Pseudo entropy and pseudo-Hermiticity in quantum field theories

In this paper, we explore the concept of pseudo R\'enyi entropy within the context of quantum field theories (QFTs). The transition matrix is constructed by applying operators situated in different regions to the vacuum state. Specifically, when the operators are positioned in the left and right Rindler wedges respectively, we discover that the logarithmic term of the pseudo R\'enyi entropy is necessarily real. In other cases, the result might be complex. We provide direct evaluations of specific examples within 2-dimensional conformal field theories (CFTs). Furthermore, we establish a connection between these findings and the pseudo-Hermitian condition. Our analysis reveals that the reality or complexity of the logarithmic term of pseudo R\'enyi entropy can be explained through this pseudo-Hermitian framework. Additionally, we investigate the divergent term of the pseudo R\'enyi entropy. Interestingly, we observe a universal divergent term in the second pseudo R\'enyi entropy within 2-dimensional CFTs. This universal term is solely dependent on the conformal dimension of the operator under consideration. For $n$-th pseudo R\'enyi entropy ($n\ge 3$), the divergent term is intricately related to the specific details of the underlying theory.


Introduction
Density matrix is a fundamental concept in quantum mechanics, used to describe the states of a given system.The reduced density matrix plays a crucial role in characterizing quantum correlations or entanglement between subsystems of the given system.One could define various relevant quantities, employed as a function of the reduced density matrix, serves as a measure of entanglement.
The quantities mentioned above can all be regarded as functions of the reduced density matrix.One could also generalize the density matrix to the transition matrix, which involves two different states |ϕ⟩ and |ψ⟩.Without normalization it can be taken as the operator |ψ⟩⟨ϕ|.Actually, in many cases we have already used the transition matrix.For example, the expectation value of an operator O in the state |ψ⟩ is ⟨ψ|O|ψ⟩ = tr(|ψ⟩⟨ψ|O) can be taken as the trace of the operator |ψ⟩⟨ϕ| with |ϕ⟩ = O † |ψ⟩.
Similar to the density matrix, the concept of reduced transfer matrices can be introduced by replacing the trace operation with partial traces for a given subsystem A in certain contexts.That is where Ā is the complementary part of A. In [14] the authors introduce the so-called pseudo entropy as a new generalization of EE, which is the von Neumann entropy of the operator T ( See also the similar quantity defined in [15].It is interesting that pseudo entropy also has a gravity dual similar as EE if the transition matrix |ψ⟩⟨ϕ| has a bulk geometry dual.To evaluate pseudo entropy we usually calculate its one parameter generalization pseudo Rényi entropy, defined as where n is an integer.The reduced transition matrix is generally non-hermitian, thus the eigenvalues of it may be complex.The pseudo Rényi entropy and pseudo entropy may also be complex number.The imaginary part of pseudo entropy can be explained as timelike entanglement [19].It is interesting that the pseudo Rényi entropy can be connected with the Rényi entropy for the superposition states of |ψ⟩ and |ϕ⟩ by a sum rule [20].There have been many recent studies related to pseudo-entropy, please refer to [21]- [42].
The class of the transition matrix that has real-valued pseudo entropy should only be a special subset of the transition matrix.Motivated by the recent works on nonhermitian physics [16,17,18], the authors in [21] find the real-valued condition of pseudo entropy can be understood by the concept of pseudo hermitian.This paper represents further research on the topices discussed above.In QFTs the states are constructed by acting operators on the vacuum states.The transition matrix usually includes two different operators, thus the pseudo Rényi entropy should include more information on the correlators than the EE.Therefore, we expect the realvalued condition of pseudo Rényi entropy can be related to some properties of correlation functions in QFTs.
In this paper we mainly focus on the excited states constructed by acting local operators in Minkowski spacetime.Some examples are discussed in the paper [21].We extend the results to more general cases and find some new properties of the pseudo Rényi entropy.We also explain our new results by pseudo hermitian condition.Besides, the pseudo entropy is usually divergent if the operators are located near the lightcone [27].We show that the divergence is universal for the second pseudo Rényi entropy, which only depends on the conformal dimension of the operator for 2-dimensional conformal field theories.But for n ≥ 3 the divergent terms depend on more details of the theory.
The paper is organized as follows.Section.2 is the general set-up, which includes the transition matrix that we will consider.In section.3we will firstly evaluate the pseudo Rényi entropy for some examples for different cases.Then we will analyse the result for general operator by using properties of correlation functions.In section.4we focus on the divergent term for operators located near the lightcone.For pseudo Rényi entropy we find the divergent term is universal, which only depends on the conformal dimension of the operator near the lightcone.Section.5 is devoted to pseudo hermitian condition for the examples that we discussed in previous section.The results can be explained by the pseudo hermitian condition.The last section is the conclusion.In the appendices we show more details of the calculations.

Transition matrix construction in QFTs
For a given subsystem A, the local operator algebra R(A) consists of the operators supported in A, where D(A) is the domain of dependence of A. Let Ã be another subsystem which is spacelike with A. Denote the algebra associated with Ã by R( Ã).By microcausility we would have [O, Õ] = 0 for O ∈ R(A) and Õ ∈ R( Ã).According to the Reeh-Schlieder theorem [43] [44], for any pure state |ϕ⟩ there exists local operators O ϕ in R(A) such that |ϕ⟩ can be approximated by O ϕ |0⟩, that is the distance between O ϕ |0⟩ and |ϕ⟩ can be arbitrary small.Therefore, we would like to consider the general transition matrix defined as where O ∈ R(A) and O ′ ∈ R(A ′ ) for subsystem A and A ′ .We would assume O, O ′ are Hermitian operators.Any operators can be written as linear combination of two Hermitian operators.If the operators O, O ′ are not Hermitian, one could rewrite the transition matrix as linear combinations of transition matrices constructed by hermitian operators.
In this paper we would mainly focus on the Rindler wedges in Minkowski spacetime.For d-dimensional spacetime the metric is ds 2 = −dt 2 +dx 2 +d⃗ y 2 , where ⃗ x ′ are coordinates of d-dimensional Euclidean space.The left Rindler wedge is defined in the region |t| < −x.The right Rindler wedge satisfies the condition |t| < x.The region t > |x| (t < −|x|) is called the expanding (contracting) degenerate Kasner universe.The Minkowski vacuum |0⟩ can be written as entangled states between the left and right Rindler wedges [45].If taking x ∈ (0, +∞) as the subsystem A, one could calculate the reduced density matrix ρ 0 A := tr Ā|0⟩⟨0|, where Ā is the region x ∈ (−∞, 0), and define the Rényi entropy of A, which are divergent due to infinite size of the subsystem.But one could focus on the difference which is usually finite.Let us consider the local operators O = O(t, ⃗ x) and O ′ = O(t ′ , ⃗ x ′ ).The transition matrix is given by We assume the operator O is hermitian.One could use replica method to evaluate pseudo Rényi entropy.We will consider the operators O and O ′ are located in different regions, e.g., O is in the left Rindler wedge and O ′ is in the right Rindler wedge.The positions of the operators play a crucial role in determining the behavior of pseudo Rényi entropy.It is obvious the pseudo Rényi entropy can be used as a tool to detect the spectra of T ψ|ϕ A .In this paper we will mainly focus on primary operator O in 2-dimensional CFTs.In the following we will firstly calculate the pseudo Rényi entropy for some special cases.Let us fix the coordinate (t ′ , x ′ ) = (0, x ′ ) with x ′ > 0, i.e., O(t ′ , x ′ ) is in the right Rindler wedge and on the time slice t = 0. We will mainly focus on the following cases: Case I: x < 0, t = 0, i.e., O(t, x) is on the time slice t = 0.
Case II: x < t < −x and t ̸ = 0, i.e., O(t, x) is in the left Rindler wedge.
Case III: −t < x < t, i.e., O(t, x) is in the expanding degenerate Kasner universe.
In Figure .1 we show the three different cases.
Figure 1: Three different cases that we consider in this paper.The operator O(t ′ , x ′ ) is fixed at the point (0, x ′ ) with x ′ > 0. O(t, x) located in the blue region (case I), light blue region (case II) and orange (case III).

Pseudo Rényi entropy
In this section we will directly evaluate the pseudo Rényi entropy by replica method for the three different cases.

Review of replica method
We will consider the transition matrix where N is the normalization constant.τ and τ ′ is the Euclidean time, later we will obtain the real time result by analytical continuation of τ .Define the coordinates w = x + iτ and w = x − iτ .The transition matrix is given by with where O(w 2i−1 , w2i−1 ) and O(w 2i , w2i ) with i = 1, 2, ..., n are operators inserted on i-sheet.In general it is hard to evaluate the 2n-point correlation functions on the manifold Σ n .We will mainly focus on subsystem A = (0, +∞) in 2-dimensional CFTs.One could use the transformation z = w 1/n , the n-sheet manifold Σ n is mapped to z-plane.The 2n-point correlation funcation is given by where z j = w 1/n j and zj = w1/n j .By definition the variation of the psuedo Rényi entropy from the vacuum state is given by In this paper we will consider the transition matrix (6) with real time.One could evaluate the pseudo Rényi entropy by analytical continuation τ = ϵ + it and τ ′ = ϵ − it ′ , where ϵ is the UV cut-off.In the final result we will take the limit ϵ → 0. With this one could obtain the pseudo Rényi entropy for the transition matrix T O|O ′ A .
For n = 2 by the conformal transformation z = w 1/2 , the coordinates We can obtain that tr(T where the cross ratio η : , which is related to the coordinate (t, x) and (t ′ , x ′ ), see the Appendix.A for details.
In the following we will start with some simple examples and show the general properties of pseudo Rényi entropy for the three different cases.Then we try to extent the conclusions for more general operators.

Two-dimensional free boson
Let us show the pseudo Rényi entropy for the transition matrix (6) with operators ∂ϕ ∂ϕ and V α := e iαϕ + e −iαϕ .
Let us consider the second pseudo Rényi entropy ∆S (2) .With some calculations we have tr(T  The plots of the logarithmic part of ∆S (2) for the operator ∂ϕ ∂ϕ.For all the plots (t ′ , x ′ ) is fixed to be (0, 10).As shown in Appendix.A the cross ration η and η would have different values for the three different cases.Taking the cross ratios into the above formula, one could obtain the result which depends on the coordinate (t, x).Some results are shown in Fig. 2. For the three cases we find the logarithmic part of ∆S (2) are all positive.
One could also calculate the ∆S (n) (n ≥ 3) by using Wick theorem for the free scalar theory.For more details of the results for free scalar theory see the Appendix.B, which is very similar with ∆S (2) .
For the operator ∂ϕ ∂ϕ we find the logarithmic part of ∆S (n) with n = 2, 3, 4 are all positive in the three cases.This implies the eigenvalues of the reduced transition matrix T ψ|ϕ A may be real or come in complex conjugate pairs.Another interesting fact is that ∆S (n) is divergent when the operator O(t, x) approaches to the lightcone.In the following examples we will also find the similar results.

Operator V α
For the vertex operator V α , The conformal dimension of this operator is h = h = α 2 2 .By using the formula for the correlation function of the vertex operator, where n i=1 α i = 0. We could obtain tr(T Taking the cross ratios intro the above equation, one could obtain the results, which are graphically represented in Fig. 3.We also show the result of ∆S (3) in the Appendix.B.In Fig. 3 we show the result for three different cases with respect to the parameter α.
For the operator V α we find that tr A T O|O ′ A n /tr A (ρ 0 A ) n with n = 2, 3 is real for the case I and case II.But the results are generally complex in case III, which is different from the operator ∂ϕ ∂ϕ.We also find near the lightcone ∆S (n) are also divergent.) 2 /tr(ρ 0 A ) 2 .

Minimal model
Let us consider another simple example: the operator ϕ (r,s) in the minimal model M(p, q).The conformal dimension of this operator is We would like to consider the operator ϕ (2,1) , the conformal dimension is To evaluate tr(T ) 2 /tr(ρ 0 A ) 2 we need the conformal block for the operator, which are given in [46] [47], where I 1 , I 2 are defined as

Summary of the examples
In the above examples the second pseudo Rényi entropy or tr(T show some general properties.For case I and case II all the examples support that tr(T A ) 2 is real, but may be negative.For case III tr(T ) 2 /tr(ρ 0 A ) 2 may be complex or real, which depends on the theory and the operators.
For all the examples we find logarithmic part of the pseudo Rényi entropy would be divergent near the lightcone.Similar behaviors have been found and discussed in [27].In the following sections we will show the divergence is universal, which only depends on the conformal dimension of the operator.
The results are shown in the following table.
Table 1: Summary of the results for tr(T 3.5 General argument

The second pseudo Rényi entropy
The second pseudo Rényi entropy is associated with conformal block (12).Using the cross ratios in Appendix.A, for case I and case II we have where * means the complex conjugation.By using the cross symmetry we have Further using ( 12) we find tr(T ) 2 /tr(ρ 0 A ) 2 should be real.However, for case III, η is a real number, η is complex.The conformal block can be expanded as G(η, η) ∼ C p F p (η) F (η) with C p being real.Using (12) we find generally tr(T A ) 2 should be complex.This suggests that the spectra of the reduced transition matrix T O|O ′ A should include complex eigenvalues.

n-th pseudo Rényi entropy for case I
One could also obtain the above results by considering the correlation functions.According to 9 and 10 where z j = w 1/n j and zj = w1/n j .To simplify the notations let us focus on n = 3 and case I.The corresponding coordinates are as shown in Fig. 5.By directly calculations one can show that the coefficients and the two point correlation functions ⟨O(w 1 , w1 )O(w 2 , w2 )⟩ Σ 1 are real.The six point correlation functions can be written as where we have defined the state  It appears that the arguments presented above do not apply to case II.In the context of case I, the 2n-point correlation functions can be mapped to the Euclidean z-plane.However, in case II, these correlation functions must undergo analytic continuation into real time.This fundamental distinction sets case II apart from case I, leading to a significantly different scenario.Addressing this issue requires further research, and we defer its resolution to future studies.

Pseudo Rényi entropy near lightcone 4.1 The second pseudo Rényi entropy near lightcone
In all the examples we find the pseudo Rényi entropy is divergent near the lightcone.In this section we would like to show the divergence of pseudo Rényi entropy is universal near the lightcone.Define the null coordinate while η is finite.The cross symmetry of conformal block is In general, conformal block G(z, z) can be expanded as where C p is the coupling constant, F(z) and F(z) are holomorphic and antiholomorphic parts of the conformal block.Further, the conformal block can be expanded as a power series in z: Using the above results one can see in the lightcone limit t + x → 0 − we have Using (12) in the lightcone limit u → 0 − we have tr(T Therefore, the leading contribution to ∆S (2) is divergent as The result is only related to the conformal dimension h of the operator.Similarly, we can see the lightcone limit in case III, that is u → 0 + .The cross ratio is also divergent Using same argument as case II, we can obtain tr(T Thus in the lightcone limit u → 0 + , ∆S (2) is also divergent as The pseudo Rényi entropy is also only related to the conformal dimension of the operator.Near the lightcone u ∼ 0, the real part of the second pseudo Rényi entropy is divergent.Both in case II and case III the divergent part is associated with the conformal dimension h of the operator.The imaginary part of the second pseudo Rényi entropy is non-universal, which depends on the anti-holomorphic cross ratio η.In case II we have shown ) 2 tr(ρ 0 A ) 2 must be real.Thus the imaginary part of the second pseudo Rényi entropy should be iπ or 0, which depends on the conformal block of anti-holomorphic part as we can see from (30).

n-th pseudo Rényi entropy near lightcone
The result in last section can be generalized to the n-th pseudo Rényi entropy.To map the manifold Σ n to z complex plane, we use the coordinate transformation z = w 1/n .The corresponding coordinates on z-plane are given by In case II near the lightcone u → 0 − , we have That is z 1 − z 3 → 0 in the limit u → 0 − .Similarly, we have z 2j−1 − z 2j+1 → 0 with j = 1, 2, ..., n − 1 .
From the expression of tr(T E,A ) n /tr(ρ 0 A ) n (22) we can see that there are two sources for the divergence in the lightcone limit u → 0 − .One is the coefficients n i=1 z The other one is from the 2n-point correlations functions, which can be determined by using operator product expansion (OPE), which has the following form: (38) where h is the conformal dimension of the operator O, p labels the operators that appear in the OPE, K, K labels the descendants, C {K, K} p are the coupling constants.
Let us consider the n = 2 as an example.The coefficients 2 i=1 z −h 2i−1 ∼ (−u) −h by using (37).In the lightcone limit we have z 1 − z 3 ∼ (−u) 1/2 , thus the OPE can be approximated by where O 0 denotes the identity operator.The leading contribution comes from the identity operator and its its anti-holomorphic descendants, such as T .Using the fact z 1 − z 3 ∼ (−u) 1/2 , the 4-point correlation function has the limit where which can be taken as summations of the anti-holomorphic descendants.The 3-point correlation function in ( 40) is expect to be finite, thus we find or equally ∆S (2) ∼ 2h log(−u), which is consistent with the discussion by using conformal block (30).We can as well utilize Operator Product Expansion (OPE) to simplify the 2n-point correlation functions mentioned in (22) as the lightcone limit u → 0 − .However, it is crucial to note that the diverge behavior significantly based on the details of the theory.
Let us go on considering the case with n = 3.In the limit u → 0 − we have z 1 , z 3 , z 5 ∼ (−u) 1/3 , thus the coefficients which is also divergent.Since z 1 − z 3 ∼ (−u) 1/3 we have the OPE In the n = 2 case, we only consider contributions from anti-holomorphic descendants of the identity operator.However, it is essential to take into account other operators, such as the stress-energy tensor T (z), as these operators in the OPE have strong correlations with O(z 5 , z5 ).This complicates the discussions significantly.
Same as the case n = 2, the identity operator and its anti-holomorphic descendants will give the contribution Unlike the case n = 2 the holomorphic descendants of identity operator will also contribute.Consider the stress energy tensor T (z) as an example, the OPE gives where in the second step we use the Ward identities and only keep the leading contributions.Similar calculations can be done for other descendants.It can shown the results are all divergent as |u| − 2h 3 .Further, we should consider the possible contributions from other primary fields and their descendants.Take the primary operator O p as an example.The contribution is given by (47) Note that z 3 −z 5 ∼ |u| 1/3 .We can further expand the product O p (z 3 , z3 )O(z 5 , z5 ).(47) we find the leading divergent term of 6point correlation function is |u| −h+ h p ′ 3 .Recall that the contribution from the identity and its anti-holomorphic descendants is divergent as

Assume the fusion rule
3 , the 6-point correlation function is divergent as |u| − 2h 3 .Combining with the divergent term from the coefficients Otherwise, if h p ′ < h, we would have ∆S 3 ) log |u|.Although the above argument can be extended to any arbitrary n, the results will also become more complicated.For the general case of n, we do not expect a simple conclusion.Instead, it depends on more specific details of the theory.

Examples for pseduo Rényi entropy near lightcone
As we show in last section, near the lightcone, the pseudo Rényi entropy should be divergent as log |u| (31) (34).In this section we would like use examples to check the results.For the second pseudo Rényi entropy the result is universal.We show that the divergent term is 2h log |u| for both case II and III, which only depends on the conformal dimension of the operator.While for n ≥ 3 the result would be more complicated, it depends on the details of theory, that is the OPE of the operators in the theory.
Firstly, consider the operator ∂ϕ ∂ϕ.The second pseudo Rényi entropy can be obtained by using (89) for case II and case III.It is easy to check the leading divergent term is ∆S (2) ∼ 2 log(−u), which is consistent with result (31) (34).For the vertex operator V α and ϕ (2,1) in the Minimal Model one could also check this directly by using the expressions in section.3.2.2 and section.3.3.It is also helpful to define the difference between the pseudo Rényi entropy and the universal divergent term, that is ∆S (2) fin := ∆S (2) − 2h log |u|. ( We show ∆S fin for the three examples in Fig. 6.For n ≥ 3 the results would be more subtle.Consider the vertex operator V α , one could obtain ∆S (3) by calculating the 6-point correlation functions.Using the notation in [14] the result is where η ij mn are the cross ratios defined as where 1 ≤ i, j, m, n ≤ 6.In the lightcone limit u → 0 − , the cross ratios would be divergent as Using the result (50) we conclude that which is consistent with (48) with h = α 2 2 .In last section we have shown the result ( 48) is based on the assumption that h p ′ ≥ h.For the vertex operator V α we have the OPE The operator V 2α may give the contribution to the divergent term of 6-point correlation function.Further, we have Note that the conformal dimensions of V α and V 3α are α 2 2 and 9α 2 2 .These two operators actually correspond the operator O p ′ that we discuss in last section.It is obvious that the condition h p ′ ≥ h is satisfied.Thus this example can be taken as a nice check of the result (48).
For the operator ∂ϕ ∂ φ we should be more careful.In the discussions of last section we implicitly assume the correlation functions cannot factor as holomorphic and anti-holomorphic parts, which is not correct for ∂ϕ ∂ φ.Near the lightcone the divergence of correlation function comes from the holomorphic field.Take n = 3 as an example, the correlation function is proportional to ⟨∂ϕ(z 1 )∂ϕ(z 2 )∂ϕ(z 3 )∂ϕ(z 4 )∂ϕ(z 5 )∂ϕ(z 6 )⟩. (56) Near the lightcone we know z 1 , z 2 , z 3 ∼ | − u| 1/3 .The leading divergent term of the above correlator is given by which is vanishing.Thus the final result is finite in the limit |u| → 0. The divergence of the pseudo Rényi entropy comes from the coefficients Thus the third pseudo Rényi entropy should be divergent as log |u|.
For n = 4 we can use similar argument as the case n = 3, the leading divergent term of the 8-point correlation function is given by Combining with the coefficients 4 i=1 z −3 2i−1 ∼ |u| −3 , we have ∆S (4) ∼ 4 3 log |u|.In Appendix.B we check this by directly calculation using Wick theorem.

Pseudo-Hermitian condition
In previous sections we study the pseudo Rényi entropy for the transition matrix (6).We mainly focus on three cases, where the location of the operator O(t, x) is different.From several examples we find the results summarized in the Table .1.For the cases I and case II the pseudo Rényi entropy is real for all the examples we consider.For case III the results are generally not real.The pseudo Rényi entropy being complex implies that some of the eigenvalues of the reduced transition matrix T ψ|ϕ A are complex.
The pseudo Rényi entropy being real implies the eigenvalues of T ψ|ϕ A are real or complex coming in conjugated pairs.
In [21] the authors point out one could understand the real-valued condition by using pseudo-Hermiticity.If an operator M satisfies where η is an invertible and Hermitian operator, we say M is η-pseudohermitian.A notable fact is that the diagonalizable operator M is η-pseudohermitian operator M if and only if the eigenvalues are real or complex coming in complex pairs.In [21] it is found the general η-pseudo-Hermitian transition matrix can be written as If η = η A ⊗ η Ā with both η A and η Ā being invertible and hermitian, one could show that the reduced transition matrix would be pseudo-Hermitian.
Thus the eigenvalues would be real or complex coming in conjugated pairs.The pseudo Rényi entropy is expected to be real.Further, if η A and η Ā are positive or negative operator, the eigenvalues are expected to be positive thus the pseudo Rényi entropy would be positive.The result of our previous examples imply the transition matrix for case I and II may be pseudo-Hermitian.Our goal in this section is to investigate whether the reduced density matrix can be written as pseudo-Hermitian form.

Translation and boost operators
In this section we would like to introduce the smearing operators with stress energy tensor which are related to our problem.A local QFT has stress energy tensor T µν which is conversed ∂ µ T µν = 0.One could construct the Hamiltonian where V denotes the whole region on the time slice t = 0, which generates the time translation.The i-th component of momentum operator is where i = x, y, z, which generates the translation on the i-the direction.The boost in the x-direction is generated by the modular Hamiltonian For our motivation we would like to introduce the similar operators located in A or Ā, that is the operators where f and g are function supported in region A. Similarly, we can define the local operators in R( Ā) by using the functions supported in Ā. Specially, we are interested in the following ones, It is well known that the generators H, P i , K should satisfy the Poincare algebra.We have Since P x , K and H are all constructed by stress energy tensor, the Poincare algebra should give some constraints on the commutators of stress energy tensor [T µν , T ρσ ].The form of the commutators can be determined up to the so-called Schwinger terms, which need to be total derivatives [48].Therefore, the commutators of H A( Ā) , P i A( Ā) , K A( Ā) may be different from the Poincare algebra (66).In this paper we will mainly focus on 2-dimensional CFTs, for which the commutators of stress energy tensor are known.For free scalar theory we also calculate the commutators, which are shown in Appendix.D.

2 dimensional CFTs
For 2 dimensional CFTs the commutators of stress energy tensor is given by where u = t − x.We have the similar commutator relation for T vv with v = t + x.Define the smearing operators T 00 (f ) := dxf (x)T 00 , T 0x (g) := dxf (x)T 0x with T 00 = T uu + T vv and T 0x = T vv − T uu .Similar as the definition (65) let us define the local operators One could evaluate the commutators of H A( Ā) , P i A( Ā) , K A( Ā) by using (69), see the appendix for details.The commutators are given by and By using BCH formula we would have Further it can be shown which means the operator η A := e πK A e iP A,x a is a Hermitian operator.

Pseudo Hermitian
Let us consider the transition matrix (6) with O(t, x) and O ′ (t, x) located in left and right Rindler wedges, respectively as shown in Fig. 7.We would like to show the transition matrix ( 6) is η-pseudo-Hermitian with the form Firstly, let us consider the case I.Note that we have x < 0 and x ′ > 0. Using the translation and boost operator, we have The transition matrix can be written as by using the fact that K|0⟩ = 0 and P x |0⟩ = 0.The operator η = e Kπ e −iPx(x ′ +x) is Hermtian and invertible (68).Thus the transition matrix (6) in this case is pseudo Hermitian.In fact we will further show η can be written as η A ⊗η Ā where both η A and η Ā are Hermitian and invertible.
In last section we define the local operators K A( Ā) and P x,A( Ā) by choosing the smearing functions f and g for the operators T 00 (f ) and T 0x (g) (70).Abviously, we have the relation P x = P x,A + P x, Ā and K = K A + K Ā by definitions.There is a subtle point for the operators P x,A( Ā) .If we calculate the commutator [P x,A , P x, Ā] by using ( 70) and (94), a boundary term located at the entanglement boundary ∂A will appear.Therefore, it seems we cannot decompose e iPx = e iP x,A e iP x, Ā .However, we will argue that the boundary will not appear if we carefully consider the process to evaluate pseudo Rényi entropy.
It is well known that the (pseudo) Rényi entropy exhibits UV divergence in quantum field theories (QFTs) and requires regularization.As shown in [49] the regularization can be taken as projection P ϵ ∂A in the Hilbert space, which removes small spatial region of thickness ϵ around the entanglement boundary ∂A.For a pure state |ϕ⟩ that is to say we would consider the regularized states In the Euclidean path integral formulation the projection can be understood as introducing a small slit with lengh ϵ around the boundary ∂A.Using this the authors in [49] derive the modular Hamiltonian and Rényi entropy for a lot of known examples.With considering the regularization the modular Hamiltonian of the subsystem A = (0, +∞) should be K A = ∞ ϵ dxxT 00 or K A = T 00 (H(x − ϵ)).Therefore, it is more properly to define the local operator P x,A = T 0x (H(x − ϵ)) and P x, Ā = T 0x (H(−x + ϵ)).One could show that [P x,A , P x, Ā] = 0. Now we could decompose the Hermitian operator as η = η A ⊗ η Ā with η A := e πK A e iP A,x (x 1 +x 2 ) and η Ā := e πK Ā e iP Ā,x (x 1 +x 2 ) .We also show η A and η Ā are Hermitian and invertible operators.According to the theorem in [?] we conclude that the eigenvalues of T ψ|ϕ A will be real or complex coming in conjugated pairs, which is consistent with the pseudo Rényi entropy would be a real number (1).Now let us consider the two operators are not on same time slice.If O(t ′ , x ′ ) is located in right Rindler wedge, we will have the following relation where O(−t ′ , −x ′ ) is the operator located in the left Rindler wedge, e −πK is the modular operaor.By further using translation operators we have Let us define the operator η ′ = e −iH(t+t ′ )+iPx(x+x ′ ) e −πK .By using the commutators (66), one could show η ′ is a non-negative operator.Futher, we could decompose the operator η ′ into local operators in A and Ā as η ′ A = e −iH A (t+t ′ )+iP x,A (x+x ′ ) e −πK A and η ′ Ā = e −iH Ā(t+t ′ )+iP x, Ā(x+x ′ ) e −πK Ā .We would have η ′ = η ′ A ⊗ η ′ Ā, where both η ′ A and η ′ Ā are Hermitian and invertible.Thus the eigenvalues of T ψ|ϕ A are expected to be real or complex coming in conjugated pairs.The pseudo Rényi entropy should be real, which is consistent with our calculations.
However, if the operator O(t ′ , x ′ ) is located in the Kasner universe, the relation (78) is no longer right.For real θ the operator e −2πiθK is the Lorentz boost, which is unitary and acts on the state O(t, x)|0⟩ as where t ′ = t cosh(θ) + x sinh(θ) and x ′ = t ′ = t sinh(θ) + x cosh(θ).Now we would like to analytically continue θ to a complex parameter.If θ = iπ, one could obtain t ′ = −t and x ′ = −x, thus we get the relation (78).But the analytical continuation is applicable only if x > |t|, that is in the Rindler wedge [44].Therefore, if O(t ′ , x ′ ) is located in the Kasner universe, the transition matrix ( 6) is no longer η A ⊗ η Ā-pseudo Hermitian.We expect the spetra of the reduced transition matrix would have be complex in general.
The pseudo Rényi entropy would be complex.

Conclusions and discussions
In this paper we investigate the pseudo Rényi entropy in QFTs.The transition matrix ( 6) is constructed by acting local operators on the vacuum.We mainly focus on three different cases, in which the locations of the operator O(t, x) are different, while O(t ′ , x ′ ) is fixed at the right Rindler wedge.In 2-dimensional CFTs we calculate the pseudo Rényi entropy for some examples, including operators ∂ϕ ∂ φ, V α in free scalar theory, ϕ (2,1) in Minimal Models.It is found the pseudo Rényi entropy would be real for the cases I and II, that is the operator O(t, x) is located at the left Rindler wedge.While the pseudo Rényi entropy is generally complex for case III, that is O(t, x) is located at the Kanser universe.The results are summarized in the Table .1.
Another interesting results in this paper is the universal divergent term of pseudo Rényi near the lightcone, i.e., O(t, x) is located near the Rindler horizon t ± x = 0.The divergent behavior is observed in the paper [27], where the authors studied the time evolution of pseudo Rényi entropy.The situation is very similar, so our results are also used to understand the divergent behavior of the time evolution of pseudo Rényi entropy.It is found the second Rényi entropy shows the universal divergent term 2h log |u|, where h is the conformal dimension of the operator O.The results are independent with the details of the theory, such as fusion rule of operators.Our results are only in 2-dimensional CFTs.The divergent behavior is closely related to the OPE of operators near the lightcone.By using the OPE of lightcone operators [50] one may obtain more universal conclusions for general theory.
Finally, we use pseudo-Hermitian condition to explain the real-valued pseudo Rényi entropy in case I and II.In the paper [21] it is shown the real-valued condition can be associated with the pseudo-Hermiticity.Some examples are already discussed in [21].We further study some examples and extend the results in [21].For the case I and II we find the operator η, thus prove the pseudo-Hermitian condition for the reduced density matrix T ψ|ϕ A .It should be noted that the η-pseudo-Hermitian condition actually ensures the eigenvalues of the operator T ψ|ϕ A are real or complex coming in conjugated pairs.Thus the logarithmic term of the pseudo Rényi entropy are expected to be real for any n.For the examples we actually only calculate pseudo Rényi entropy for n = 2, 3. Therefore, we could predict that the logarithmic term of n-th pseudo Rényi should be real for all the examples with the transition matrix (6) in case I and II.One could check this prediction in more examples.
The pseudo Rényi entropy includes more information of the theory.It can be a useful probe to detect the correlation functions, symmetry of the underlying theory.There are many interesing directions that are worth to explore in the near future.It is still unclear why the logarithmic term of the pseudo Rényi entropy becomes complex for case III.It should be associated with the causality, which gives non-trival constraints on the correlators [51].The divergent term of the pseudo Rényi entropy is also mysterious.For the Rational CFTs one could understand the time evolution of Rényi entropy by using quasi-particles picture.This picture cannot be applied for the pseudo Rényi entropy.It is hard to image how the appearance of divergence by the quasi-particles picture, let alone the pseudo Rényi entropy may be complex.Recently, the authors in [20] find a sum rule for pseudo Rényi entropy.The pseudo Rényi entropy is associated with the Rényie entropy of the superposition state.By using this sum rule one may make more physical understanding of pseudo Rényi entropy in the quasi-particles picture.
Note that near the light cone t ∼ −x the cross ratio η would be divergent.
B More results of pseudo Rényi entropy for free boson theory Consider the operator ∂ϕ ∂ϕ.By taking the cross ratio (84) into ( 13) we obtain the result for case II tr(T which is always positive.Similarly, taking (84) into ( 13) we obtain the result for case III.The result is same as (89).
For n-th pseudo Rényi entropy we can also directly evaluate the results by using Wick theorem.We obtain the results of n = 3 and n = 4 for case II as follows.

tr(T
and tr(T We can easily read the divergent behavior for the above two expressions near the lightcone.For n = 3 we find tr(T  The plots of the logarithmic parts of ∆S (3) and ∆S (4) for the operator ∂ϕ ∂ϕ.For all the plots (t ′ , x ′ ) is fixed to be (0, 10).In the main text we show the second pseudo Rényi entropy results for the operator V α with some fixed parameters α.Here we show the plot of the second pseudo Rényi entropy as a function of α.For the operator V α in n ≥ 3, we can still calculate ∆S (n) by using the formula (14), we show the expression for n = 3, α = 1 2 in case II.By using the commutator (69) we have

D Commutator of local operators for free scalar
The Lagrangian density is given by The canonical stress energy tensor is Similarly, we also have [T 00 (f ), T 0i (f ′ )] = −i d 3 xf (⃗ x) d 3 x ′ f ′ (⃗ x ′ )∂ 0 ϕ(0, ⃗ x)∂ 0 ϕ(0, ⃗ x ′ )∂ i δ(⃗ x ′ − ⃗ x) where the term in the last line is total derivative, which can be written as a boundary term using integral by parts.
Let us consider the commutators between the local operators and ϕ, π.With some calculations for operator ϕ located in the right Rindler wedge we have e iθK ϕ(0, x, y, z)e −iθK = ϕ(t(θ), x(θ), y, z), where t(θ) = t cosh θ+x sinh θ and x(θ) = x cosh θ+t sinh θ.K A generate the boost in x-direction.If taking θ = iπ, we have t(iπ) = −t and x(iπ) = −x, that is the operator is mapped into the left Rindler wedge by e −πK .P x generates the translation along x-direction, we have e iPxa ϕ(0, x, y, z)e −iPxa = ϕ(0, x + a, y, z), if ϕ is located in the region A.

Figure 3 :
Figure 3: The plots of the logarithmic part of ∆S (2) for the operator V α with respect to spatial position x in three cases.For all the plots we fix x ′ = 10.(a) is for Case I, t = 0, x ∈ (−200, 0).(b) is for Case II, t = −100, x ∈ (−240.− 100).(c) and (d) are for Case III, we fix t = 100, x ∈ (−100, 100).The results are complex, (c) and (d) are respectively the real and imaginary part of tr(T O|O ′ A
By using the Baker-Campbell-Hausdorff (BCH) formula one could show e −Kπ P x e Kπ = −P x , e −Kπ He Kπ = −H.(67) Define the operator η := e Kπ e iPxa for the real parameter a.It can be shown η is Hermitian, e Kπ e iPxa † = e −iPxa e Kπ = e Kπ e iPxa .

Figure 7 :
Figure 7: The case that operators O(t, x) and O(t ′ , x ′ ) are located in left and right Rindler wedge.

Figure 9 :
Figure 9: The plots of the logarithmic part of ∆S (2) with respect to the parameter α in three cases.We fix x ′ = 10.(a) is for Case I, x = −20, α ∈ (0, 4), the results are real.(b) is for Case II, the result is also real.(c) and (d) are plots for the real and imaginary parts for Case III.We take t = 15 and x = −5.