Holographic quantum distances and replica trick

This paper gives concrete examples to exhibit how to use the replica trick to calculate the quantum (quasi-)distances holographically. First, we consider the fidelity and relative entropy between thermal states that are dual to the Schwarzschild-AdS black holes. Then we generalize our method into the RN-AdS black holes by adding a U(1) gauge field. We also investigate the fidelity between states excited by scalar operator in probe limit. In this case, it is surprising that the fidelity in standard quantization will suffer from new UV divergence though the usual holographic renormalization has been applied. We call for deep understanding for such divergence in the future. We also discover a holographic method to check whether the density matrices of two holographic states are commutative.


Introduction
In recent years, quantum information theoretic considerations play an important role in the study of AdS/CFT or more general gauge/gravity duality [1][2][3].For instance, Ryu-Takayanagi (RT) formula [4,5] quantifies the entanglement entropy of the boundary conformal field theories (CFTs), which is given by the area of a codimension-2 minimal surface in the dual bulk spacetime.Under some reasonable assumptions, RT formula can be obtained from the more general Rényi entropy [6][7][8][9] S n ≡ 1 1 − n ln trρ n (trρ) n by applying the replica trick.Here the index n labels the number of the replicated boundary CFTs and ρ is the density matrix of the entangling region.The n is originally assumed to be a positive integer value.In [8], Lewkowycz and Maldacena argued that, for integer index n > 1, the CFTs on a branched cover 1 M n over the entangling region still have 1 See Ref. [10,11] for how to construct Mn by taking n copies of M1.The key points here are making suitable identifications for quantum fields between these copies and obeying right boundary conditions.The latter, in other words, requires us to regulate the path integral of Mn by imposing the right field configurations at the boundaries.
corresponding interior gravity duals B n like the original Euclidean spacetime B 1 does.In the gravity side, there exists a solution B n that satisfies the Einstein equation along with boundary condition ∂B n = M n .Based on the holographic duality, one should expect that the partition functions of two sides' theories are equated each other where Z[M n ] = trρ n .Note that we are working in the large N limit such that the bulk geometry can be taken as a solution of Einstein gravity.Under the saddle point approximation, the partition function of bulk geometry can be written as the exponential function of corresponding Einstein-Hilbert action Following Rényi entropy's definition, the direct calculation indicates that Rényi entropy is proportional to the gravitational action of corresponding bulk geometry.If it is possible to analytically continue the Rényi entropy S n to noninteger2 n, the von Neumann entropy that is usually referred as the entanglement entropy in quantum information theory can be recovered from S n at the n → 1 limit.The entanglement entropy concerns about the entanglement property of only one quantum state but not involves the relationship between two states.There are other quantities in quantum information [12], which measures how close two quantum states are.In other words, these quantities label some kind of "distance" between quantum states.For example, given quantum states ρ and σ, the trace distance is defined as where tr|A| ≡ i λ i and λ i is the eigenvalue of For n = 1, the above formula is just the trace distance.There is other special choice that is widely applied.By choosing n = 2, Hilbert-Schmidt distance can be obtained, which is most convenient D n for calculations.The second families of quantum distance are based on the fidelity (1.5) The fidelity is not a distance but it can be used to define other measures of distance: theFubini-Study distance D F [ρ, σ] and the Bures distance Due to the well-defined calculations in finite dimensional Hilbert spaces and the preservation under unitary transformations, trace distance and fidelity are widely used within the quantum information community.However, their analytical calculations are extremely difficult in quantum field theory.Ref. [13] makes the first step towards this issue.By virtue of the certain correlation functions for the vacuum or the thermal state, author of Ref. [13] develop a replica trick to calculate the fidelity for (1 + 1) dimensional CFTs.Using similar method, Refs.[14][15][16] also developed replica method to compute the trace distance for a class of special states for single short interval in 1+1 dimensional CFTs.Recently, Ref. [17] proposes how to use replica trick to compute Rényi mutual information in 1+1 CFTs.Though in higher dimensions the replica trick is still effective, the correlation functions of interacting theories are highly nontrival to compute.Similar to holographic descriptions of entanglement, one would wish there exist a holographic duality to calculate trace distance and fidelity.Actually Ref. [18] argues that the fidelity between a state and its infinitesimal perturbation state is approximately given by a volume of maximal time slice in AdS spacetime.However, when the difference between two states is not infinitesimal, the gravity dual of fidelity given by Ref. [18] is not reliable.In addition, the holographic proof on the proposal of Ref. [18] is absent.See [19][20][21][22][23] for recent progress related to replicas in quantum information under the holographic setup.
Although it seems quite difficult to solve this problem completely, this paper uses some concrete examples to exhibit how to calculate quantum distances in holography.As a basic perspective in holographic picture, Maldacena argues that the thermal states in the boundary CFT correspond to Schwarzschild-AdS black holes.Firstly, similar to the deduction of Rényi entropy by employing the replica trick, we analytically calculate the fidelity and relative entropy between thermal states in holography.Next, we want to check our method in the presence of bulk matters.Here we consider two concrete examples.In the first one we consider that there is a conserved charge in the boundary theory so the bulk theory is the Einstein-Maxwell theory.In the second example, an Einstein-scalar theory is constructed here to gain the insight into the fidelity between states excited by scalar operator.In probe limit, we derive the analytic expression of fidelity in this case.Surprisingly, we find that the fidelity of probe limit is well defined only in the alternative quantization.The fidelity in standard quantization will suffer from new UV divergency though the usual holographic renormalization has been applied.To the best of our knowledge, such kind of divergency has not been discussed well in the study of holography, and so we call for deep understanding in the future.In a physical system described by quantum mechanics, quantum states are generally the superposition of several eigenstates under a observable Hermitian operator.It is more troublesome but feasible to check the commutativity of two quantum mechanics states.As we claimed before, quantum states in infinite dimensional Hilbert spaces are more tricky.Without more assumptions, it is difficult to directly write the density matrix for a general quantum state in quantum field theory.So it is nontrivial question to determine the commutativity of two quantum states in quantum field theory.At the end of this paper, we propose a method to answer this question partially if two quantum states are holographic.We will see whether two holographic states are commutative or not is related the contribution of corresponding bulk theory's action.
The organization of this paper is as follows.In section 2, we consider the fidelity and relative entropy between thermal states with different temperatures.In section 3, under the same procedure, we generalize the above method into the grand canonical ensemble, where we consider the fidelity and relative entropy of the states which has different chemical potentials.In section 4, we construct an Einstein-scalar theory in asymptotically AdS spacetime which is dual to states excited by scalar operator, and derive the semi-analytic formula of fidelity in the probe limit.Particularly, in the standard quantization, we show that the fidelity will suffer from a kind of new UV divergence even though we have added usual counterterms into the action.We will also show that such kind of new divergence will not appear in alternative quantization.In the section 5, we will discuss how to check the commutativity of two density matrices holographically.

Holographic quantum distance between thermal states
In this section, we show how to holographically get the quantum distances between thermal states.In addition, we will explain the main idea about how to use replica trick to holographically compute the expression such as tr(ρ Our starting point is that thermal states in CFT d are dual to Schwarzschild-AdS d+1 black hole [24].Let us start from the properties of the thermal state.Thermal state can be used to describe the quantum system which is in contact with a heat bath at temperature T = 1/β.In the canonical ensemble at temperature T , the density matrix of thermal state reads Here H is the Hamiltonian of this system and Z(β) = tre −βH is the partition function.To illustrate how use replica trick to compute the quantum distance here, we will consider two thermal systems, of which the density matrices are In following we will first introduce how to compute the trace of product ρ 1 ρ 2 holographically, This is product will play the essential role in our computations of quantum distances later on.
Note that for general two different density matrix ρ 1,2 , we can not guarantee their product ρ 1 ρ 2 is hermitian.Then, after normalized, the ρ 1 ρ 2 /tr(ρ 1 ρ 2 ) is not a density matrix unless satisfying the condition [ρ 1 , ρ 2 ] = 0.However, for thermal state, ρ 1,2 commute with each other since they own the same Hamiltonian H.The main argument here is that ρ 1 ρ 2 can be regarded as an un-normalized density matrix of a new thermal state with the inverse temperature β = β 1 + β 2 , as depicted in Fig. 1.Note that ρ 1 ρ 2 is not properly normalized.
The schematic diagram about the replica trick between thermal states.Here we consider ρ 1 ∝ e −2βH and ρ 2 ∝ e −βH .Then, the inverse temperature of replicate thermal state is 3β.
From the bulk perspective, the gravity dual of ρ 1 ρ 2 would be a Schwarzschild-AdS black hole with new inverse temperature β.See appendix A for the convention we use here.According to the relation (A.7) between temperature T and horizon radius r h , this new black hole's horizon radius r h can be in principle solved.For convenience, this paper will focus on planar black hole (black brane).For planar Schwarzschild-AdS black hole denoted by k = 0, Then we can get the new horizon radius rh through r 1,2 , that satisfies (2.5) Based on the above calculation, we conclude that the gravity dual of ρ 1 ρ 2 is Schwarzschild-AdS black hole with smaller horizon radius rh .The product between thermal states gives us more "smaller" black hole as depicted in Fig. 2.
The gravity dual of the replicate system by multiplying n thermal states.For general replicate thermal state ρ 1 ρ 2 • • • ρ n , its gravity dual is a Schwarzschild-AdS black hole with horizon located at rh = 1/(r −1 After a long journey, we are ready to show how to calculate tr(ρ 1 ρ 2 ) in Einstein-Hilbert gravity.From the basic holographic dictionary in Ref. [24], we have the relation that is a most simplest nontrival example of relation (1.1).Recalling the relation (1.2), here we have Ref. [25] shows that, the full Euclidean gravitational action I[β] in d + 1 spacetime dimensions should have three contributions Here (2.9) The first term is the Einstein-Hilbert action with cosmological constant −d(d−1)/2L Under such holographic description and the saddle point approximation, the trace of ρ 1 ρ 2 can be expressed in geometric quantities tr(ρ Above calculation can be generalized into more general case such as tr(ρ This gives us a formula to compute the trace of ρ 1 . . .ρ n from holography.
To check our result, one can calculate the Rényi entropy for thermal state ρ by taking n copies.According to the definition of Rényi entropy, we can obtain that . (2.13) Following the spirit of the replica trick, then we analytically continue n into unit one We can see that the Bekenstein-Hawking entropy formula can be precisely recovered.This is an check of self-consistence on our method.Taken the lessons we learned above, we will consider various quantum distances between thermal states in the remaining content.More specifically, we will employ the method of constructing Rényi entropy, i.e. replica trick, to calculate these quantum distances.

Fidelity
Given two thermal states ρ and σ in the same Hilbert space, we now is ready to calculate the fidelity Fi(ρ, σ) =tr √ ρσ √ ρ in holography.Since the dimension of Hilbert spacetime is infinity in the field theory, the fidelity of two different states will almost vanish and it will be more useful to consider the logarithmic fidelity ln Fi(ρ, σ).Since the square root is involved into the trace, we cannot directly compute the fidelity by using integer-order replica shown in above example.To overcome this issue, we now define a generalized n-order log-fidelity for positive integer n Then, we can analytically continue n into real number.If ln Fi n (ρ, σ) is analytic for n > 0, the fidelity can be recovered Generalizing the procedure for trρ 1 ρ 2 into tr(ρ that is an inverse proportional function of copies number n.As just claimed before, we then analytically continue n into 1  2 to obtain the fidelity here rh take the value rh = 2r 1 r 2 /(r 1 + r 2 ).As a consistent check, we can confirm that

Relative entropy
Relative entropy is a measure of the distinguishability of two states.It will not suffer from the ultraviolet divergence that carries no information about the state.Given two density matrices, the relative entropy is defined as Here ρ and σ are both properly normalized.The first term on the right hand of equal sign is just the entanglement entropy of ρ.In order to compute the second term in holography, we can imitate the construction from Rényi entropy to entanglement entropy tr(ρ ln σ) = lim Still considering two thermal states ρ 1 and ρ 2 , tr(ρ ) is given by The second term in Eq. (2.19) can be recovered by analytically continue n into 1 Combining the above result with entanglement entropy of ρ 1 , the relative entropy between thermal states ρ 1 and ρ 2 is given by (2.23) It should be noted that the relative entropy vanishes when ρ 1 equates to ρ 2 , which is a expected result no matter what method is used here.
3 Add some matter: Maxwell field as the first step In this section, we want to generalize our replica trick into the charged black hole.For boundary field theory, we consider the grand canonical ensemble instead of the canonical ensemble.Given the conserved charge Q, we can do the similar construction for the matrix density of the states Here µ is the chemical potential.The gravity dual of such state (3.1) is charged black hole in AdS spacetime.Since charge Q is conserved, it should satisfy [Q, H] = 0.By virtue of such commutativity, we will find that the density matrices of different temperatures and charges are also commutative to each others.Thus we can generalize our above replica trick from neutral system into the charged one.Considering states ρ 1 , ρ 2 Following the spirit of the previous section, we reconsider ρ 1 ρ 2 as a new state We can see that the replicate state ρ 1 ρ 2 have the same construction with the original two states.In holography, this claim means the replicate state ρ 1 ρ 2 prepared by (3.2) is still dual to AdS-RN black hole with new parameters β, μ defined by (3.3).
In gravity side, considering the Einstein-Maxwell theory, the total action (2.8) need to add the contribution from the Maxwell field If we just consider the RN black hole, the form of the metric coincide with the neutral one (A.1)but with Here µ is the chemical potential of the black hole and r h is the largest root of f (r) = 0.
For RN black hole, the Maxwell field strength tensor has a form Here Q is the total charge of RN black hole.Notice that F µν F µν = 2Q 2 /r 2(d−1) , the direct calculation for I Maxwell reads {r h , β, µ} is not independent of each other.For k = 0, we have a relation among {r h , β, µ}: According to the replica trick (3.3), we should consider the gravity dual of replicate state ρ 1 ρ 2 as a RN-AdS black hole with thermodynamic parameters β, μ .By employing the holographic dictionary we can use the on shell action to express tr(ρ 1 ρ 2 ) For instance, if we set In general the action Z(β, µ) will be a complicated nonlinear function of β and µ.For simplicity of calculation, we can ignore the back reaction to bulk geometry from the Maxwell field if µ ≪ 1.In this sense, Then we obtain that Using the fact that rh is the half of r h and neglect the terms smaller than O(µ 2 ), it is straight to calculate tr(ρ 1 ρ 2 ) (3.13) Finally, we generalize our result into n copies and calculate the Rényi entropy (3.14) By taking the limit n → 1, the Von Neumann entropy can be recovered One can also get the above result according to Bekenstein-Hawking formula.Since we consider the chemical potential µ as a infinitesimal, the horizon (3.8) of RN-AdS black hole denoted by r RN can be separated into two parts Here r h is the largest root of f (r) = 0 for µ = 0. Following the Bekenstein-Hawking formula, the entropy is described by the area of horizon.Up to the order O(µ 2 ) we obtain (3.17) We see that two formulas (3.15) and (3.17) are same, which is what we expect and shows the consistence of our methods.

Fidelity between states excited by scalar operator
In this section, we consider how to compute the quantum distance between states that are excited by scalar operator.In holography, the scalar operator O ϕ in boundary CFT is dual to a bulk scalar field ϕ.Considering a (d + 1)-dimensional Einstein-scalar theory, such theory's action is The parameter m is the mass of the scalar field.Different from the Minkowski spacetime, m 2 is negative in AdS spacetime.Despite this, m 2 can not be too negative but above the Breitenlohner-Freedman bound m 2 BF ≡ −d 2 /4L 2 .For dual boundary CFT, the corresponding conformal dimension ∆ O for scalar operator is The "+" corresponding to standard quantization and "−" stands for alternative quantization.We can get the equations of motion following the action (4.1) In most general cases, we should solve this coupled gravitational system since the backreaction of the scalar field can not be neglected.In principle, given the proper boundary conditions, these equations can be solved analytically or numerically.We consider the thermal states in the canonical ensemble.However, we can neglect the backreaction of the scalar field in the bulk, if the change of the total energy ∆E induced from the scalar operator O ϕ is much smaller than the mass of black hole.This is the so-called probe limit.In the probe limit, the gravity and scalar field will decouple.That means in contact with a heat bath at β, the bulk geometry remains to be Schwarzschild-AdS black hole.Then, we just analyze the behavior of scalar field in this fixed black hole background.What's more, we only need to explore the additional contributions from the scalar field to holographic quantum distance.In this paper, we focus on the fidelity between states excited by scalar operator since it is easy to obtain.The main reason why it is tricky to calculate other quantum distances will be explained soon.

Set up
In this section, we adopt the Poincaré coordinate z, τ, x i .Under this gauge, the n-copies Euclidean Schwarzschild-AdS black hole's metric reads Here the factor 2n is induced from the definition of fidelity where tr(ρσ) n is involved.Based on our previous discussion, for two states ρ and σ with same inverse temperature β, the dual black hole to compute tr(ρσ) n will have inverse temperature 2nβ.Then, we write the equation of motion for scalar field in this background: More specifically, where a prime/dot denotes the derivative with respect to z or τ , respectively.In general case, the Eq.(4.5) does not have analytical solutions so we have to solve this equation numerically.
Here we assume the boundary can keep the asymptotic AdS behavior such that Other requirement is that ϕ should be continuous near the horizon z h .Here φs (τ ) and φv (τ ) are independent of z.In holography, both φs (τ ) and φv (τ ) can be considered as a source if the mass parameter satisfies m 2 BF < m 2 < m 2 BF + 1/L 2 .It needs to note that φs and φv will be both nonzero if the scalar field is not zero in the bulk.To keep the asymptotically AdS structure and make the probe limit self-consistent we have to require ϕ 2 → 0 as z → 0. This requires that we have to set d > ∆ and so we have to set: The selection of source corresponds different dual field theories, see Ref. [30].In gravity side, this selection determines how to set the boundary condition for Eq.(4.5).For instance, if considering φs (τ ) as source, we must fix φs (τ ) at z → 0 correspondingly.This is called standard quantization scheme for the scalar field.While in this paper, we employ the alternative quantization scheme that considers φv (τ ) as the source.The reason why we choose "non-standard" quantization scheme will be explicitly explained.It should be noted that the scalar field obeys second order equation of motion (4.5), but we only give one boundary condition such as fixing φv (τ ) in alternative quantization scheme.However, we must discard one of the independent solutions since this solution blows up at the horizon z h .Hence, the regularity of Euclidean spacetime makes the bulk scalar field uniquely specified.The next step is to solve Eq. (4.5) since the final result for action (4.1) is dependent of φv (τ ) and φs (τ ).Then we can assume the general solution of ϕ(τ, z) by the method of separation of variables Substituting this into Eq.(4.5), we can get the asymptotic behavior of general solution for scalar field T 0 (τ ) = A 0 , T j (τ ) = A j cos w j τ + B j sin w j τ , and the radial equation Here ϕ s,v is denoted by the asymptotic behavior of R j (z).The frequency w j take some discrete values given by where τ is the period in boundary condition (4.6).According to Fourier transformation, coefficients A 0 , A n , B n are determined by φv (τ ) sin w j τ dτ (4.12)

Fidelity in the probe limit
In following we will first focus on alternative quantization, i.e. treating the coefficient ϕ v as the source term.As previously defined, given two density matrices ρ and σ that are both excited by scalar operator O ϕ with different values of source φv , a generalized n-order log-fidelity for positive integer n is Then, we can analytically continue n into 1/2 to recover the fidelity Next we construct the corresponding distribution φv (τ ) for tr(ρσ) n in the bulk, see Fig. (3) for illustration of tr(ρσ) 2 .In the bulk gravity side, the boundary condition for scalar field ϕ will be dependent of euclidean time Here ρ v and σ v are constants.We already set β 1 = β 2 = β.To simplify the discussion, in this paper we just set σ v = 0.In principle, we need to know the value of bulk scalar field in every spacetime point.Since the total action (4.1) is a spacetime volume integral.However, according to Gauss theorem and using the equation of motion, the on-shell action for scalar field can be expressed by the boundary term Here n µ is the outward point normal vector of AdS boundary ∂B n .In Poincaré coordinate, the only nonzero component is n r = −z f (z) , (4.17) so this boundary integral can reduce to Note that the above action is not regulated.In principle, according to spacetime dimension d and conformal dimension ∆, we will correspondingly select different forms of counter term [26][27][28][29].In this paper, we will focus on 4-dimensional Einstein-scalar theory while fixing m 2 = −2.For {d = 3, ∆ = 2}, the corresponding counter term is Therefore, the regulated scalar field action Ĩρσ can be easily calculated Here Before we go further, it needs to emphasize that divergency of "naked" action arises only when we approaches to UV boundary z → 0. The counterterm then is applied to cancel such divergency.We will see later that, since the boundary is inhomogeneous, there is an other different divergency of on-shell action appears in standard quantization even if we do not take the limit z = 0.This kind of new divergency cannot be cancelled according to the known holographic renormalization schema [26][27][28][29].
We now return to Eq. (4.20).Notice that Ξ is dependent on the copies number n, since the radial equation is related to n, i.e. f (z) = 1/L 2 − (2nz) d f 0 .This property is very different from the usual cases that φv is independent of the Euclidean time τ .If Ξ[ρ v , 1  2 ] is convergent, then we would analytically continue n into 1/2.In other words, we need to solve the radial equation (4.10) with f (z) = 1/L 2 − z d f 0 .Finally, the log-fidelity reads ] .
Here Ĩρ is the regulated action for trρ Note again that Eq. (4.10) can be solved for arbitrary real number n.Thus, we can numerically compute the value of Ξ[ρ v , n] in the Eq. ( 4.21) and then obtain the fidelity numerically.

Numerical result
After the previous derivation for fidelity, we then show the corresponding numerical process and result.The only remaining unsolved equation is the radial equation We need to get the value of ϕ for every frequency w j .For j = 0, the frequency is vanished, i.e. w 0 = 0.The general solution for R 0 is just the static case Continuity condition requires that R 0 should be finite at horizon z h which satisfies R 0 (z h ) can be expressed as

.28)
Notice that R 0 (z h ) is singular since Γ(0) is infinite.We need to eliminate this singularity.Extract Γ(0) outside the square bracket Therefore, the square bracket must vanish and the ratio ϕ For j = 1, 2, 3, . . ., Eq. (4.24) does not have analytical solutions so we have to solve the radial equation numerically.Notice that Ξ[ρ v , n] is the sum of an infinite series.Although we cannot analytically get the explicit expression of ϕ v , the convergence or not of Ξ[ρ v , n] can be obtained from the comparison with the scalar field in thermal AdS spacetime given f 0 = 0 in metric (4.3).Considering the scalar field with the same boundary condition (4.15) in thermal AdS spacetime, the corresponding radio ϕ v can be analytically solved: Notice that in thermal AdS spacetime Ξ[ρ v , 1 2 ] thermal AdS is convergent since the general term is proportional to 1/j 3 .To determine whether Ξ[ρ v , 1  2 ] is convergent, we give a concrete numerical result to check the change of δ j with j δ j ≡ ϕ -2 Figure 4: The curve-fitting result of δ j .Here we use a logarithmic function b + ln j a to fit their difference.
Here we fix the following parameters From above parameters, we can obtain β = 4π/3 and ∆ = 2.We solve the equation (4.24) numerically and then read the radio ϕ v from the asymptotic behavior of the radial equation.
According to the fitting result, i.e.Fig. ( 4), we conclude that the difference between thermal AdS is j a order.Since a ≈ −3.81678, the difference between two series is convergent.It should be noted that the numerical result of a is very close to −4.In practice, we can confirm that with a analytical approximation method.See appendix B for more details about the solution for radial equation.Then, according to the comparison test, Ξ[ρ v , 1  2 ] must be convergent under the alternative quantization scheme since in this scheme Ξ[ρ s , 1  2 ] thermal AdS is convergent.The final numerical result reads

Comment on standard quantization
We have shown that the on-shell action of replica spacetime is convergent in alternative quantization.However, under the standard quantization the situation is very different.In this case, we still have Eqs.(4.20)-(4.23),but the ρ v is replaced by ρ s and Ξ should be replaced by We will find that Ξ[ρ s , 1 2 ] thermal AdS is divergent.Specifically, Ξ[ρ s , 1  2 ] thermal AdS is a harmonic series under the standard quantization since the general term in Ξ[ρ s , 1  2 ] thermal AdS becomes The high frequency scalar modes with w j ≫ 1 will be not suppressed enough and so will play more and more important roles.When it comes to the computation of action, this effect will cause the infinity of Ξ[ρ v , n] in the standard quantization scheme.While under alternative quantization scheme, these UV contributions from high frequency modes are suppressed enough.It should be noted that this behavior is quite common when considering inhomogeneous boundary condition for bulk matter.It needs to emphasize that such as divergency appearing in standard quantization cannot be removed by using the usual so called "holographic renormalization", since one can realize that this divergency will appear even we move the AdS boundary into finite z.We note similar thing was also noticed by Ref. [31], which finds that generalized susceptibly in terms of momentum integration will be divergent in standard quantization.One may seek to regulate this infinity in standard quantization scheme.For instance, given a UV cutoff k, we seem to be able to extract divergence from the harmonic series here γ is Euler-Mascheroni constant.So the divergence here is logarithmic divergence.When the mode j is high enough, the probe limit and even including the classical gravity approximation will be broken, so we should take the full backreaction and even full UV quantum gravity theory into account.If we believe the theory is UV complete, then "unknown" high energy physics should just cancel such a logarithmic divergency.Then we will throw away ln k, which will be interpreted as the contribution from the deformation of spacetime due to these high frequency modes.However, this procedure is only effective for the set of parameters we have selected here.When we fine-tune these parameters, such as m 2 , the above procedure is broken and we need to regulate such infinity again in this new set of parameters.From the Eq.(B.15) in appendix B one can see that, for large mode j.Thus, in standard quantization, we can find from Eq. (4.35) that Since 2∆−d = √ d 2 + 4m 2 L 2 > 0, we see that Ξ will be divergent.It is still an open question for us about how to regularized and renormalize such a divergency systematically.We leave the general regularization under the standard quantization scheme for an important future problem.

Commutativity of ρ and σ
In this section, we will gives a holographic method to check if the density matrices of two holographic states are commutative to each others.In quantum field theory, we can't usually write the density matrix of quantum states precisely.Assume that the quantum state ρ is excited by some operator O ρ (x) and other quantum state σ is excited by operator O σ (y).Here x, y label the points in the background spacetime.It should be noted that even employing the same operator, the commutator [O(x), O(y)] is not necessarily vanish for different spacetime points.However, as mentioned before, there exist a gravity dual for the trace of density matrix's product.Based on the holographic description, is it possible to determine whether two operators are commutative?In this section, we propose a feasible strategy to answer this question.
Considering two density matrix ρ and σ in the same Hilbert space, the commutativity of ρ and σ indicates ρσ = σρ . (5.1) One may think we can use their trace to judge whether they are commutative, but the property of trace tr(AB) = tr(BA) rejects this idea.Instead, considering a auxiliary matrix A A = ρσ − σρ , ( if A = 0 then we can get tr(ρσρσ) = tr(ρ 2 σ 2 ) . (5.3) This result is so straightforward since we can change the position of two density matrix arbitrarily.Conversely given the condition (5.3), [ρ, σ] = 0 is expected to be proved.Firstly, it is not hard to find that tr(ρσρσ) = tr(ρ 2 σ 2 ) =⇒ tr(A 2 ) = 0 . (5.4) Since A = −A † , A can be diagonalized with pure imaginary eigenvalue {|n⟩} is a complete orthonormal set.tr(A 2 ) = 0 indicates that a n = 0 for all n, by virtue of anti-hermiticity we actually obtain that From the perspective of holography, there are two bulk geometries corresponding to boundary condition ρσρσ or ρ 2 σ 2 , as depicted in Fig. 5.In order to determine whether states ρ and σ are commutative, we should calculate these bulk geometries' partition function respectively.Recall that under the saddle point approximation, the partition function of bulk geometry can be written as the exponential function of corresponding action.So we only need to check whether their corresponding geometric action are equal, i.e.

S ρσρσ
(5.7) Furthermore, in probe limit we only need to compare the sector of scalar field, (5.8) Without losing generality, we continue to use the scalar field model in previous section.
In order to compute the fidelity between ρ and σ, we already calculate the action of bulk gravity that corresponds to the boundary field (ρσ) n .For a detailed discussion, we will present such result in the previous section again (5.9) Here, we need to set n = 2 to recover Ĩρσρσ from the expression (4.20).Then we need to calculate the action Ĩρ 2 σ 2 for boundary field configuration ρ 2 σ 2 .Following the same process in previous section, we calculate the action Ĩρ 2 σ 2 but with boundary condition that is different from (4.15)Here the red line labels the distribution of state ρ and blue dashed line labels the state σ.For example, figure 5a shows schematically that the bulk geometry with the boundary field configuration ρσρσ.
It is straightforward that period τρ 2 σ 2 is two times than τρσρσ .And it can be shown that Ĩρ 2 σ 2 has the same form as Ĩρσρσ However, it should be noted that ϕ v has different values in formula (5.9) and (5.11).More specifically, we know that w (j) ρσρσ = 2πj/τ ρ 2 σ 2 and τρ 2 σ 2 = 2τ ρσρσ = 4β . (5.12) Then we can obtain that w  .13)According to the above discussion, Ξ for Ĩρ 2 σ 2 can be easily seen that So we conclude that Ĩρσρσ ̸ = Ĩρ 2 σ 2 and ρ, σ are not commutative to each other in our case 3 .Another example is considering the Maxwell field F µν probing the thermal states which are corresponding to AdS-Schwarzschild black hole.In the probe limit, Maxwell field F µν is decoupled with gravity sector g µν .Therefore, the only different contribution of tr(ρσρσ) and tr(ρ 2 σ 2 ) is the contribution from the Maxwell filed: (5.15) As we have calculated in section 3, the field strength is determined by

.16)
Here r h is determined by πL 2 /dβ.In this case, the chemical potential is time dependent.For tr(ρσρσ), the boundary condition of µ(τ ) is (5.17) Here µ ρ and µ σ are constants.Although µ(τ ) is time dependent, the field strength has the same nonvanishing component F rt .The direct calculation for I Maxwell for both tr(ρσρσ) and tr(ρ 2 σ 2 ) reads (5.18) So we check the consistency of conclusion from section 3 in the probe limit, which shows any two charged thermal state in the same Hilbert space are commutative.As shown in the previous two examples, this gives us a universal4 holographic approach to check if the density matrices of two holographic states are commutative to each other.As we investigated in this section, the commutativity of ρ and σ are holographically determined by the difference between two bulk geometries dual to ρσρσ or ρ 2 σ 2 .

Summary
To summary, this paper uses the holographic approach to calculate the fidelity and relative entropy in some concrete examples.By employing the replica tick, we can analytically calculate these quantum distance between thermal states.The key point here is that for two thermal states, the product of their respective density matrices is still a thermal state.In gravity side, this indicates that the gravity dual of the replicated thermal state is continue to be Schwarzschild-AdS black hole.We also investigate our approach under the presence of matter field.By the virtue of U (1) invariance for the Maxwell field, these conclusions for the thermal state can be generalized into the charged system within the grand canonical ensemble.The similar step can be also generalized into other cases.Without loss of generality, an Einstein-scalar theory is constructed here to gain the insight into the process of calculating the fidelity.For bulk matter, the replicated boundary field will cause the inhomogeneous boundary condition for the equation of motion.In general, the physical system we consider here will be time-dependent since the bulk scalar field has inhomogeneous boundary condition.For simplification, we ignore the backreaction from the scalar field to the metric.In other words, the question becomes how to solve the scalar field under the background of fixed Schwarzschild-AdS black hole.Then, we solve the corresponding scalar field's equation of motion in the probe limit.According to the standard holographic renormalization, we derive the analytic expression of the fidelity.The contribution from the scalar operator is a sum of an infinite series.Then we give a numerical result to intuitively feel this contribution here.It is found that on-shell action is divergent under the standard quantization scheme while convergent under the alternative quantization scheme.We observed that this divergency is caused by high frequency modes in Fourier space.While these UV contributions from high frequency modes are suppressed under the alternative quantization scheme.We have concluded that it is a quite common behavior in terms of the bulk matter under inhomogeneous boundary condition.At the end of this paper, a holographic method is discovered to check whether the density matrices of two holographic states are commutative.As examples, we use scalar and Maxwell field to demonstrate this method in the probe limit.The result is expected that two holographic states excited by scalar operator are generally not commutative and two charged thermal states are commutative.However, it should be noted that we do not really solve the Einstein field equation with full backreaction from time-dependent chemical potential.To the best of our knowledge, the exact bulk solution for time-dependent chemical potential is still unclear and worth to explore.
Although this paper studies the fidelity between the states that are excited by a scalar operator, it is also interesting to generalize the procedure into cases of other operators.Further, it is meaningful to generalize our method to study the quantum distance in the subsystem that is a more practical physical system.As an important future problem, more general regularization under the standard quantization scheme is needed to be explore.In the presence of the matter field, the probe limit is very critical for our calculation.In principle, someone can numerically calculate the quantum distance without neglecting the backreaction of the matter field.What is more, the divergency here though is also a UV divergency, it is not caused by AdS boundary and so the usual holographic renormalization cannot not be used to remove such divergency.We leave this problem in the future and call for deep understanding.
In this paper, we assumes horizon locates at z = z h = 1 and w j = dj 2n .At the horizon we have following asymptotic solution Near the AdS boundary we have following asymptotically behavior: 3) The horizon at the Euclidean black hole will be a smooth tip and scalar field should be finite here.Thus, we then have following boundary condition In principle, Eq. (B.6) has no analytical solution.However, we can find an analytically approximated solution, which approaches to the exact solution when w j becomes large enough.We note that V (s) → V (0) (s) = m 2 + d − 1 s 2 (B.9) when w j ≫ 1.We then use V (0) (s) to replace the potential V (s) and obtain the analytical solution Z

Figure 3 :
Figure 3: The schematic diagram about the corresponding bulk geometry with the boundary field configuration ρσρσ.Here the red line labels the distribution of state ρ and blue dashed line labels the state σ.Here subscript v follows from treating the coefficient ϕ v as the source term under alternative quantization.As setting β 1 = β 2 = β, we can see that the period τ = 2β.

Figure 5 :
Figure 5:The schematic diagram about two boundary conditions relate to the commutativity of ρ and σ.Here the red line labels the distribution of state ρ and blue dashed line labels the state σ.For example, figure5ashows schematically that the bulk geometry with the boundary field configuration ρσρσ.

ρ 2 σ 2 .for 2 ,v
In previous section, we already get the value of ϕ Ĩρσρσ and j can only take the odd term, i.e. j = 1, 3, 5, 7,• • • .Since w (j) ρσρσ = 2w (j)ρ 2 σ 2 , based on radial equation (4.24) for Ĩρσρσ , we only take j = 1 • • • to get the exact value of Ĩρ 2 σ 2 .Why can we do this because ϕ in radial equation (4.24) changes continuously with the continuous value of parameter j.Considering the thermal AdS background, Ĩρσρσ is related to the sum of series(4.21)