functions, as opposed to those for pure states. The special properties of holographic CFTs allow us to introduce a CFT counterpart of the entanglement wedge, as we will explain in this paper for various examples. We call these geometrical structures in holographic CFTs CFT wedges, which we introduce below.

Consider an information metric (here, specifies the type of distance measure) for a reduced density matrix of a locally excited state given by Eq. (1.1), regarding the operator insertion point as the parameter in Eq. (2.20). The information metric has the components with , and depends on the location . Since the restriction to 2d CFTs is not necessary in this subsection, we have in mind holographic CFTs in any dimensions below.

In this setup, we introduce the geometrical structure in a CFT, which we call the CFT wedge for the subsystem , as follows: (2.30)0 ; \nonumber \\ \mbox{if}\ X\notin C^{(\#)}_A, \mbox{ then } G_{\# ij}(X)\simeq 0. \label{CFTw} \end{eqnarray} $$]]>

In the case of the Bures metric, we can write this equivalently in terms of fidelity as follows: (2.31)

Also, for the distance measure we can express the CFT wedge by (2.32)

Note that the sharp geometrical structures in Eqs. (2.30), (2.31), and (2.32) only appear in holographic CFTs, where we take the limit as in Eq. (1.2). The non-vanishing information metric in Eq. (2.30) scales as . For generic CFTs, such as free field CFTs, we only find smeared behaviors, which prohibit us defining a CFT wedge, though qualitatively the behaviors of distance measures are often similar. In other words, the sharp CFT wedges emerge only when we consider holographic CFTs.

We would also like to stress that the CFT wedges can depend on the choice of distance measures. Indeed, as we will see later, for generic setups, and can differ. In the end, we argue that the correct choice which probes the low-energy states in AdS/CFT (i.e. the code subspace) will be the Bures metric. We will comment more on this point in the final part of this paper.

We choose the subsystem to be an interval at . The extremal surface in the bulk AdS is given by the semicircle . Therefore, the entanglement wedge is given by (3.1)

Note that this is also identical to the causal wedge [45].

From the viewpoint of CFTs, we consider an excited state by inserting a local operator at and define the reduced density matrix in Eq. (1.1). We regard the location of the insertion point as the parameters of . Having in mind the AdS/CFT duality, the geodesic which connects and intersects the time slice at the point given by . Therefore, if the entanglement reconstruction is correct, the CFT wedge, based on a proper distance measure, should coincide with , or, equally, (3.2)

Accordingly, the information metric should vanish if the intersection is outside the CFT wedge, i.e. (3.3) \frac{L^2}{4}, \label{outent} \end{eqnarray} $$]]> while it is non-vanishing inside the wedge, Eq. (3.2).

In this section we focus on calculating the CFT wedge for the measure of Eq. (2.7).

Let us calculate of Eq. (2.7) for the two density matrices (3.4)

To calculate Tr, consider the conformal transformation¹ (3.5) which maps two flat-space path integrals for and into a single plane. The coordinate of the latter (single plane) is written as . The insertion points of the local operators and are given by (3.6) for , and (3.7) for . Refer to the upper two pictures in Fig. 2. The transformation in Eq. (3.5) maps these four points into , , , and given by (3.8)

It is important to note that the boundaries of the CFT wedge of the original two flat planes are mapped into the diagonal lines , as depicted in Fig. 2. As we will see soon, this leads to the CFT wedge structure in the distinguishability.

The trace Tr is now expressed as a correlation function on the -plane: (3.9) where denotes the normalized correlation function such that , and we also write the vacuum partition function on an -sheeted complex plane by .

Thus, we obtain (3.10) where is the (normalized) four-point function (3.11)

Because we have the relations (3.12) we can simplify Eq. (3.10) as follows: (3.13)

We now study this quantity for both a holographic CFT and a free scalar CFT.

First, let us evaluate Eq. (3.13) in holographic CFTs. We assume the range in Eq. (1.2) of conformal dimension . In this case, the large- (or large-) factorization property justifies the generalized free field approximation [47].

That is, in the large- limit, the leading contribution to the correlation function in Eq. (3.11) is given by a simple Wick contraction based on the two-point function (3.14)

The generalized free field prescription leads to a simple expression of the four-point function: (3.15) where in the final line we remember that and . In the right-hand side of Eq. (3.15), the first term comes from the Wick contraction , which we call the trivial Wick contraction. The second term arises from the other Wick contraction, , which we call the non-trivial Wick contraction.

First, consider the case where the local operator is inserted outside the CFT wedge, Eq. (3.3). This is mapped into the uncolored region in Fig. 2 given by the wedge region . When both and are outside the wedge, the lengths and are shorter than . Therefore, the four-point function in Eq. (3.11) is approximated by the first term, which comes from the trivial Wick contraction. Therefore, we finally obtain (3.16)

This tells us that we cannot distinguish between and when the local excitations are outside the CFT wedge.

Next, we turn to the case where both and are inside the CFT wedge, Eq. (3.2). In this case, the lengths and are larger than . Therefore, the four-point function in Eq. (3.11) is approximated by the second term, which comes from the non-trivial Wick contraction. Therefore, we finally obtain (3.17)

Since we always have and take the limit , this quantity is vanishing except when : (3.18) (3.19)

Finally, when either of or is inside the CFT wedge, we find that is vanishing: (3.20)

These behaviors in Eqs. (3.16), (3.18), (3.19), and (3.20) confirm our expectations in Eq. (2.32), and this shows that the CFT wedge agrees with the entanglement wedge in AdS/CFT in the present example. Refer to Appendix A for more detailed calculations of in this example.

We also plot the profiles of in the left columns of Figs. 3 and 4. The left graph in Fig. 3 shows as a function of when is fixed inside the CFT wedge. We observe a clear peak at , which will be highly localized in the limit . In the left graphs of Fig. 4 we fixed outside the CFT wedge. We can observe a clear entanglement wedge structure, where we have inside and outside.

To understand how the properties of holographic CFTs are relevant to the emergence of entanglement wedges in the gravity duals, consider the free massless scalar CFT ( CFT) in two dimensions. We choose the operator to be (3.21) where and are chiral and anti-chiral massless scalar fields. Note that the conformal dimension of the above operator is . In this case we obtain (3.22)

We can easily estimate Eq. (3.13) analytically, and obtain (3.23) for any values of and . Note that in these excited states, we always have as they do not generate entanglement between the left and right moving modes [29, 30, 46].

Thus, in this free scalar CFT there is no sharp CFT wedge structure, as expected for non-holographic CFTs. The numerical plots are in the right columns of Figs. 3 and 4. Even though we can observe a peak when is inside the CFT wedge (see Fig. 3), which is similar to the holographic case, we do not find any sharp CFT wedge when is outside the wedge (see Fig. 4). In this way we can conclude that there is no emergence of the entanglement wedge in CFT, as expected.

So far we have assumed that both and are created by the same local operator , as in Eq. (1.1). It is also instructive to consider the case where and are created by two orthogonal operators and respectively (each chiral conformal dimension and ) such that the two-point function vanishes. We would like to calculate in this case. Again, we can use the expression in Eq. (3.10) as (3.24) where we can write , , , , etc.

Now we evaluate this in holographic CFTs, by applying the large- factorization (generalized free field prescription). First of all, we can always estimate (3.25)

Depending on whether is inside or outside the CFT wedge, Eqs. (3.2) or (3.3), we find (3.26)

Thus, we can evaluate Eq. (3.24) as follows: (3.27)

This nicely fits with the entanglement wedge structure in AdS/CFT: we can distinguish two different operators inside the wedge, while we cannot outside. In particular, since this analysis can be applied to the case when is the identity operator, cannot be distinguished from the vacuum one (no insertions of operators), if the insertion of is outside the wedge.

Let us focus on a holographic 2d CFT. The leading contribution is again given by the generalized free field prescription. When and are outside the CFT wedge, Eq. (3.3), we can approximate the -point function as (4.12)

In this case we get the trivial Bures distance (4.13) where we note that in this limit. Thus, the Bures metrics all vanish in the outside wedge case.

On the other hand, when and are inside the CFT wedge, Eq. (3.2), we can approximate (4.14) where we regard . Thus, we have (4.15)

In the limit (), we find (4.16) where and are related by in the limit. By assuming that is infinitesimally small, we obtain the Bures metric (4.17)

Interestingly, this Bures metric coincides with that for the pure state in Eq. (2.28). Therefore, it is proportional to the metric on a time slice of AdS. Remember that the original Euclidean time coordinate can be regarded as the radial coordinate via the intersection between the geodesic and the time slice, as in Fig. 1. This agreement between the information metric with the bulk metric is natural if we think that the distinguishability in the quantum estimation theory is related to the bulk locality resolution. At the same time, the agreement between the Bures metric for with local excitation inside the CFT wedge and that for the pure state tells us us that we can perfectly reconstruct the information in the entanglement wedge from . This supports the entanglement wedge reconstruction.

Next, we turn to a holographic CFT dual to the Euclidean global AdS, (4.18)

This is a 2d holographic CFT with the space coordinate compactified on a circle, . We choose the subsystem to be the interval at .

By performing the conformal transformation with , we find (4.19)

This leads to the following Bures metric inside the CFT wedge: (4.20)

Since the geodesic in global AdS which connects the two points and at the boundary looks like (4.21) where is the intersection point of the time slice and this geodesic in the bulk AdS, by taking the boundary limit we find the relation (4.22)

By relating the boundary point to the bulk point on the time slice using this relation we can rewrite the metric in Eq. (4.20) as (4.23) which agrees with the time slice metric of the global AdS, Eq. (4.18).

Consider a holographic CFT dual to the Euclidean Ba nados–Teitelboim–Zanelli (BTZ) (with a non-compact horizon), (4.24)

This is given by a 2d holographic CFT, with the space coordinate compactified on a circle, .

By performing the conformal transformation with , we find the following result in the case of the non-trivial Wick contraction: (4.25)

Note that we limit the range of to .

This leads to the following Bures metric inside the wedge: (4.26)

The geodesic in BTZ which connects the two points and at the boundary looks like (4.27) where is the intersection point of the time slice and this geodesic in the bulk. Note that Eq. (4.19) in the global AdS and Eq. (4.27) in BTZ are related by the familiar coordinate transformation (4.28)

By taking the boundary limit , we find the relation (4.29)

By relating the boundary point to the bulk point on the time slice using this relation, we can rewrite the metric in Eq. (4.26) as (4.30) which agrees with the time slice metric of the BTZ, Eq. (4.24).

Moreover, we can also confirm that the CFT wedge in this case agrees with the entanglement wedge in BTZ as follows. The condition for the non-trivial Wick contraction is |z+\bar{z}|$]]>, where (4.31)

This leads to the condition (4.32)

On the other hand, the geodesic which connects and (on the slice ) in the BTZ geometry is (4.33) where (4.34)

This coincides with the border of Eq. (4.32) via the relation between and given by Eq. (4.29).

Next, we consider the Bures distance , where and are defined by locally excited operators and that are orthogonal to each other. Let us work out the behavior of the Bures distance by computing introduced in Eq. (4.11) and taking the limit . Using the expression in Eq. (4.9), we eventually find: (4.35)

Here, we used the assumption that and noted that the inside CFT wedge region is given by . By taking the limit, the fidelity behaves as follows: (4.36)

The above behaviors precisely agree with what we expect from the entanglement wedge reconstruction.

It is useful to compare the previous Bures metric in holographic CFTs with that in free scalar CFT. Consider a free scalar CFT and choose the primary operator to be as in Eq. (3.21) with to simplify the calculations. As we explain in Appendix B, in this case we can analytically evaluate , and eventually we find the fidelity (4.37) where . Several profiles of the fidelity are plotted in Fig. 6.

The Bures metric for the free scalar can be found as (4.38)

This metric is plotted in Fig. 7. Note that we cannot find any sharp structure of a CFT wedge, as opposed to the holographic CFT. However, in the limit , we find the metric for .

We employ the following conformal transformation (analogous to the one in Refs. [53–55]) which maps a complex plane (the -plane) with two slits along and into a cylinder (coordinate ): (6.2) where we have introduced (6.3)

Note that we have (6.4)

Also notice that we are considering the analytical continuation of the integral given by the Jacobi elliptic function: (6.5)

It is useful to note the relation (6.6)

Consider the calculation of Tr, where and . Each of and is described by the path integral on the complex plane with two slits. We can compute Tr as the partition function on the space obtained by gluing the two complex planes along the slits. This is conformally mapped into a torus. This torus is constructed by gluing two cylinders: one of them describes and is obtained by performing the transformation in Eq. (6.2). The other corresponds to and is obtained from another transformation, . These conformal maps take the original two-sheeted geometry into a torus, as depicted in Fig. 9. The horizontal and vertical length of the torus are given by and , respectively.

Finally, we find that is given by the same formula as in the single-interval case, Eq. (3.13), where is the torus four-point function. In the next subsection we study the CFT wedge geometry by focusing on holographic CFTs.

In holographic CFTs, we need to distinguish two phases depending on the moduli of the torus [26]:

In phase (i), the entanglement wedge gets connected because (2s+l)l$]]>, i.e. S_{A_1A_2}$]]>. In this case, the AdSCFT duality tells us that the entanglement wedge in the Poincaré AdS, Eq. (1.3), looks like (6.7) on the time slice . In terms of the location of the local operator insertion, the corresponding CFT wedge is expected to be (6.8)

On the other hand, in phase (ii) the entanglement wedge gets disconnected as , i.e. . In this case, the entanglement wedge in the Poincaré AdS, Eq. (1.3), is found to be , where (6.9)

The corresponding CFT wedge reads (6.10)

Now let us work out the CFT wedge from the calculation of in holographic CFTs. The two-point functions on the torus in phases (i) and (ii) behave like (6.11) where we assumed that takes the smallest value among all integer at for phase (i) and takes the smallest value among all integer at for phase (ii).

This expression for the two-point functions in Eq. (6.11) follows from the standard fact in AdSCFT that the gravity dual of the torus is given by a solid torus. We can construct the dual solid torus by filling the inside of the torus such that the circle Re (or Im) shrinks to zero size in the bulk when we consider phase (i) (or (ii)). This is due to the well-known Hawking–Page phase transition [56], and matches perfectly with the large- CFT analysis [26].

We can rewrite the value of in holographic CFTs using the generalized free field approximation: (6.12) where (6.13)

The locations and of the operator insertions are depicted in Fig. 9, explicitly obtained via the map in Eq. (6.2) from the original insertion locations and in the double-sheeted geometry which describes the path integral of .

When the true minimum is the first term in Eq. (6.13), i.e. the trivial contraction, we simply find and we cannot detect the local operator insertions. On the other hand, if the other one is favored as the minimum (i.e. the non-trivial contraction), then becomes a non-trivial function of the locations of operator insertions.

The condition that the non-trivial contraction is favored is given by (6.14) in the connected case (i), and by (6.15) in the disconnected case (ii).

We plot the parameter region of , where the non-trivial contraction is favored, in Fig. 10 for the connected phase (i) and Fig. 11 for the disconnected phase (ii).

In both cases, the region is very close to the true entanglement wedge, Eq. (6.8). The deviation is interestingly very small (within a few percent), and is sketched in Fig. 12. The wedge derived from in the holographic CFT can be larger or smaller than the true entanglement wedge in AdS/CFT, depending on the situation. Notice that these deviations are leading order in our computational scheme, i.e. expansions, and thus we cannot regard them as quantum corrections in gravity. Rather, it is an essential feature of the Rényi-like measure . We will comment on possible interpretations of this phenomenon in later subsections.

We also explicitly plot the values of as a function of (the location of operator insertion of ) when we fix (the location of operator insertion of ) for both the connected (upper two pictures) and the disconnected (lower two pictures) cases in Fig. 13. In both plots, the left graphs show the plots when we fix to be inside the wedge. In this case we find a sharp peak of , which reaches the maximum only when . In the right graphs, is outside the wedge. We see that when is also outside the wedge, while we have when is inside the wedge. All of these agree with the expectation from AdS/CFT, neglecting the small deviation previously discussed.

It is instructive to also consider the behavior of CFT wedges for the reduced density matrix , where is the complement of the subsystem . We again focus on CFT wedges based on . The calculation of Tr is very similar to the previous one of Tr, as depicted in Fig. 14. The only, but very important, difference is that the locations of and are flipped. Therefore, the conditions of non-trivial Wick contraction are simply opposite to each other: when we need to take the non-trivial one for Tr, we need to take the trivial one for Tr, and vice versa. Therefore, the CFT wedge for is just the complement of that for .

This relation helps us to understand the behavior in Fig. 12. First of all, when the CFT wedge for is disconnected, it is clear that the CFT wedge should be larger than or equal to that for the union of the CFT wedges and , as the information included in is greater than that in the union of and . This explains the right picture of Fig. 12. Also, this requirement is trivially satisfied in the left picture.

To better understand the left picture in Fig. 12, let us consider the complement of , i.e. . Since the wedge of is disconnected when that for is connected, we can apply the same rule, i.e. should be larger than or equal to the union of and . As we just showed, we also know that is the complement of . These two facts lead to the behavior of the left picture in Fig. 12.

In the double-interval case we found that the CFT wedge defined by the distinguishablity measure does not precisely agree with the expected entanglement wedge from AdS/CFT, though the deviations are very small. This motivates us to study CFT wedges for the Bures distance , Eq. (2.11), or equally the fidelity , Eq. (2.1), which is expected to be the ideal distinguishablity measure. As we will see below, the CFT wedge for precisely agrees with the correct entanglement wedge.

The fidelity can be computed from the analytical continuation of , Eq. (4.1), via a replica-like method. Even though it is very difficult to evaluate for general integers and , we can heuristically obtain analytical results in the limit and as follows. First, notice the useful property shown in Refs. [27, 28] that the vacuum replica partition function of a holographic CFT with can be approximated by² (6.16)(2s+l)l, \\ Z_{\Sigma_k ([0,s])} Z_{\Sigma_k ([l+s,l+2s])}, & \mbox{(ii) disconnected phase: } s^2<(2s+l)l, \\ \end{array} \right.\\ \end{aligned} \end{equation} $$]]> where means the -sheeted manifold with a cut along the interval , and is the vacuum partition function on that manifold.

Indeed, the limit of fidelity and corresponds to , as is clear from the relation in Eq. (4.4). Therefore, we can factorize the computation of the fidelity into two correlation functions, each of which includes a single interval. In this sense the calculations are reduced to the fidelity in the single-interval case, which we already worked out, e.g. in Eqs. (4.13) and (4.16). A CFT wedge in the single-interval case is bounded by a semicircle, which agrees with the correct entanglement wedge.

We can illustrate this factorization from another viewpoint. If one wants to probe the disconnected entanglement wedge , one may consider the conformal transformation in Eq. (4.3) with . This leads to the geometry shown in Fig. 15, which has “cuts” associated with the slit (the red solid lines in the figure). Although these cuts give non-trivial contributions to the -point function in general, these contributions can be neglected in the limits . Therefore, we can evaluate this -point function in the same way as the single-interval case, which means that the result just reduces to Eq. (4.16).

In this way, owing to the factorization in Eq. (6.16), we can conclude that the CFT wedges calculated from the Bures distance (or equally fidelity) coincide with the expectations from the entanglement wedges: Eq. (6.8) in the connected case and Eq. (6.10) in the disconnected case. It is also clear that the Bures metric in the double-interval case also agrees with the AdS metric as in the single-interval case, when the locations of operator insertions are inside the wedge.

So far, we have seen the calculations of two distinguishability measures and in the double-interval case. Entanglement wedges in AdS/CFT are precisely reproduced from the latter, i.e. the fidelity, while the former predicts CFT wedges which are slightly distorted from the actual entanglement wedges. Here, we discuss why CFT wedges depend on the choice of these distinguishability measures.

First, remember that is essentially the calculation of Tr, and the fidelity is equal to Tr. In this sense the total power of the density matrices (for this we identify and ) is two for the former and one for the latter.

A measurement of a physical quantity is described by . In the classical gravity limit of AdS/CFT, we restrict the operators to low-energy ones. Therefore, we expect that the entanglement wedge should be determined by the distinguishability of low-energy states (or so-called code subspaces [19, 20]).

In this sense, the quantity Tr goes beyond the low-energy approximation as is a highly excited operator. A reduced density matrix can be expressed as in terms of the modular Hamiltonian . For a CFT vacuum, for example, is given by an integral of the energy stress tensor. Therefore, includes an infinite number of energy stress tensors, which are clearly outside of the low-energy states.

On the other hand, when we calculate the Bures metric the fidelity distinguishes low-energy states when is very close to . We argue that the above different property of distinguishing states causes the difference of CFT wedges between and . This also explains why the latter agrees with the expectation from the actual entanglement wedge in AdS/CFT. We will explore differences of CFT wedges for various other distance measures Sect. 9.

We expect that the state in Eq. (7.1) is dual to half of the eternal BTZ geometry [59]. In the Euclidean setup, it is identical to the geometry given by the metric in Eq. (4.24). In the AdS/BCFT (see Refs. [23–25] for details), the gravity dual of a BCFT state is found by adding a boundary surface into an AdS space, which extends to the bulk.

There are two phases in the holographic calculation of the entanglement entropy which follows from the prescription of AdS/BCFT: (a) the connected geodesic is favored, and (b) the disconnected geodesics which end on the horizon are favored. Accordingly, the geometry of the entanglement wedge changes between (a) and (b). Since the length of the connected and disconnected geodesic is computed from the explicit form of the geodesic, Eq. (4.33), as (7.5) (7.6) where the constant is related to via (7.7) phases (a) and (b) correspond to the regions (7.8)1\ \ \leftrightarrow\ \ \sinh\left(\frac{\pi L}{\beta}\right)<1\ \ \leftrightarrow\ \ \kappa=e^{-\frac{2\pi L}{\beta}}>3-2\sqrt{2} ; \nonumber \\ && \mbox{phase (b) \Gamma_{\rm dis}:}\quad \sinh\rho_*<1\ \ \leftrightarrow\ \ \sinh\left(\frac{\pi L}{\beta}\right)>1\ \ \leftrightarrow\ \ \kappa=e^{-\frac{2\pi L}{\beta}}<3-2\sqrt{2}. \qquad \end{eqnarray} $$]]>

This is the same condition we encounter in the case of a double interval. This is not a coincidence, and indeed we find that the ratio of the horizontal length and vertical length of the cylinder of the coordinate in Fig. 16 is given by , which is the same ratio as appears in Fig. 9. Indeed, it is a cylinder with circumference and length . Via the doubling trick this can be extended as a torus, with the periodicities given by and .

In this way, the reduced density matrix analysis provides the phase transition of the entanglement wedge at the correct value of subsystem size. The expected entanglement wedge geometry from AdS/BCFT is shown in Fig. 17.

Now we come back to the evaluation of . This is given by the four-point functions as (7.9) where denotes the four-point function on the cylinder in the coordinate, (7.10)

Note that this four-point function is defined on the cylinder.

In the generalized free field prescription, we can evaluate this four-point function via Wick contractions. There are three possible Wick contractions: (i) trivial contraction, (ii) non-trivial contraction, and (iii) boundary contraction, as depicted in Fig. 18. The third one is new, and is the contraction between each point of () and its mirror point due to the presence of the boundary.

In phase (a) we have \pi$]]>, and thus the state is dual to the BTZ black hole on an interval , where Im is the Euclidean time. Therefore, the two-point function behaves as (7.11)

In phase (b), since the state is dual to a global AdS on an interval , where Im is the Euclidean time. Therefore, the two-point function behaves as (7.12) where .

It is obvious that we obtain (i.e. and are indistinguishable) when contraction (i) or (iii) is favored. We can distinguish and , i.e. , when the non-trivial contraction (ii) is favored. The condition that the non-trivial contraction (ii) is favored is: (7.13) when is very large.

In phase (a) they are equivalent to the condition (7.14) (7.15)

We numerically plot this region in the left panel of Fig. 19. If we ignore the boundary contributions, this CFT wedge is very close to the actual entanglement wedge from AdS/CFT as depicted in the right panel of Fig. 19. This small deviation is because we are actually employing the measure which has the unwanted property that it is also sensitive to high-energy states. In other words, if we utilize the Bures metric instead, we can reproduce the expected CFT wedges which agree with the entanglement wedges. This situation is the same as that discussed in Sect. 6.6 for the example of double intervals.

In phase (b), they are equivalent to the conditions (7.16) (7.17) (7.18)

This region is plotted numerically in Fig. 20. The resulting CFT wedge is largely different from that expected from the entanglement wedge. However, if we are allowed to ignore the boundary contribution (i.e. the constraint to the green region), this CFT wedge is the same as the actual entanglement wedge from AdS/CFT. In other words, we can reproduce the correct geometry of the entanglement wedge only when the one-point function vanishes. This is because in this case the boundary contraction (iii) is not allowed. If the boundary one-point function does not vanish, then we get the smaller wedge from the holographic CFT rather than the correct entanglement wedge; see Fig. 21.

Even though when the CFT wedge does not agree with the entanglement wedge in AdS/CFT, this discrepancy is present even when is the total system (i.e. the pure state). In other words, we cannot probe points near the black hole horizon by two-point functions dual to the geodesic which connects two boundary points. This is simply because the two-point function gets factorized into one-point functions when the points are close to the boundaries of BCFT. Therefore, this means that we cannot employ our original idea that we probe the bulk geometry by two-point functions when does not vanish. In this sense, we should not think the above discrepancy shows that the CFT predicts an entanglement wedge which differs from the AdS/CFT prediction. Rather, we need to find a better CFT quantity which can probe the bulk geometry.³

The entanglement wedge in AdS/BCFT which ends on the boundary surface as in the upper right picture of Fig. 17 plays a crucial role in a recent explanation of the black hole information paradox [61–66], where a region of the entanglement wedge near the boundary surface is called the islands. When , our arguments above support the entanglement reconstruction relevant to this interesting problem.

The thermofield double (TFD) state also provides a closely related but different setup of AdS/CFT. It is given by the pure state in the direct product of two identical CFT Hilbert spaces, : (7.19) where is the energy eigenstate with energy , and is the thermal partition function. When we trace out either one of the Hilbert spaces, the reduced density matrix coincides with the canonical distribution. As discovered in Ref. [67], this pure state is dual to the eternal AdS black hole. In AdSCFT, the dual geometry is given by the eternal BTZ solution, which is obtained by continuing the Lorentzian geometry inside the horizon and which has two asymptotically AdS boundaries. The two boundaries correspond to the first and second CFTs. In the well-known path integral formulation, the state in Eq. (7.19) is described by a strip with width in the Euclidean time direction, while the space direction is an infinite line. The boundary conditions on the two boundaries of the strip are arguments of two CFTs, which in total represent the wave functional of the TFD state.

Let us choose subsystem in the first CFT at and subsystem in the second CFT at . For simplicity, we choose and to be symmetric with respect to the middle line . In this setup, if we artificially take a quotient , then we get back to the previous example of the global quantum quench, Eq. (7.1). Thus, the mathematical structures are very similar.

Consider the CFT wedge for the union of these two subsystems in the TFD state. The entanglement wedge from CFT is simply given by doubling that for the global quench (see Fig. 17) across the horizon, utilizing the symmetry.

The calculation of the measure in CFT can be done by doubling the cylinder into a torus, as depicted in Fig. 16, where the dotted green circle represents the subsystem . Therefore, we find that the phase transition structure, i.e. the connected phase (a) and the disconnected phase (b), is identical. Moreover, the CFT wedge is determined by the condition that the non-trivial Wick contraction is favored over the trivial one. Notice that boundary contractions are not allowed as we do not have any boundaries in our CFT, as opposed to the previous example. Because of this, we find that the CFT wedge in the connected phase agrees with the entanglement wedge up to a very small deviation, which can be confirmed in the right picture of Fig. 19. In the disconnected phase, the CFT wedge perfectly agrees with the entanglement wedge, as confirmed from Fig. 21. This small deviation for the connected case is again due to the measure , and should be absent in the CFT wedges for the Bures metric, as in Sect. 6.6 for the example of double intervals.

Let us start with the simple example where the subsystem is given by the half-plane 0$]]> at . A path integral calculation of the quantity in Eq. (2.7) can be obtained as a natural generalization of our previous analysis in two dimensions, and is depicted in the upper pictures of Fig. 22. To proceed, it is useful to introduce a polar coordinate as follows: (8.1) where are the same as before. The metric looks like (8.2)

By using this polar coordinate, we can express the trace Tr as a path integral on the space illustrated in the lower picture of Fig. 22. Since the two spaces R are glued to each other along , the periodicity of is now .

The gravity dual is given by the topological black hole (refer to Ref. [68]): (8.3) where . The smoothness of the geometry determines the periodicity of as (8.4) where is the outer horizon .

We take the periodicity to be . This leads to (8.5)

We can evaluate two-point functions in the holographic CFT from this geometry by applying the standard formula in AdS/CFT: (8.6) where is the geodesic distance between the two points and in the gravity dual. Note that even though the two-point functions on R are universal in higher-dimensional CFTs, that is not true for two-point functions on a curved manifold. Therefore we need the evaluation of two-point functions using the gravity dual.

We consider geodesics described by the form , where and take fixed values. The geodesic equation in the metric of Eq. (8.3) looks like (8.7) where is the minimum value of on the geodesic (or equally the turning point). By integrating the solution to this equation as (8.8) we can find the geodesic length between two boundary points and ; is the cutoff at the AdS boundary and is written as in terms of the CFT cutoff . The geodesic length is a function of the time difference , and they are parameterized by as (8.9) (8.10)

Now let us consider the evaluation of . As in the two-dimensional CFT case we apply the large- factorization, namely generalized free field calculation. Then, the non-trivial Wick contraction is favored when L_{bc}$]]>, where the points , , and are the AdS boundary points , , and . Since is a monotonically increasing function of , we find that the non-trivial Wick contraction is favored when L_{bc}$]]> holds, i.e. (8.11)T_2-(2\pi-T_1). \label{condgh} \end{equation} $$]]>

When we calculate the information metric we assume that and are almost the same position in each R. This means that (look at the bottom picture of Fig. 22). In this way, the condition of non-trivial Wick contraction, Eq. (8.11), leads to (8.12)

In the original coordinates of , this is equivalent to (8.13)0. \end{equation} $$]]>

This reproduces the correct entanglement wedge of the half-plane .

In the Bures distance limit, the replica number is finally taken to be . Therefore, we do not need to worry about the curved space complications and the two-point function takes the standard universal form: (8.14)

In the same way as the two-dimensional CFT case, we find, in the limit , (8.15) where the two-point functions are given by Eq. (8.14).

Thus, the final Bures information metric is computed as (8.16)

This does indeed agree with the time slice metric of a -dimensional Poincaré AdS.

Next, we turn to spherical subsystems. Consider a holographic CFT on R. In polar coordinates, the metric takes

We take the subregion to be inside the spherical region defined by . To apply the replica method, we use the map [68]

After this coordinate transformation, the metric looks like which is conformally equivalent to . The S direction represents the Euclidean time coordinate and its period is , and in this map the original surfaces and will transform to and , respectively.

The gravity dual of the above space is a topological black hole with the metric (see Ref. [68])

Around the event horizon we can approximate , where is the larger solution of . After substituting this form if we require that this spacetime is the regular solution to the Einstein equation, i.e. we do not admit any conical singularity, the inverse temperature is fixed as .

Now, let us consider calculating ; is a state in which operators and are inserted, and is a state in which operators and are similarly inserted. If we apply the replica method to evaluate the correlation function, we have to consider geodesics in the topological black hole which connect two boundary points, and to choose the mass parameter in Eq. (8.17) such that the periodicity of is . However, as in the previous calculation, the geodesic length is monotonic with the difference of boundary time coordinates, and hence we only have to specify the difference of instead of calculating the length of the geodesic directly.

As in the previous argument, we find that non-trivial contraction is favored when (8.17)

This condition is equivalent to (8.18) which indeed perfectly reproduces the expected entanglement wedge in AdS.

Correlation functions on are related to those on R by where is a conformal factor.

In the above form we just care about Jacobian and conformal transformations of the correlation functions, whose explicit forms are given by

Then, the Bures distance becomes where (8.19)

Above, we neglected the spherical part for simplicity; however, we can treat it in a similar way and thus can derive the full Bures metric: (8.20)

By considering a geodesic which connects at the AdS boundary and the middle point and , the relation between and is found as (8.21)

This maps the Bures metric of Eq. (8.20) into the time slice metric of AdS: (8.22) up to a constant factor.

The affinity is defined by Eq. (2.5), and the Hellinger distance is introduced as in Eq. (2.16), accordingly. The affinity for our density matrix in Eq. (1.1) in 2d CFTs with a single-interval can also be evaluated by the analytic continuation of the replica correlation function as (9.1) where the correlation function is the same as in Eq. (4.5) with , and (9.2)

The normalization is given by (9.3)

The partition function can be evaluated in a similar manner to the fidelity. For example, the partition function for the single-interval case is (9.4) where the CFT wedge for the affinity is the same as that for the fidelity. In this example we find that (9.5)

Actually, the same relation also holds for the double-interval case. CFT wedges of affinity in both single- and double-interval cases coincide with those of the fidelity, and therefore agree with the actual entanglement wedge in AdS.

From the property in Eq. (2.19), we have (9.6)

Therefore, the trace distance has the same transition point as the fidelity, which perfectly matches the entanglement wedge. It would be interesting to check this conclusion from a direct calculation in holographic CFTs.

The quantum Chernoff bound is largely discussed as another distinguishability measure. It was first introduced in Ref. [69] as (9.7) where is the quantum Rényi overlaps [70], (9.8)

The partition function is the same as Eq. (4.5) with , and (9.9)

Note that this quantity is bounded from above by , which is saturated if , and from below by , which saturates if . One important property is that the Chernoff bound gives bounds on the affinity and the fidelity as (9.10)

Combining with Eq. (9.5), one can easily find, for the single- and double-interval cases, (9.11)

We can directly check this equality by evaluating the replica partition function. Note that this equality holds if both density states and are pure states, that is, (9.12)

Note that we also have the following bounds on the trace distance: (9.13) which is consistent with our conclusion that the quantum Chernoff bound also plays a role as a probe of the correct entanglement wedge.

In general cases, it is hard to get fidelity and affinity due to the complication involved in evaluating the square root of a density matrix. Instead, we can rely on super-fidelity, which is defined by (9.14)

This quantity involves only products of density matrices, which greatly simplifies its evaluation, in sharp contrast with the fidelity. The super-fidelity does not satisfy the property .

The point is that the super-fidelity gives the upper bound on the fidelity as [71, 72] (9.15)

The equality is satisfied when . From this inequality, one can find that directly implies , which means that the super-fidelity is another similarity measure.

Let us focus on holographic CFTs. In fact, one can immediately find that , which means that the super-fidelity reduces to the trivial upper bound in the large- limit. Therefore, we cannot distinguish our two states by making use of the super-fidelity in holographic CFTs. Note that in CFTs with finite , this also gives a non-trivial bound.

A generalization of the fidelity, -fidelity [36] is defined by (9.16) where we introduce (9.17)

The fidelity coincides with . By using the -fidelity, the lower bound on is given by the measure , Eq. (2.7): (9.18)

Therefore, we cannot utilize the 2-fidelity as a probe of the entanglement wedge in general.

The quantum Jensen Shannon divergence (QJS divergence) is defined in Ref. [73]⁴ as (9.19) where is the von Neumann entropy. This quantity can also be seen in quantum information theory, where it is called the Holevo information. As shown in Ref. [73], it shares many relevant physical properties with the relative entropy. Since the relative entropy is well defined only in some restricted situations, the QJS divergence is more useful as a distinguishability measure. The QJS divergence also satisfies the inequality (which comes from the bound on the Holevo information [38]) (9.20) where the lower bound is saturated if and only if .

For two neighboring density states, this quantity can be approximated by the fidelity as (9.21)

Through this relation, we can conclude that the QJS divergence can also probe the entanglement wedge in a similar way to the fidelity.

Finally, we would like to compare the results of the above distinguishability measures in addition to and the fidelity . CFT wedges defined by the measures reproduce the correct entanglement wedges for 2d holographic CFTs. On the other hand, CFT wedges deviate from the correct entanglement wedges when we employ the measures . This is summarized in Table 1.

The fundamental properties of these measures are listed in Table D.1 in Appendix D. By comparing this table with the previous one, we notice that property (ix), i.e. monotonicity under completely positive trace-preserving (CPTP) maps, seems to be responsible for reproducing correct entanglement wedges.⁵ At the same time, another common property for the coincidence between CFT wedges and entanglement wedges is that the total power of and is one in the trace, as we emphasized in Sect. 6.6. This requirement comes from probing only the low-energy states dual to the classical gravity. On the the other hand, for the other measures the total power of and is two. In this sense the former look analogous to the von Neumann entropy, while the latter seem analogous to the second Rényi entropy. In summary, our results in this paper suggest that these two properties are necessary for a distinguishability measure in holographic CFTs to reconstruct the correct entanglement wedges.⁶

It is interesting to note that there are other similarity measures which satisfy property (ix), for example the relative entropy. For this reason, we can expect that this quantity can also probe the entanglement wedge. It would be interesting to investigate whether the relative entropy can actually detect the entanglement wedge; this is left for future work.