On the real-time evolution of pseudo-entropy in 2d CFTs

In this work, we study the real-time evolution of pseudo-(R\'enyi) entropy, a generalization of entanglement entropy, in two-dimensional conformal field theories (CFTs). We focus on states obtained by acting primary operators located at different space points or their linear combinations on the vacuum. We show the similarities and differences between the pseudo-(R\'enyi) entropy and entanglement entropy. For excitation by a single primary operator, we analyze the behaviors of the 2nd pseudo-R\'enyi entropy in various limits and find some symmetries associated with the subsystem and the positions of the insertion operators. For excitation by linear combinations, the late time limit of the $n$th pseudo-R\'enyi entropy shows a simple form related to the coefficients of the combinations and R\'enyi entropy of the operators, which can be derived by using the Schmidt decomposition. Further, we find two kinds of particular spatial configurations of insertion operators in one of which the pseudo-(R\'enyi) entropy remains real throughout the time evolution.

Also recently, a new quantity associated with a bulk minimal area surface, called pseudoentropy, has been introduced in [20] under the framework of AdS/CFT correspondence. Given a total system S, one can define the pseudo-entropy associated with a subsystem A in terms of 1 the corresponding pseudo-Rényi entropy where the matrix T 1|2 A involved two nonorthogonal quantum states |ψ 1 , |ψ 2 ∈ H S , named reduced transition matrix, is the partial trace of the transition matrix T 1|2 , 4 The pseudo-entropy of subsystem A is obtained by taking the limit of n → 1 for S (n) A . The reduced transition matrix is not Hermitian in general; hence the pseudo-entropy usually takes complex values. However, the results in the qubit system suggest that the real part of pseudo-entropy can be used to characterize the number of distillable Bell pairs averaged over the histories between the initial and final state [20].
More intriguingly, it was recently found in [21] that the real-time evolution of the real part of pseudo entropy follows the Page curve [22] under some field-theoretic settings based on the black hole final state proposal [23]. 5 Hence pseudo-entropy, like entanglement entropy, can reflect certain underlying correlation structures. Refer to [25][26][27][28][29][30][31][32] for other related developments of pseudo-entropy.
The main purpose of this paper is to study the real-time evolution of the real part of pseudoentropy for locally excited state generated by a single primary operator or linear combination of a bunch of them in various 2d CFTs. Unlike the case in [21], our investigation can be regarded as a pseudo-entropy extension of the real-time evolution of the entanglement entropy after such local operator excitations [33]. In recent years, the time evolution of entanglement entropy for locally excited states has been widely studied, including rational CFTs [34][35][36], irrational CFTs [37], large-c CFTs [38,39], boundary CFTs [40], warped CFTs [41], CFTs at finite temperature [42], multiple local excitations [43], and holographic duals of the local excitations [44][45][46][47]. In 2d rational CFTs (RCFTs), it was found that the variation of nth Rényi entropy for locally primary excited states saturates to a constant equal to the logarithm of the quantum dimension of the local operator's conformal family [34][35][36]. Such saturation get well interpreted in the picture of propagation of quasiparticles pairs [33,48]. On the other hand, it was found in large-c CFTs [38,39] that a characteristic feature called scrambling of entanglement would scramble the information of non-perturbative constants like quantum dimensions and lead to a logarithmically diverged Rényi entropy [38,49].
Since pseudo-entropy is a straightforward generalized concept of entanglement entropy, we shall study the time evolution behavior of pseudo-entropy for locally excited states in various 2d CFTs. We set up various situations to calculate the pseudo-entropy of locally excited states in 2d rational CFTs and large-c CFTs and to explore universal properties of pseudo entropy of locally excited states. This paper is organized as follows. Section 2 outlines the standard replica method to compute the nth pseudo-Rényi entropy for locally excited states. In Section 3 we mainly focus on the case of n = 2. We first study the limiting behaviors of real-time evolution of 2nd pseudo-Rényi entropy in rational CFTs and large-c CFTs. We then numerically analyze the full-time evolutions of 2nd pseudo-Rényi entropy in some specific interacting theories. In Section 4, we extend the above analysis to the nth pseudo-Rényi entropy. We end in Section 5 with conclusions and prospects. Some useful formulae and calculation details are presented in the appendices.

.1 Setup for local excitations and S (n)
A from replica method In this section, we review the replica calculation for the pseudo-Rényi entropy [20], which is almost same with that for the ordinary Rényi entropy [6]. Consider a 2d CFT dwells on a plane Σ 1 with coordinates {τ, x} (ds 2 = dτ 2 + dx 2 ). We are primarily interested in the cases that |ψ 1 , |ψ 2 are states defined by acting various operators on the ground state |Ω , where N j is normalization factor and O j,i (−τ j,i , x j,i ) ≡ e −Hτ j,i O j,i (x j,i )e Hτ j,i is operator located We can write down the corresponding transition matrix (2) in terms of the path integral language as follows Here the dashed lines represent the free boundaries, and the stars denote the insertion points of the operators. The reduced transition matrix of the subsystem A is obtained by "stitching up" the upper and lower edges of A c (1,1) early time a2 a3 a4 Substituting (6) into (1), the path integral representation of S

(n)
A is given by early time a2 a3 a4 In the above, S

(n)
A;vac stands for the nth Rényi entropy of A when the total system is in vacuum state, and Σ n is a n-sheeted Riemann surface constructed by gluing n sheets Σ 1 together at subsystem A. The subscript of O (k) denotes that the operator O is living on the kth sheet of Σ n . Since S

(n)
A;vac does not carry any information about excitations, we shall focus on the excess ∆S A;vac hereafter.

The excess of second pseudo-Rényi entropy ∆S
(2) A Let us first concentrate on the simplest case that n 1 = n 2 = 1, n = 2. In the meantime, we are mainly interested in the case where two inserted operators are the same. Now (4) can be reduced to and the corresponding excess of pseudo-Rényi entropy is given by The above expression is reduced to two-and four-point functions that we know for precisely respectively, where (η,η) ≡ z 12 z 34 /(z 13 z 24 ),z 12z34 /(z 13z24 ) are cross ratios and c 12 is normalization factor. Since we can apply the conformal transformation to map Σ n to Σ 1 , we obtain the four-point function on Σ 2 by applying the above conformal where we have set as shown in figure 1. Combining (14) with (10), the excess of the second pseudo-Rényi entropy 3 Real-time evolution of Re[∆S (2) A ] The real-time evolution of the pseudo-Rényi entropy for locally excited states can be regarded as a generalization of that of the Rényi entropy for locally excited states [33-38, 40, 42, 50]. In rational CFTs, it is known that the excess of the Rényi entropy saturates to a constant equal to the logarithm of the quantum dimension of the inserted primary operator [33,34]. A similar result for pseudo-Rényi entropy is found in free CFT in [20]. However, [20] only consider the real-time evolution of pseudo-Rényi entropy with different insertion time and the same insertion spatial coordinates. Richer evolutionary structures seem to lie in another insertion configuration of operators-two operators with different spatial coordinates and the same insertion time. In this section, we mainly explore this insertion configuration and give a general argument in the light of [34].

∆S
(2) A for two primary operators with different spatial coordinates In the following, we explore the case where the time coordinates of the two inserted operators are the same, but the spatial coordinates are different. That is, we are considering the following real-time dependent transition matrix It amounts to perform the following analytic continuation for τ 1 , τ 2 in (8): wherein > 0 is an infinitesimally small regularization parameter to suppress the high energy modes [48].

Early time, middle time, and late time behaviors
Let us first study some limiting behaviors of the pseudo-Rényi entropy to obtain some generic conclusions. The procedure is similar to that of entanglement entropy [34,38]. For the subsystem A of infinite length, we are mainly interested in the early time and late time limits of pseudo-Rényi entropy, while for the subsystem A of finite length, we are also interested in the middle time limit. 7 The subsystem A = [0, ∞): Consider the case of A = [0, ∞), in which we are mainly interested in the early (t → 0) and late (t → +∞) time limits. According to the expression of the 2nd pseudo-Rényi entropy (16), it's helpful to study the early and late time behaviors of η andη firstly, which we summarize in the table 1. 8 We can see from the table that for general (η,η) Early time (t → 0) These coincide with the second Rényi entropy results in [36]. Another intriguing case is to set x 2 = −x 1 = 0, where the early time limit of cross ratios is reduced to We next follow the arguments in [34,38] to cope with the function G ( η,η) in (16). In general CFTs, G(η,η) can be expressed as follows using the conformal blocks [51] G(η,η) = where C p O † O is the coefficient of the three-point function O † Oφ p and the index p corresponds to each φ p of all Virasoro primary fields. We can normalize G(η,η) such that the two-point function (10) has a unit normalization c 12 = 1, and it leads to the following behavior of F O (p|η) The above behavior indicates that as η goes to zero, the identity operator dominates the contribution in the summation of (21). Moreover, with the bootstrap relation we obtain behaviors of G(η,η) in two limits for general 2d CFTs lim η→0 η→0 The above results correspond precisely to the early time behavior of the G(η,η) function when the spatial positions of the two inserted operators tend to coincide.
The late time behavior of G(η,η), according to the table 1, requires some knowledge of the behavior of conformal block in the limit η → 1. For rational CFTs, the fusion transformation [52,53] can be exploited to give the expression of F O (p|η) in η → 1 limit and thus fix the leading-order (24) and (26), we find the following expression for the second pseudo-Rényi entropy It is noted that the late limit of both 2nd pseudo-Rényi entropy and Rényi entropy saturates to log d O , which indicates that the quasiparticle pair picture seems to be preserved in the pseudo-entropy.
We next move to analyse large-c CFTs. For large-c CFTs, in the limit ∆ p /c 1, the conformal block has the following universal form [54,55] is the hypergeometric function. Whereas, the authors in [38] argued that the above approximation fails when η is very close to 1 such as

is a certain positive constant) is exponentially large and in RCFTs
When η is very close to 1, the leading order of Furthermore, following the arguments in [38], the summation in Eq. (21) can be approximated by counting the contribution of the conformal vacuum block 8 when we are considering large-c theroies with gravity duals. Eq. (22) with Eq.(28-30) together lead to another type of the late time limit of G(η,η) Substituting (24) and (31) into (16) respectively, We obtain three distinct stages of evolution of the second pseudo-Rényi entropy.
Like Rényi entropy [38], the second pseudo-Rényi entropy has an intermediate process of logarithmic time evolution. If we take the real part of ∆S (2) A , as we are interested in, the corresponding logarithm evolves as follows Re ∆S which matches the result in [38] when two space points coincide. It's rewarding to mention that when we take the large-c limit first, like in holography, D 0 O will go to infinity, and ∆S Once again, we encounter a complicated early time behavior, and it can be simplified by taking Furthermore, we find that another interesting class of space configurations, |x 1 |, |x 2 | L, can also reduce the early time results (34), To analytically extract the middle time t ∈ [u, v], u = min |x 1 |, behavior of cross ratios, let us consider the large L limit, that is, L |x 1 − x 2 |. This is because one can expect that when L |x 1 − x 2 |, the middle time behavior of ∆S (2) A will tend to the late time behavior of ∆S (2) A of the infinite subsystem. Consider two special spatial configurations that satisfy the constraint: i.
The above two configurations correspond to the situation where operators live concentratedly near the left and right boundaries of A, respectively. 10 In these cases, the value of the cross ratios at a typical middle time, t = L/2, can be calculated We obtain a middle-time behavior similar to the late-time behavior (86) for the infinite subsystem. For more general cases, the numerical calculation shows that Combining with the previous discussion of G(η,η), according to Eqs. (36), (40), and (35), we get the picture of the evolution of ∆S (2) A under some constraints in RCFTs 11

Examples in 2d CFTs
In the previous subsection, we have studied several limiting behaviors of the second pseudo-Rényi entropy. However, there are still some mysteries about the evolution of ∆S (2) A that limit analysis is infeasible to solve: 1. The intermediate process of the evolution of Re[∆S (2) A ] from an initial value to log d O in RCFTs. 10 One may also interested in the opposing situation that the operators are scattered at both ends of A. The middle time behavior of cross ratios in this case is found to be 2. Is there anything special about Re[∆S (2) A ] evolution in certain symmetric spatial configurations (such as In this subsection, we will resort to numerical analysis to uncover the whole time evolution picture of ∆S (2) A under several specific 2d CFT models. We expect that the above problems will be answered to some extent in these concrete models. Before entering into the numerical study, we first point out some model-independent symmetries of the second pseudo-Rényi entropy, which are reflected in the following examples.
Symmetries for ∆S (2) A : Re-examining the cross ratios of finite and infinite subsystem ( (92) and (85),respectively), one can find some hidden symmetries of them, where " * " denotes complex conjugate. Further, it's easy to show that the above symmetries may be extended to ∆S (2) A when the G(η,η) function has the following properties Combining (16), (42), and (46), one obtains the first symmetry of ∆S Combining (16), (23), and (43)(44)(45), the second symmetry of ∆S There are some physical or mathematical understandings that may explain the appearance of in terms of the symmetry of the system. In addition to the basic property A ) of the nth pseudo-Rényi entropy [20], we obtain (49); Eq.(47) can be interpreted by the fact that The above argument suggests that these symmetries hold not only for the 2nd pseudo-Rényi entropy, but also for any order.
We shall make a numerical examination on them in section 4.
On the other hand, one can see that there are two special space configurations - , that are screened out by these symmetries. Taking as an example, the operation of swapping x 1 and x 2 is equivalent to the spatial reflection operation, which means that ∆S Since one can expect G(η, η * ) to be greater than 0, we obtain a real second pseudo-Rényi entropy evolution in this insertion configuration, whose correctness is verified in subsequent examples.
Finally, when only paying attention to the real part of ∆S (2) A , the above results show that the evolution of Re[∆S (2) A ] may be "4-fold degenerate", Re[∆S Hence we may choose to label each space configuration with the following parameters A (η,η) = log On the other hand, utilizing the identity σ(z 1 ,z 1 )...σ(z 2n ,z 2n ) As shown in figure 2(a), since the relative size between the spacing of two operators and the 12 The absolute value here is |η| ≡ √ η · η * . 13 However, due to the symmetries of G(η,η), the above two cases give the same value of Re[S (2) A ]. For the most special case of x m = 0, which we have already encountered in the limit analysis (20), Re[S (2) A ] behaves exactly like the second Rényi entropy [50]. In fact, we find ∆S (2) A is real in this case, which is consistent with our previous symmetry analysis. Next, we categorize the case of Re[S (2) A (η,η)] is equal in the above cases, on account of the symmetries of G(η,η). We then gradually increase the spacing between operators such that the constraint l L no longer holds (see figure 2(c)). We can see that the intermediate behavior of log d O gradually vanishes as l increases, but the peaks of the humps seem to remain the same. Another interesting case, as we have discussed in symmetry analysis, is to fix x m = L/2 and then gradually increase l. As shown in figure 2(d), we do obtain a real pseudo-Rényi entropy. Meanwhile, the result shows that the middle time behavior of S A tends to zero instead of log d O , and the time to saturation of middle time behavior also shifts from 1 2 L to Example II-Minimal model: Another simple example is the excitation of (2, 1) operator φ (2,1) in the minimal models M(p, p ) with p > p . The conformal dimension and quantum dimension of the φ (2,1) are well-known to be ∆ (2,1) = 3p 4p − 1 2 and d (2,1) = −2 cos πp p , respectively. In addition, it has a relatively simple four-point function [57,58] that satisfies Eqs. (45) and (46), where the functions I 1,2 are defined as follows The normalization factor c 12 in (16)   In these two cases, it can be found that Re[∆S (2) A ] saturates to the theoretical value log d (2,1) in the middle and late time, respectively, which coincides with the case of free scalar. Whereas A no longer remains real in the full-time evolution, which is manifest to seen in figure 3(b). 14 The trace of (T 1|2 A ) 2 is negative over an interval except for the Ising model, which results in a complex ∆S (2) A ; ii). The evolution of pseudo-Rényi entropy in the case of x m = 0 no longer behaves like that of Rényi entropy; Figure 3(d) exhibits the evolution of ∆S (2) A under the second symmetric space configuration x m = L/2 in the case of A = [0, L]. As predicted by the symmetry analysis, we can see that ∆S (2) A is real throughout the time evolution. Notice that (η,η) takes the value of (1/2, 1/2) at the peak or valley of the middle time evolution.
Example III-Wess-Zumino-Witten model: The last example we would like to explore is the excitation of g α β (z,z) operator in a Wess-Zumino-Witten (WZW) model with affine Lie 13 We have φ (2,1) (z 1 ,z 1 )φ (2,1) (z 2 ,z 2 )φ (2,1) (z 3 ,z 3 )φ (2,1) (z 4 ,z 4 ) → c 2 12 |z 12 | −8∆ (2,1) in the limit of z 12 = z 34 → 0. 14 Note that the complex ∆S (2) A is not contradictory with the previous symmetry analysis, because by symmetry analysis we can only prove that Tr[(T       where We next explore the full-time evolution behavior of ∆S (2) A utilizing the above information. Particularly, we are mainly concerned with the behavior of ∆S  (2) A ] determined by the relative size of the rank N and level k. When N ≤ k (see figure 4(a) and (b)), we find that Tr[(T 1|2 A (t)) 2 ] is always greater than 0, thus ∆S (2) A remains real in the full-time evolution. A rather fascinating situation is that N = k, since again we observe a pseudo-Rényi entropy evolution identical to that of Rényi entropy. When N > k, the 2nd pseudo-Rényi entropy evolution behavior is similar to that in the minimal model. Since Tr[(T 1|2 A (t)) 2 ] is less than 0 near t = l/2, we have a complex pseudo entropy of in a certain interval. Figure 4(d) depicts the behaviors of S (2) A under the second symmetric space configuration x m = L/2 in A = [0, L]. Notice that no matter k is greater than N or not, we obtain a pseudo-Rényi entropy that remains real throughout time evolution. summaries of the above results: Let us take a short stay to briefly summarize the above results and try to answer the questions posed at the beginning of the section.
1. In general, there are one or two hump evolutions (for example, figure 2(a), (b)) between the early time and late time evolution of ∆S (2) A , and the value of ∆S (2) A at the peaks of humps is ∆S A , which is consistent with all the numerical results.
3. In the e i 2 φ +e − i 2 φ -excitation of free scalar and g α β -excitation of SU(N ) k WZW models (N = k), we observe that the 2nd pseudo-Rényi entropy exhibits the same behavior as Rényi entropy in the case of

Linear combination of operators
In the previous subsection, we study the real-time behaviors of ∆S (2) A for states excited by the same primary operator. It's not so straightforward to extend the results to two different primary operators. Simply substituting one of the operators would probably make a non-normalizable transition matrix since the two-point function for two different primary operators is likely to be zero. One feasible way is to consider the linear combinations of operators. 15 Consider a real-time dependent transition matrix T ψ|ψ (t) = e −iHt |ψ ψ |e iHt ψ |ψ consisting of two quantum states |ψ and |ψ , In the above, O(x, ) ≡ e H O(x)e − H , O p are primary or descedant operators that are orthogonal to each other in the sense of the two-point function, C p (C p ) are superposition coefficients used to give a non-zero inner product. 15 We thank Tadashi Takayanagi for bringing this idea to our attention. A to take the following form where w = x−i ,w =x+i , and S (n) [O p ] is the late time limit of the difference of entanglement . It is difficult to utilize replica trick to prove Eq.(65). Nevertheless, we can provide a quantum mechanical derivation from another perspective, which as far as we know was first introduced in [43]. 16 We next numerically examine the correctness of Eq.(65) using the replica trick in the concrete model.

Example in critical Ising
We would like to compute ∆S (2) A of linear combination operators in the critical Ising model to examine Eq.(65). There are three primary operators in the Ising model at a critical point, namely the identity I, the spin σ, and the energy density ε. The fusion rule of them is wellknown, For simplicity, below, we consider the combination of σ and I as a typical example.
(z,z) on Σ 1 [34] to complete the analytic continuation of time, We start with the case of {C p } = {C p } andx = x. An efficient way is to set C σ = q ∈ [0, 1], C I = 1−q,C σ = q k ,C I = 1−q k , and obviously what we will obtain when k = 1 is pseudo-Rényi entropy rather than Rényi entropy. Figure 5(a) shows the behavior of the late time limits of ∆S (2) A when we adjust the mixed coefficient q. We can see that the late time limits of ∆S A , see figure 5(b). We find that although ∆S (2) A saturates to a real value, globally ∆S (2) A is complex in all cases except k = 1 (the case of Rényi entropy). The other interesting case we shall investigate is that only the information of the distance of two space points l ≡ |x − x| is related. We then plot the change of saturation value of ∆S (2) A with q under different l, as depicted in figure 5(c). Once again, we find that the theoretical value (solid lines) given by Eq.(65) is consistent with the numerical result (square points) given by replica trick. On the other hand, when the insertion point is symmetric about the origin, i.e.x = −x, ∆S (2) A is found to be real throughout the the time evolution (see figure 5(d)), which is consistent with the previous symmetry analysis.

General arguments and examples on ∆S
In the previous section, we study the real-time behavior of ∆S (2) A for two insertion operators with different spatial coordinates. Meanwhile, we propose a formula to describe the late time limit of ∆S (n) A of linear combination operators. However, as shown in appendix B, its rationality also depends on the behavior of nth pseudo-Rényi entropy of a single primary operator insertion.
Therefore, in this section, we shall quest for the properties of ∆S (n) A for two operators with different space points in the light of the results of ∆S (2) A that we have found before.

Late time limit of ∆S
where On the other hand, we find that at the late time (t → ∞) The above results enable us to factorize the 2n-point function O † (z 2n ,z 2n )...O(z 1 ,z 1 ) Σ 1 into n-point functions by using the fusion transformation (25) n − 1 times (see figure 6), Substituting (73)  (76)

Symmetries of ∆S (n)
A : The second intention in this section is to investigate whether the symmetries found in second pseudo-Rényi entropy, i.e. Eq. (47)(48)(49), still hold in higher-order or not. It may be difficult to verify analytically, but the numerical examination is easy to take.
One good object of study is the σ-excitation in the critical Ising model, since the 2n-point function of the spin operator σ is well-known [56,60], With the help of (71) and (77), we can study the evolution behavior of ∆S   Figure 7 demonstrates all situations that we are interested in. We find that the symmetries (47)(48)(49) also hold in the higher-order pseudo-Rényi entropy. It can be clearly seen from figure   7(b) and (d). Because we know that the establishment of(47-49) may bring about a real ∆S (n) A evolution. Another interesting finding is that (b) shows that the evolution of higherorder pseudo-Rényi entropy of σ-excitation under the first symmetric space configuration still maintains the evolution pattern described by (63). 17 We can also explore the asymmetric cases, as shown in (a) and (c), and there are two points worth noting: i). The higher-order pseudo-Rényi entropy in asymmetric space configuration still has hump evolution, and its peak value changes with n; ii). Figure 7(c) suggests that after the relative sizes of L and l are fixed, the middle time behavior of log d will gradually disappear with the increase of n.

Conclusions and prospect
In this work, we study a generalized version of entanglement entropy and Rényi entropy, which are so-called pseudo-entropy (PE) and pseudo-Rényi entropy (PRE), respectively, in 2d CFTs.
In particular, the real-time evolution of PRE associated with two locally excited states has been evaluated in various 2d CFTs, e.g., free bosonic field theory, critical Ising model, WZW model 17 Due to the increasing computational complexity, we verify this point up to n = 7.

23
as well as large-c CFTs. These locally excited states are generated by acting local operators on the vacuum state, and these operators can be a single primary operator or a linear combination of them. Some fascinating behaviors of PRE evolution are found as follows: For the reduced transition matrix generated by two primary operators with different spatial coordinates (17), we show that when subsystem A has an infinite length, the late time value of 2nd PRE is logarithmically divergent in large-c CFTs (take the large c limit first). The late time value of nth PRE saturates to log d in RCFTs (for Example, see figure 7(a)), where d is the quantum dimension of the corresponding primary operator. Whereas, when subsystem A has a finite length, we show that the middle time behavior of log d of PRE in RCFTs gradually disappears as the distance between operators or the order number n increases (see figure2(c) and figure 7(c) respectively). Unlike the entanglement entropy, we find that in general, there is a hump during the evolution between the early and late time evolution of the nth PRE (for example, see figure (7)(a)), and for n = 2, its peak value can be found as ∆S (2) A (η,η) = ∆S (2) A (1/2, 0), where (η,η) are cross ratios. On the other hand, for excitation by a linear combination of operators, using Schmidt decomposition, we find that the late time limit of the nth pseudo-Rényi entropy is governed by the formula (65). A prominent property that distinguishes linear combination excitation from single primary operator excitation is that the late limit of PRE under linear combination excitation is not necessarily the same as that of Rényi entropy (see, for example, figure 5(c)).
This means that for the case of a single primary operator, the initial information about the positions of the insertion operators is lost in the long-time evolution. In contrast, for the case of linear combinations, the late limit of the pseudo-Rényi entropy still contains the initial information of the operator positions. It would be interesting to explore whether it is possible to recover the initial data by using the late time limit of pseudo-Rényi entropy.
Finally, building on the analysis of the cross ratios, we uncover three kinds of symmetries for the 2nd PRE (47)(48)(49), which naturally screen out two kinds of special space configurations of insertion operators-x 1 = −x 2 for subsystem A = [0, ∞) and x 1 = L − x 2 for subsystem A = [0, L]. We show that the trace of T It will be an attractive research direction for us in the future to fully clarify the condition that PRE remains real in the time evolution process and the condition that PRE behaves as Rényi entropy in the first symmetric space configuration. Furthermore, it's also interesting to make a higher-dimensional generalization of our results and dig out the possible corresponding holographic counterpart.
According to the analysis in section 4, we know that the late time limit of ∆S (n) T φ|φ A (t) is equal to lim t→∞ ∆S (n) Tr A c |φ(t) φ(t)| , and the latter we already know is equal to (97) [43].
On the other hand, following the logic in [43], it's natural to expect that lim t→∞ ∆S (n) Substituting Eq.(103) into Eq.(100) and taking some algebra, we finally obtain which, in the light of the logic in [43], just corresponds to the late time limit of nth pseudo-Rényi entropy of A. Let {C p } = {C p } = 1 andx = x, it can be readily found that (104) is reduced to the Eq.(2.26) in [43].