Pseudo-entropy for descendant operators in two-dimensional conformal field theories

We study the late-time behaviors of pseudo-(R\'enyi) entropy of locally excited states in rational conformal field theories (RCFTs). To construct the transition matrix, we utilize two non-orthogonal locally excited states that are created by the application of different descendant operators to the vacuum. We show that when two descendant operators are generated by a single Virasoro generator acting on the same primary operator, the late-time excess of pseudo-entropy and pseudo-R\'enyi entropy corresponds to the logarithmic of the quantum dimension of the associated primary operator, in agreement with the case of entanglement entropy. However, for linear combination operators generated by the generic summation of Virasoro generators, we obtain a distinct late-time excess formula for the pseudo-(R\'enyi) entropy compared to that for (R\'enyi) entanglement entropy. As the mixing of holomorphic and antiholomorphic generators enhances the entanglement, in this case, the pseudo-(R\'enyi) entropy can receive an additional contribution. The additional contribution can be expressed as the pseudo-(R\'enyi) entropy of an effective transition matrix in a finite-dimensional Hilbert space.

Recently, a new entanglement measure, called pseudo-entropy, was proposed in 19 as a generalization of entanglement entropy.Specifically, pseudo-entropy is a two-state vector version of entanglement entropy, defined as follows.Given two non-orthogonal states |ψ⟩ and |φ⟩ in the Hilbert space H S of a composed quantum system S = A ∪ B, we first construct an operator called the transition matrix acting on H S 19,20 , The pseudo-entropy of subsystem A, then, is obtained by calculating the von Neumann entropy of the reduced transition matrix T a) Electronic mail: hesong@jlu.edu.cnb) Electronic mail: yangjie@cnu.edu.cn c Electronic mail: yuxuanz20@mails.jlu.edu.cnd) Electronic mail: zzx23@mails.jlu.edu.cn Generally, the reduced transition matrix is non-Hermitian, requiring careful consideration when discussing pseudo-entropy in systems with infinite-dimensional Hilbert spaces (such as in QFTs) since taking the logarithm of a generic operator requires choosing a radial line in the complex plane that does not intersect the spectrum. 21To avoid dealing directly with the logarithm of the non-Hermitian matrix, in practice, one usually computes a quantity called pseudo-Rényi entropy, instead of pseudo-entropy and the branch of the logarithm function is chosen to be −π < Im[log(z)] < π.For n ∈ N + , n ≥ 2, (3) admits an alternative expression: where are the eigenvalues of T ψ|φ A , directly following from a Jordan decomposition of T ψ|φ A .In this paper, we mainly focus on the pseudo-Rényi entropy for general n (n ≥ 2).When discussing pseudo-entropy, we refer to the Shannon entropy defined by the eigenvalues of For finite-dimensional systems, it is clear that (2) equals (5), and the latter can be obtained from taking an analytic continuation of n → 1 on (4).However, as previously mentioned, for infinite-dimensional systems, the definition of pseudo-entropy in (2) may not be well-defined in general.Pseudo-entropy was originally proposed from the study of the generalization of holography entanglement entropy 19 .In the AdS/CFT context, the pseudo-entropy of a boundary subsystem is proposed to be dual to the area of a minimal surface in a Euclidean timedependent AdS space 19 .In addition, it is found that pseudo-entropy is closely related to postselection experiments in quantum information 19,22 (i.e., in addition to the initial state, the system's final state is also specified 23 ).There are also many research interests and prospects driving the study of pseudo-entropy in QFTs [24][25][26][27][28][29][30] .See 20,[31][32][33][34][35][36][37][38] for other related developments of pseudo-entropy.
Nonequilibrium dynamics in quantum many-body systems is a subject of intensive research 39,40 .One of the recurring themes is how quantum entanglement arises and propagates in non-equilibrium processes, known as entanglement dynamics.Research shows that chaotic quantum many-body systems can non-locally disrupt quantum information.The scrambling of quantum information will at least lead to the loss of local initial state information and lead to thermalization [41][42][43] .A typical non-equilibrium process in quantum many-body systems is quantum quench 44,45 .The process usually involves two steps: first prepare an initial state |ψ⟩, which can be the ground state of a certain Hamiltonian H, and then evolve it with a different Hamiltonian H ′ .One can also quench the system by a local perturbation (generally called a local quench 46,47 ), for instance acting on a local operator (generally called a local operator quench 48,49 ).Then the entanglement dynamics are diagnostic about the nature of this excitation.
The present paper aims to study the properties of pseudo-(Rényi) entropy of states obtained by acting on the vacuum with a descendant of a local primary operator (also referred to as descendant states in this paper) in two-dimensional conformal field theories (2D CFTs).Our study can be traced back to the research on entanglement entropy in local operator quantum quenches in 2D CFTs [49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64] . 65The local operator quench exhibits broad applicability in measuring scrambling and thermalization effects in CFTs with large central charge 53,54,58,66,67 , which can be regarded as a manifestation of quantum chaos, as well as in probing the bulk geometry 68 and characterizing bulk dynamics 63,[69][70][71] in the context of AdS/CFT correspondence-an essential avenue for comprehending quantum gravity.It is found that the excess of Rényi entropy of the local primary or descendant excited states in rational conformal field theories (RCFTs) saturates to a constant equal to the logarithm of the quantum dimension 72 of the local operator's conformal family 51,56,57 .Such saturation is well explained by the picture of quasiparticle pair propagation 49 .The related research has been extended to the pseudo-entropy in parallel 30 .Specifically, when studying the real-time evolution of the pseudo-Rényi entropy, such as the 2nd pseudo-Rényi entropy, for locally primary excited states in RCFTs, the conformal block at early times relies on the spatial positions of two identical primary operators, leading to a model-dependent pseudo-Rényi entropy.Nevertheless, the pseudo-Rényi entropy shows a universal behavior at late times, which only depends on the quantum dimension of the primary operator, just like the entanglement entropy.The result suggests that the picture of quasiparticle pair propagation is preserved in the pseudo-entropy.We generalize the previous study 30 on the pseudo-(Rényi) entropy to descendant operators in this paper to understand the intricate connections between fusion rules and entanglement properties 73 , where fusion rules play a fundamental role in characterizing algebraic and structural properties of a CFT 74,75 .The algebraic and structural properties would be encoded in the dynamics of entanglement.Specifically, we would like to explore the late-time behavior of pseudo-Rényi entropy of two descendant operators in RCFTs.We construct the transition matrix using two locally excited states created by the operator and evaluate the pseudo-Rényi entropy using the replica method 5 and conformal mapping.In (6), O(x) is a primary operator in Schrödinger picture with chiral and anti-chiral conformal dimension ∆, L −n ( L−n ) are holomorphic (anti-holomorphic) Virasoro generators, and α {ni},{nj } ∈ C are superposition coefficients.Since the two-point function between descendant operators of different levels does not vanish, the transition matrices we are permitted to construct have more degrees of freedom than the cases of the primary operator. 76It is interesting to see whether the late-time behavior of the pseudo-(Rényi) entropy of subsystems corresponding to these transition matrices has contributions other than the quantum dimension.The rest of this paper is organized as follows.In section II, we briefly review the replica method for locally excited states in 2D CFTs and provide our convention and some useful formulae for the later calculations.In section III, we mainly focus on the late-time behavior of the 2nd pseudo-Rényi entropy of locally descendant excited states.For simplicity, we study the cases in that a single holomorphic Virasoro generator generates the descendants.More general and complicated situations are discussed in section IV, where we derive the late-time behavior of the k-th pseudo-Rényi entropy for the generic descendant states.We end with conclusions and prospects in section V. Some calculation details are presented in the appendices.

A. Replica method with local operators
Our focus is on the pseudo-Rényi entropy of locally excited states created by acting with the operator V α (6) on the ground state in RCFTs, and the subsystem A under consideration in this paper is always taken to be the interval [0, ∞).In this scenario, the pseudo-Rényi entropy can be formulated in the path integral formalism using the replica method.Given an RCFT that lives on a plane and has a vacuum state |Ω⟩, we first prepare two locally excited states using V α to construct a real-time evolved transition matrix T 1|2 (t), Notice that an infinitesimally small parameter ϵ has been introduced to suppress the high energy modes 44 .We can obtain the reduced transition matrix of subsystem A at time t by tracing out the degrees of freedom of A c (the complement of A), . It turns out that the excess of the n-th pseudo-Rényi entropy of A with respect to the ground state, defined as ∆S (n) using the replica method.In (8), Σ n denotes a n-sheeted Riemann surface with cuts on each copy corresponding to A, and (w 2k−1 , w2k−1 )and (w 2k , w2k ) are coordinates on the kth-sheet surface.The term in the first line is given by a 2n-point correlation function on Σ n , while a two-point function gives the one in the second line on Σ 1 .We have

B. Convention and useful formulae
The 2n-point correlation function on Σ n in Eq.( 8) can be evaluated with the help of a conformal mapping of Σ n to the complex plane Σ 1 .We can then map Σ n to Σ 1 using the simple conformal mapping Let us first focus on the case of n = 2.The calculation of ∆S (2) (T 1|2 A (t)) is related to the four-point function known pretty well for exactly solvable CFTs.In our convention, using Eq.( 10), the four points z 1 , z 2 , z 3 , z 4 in the complex plane are given by The key point is that one should treat t ± iϵ as a pure imaginary number in all algebraic calculations and take t to be real only in the final expression of the pseudo-Rényi entropy.
To evaluate the four-point correlation function, it is useful to focus on the cross ratios 30 where z ij = z i − z j , and a useful relation is For the cross ratios (η, η), as shown in 30 , we have For general n-th pseudo-Rényi entropy, the 2n points z 1 , z 2 , ..., z 2n in the z-coordinates are given by

III. SECOND PSEUDO-R ÉNYI ENTROPY ∆S
(2) In RCFTs, it is known that the excess of the Rényi entropy for the primary/descendant operator saturates to a constant equal to the logarithm of the quantum dimension of the inserted primary operator 51,56,57 .To study the entanglement entropy of local operators, one needs to use two identical operators with the same spatial coordinates to generate the density matrix.However, as mentioned in the introduction, pseudo entropy provides us with greater flexibility-we can use descendant operators of different levels, and with different spatial coordinates, to construct the transition matrix.This section will explore the 2nd pseudo-Rényi entropy for some specific descendant operators.

A. ∆S
(2) Let us initially examine the simplest scenario that deviates from the previous studies 30 : ).The 2nd pseudo-Rényi entropy, according to (8), is related to a four-point function on Σ 2 , For the first descendant operators, the transformation law under the conformal mapping w = z 2 is given by where the prime denotes the derivative with respect to z or z.Then the four-point function in (17) can be written in terms of correlators on the plane as where we use the notation O(i) ≡ O(z i , zi ).Due to the conformal symmetry, we can express the four-point functions involved in (19) as follows where Under the conformal mapping between Σ 2 and Σ 1 , we have At late times (t → ∞), as shown in 30 , η and η approach 1 and 0, respectively, which leads to the following late time behavior of G(η, η) for RCFTs where d O is so called quantum dimension and by using modular S matrix S ab this is given by d Oa = S 0a /S 00 75,77 .Hence we can obtain On the other hand, the two-point function in (17) is Substituting ( 22), ( 24) and ( 25) into ( 17) and setting z 3 = −z 1 , z 4 = −z 2 , we obtain, at late times, In going from the second to the third line, we use Eq.( 11) and perform the Laurent expansion at infinity.The late-time limit of the 2nd pseudo-Rényi entropy is thus given by In this simplest case, the late-time behavior of the 2nd pseudo-Rényi entropy of L −1 O with O is the same as that of the primary operator O.Note that the four-point functions in the plane in Eq.( 19) are also encountered when studying the entanglement entropy of L −1 O 57 .However, they are discarded as sub-leading terms.Our finding shows that these sub-leading correlators can also reproduce the result of log d O , as long as we consider the pseudo-Rényi entropy instead of the Rényi entropy.

B. ∆S
(2) We next consider a more complicated case that V α is a general n-level descendant associated with the Virasoro generator L −n , and V β is still a primary.The two-point function of V α and V β reads 78 We then compute the four-point function on Σ 2 .Under the conformal transformation, the level n descendant transforms as The ellipsis stands for operators with lower conformal dimensions, contributing to lowerorder singularities in the correlation functions, that is, we have at late times.We next pick out the most singular terms of the four-point function on the z-plane in (30).According to ( 14) and (A1) in appendix A, the leading contribution at late Again, the ellipsis represents the terms that give rise to lower-order singularities in the correlation functions.Combining ( 28)-( 31) and taking the limit t → ∞, the leading-order behavior of exp{−∆S (2) The ellipsis here denotes the sub-leading terms that vanish as t goes to infinity.Hence the late-time limit of the 2nd pseudo-Rényi entropy of the transition matrix constructed by a primary O and its n-level descendant A for Vα = L−nO, V β = L−mO In this subsection, we use the conformal block and operator product expansion (OPE) to show the phenomenon discovered in previous subsections is true for a general case: In terms of 78 , the two-point function of V α and V β reads 79 The late-time behavior of the four-point function on Σ 2 of (8) can be derived according to ( 29) We can next pick out the most singular terms of the four-point function on the z-plane in (34).According to (14) , the leading contribution at late times in and its complete result is given by (B8) in appendix B. Combining (33) and (B8) and taking the limit t → ∞, the leading-order behavior of Again, the ellipsis denotes the sub-leading terms that vanish as t → ∞.The late-time limit of the 2nd pseudo-Rényi entropy of the transition matrix constructed by a m-level descendant operator L −m O and a n-level descendant operator L −n O is log d O , being consistent with the studies in previous sections.

IV. k-TH PSEUDO-R ÉNYI ENTROPY FOR GENERIC DESCENDANT STATES
In the previous section, we find that the 2nd pseudo-Rényi entropy corresponding to L −n O and L −m O is the same as the 2nd pseudo-Rényi entropy of the corresponding primary operator O at late times, i.e., the logarithm of the quantum dimension of the primary operator O.In this section, we shall investigate the k-th pseudo-Rényi entropy for general descendant states and take k → 1 to obtain the corresponding pseudo-entropy.
We begin with studying the case discussed above: (29), the 2k-point function on Σ k at late times can be reformulated as the 2k-point function on Σ 1 as follows, where is the leading factor coming from the conformal transformation between correlation functions on Σ k and correlation functions on Σ 1 and the ellipsis denotes terms contributing to lower-order singularity in the correlation functions.
Based on ( 16), it can be found that during the late time, 2k holomorphic coordinates and 2k anti-holomorphic coordinates approach each other in distinct pairings 30 .
Hence, at late times, the most divergent part of the 2k-point correlation function on the plane in (36) arises from the OPE of O (−n) (2j + 1)O (−m) † (2j + 4), i.e., where D 2i+1,2i+4 is a derivative operator that only contains constants related to the information of two descendant operators and derivatives coming from the most singular part of the OPE of O (−n) (2i + 1)O (−m) † (2i + 4), i.e., D 2i+1,2i+4 = D(∂ 2i+1 , ∂ 2i+4 ; m, n, c, ∆).See appendix B for a concrete example of the D-operator.We need to pick up the proper channel to expand the 2k-point function into the holomorphic and the anti-holomorphic part, as graphically shown in figure 1.In each channel, only the identity operator contributes to the final result.Hence, the 2k-point function breaks up into k two-point functions for the holomorphic part (and k for the anti-holomorphic part).= ⟨O(z 1 )O(z 2 )⟩⟨O(z 1 )O(z 2 )⟩.In the last line, the fact that D 2i+1,2i+4 is a linear operator, and coordinates z i and z j are independent for i ̸ = j has been applied.
Changing back into the w-coordinate, with the leading divergent term being transformed homogeneously and keeping the most divergent term, we find that By utilizing Eq.( 16), ( 29) and (38), we find that in the late-time limit, the correlation functions of both holomorphic and anti-holomorphic two-point functions on Σ k are equal to those on Σ 1 , up to a unitary factor.
Substituting ( 42) into ( 41), the 2k-point function on Σ k is reduced to where we use the relation between the quantum dimension and the fusion matrix: Finally, in accordance with Eq.( 43), the excess of the k-th pseudo-Rényi entropy of L −n O and L −m O at late times can be deduced as equal to

B. ∆S (k)
A for Linear combination of descendant operators Let us consider two linear combination operators constructed by operators in O's conformal family. where ) are required to be dimensionless, all V i (w, w) V ′ i (w, w) should have the same mass dimension, denoted as N (N ′ ).This indicates that {K i } and {K ′ i } satisfy Firstly, the two-point function of V α and V † β on Σ 1 is given by Similar to the previous subsection, in the above, we formally decompose the operator L −{Ki} L−{ Ki} O(w, w) into a holomorphic operator L −{Ki} O(w) and an antiholomorphic operator L−{ Ki} O( w) in the sense of the two-point function.c 0 and c0 in Eq.( 47) are respectively the coefficients of the holomorphic and antiholomorphic two-point correlation function.
We then cope with the 2k-point function on Σ k .At late times, the 2k-point function is given by In the above derivation, from the first equation to the first tilde, we follow the approach outlined in the preceding subsection: first, we map the 2k-point function on Σ k to the plane through conformal transformation w = z k and extract its leading behavior; subsequently, using Eq.( 38) and fusion transformation k − 1 times, we decompose the leading 2k-point function on the plane into k holomorphic two-point functions and k antiholomorphic twopoint functions, and finally, map the 2k two-point functions back to Σ k .From the first tilde to the second tilde, we utilize a late-time relation similar to (42) as follows: which can be readily derived using Eq.( 16), ( 29) and (38).Upon substituting Eq.( 47) and ( 48) into the k-th pseudo-Rényi entropy expression (8) and attempting to eliminate w 1,2 , we encounter some subtleties.Specifically, after analytic continuation, the expressions for w 1,2 and w1,2 (9) imply that when x 1 ̸ = x 2 , we have w 1 − w 2 = w1 − w2 = x 1 − x 2 in the limit ϵ → 0. Consequently, based on the initial constraint (46), we can extract the power of x 1 − x 2 in (48) from the summation, which is equal to (x 1 − x 2 ) −k(4∆+N +N ′ ) (For Eq.( 47) it is (x 1 − x 2 ) −(4∆+N +N ′ ) ).The late-time excess formula of the k-th pseudo-Rényi entropy is thus given by lim t→∞ However, things become slightly different when x 1 = x 2 .This is because in this case, w 1 − w 2 = −( w1 − w2 ) = −2iϵ.When attempting to eliminate the normalization parameter ϵ by dividing Eq.( 48) by the k-th power of Eq.( 47), we will be left with a negative power in both the numerator and denominator summations, leading to another late-time excess formula for the k-th pseudo-Rényi entropy.
Indeed, one can absorb such negative power in Eq.( 51) into the coefficient of the holomorphic two-point function, to obtain a formula similar to Eq.( 50). where When we delve into a detailed analysis of these two formulas, we may find that Eq.( 51) (or Eq.( 52)) is compatible with the late-time excess formula for entanglement entropy given in 57 .This can be verified by simply removing the prime from Eq.( 51) (or Eq.( 52)).However, generally speaking, since Eq.( 50) is not equal to Eq.( 51), Eq.( 50) cannot be reduced to the formula for entanglement entropy.We verify the discontinuity of the pseudo-Rényi entropy in these two cases (i.e., x 1 = x 2 and x 1 ̸ = x 2 ) through numerical calculations in the critical Ising model, see figure 2. Mathematically, we can attribute this discontinuity of pseudo-Rényi entropy to the noncommutativity of the limits as ϵ → 0 and x 1 → x 2 .It would be interesting to comprehend this point from a physical perspective.
More importantly, regardless of the case (whether x 1 = x 2 or x 1 ̸ = x 2 ), the late-time excess of the pseudo-Rényi entropy of two linear combination operators is composed of two parts.The first part is the logarithm of the quantum dimension of the corresponding primary operator, which reflects the entanglement properties of the primary/descendant operators used to construct the linear combination operators.The second part involves the coefficients of the superposition C i , the coefficients of the holomorphic and antiholomorphic two-point functions, which reflect the additional entanglement generated by the process of linear combination.We can express this part of the additional entanglement contribution as the entanglement entropy (pseudo-entropy) of an effective density (transition) matrix in a finite-dimensional Hilbert space.Taking Eq.( 50) as an example (Eq.( 51) shares a similar treatment), we use the superposition coefficients, the coefficients of holomorphic and antiholomorphic two-point functions to define the following M × M matrix T eff , We refer to T eff as an effective transition matrix, because T eff is usually non-Hermitian, and we will see that it characterizes the additional pseudo-Rényi entropy at the late time.With the help of T eff , Eq.( 50) can be equivalently written as From the above equation, it is clear that the additional pseudo-Rényi entropy generated by the linear combination process equals the pseudo-Rényi entropy of an effective transition matrix in a M -dimentional Hilbert space, and the additional pseudo-entropy thus is equal to −tr[T eff log T eff ].
It is evident that the late-time additional contributions for all levels of pseudo-Rényi entropy resulting from a linear combination are zero only when T eff possesses a single nonzero eigenvalue of 1.We show that the physical origin of these additional corrections is attributed to the mixing of holomorphic Virasoro generators and antiholomorphic Virasoro generators.Considering the holomorphic and antiholomorphic Virasoro generators in V α(β) appear in the form of product (not mixed).Based on Eq.( 53), we can write down the matrix element of T eff Evidently, T eff under this scenario takes the form of the pure state transition matrix (1), resulting in a single non-zero eigenvalue of 1 for T eff .Therefore, we prove that the linear combination process does not generate any additional correction in this case.To provide a heuristic understanding of this results, we may draw an analogy to 49 : when enhancing the entanglement.Finally, we consider the late-time excess formula (50) for the 2nd pseudo-Rényi entropy to show the phenomenon of "mixing enhances the entanglement".
The two-point function is Formula ( 58) is easy to check.Here, we replace x 1 + t and x 2 + t into w 1 and w 2 in the final result.The four-point function at the late time is From ( 58) and ( 59), we have In this case, the correlation function of V α and V β can not be divided into the prodlog -log - A due to the mixing of holomorphic and anti holomorphic Virasoro generators.The horizontal axis is the conformal dimension of the primary operator O.
uct of the holomorphic part and antiholomorphic part, and ∆S (2) contains an extra correction log 1 −  3 (the orange curve).Note that the extra correction will be log 2 when we consider another late-time excess formula (51), reproducing the result of entanglement entropy in 57 .
The two-point function is The four-point function at the late time is Combine ( 61) and ( 62), the 2nd pseudo-Rényi entropy is given by Notice that there is an additional correction depending on the conformal weight of the corresponding primary operator and its relation with the conformal weight can be seen in figure 3 (the green curve).For two general linear combination operators, its pseudo-Rényi entropy may also acquire extra correction depending on the central charge and conformal weight of the theory at the late time, and one can calculate the extra correction in general cases.

V. CONCLUSION AND PROSPECT
In this paper, we investigate the pseudo-Rényi entropy of local descendant operators in RCFTs, extending the previous studies in 305157 .In 3057 , it has been found that the latetime excess of the pseudo-Rényi entropy of two primary states and the Rényi entropy of a descendant state equal to the logarithmic quantum dimension of the primary operator in RCFTs.It is a natural question to consider the pseudo-Rényi entropy of the descendant states.
Firstly, we show that in some special cases: being primary, the late-time excess of the 2nd pseudo-Rényi entropy ( 8) is still logarithmic of the quantum dimension of the primary operator.Using the conformal block and operator product expansion, we compute the 2nd pseudo-Rényi entropy constructed by two descendant operators with different Virasoro generators.We show that their 2nd pseudo-Rényi entropy is the same as their primaries for such states.Although the calculation looks quite complicated, the leading divergent terms in the late time limit are simple, behaving as the one for primary operators.
Further, we compute k-th pseudo-Rényi entropy with two descendant operators L −n O and L −m O.We extract the most divergent term of the 2k-point function on Σ k with an overall factor F (36), and then associate the 2k-point function of descendant operators with the 2k-point function of primary operators (39) with some derivative operators of the form We find the 2k-point function breaks up into k two-point functions for the holomorphic part(and k for the anti-holomorphic part).The two-point function only depends on the conformal weight and some constant (42).As a result, in this case, the pseudo-entropy of the descendant operators is the same as primaries.Finally, we discuss the most generic descendant operators, which are two linear combination operators constructed by operators in O's conformal family We derive the formula for k-th pseudo-Rényi entropy of linear combination operators at the late time which is pretty different from the formula derived when For convenience, we also introduce an effective transition matrix T eff to simplify the formula (66) (Eq.( 67) shares a similar treatment).Using the formula (50), we find that the pseudo-Rényi entropy of linear combination operators is generally different from that of the primary operator O.The pseudo-Rényi entropies are the same as the ones of the primary when the correlation function of V α and V β can be divided into the product of the holomorphic part and the anti-holomorphic part.A typical example is Otherwise, there is an extra contribution due to the mixing of the holomorphic and antiholomorphic Virasoro generators.A typical example of extra contribution is In general, the k-th pseudo-Rényi entropy for two linear combination operators at the late time only depends on the quantum dimension and the contribution from a finite-dimensional Hilbert space, Noticing the current results in RCFTs, one can directly calculate the pseudo-entropy of generic local operators in Liouville CFT, holographic CFTs, non-diagonal CFTs, etc.Since the spectra in such theories have different structures, the associated pseudo-entropy will be highly different from those in RCFTs.In particular, since holomorphic and antiholomorphic conformal blocks have different structures in non-diagonal CFTs, the late-time behavior of the entanglement entropy and pseudo-entropy associated with locally excited states will not be the same as the ones demonstrated in the current paper.We would like to leave them to future work.
So we can firstly expand O(1)'s Virasoro generator, The correlation function with four Virasoro generators is deformed into correlation functions containing no more than 3 Virasoro generators.We can then expand O(4)'s Virasoro generator, Form (B1) and (B2), we can read the exact form of D 1,4 introduced in (39) We can expand O(2)'s Virasoro generator and O(3)'s Virasoro generator in a similar way, Finally, we can have where Combining (B1),(B2),(B4) and (B5), we have (B6) The correlation function of four descendant operators becomes the correlation functions of their corresponding primary operators with some constants and derivatives.
For i ̸ = j ̸ = k ̸ = l, we can have (B7) From (B6), (B7), ( 19), (15) and (34) we derive the leading behavior of ⟨O where we fomally decompose the operator O(z, z) into a product of a holomorphic operator O(z) and an anti-holomorphic operator O(z), in the sense of the two-point function ⟨O(1)O(2)⟩ = z −2∆ 12 z−2∆ 12 which is independent of the level k and consistent with the results of the 2nd pseudo-Rényi entropy in the previous sections.Based on the above results, we can conclude that the latetime excess of the pseudo-entropy of L −n O and L −m O is consistent with the entanglement entropy of L −n O and also equals log d O .

1 FIG. 2 .
FIG.2.The late-time excess of the 2nd Rényi entropy (in blue) or the 2nd pseudo-Rényi entropy (in orange) of the linear combination operator (C1∂ + (1 − C1) ∂)ε, where ε is the energy density operator in the critical Ising model.we have dε = 1.The hollow circles represent the numerical data obtained by using the known four-point function of ε, while the solid lines represent the theoretical result obtained by using Eq.(51)(or (52)) (corresponding to the blue line) and Eq.(50) (corresponding to the orange line).It should be noted that when the linear combination operator is the equally-weighted sum of L−1ε and L−1ε, i.e., C1 = 1/2, the late-time excess of the Rényi entropy (x1 = x2) is log 2, while the late-time excess of the pseudo-Rényi entropy (x1 ̸ = x2) is log18 17 ≈ 0.057.
be decomposed into a holomorphic operator i C i L −{Ki} O(w) and an antiholomorphic operator L−{ K} O( w) in the sense of the two-point function ⟨V α V † β ⟩, producing a product state i C i L −{Ki} O H ⊗ L−{ K} O H in the Verma module H⊗ H when acting on the vacuum state.However, when V α(β) (w, w) = i C i L −{Ki} L−{ Ki} O(w, w), where L −{Ki} L−{ Ki} O(w, w) can each be decomposed into a holomorphic operator L −{Ki} O(w) and an antiholomorphic operator L−{ Ki} O( w) in the sense of two-point functions, V α acting on the vacuum state produces an entangled state i

FIG. 3 .
FIG. 3. Additional correction of the late-time ∆S

1 2 (
1+4∆) 2 besides log d O .The relation between extra correction and the conformal weight has shown in figure