Entanglement entropy in (1+1)D CFTs with multiple local excitations

In this paper, we use the replica approach to study the Rényi entropy SL of generic locally excited states in (1+1)D CFTs, which are constructed from the insertion of multiple product of local primary operators on vacuum. Alternatively, one can calculate the Rényi entropy SR corresponding to the same states using Schmidt decomposition and operator product expansion, which reduces the multiple product of local primary operators to linear combination of operators. The equivalence SL = SR translates into an identity in terms of the F symbols and quantum dimensions for rational CFT, and the latter can be proved algebraically. This, along with a series of papers, gives a complete picture of how the quantum information quantities and the intrinsic structure of (1+1)D CFTs are consistently related.


Introduction
Information theory provides us with a new view on the structure of quantum field theory (QFT). Recently many attempts have given us more insights into the relations between the two, e.g., [1]- [13]. For example: the entropic g-function [7] for 1+1 dimensional quantum field theories can be derived from the relative entanglement entropy, the quantum null energy condition can be obtained [5,6] from the inequalities of entanglement entropy, and authors of [8][9][10][11][12][13] use quantum information quantities to set up criterion of Eigenstate Thermalization Hypothesis (ETH) in order to classify the chaotic behaviors of CFTs.
Among all the quantum information quantities, we will be interested in the Rényi and entanglement entropies of locally excited states in (1+1)D conformal field theory(CFT). The n-th Rényi entanglement entropy for a subsystem A is defined by S (n) A = log Tr[ρ n A ]/ (1 − n), where ρ A is the reduced density matrix of A.The subsystem A is chosen to be the half plane x > 0 in this paper, for simplicity. The locally excited states are defined JHEP05(2018)154 by inserting operators on the vacuum of the theory, in the form O |0 , where O can be a primary or descendant operator, or even the product or linear combination of different operators. The former cases have been extensively studied in the literature [9][10][11][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], while the latter is the focus of the current paper.
We mainly study the variation of S (n) A between the excited states and the ground states, where the excited states are obtained by acting general product of different primary operators or linear combination of different operators. That is the state |ψ m : is a primary or descendant operator located at point x i . We will consider the time evolution of the variation of n-th REE, denoted by ∆S (n) A . In the limit t → ∞, we will show that the variation of Rényi entropy of state |ψ m satisfies the following sum rule (1.1) This sum rule tell us ∆S where the entries of rank-three tensor N k ij are non-negative integers. For simplicity, we consider the case m = 2, the state |ψ L = O 1 (x 1 )O 2 (x 2 ) |0 . In (1+1)D CFT, we can rewrite O 1 (x 1 )O 2 (x 2 ) as a linear combination of OPE blocks [35], i.e., where h 1 , h 2 are conformal dimension of operator O 1 , O 2 , C 12k is the coupling constant for 3-point function, and O k (x 2 ; x 1 ) is a non-local operator, in the sense that the two points x 1 and x 2 can have a nonlocal distance [36]. Here the sum is over all the possible fusion channels. So we can define an equivalent state to |ψ L , The Rényi or entanglement entropy of state |ψ R is denoted by S R . As a result, S R depends on the operator O k (x 2 ; x 1 ) and their linear combination coefficients explicitly. Due to eq. (1.2), the entanglement entropy S L of the state |ψ L should be equal to S R . Then the constraint S L = S R provides a connection between different data of the theory. For (1+1)D rational CFTs, S L is only associated with the quantum dimension of operators O 1 and O 2 which has been obtained in [14], while S R depends on the quantum dimension of O k and the fusion coefficients. It is difficult to get the complete form S R by replica trick. In this paper, we use the Schmidt decomposition approach to obtain the late time behavior of S R . The constraint S L = S R will then leads to an identity (eq. (3.47) in the main context), which can be proved using algebraic relations of F symbols and quantum dimensions. We examine Minimal models M(p, p ) as typical examples.
The layout of this paper is as follows. In section 2, we will give the general set-up. For the locally excited state with many primary operators inserted, we prove the sum rule (1.1). For the case of linear combination of different operators, we also obtain the Rényi entropy JHEP05(2018)154 by Schmidt decomposition. In section 3, we focus on the S L = S R in rational CFTs and obtain the identity. Minimal model examples are discussed in detail. In section 4, we prove the identity. In section 5, we discuss the extension of the above analysis to large-c CFTs, and the relation with (2+1)-D topological orders.

Entanglement of locally excited states
As reviewed in the introduction, the locally excited states we will focus on are of the form where |0 is the vacuum of (1+1)D CFT, and O can be a primary operator, a descendant operator, or the products or linear combinations of different operators. The former two cases have been studied in papers [14,16]. In this section we will study latter two more complicated situations: 1. O is the product of primary operators.

O is linear combination of different operators.
We will mainly focus on rational CFTs, for which the result is robust. The first case has already been studied in paper [23] in rational CFTs. We slightly generalize the result to other (1+1)D CFTs and give the sum rule. As far as we know the second case has not been discussed in literature.

Product of primary operator
where O i (l i , 0) are primary operators located at x = −l i (l i > 0). We regularize the state by introducing a UV cut-off as usual, and N ( ; l 1 , l 2 , . . . , l m ) is the normalization constant. We shall further assume the distance between different operators |l i − l j | (i = j). At time t, the state becomes In the following we will first consider m = 2 and O 1 = O 2 = O, it will be straightforward to generalize to arbitrary m. We would like to study these locally excited states by calculating the entanglement entropy or Rényi entropy of the subsystem A := {x > 0}. By using the definition of Rényi entropy and the replica trick, we find the difference between the excited state |ψ(t) 2 and ground state as

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where and (w s,i ,w s,i ), (w s,i ,w s,i ) (i = 1, 2 and s = 1, 2, . . . , n) are the replica of (w i ,w i ) and (w i ,w i ) on the s-th sheet of R n . The denominator is the four point correlation function on complex plane C, which is related to normalization constant N ( ; l 1 , l 2 ). In the limit → 0, we have where ∆ O is the conformal dimension of operator O. Notice we have used the assumption |l 1 − l 2 | . To calculate the correlators on R n we could apply the conformal transformation w = z n , which maps R n to the complex plane C. The correlation function on R n is mapped to where C n is a constant of O(1), and the coordinates (w s,i ,w s,i ), (w s,i ,w s,i ) are mapped to z s,1 = e 2πis/n (−l 1 + t + i ) 1/n ,z s,1 = e −2πis/n (−l 1 − t − i ) 1/n , z s,1 = e 2πis/n (−l 1 + t − i ) 1/n ,z s,1 = e −2πis/n (−l 1 − t + i ) 1/n , z s,2 = e 2πis/n (−l 2 + t + i ) 1/n ,z s,2 = e −2πis/n (−l 2 − t − i ) 1/n , z s,2 = e 2πis/n (−l 2 + t − i ) 1/n ,z s,2 = e −2πis/n (−l 2 − t + i ) 1/n . (2.8) In this paper we are mainly interested in the result in the late-time region t l i . We find As we can see from (2.6), the numerator of (2.4) is divergent of O(1/ 8n∆ O ). Only the most divergent term in the numerator of (2.4) will contribute to the final result. From (2.8) we also find for i = j (i, j = 1, 2; s, t = 1, 2, . . . , n). Therefore, the most divergent term comes from the correlation between O(z s,i ,z s,i ) and O(z s,i ,z s,i ), which means

Linear combination of operators
In this subsection we would like to explore the entanglement properties of a linear combination of different operators. For a series of operators O p , which could be primary or descendant operators, we further assume they are orthogonal to each other in the vacuum in the sense that 0| O p O p |0 = 0 if p = p . The state we would like to explore is then where the state is local at point x. We follow the same regularization methods as before by defining

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where is the cut-off, H is the Hamiltonian of CFT, and N ( ) is the normalization constant. In (1+1)D CFTs, we assume x = −l. The normalization constant N ( ) is , (2.17) where w 1 := −l + i ,w 1 := −l − i , w 2 := −l − i andw 2 := −l + i . One could consider the time evolution of state (2.16), |Ψ(t) = e −iHt |Ψ . We expect the entanglement entropy of state |Ψ(t) has the following form in large t limit: 1 where S p is the entanglement entropy of A for state O p |0 , and λ p is defined as .
This can be understood as the probability of state |p in the superposition state (2.16).
To prove above formula, let's consider a general form like (2.16), where we normalize p λ p = 1 and assume p| p = δ p,p . Generally |p is an entangled state if we divide the Hilbert space into two sub-Hilbert space H p ⊗H p . By Schmidt decomposition we could write where |p ip and |p ip are orthonormal basis of two Hilbert spaces, and α ip are the real coefficients. In this basis EE of |p is One could calculate the reduced density matrix of state |ψ ψ|, With some algebra, this becomes The n-th Rényi entropy is

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which can be expressed as is the Rényi entropy of the state |p . Taking the limit n → 1 of S (n) we will obtain the entanglement entropy (EE), We could write (2.16) as the form (2.20), |Ψ = λ p |ψ p , with λ p defined as (2.19), 28) and

Identity from the constraint
In this section we would like to discuss the constraint S L = S R as we have mentioned in the introduction.

General discussion
Before we go on to the details of calculations, let's explain the idea behind the constraint S L = S R and our motivations. We will study the time evolution of the state |ψ L : L depends on t, we expect it will approach to a constant in the large t limit. Using the sum rule we have derived in section 2.1, we only need to know the results for states O 1 (x 1 )|0 and O 2 (x 2 )|0 .
On the other hand we could rewrite O 1 (x 1 )O 2 (x 2 ) OPE blocks (1.2). Note that (1.2) is an operator equality, so we may define a state |ψ R (1.3) by the OPE blocks. |ψ R and |ψ L can be seen as same states in the Hilbert space but with different basis. This fact immediately leads to the constraint S (n) R as well as S L = S R . In the following we mainly focus on S L = S R . More importantly, |ψ R explicitly depends on the CFT data associated with the coupling constant C 12k for the three point function 15) we discuss in section 2.2, therefore the final expression (2.26) for S R will depend on C 12k . However S L is given by the sum of the REE for O 1 (x 1 )|0 and O 2 (x 2 )|0 , which include different CFT data. The constraint S L = S R actually can be seen as a bridge between different CFT data.
Of course this constraint should be consistent with other constraints imposed by symmetry, such as crossing symmetry, modular invariance on torus, since here we only use the OPE of local operators, which is expected to be true for CFTs.
In this section we will mainly focus on RCFTs. On the one hand, our calculations for S L = S R can be seen as a check on the consistency of the replica method to calculate REE for locally excited states. On the other hand it may give us more insight on the JHEP05(2018)154 physical explanation of local excitation. For RCFTs we know the REE is log d O for the state O|0 [14]. But it is still not clear why the quantum dimension d O appears. It is expected this should be related to the topological entanglement entropy for anyons in (2+1)D [32,33]. Our results give more support on this. We will briefly discuss their relation in section 5.1.

The states
We continue discussing entanglement properties of the state with w 1 =w 1 = −l and w 2 =w 2 = 0. We have shown in section 2.1 that the entanglement entropy for subsystem can be expanded as follows in (1+1)D CFTs, can be fixed with the help of Virasoro algebra. The right hand side of (3.3) seems complicated, but it should exhibit the same conformal properties as the left hand side [35]. Let's denote (3.4) Under conformal transformation w = w(z),w =w(z), the left hand side of (3.3) transforms as O p (w 2 ,w 2 ; w 1 ,w 1 ) should transform by the same law as (3.5). We could define a state |ψ R can be seen as locally excited state created by a linear combination of primary and descendant operators, which are labeled by p. We have discussed the entanglement entropy of this kind state above. This state depends on the details of the fusion rule of O × O and the corresponding structure constants. Although the expression for entanglement entropies of |ψ L and |ψ R look different, they should be equal due to the consistency of OPE. This equality, as we will see later, leads to an algebraic identity.

Normalization
Let's first discuss the normalization of state, which are closely associated with the entanglement entropy. From the definition (3.1) we obtain where For the state |ψ R , we rewrite it in the standard form (2.20).
|ψ R can be rewritten as 12) where N p ( ) is the normalization constant of state |p . We can further simplify λ p as where we take the limit z,z → 1 because we would finally take → 0 which leads to z,z → 1. λ p will become a real number between 0 and 1, which can be interpreted as the probability.

Rényi entropy of the state |p
As we can see from (2.25), (2.27), to calculate the Rényi or entanglement entropy one need to know the S (n) p besides λ p . The state |p can be considered as a locally excited state by the following descendant operators, with

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where α k and α k are dimensional parameters. In paper [16] the authors have calculated the entanglement entropy of locally excited state by descendant operators for rational CFTs. However, they only consider linear combination of descendant operators with fixed conformal dimensions, i.e., where K := i k i andK := i k i are some constant. By definition (3.4), the states 2 considered in this subsection is quite different from that in [16]. But O p (w 2 ,w 2 ; w 1 ,w 1 ) is organized as a special form such that it should satisfy the transformation law (3.5). This allows us to use the replica trick as before to calculate the Rényi entanglement entropy. O p (w 2 ,w 2 ; w 1 ,w 1 ) can be seen as a non-local operator associated with the coordinates (w 1 ,w 1 ), (w 2 ,w 2 ). Consider the state |p(t) = e −itH |p , where The normalization constat N p is given by In the limit → 0, w,w → 1. In this limit we expect the conformal block F p (w) ∼ (1−w) −2h ∼ −4h , where we only keep the most divergent term. 3 Now we could use the replica method to calculate the Rényi entropy for subsystem A, with x > 0. We could express the difference of Rényi entropy between state |p(t) and vacuum state ∆S

21)
2 Here we consider the state is a summation of all possible descendant states. 3 We will take some examples to illustrate this phenomenon in the following subsections. In rational CFTs Fp(w) = q FpqFq(1 − w) , the leading contribution comes from q = 0, thus Fp(w) Fp0(1 − w) −2h .

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where (w s,i ,w s,i ) and (w s,i ,w s,i ) (i = 1, 2 and s = 1, . . . , n) are the replica coordinates on the s-th sheet of R n . We could make a conformal transformation w = z n , so that R n is mapped to the complex plane C. By using the transformation law of O p , which is same as (2.7), we have n s O † p (w s,2 ,w s,2 ; w s,1 ,w s,1 )O p (w s,2 ,w s,2 ; w s,1 ,w s,1 ) where Firstly, let's consider t < l i , as we can see from (2.8), Therefore, the leading contribution is given by The L is a differential operator as a function L(z s,1 − z s,2 , z s,1 − z s,2 ) because of the form (3.4). The action of anti-holomorphic operatorL on the anti-holomorphic partial wave is the same as that of L.

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To simplify the notation let's consider n = 2,and the generalization to arbitrary n is straightforward. For n = 2, we have O † p (z 1,2 ,z 1,2 ; z 1,1 ,z 1,1 )O p (z 1,2 ,z 1,2 ; z 1,1 ,z 1,1 )O † p (z 2,2 ,z 2,2 ; z 2,1 ,z 2,1 )O p (z 2,2 ,z 2,2 ; z 2,1 ,z 2,1 ) We will explain the above statement more clearly. In the first equality, we write the correlation function of O p as correlation function on primary operators O p with some differential operator. In the second equality, we write the correlation function of O p as conformal blocks, |m denote the m-th Virasoro module. In the third equality, we transfer the expansion into t-channel. Here we assume the theory is a rational CFT, so that different expansion is related to each other by the fusion matrix F p mn . In the fourth equality, we act the differential operators on the correlator again. The operators appeared in the correlator are the corresponding descendant operators O p . Note that since we have changed the position of coordinates in the third equality, the descendant operators O p will also change according the right order of coordinates. Finally in the fifth equality, we keep the leading contributions. Since we have the relation (2.9), only the identity channel gives the most dominant contributions. In the last step we rearrange the holomorphic and antiholomorphic part together.
One could calculate the final quantity in (3.27) by (3.10). Taking the result into (3.21) we find ∆S (2) A,p (|p(t) ) = − log F p 00 = log d p . It is straightforward to generalize the statement into arbitrary n.

The induced equality from entanglement entropy
Using the result (2.27), we obtain the entanglement entropy S R for subsystem A (x > 0) in late-time limit, The solution of above equation is λ p = d p /d 2 O . Therefore we obtain the following identity: Conformal blocks have the following transformation rule for rational CFTs, where F qp is the fusion matrix [37,38].
In the limit z,z → 1, we have The leading contribution is q = 0. Thus (3.30) can be further simplified to the relation between fusion matrixes and quantum dimensions of operators (3.33) The Rényi entropy S where we use the equality of quantum dimensions p d p = d 2 O . Therefore, we again obtain a consistent result S (n)

More general cases
We have considered the product state where we still assume |l 1 − l 2 | . The sum rule will be still right, for t > l i , we have

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On the other hand we have OPE Define the state The normalization constant N ( ; |φ R ) is same as N ( ; |φ L ), which is given by For the OPE block we have the normalization We could rewrite |φ(t) R as the standard form (2.16), It is also straightforward to generalize to the general product state (2.3).

Some examples
In this subsection we will show some examples to check the relation (3.30), (3.33) and (3.47).

Free massless scalar field
Consider the vertex operator V α = e iαφ , which has the fusion rule V α ×V β = V α+β . So there is only one fusion channel, the result is consistent with the fact the quantum dimension of V α is one. Ising model [39] at critical point has three primary operator I, and σ, which satisfy the fusion rule, × = I, σ × σ = I + . (3.51) The quantum dimension of is 1, × has only one fusion channel, which is trivially consistent with the result (3.30). The four point correlation function [40] σ(z 1 ,z 1 )σ(z 2 ,z 2 )σ(z 3 ,z 3 )σ(z 4 ,z 4 ) (3.52) One could check

Proof of identity (3.47)
In this section, we prove the identity shown in eqs. (3.31), (3.47) using the language of modular tensor category. We will see that only the "tensor" part of the category is involved. We start with reviewing some relevant concepts: a tensor category C is a set of data {Obj(C), d, N, F } that satisfy some consistency conditions. The set Obj(C) consists of superselection sectors a, b, c · · · . Quantum dimension d a assigns a real number to each sector a ∈Obj(C), and the rank-three tensor N c ab describes fusion rules between the sectors: Each entry N c ab is a non-negative integer counting the number of different channels that a and b can be combined to produce the c. In rational CFTs, the fusion is finite which means c N c ab is a finite integer. The quantum dimensions are consistent with the fusion rules, Each fusion product a×b → c has an associated vector space V c ab and its dual splitting space V ab c . The dimension of this vector space is dimV c ab = N c ab . There are two different ways to fuse a, b and c into d, related by associativity in the form of the following isomorphism: Finally, we introduce the F tensor. We will use the following graphical representation in figure  . (4.5) Additionally, we have the useful resolution of identity as shown in figure 3. For a tensor category, we should further require the F -moves to satisfy the Pentagon equation corresponding to the associativity conditions involving five external legs in total. For a modular tensor category, a consistent braiding structure, the Hexagon identity and modularity of the S-matrix are required. We will omit the further details since they are not necessary for the proof. Now we give the proof of the desired identity. For simplicity of narration, we assume the fusion rules are multiplicity-free, i.e. N c ab ∈ {0, 1}, so that the indices α, β, · · · on the vertices can be omitted. The most general case can be recovered straightforwardly by adding them back and perform summations over these indices when appropriate.
In figure 2 The identity to be proved can then be rewritten as We notice that in the graphical representations, one has freedom to add trivial lines 0 anywhere in any graph, as it has no physical consequence. Upon adding a trivial line 0 on the left hand side in the resolution of identity to connect the a and b lines, identifying a = b = O, c = p and comparing with the definition of F symbols, one observes that the coefficients on the right hand side of the resolution of identity in figure 3 From the unitarity of the F symbols in eq. (4.5), we have F −1 0p = F † 0p . Since the labels O are self-dual, one can rotate the external legs as in figure 4, leading to Plugging in the above value for F p0 to both sides of the target identity (4.6), we obtain Using (4.2) by identifying a = b = O and c = q, one immediately observes that l.h.s.=r.h.s. in (4.9). A parallel proof will follow if one consider a slightly more general case where the four external legs are not all the same. The identity would have the form identity to prove (4.6) is derived under the physical constraint S L = S R , namely the two procedures, doing OPE and calculating entanglement entropy, are interchangeable. In other words, the entanglement should be consistent with OPE. From the categorical point of view taken in this section, the entanglement stems from quantum dimensions, while the OPEs are fusion rules. Since the same algebraic structure is shared by anyons and quasiparticles (local operators) in RCFTs, we can use the language of anyon to prove (4.6). In this sense, we do show the quasi-particles of locally excited state in rational CFT follows the same rule as anyons. This can be seen as an example to realize anyons in RCFTs.

Conclusion and discussions
In this paper, we begin with same 1+1 dimensional setup with [14] and study the late time behavior t → ∞ of Rényi entropy of the two equivalent locally excited states defined by l.h.s. and r.h.s. of eq. (1.2) and obtain the Rényi entropy of a subsystem x > 0 in (1+1)D CFTs. In the limit t → ∞, we prove that S L satisfies with a sum rule (1.1) by replica method and showed that S L depends on the information of individual operator O i in l.h.s. of eq. (1.2). In general, S R is hard to obtain by replica method. In the late time limit, we derive S R of the excited states involving in r.h.s. of eq. (1.2) by making use of Schmidt decomposition. It is associated with the fusion channels and conformal block presented in r.h.s. of eq. (1.2). The constraint S L = S R leads to an identity in (1+1)D CFTs. We studied the S L , S R in rational CFTs as examples and proved the relation (3.30), (3.47).
From S L = S R with late time limit in our setup, we indeed used crossing symmetry to obtain the entanglement entropy. Namely, we have made use of bootstrap equation from s channel conformal block to t channel conformal block.

Bulk-edge correspondence
We have seen that the modular tensor category language was used in section 4 to prove the identity 3.47. On the other hand, anyons in topological orders share the same algebraic structure of modular tensor categories, see for example [43,44]. As noted in [47], the non-chiral rational CFT can be viewed as the edge theory of (2+1)-D chiral topological order in a strip [45,46]. Insertion of operators in the rational CFT can be explained in the bulk theory. Roughly, inserting of a primary operators O a at spacetime (x, t) in (1+1)D rational CFT corresponds to creating a pair of anyons labeled as (a,ā) at earlier time. The state |ψ 2 = O a (−l, 0)O a (0, 0) |0 can be viewed as creating two pairs of anyons in the bulk at some time t < 0, and they pass the boundary at spacetime (−l, 0) and (0, 0). We can specify the possible values of the total charge of the two anyons by the fusion rule a × a = N p aa p. In the CFT side this is just the OPE of two operators O a . Calculations of entanglement entropies with two pairs of anyons has been carried out in (2+1)D [48], where the result shares similar structure as above. It would be interesting to look at the general correspondence between the entanglement properties in the bulk and on the boundary.

Large-c CFTs
We mainly focus on the (1+1)D rational CFTs in previous sections. In rational CFTs, the spectrum and fusion rules are relatively simpler than the irrational ones, such as CFT with a gravity dual or Liouville CFT. In rational CFTs, we can analytically calculate the Rényi entropy of locally excited states and explain the evolution behavior by quasi-particles picture.
In this section, we would like to briefly discuss the constraint S L = S R in the CFTs with a gravity dual, or large-c CFTs. Generally the time evolution of Rényi entropy can be very different from the rational ones [27,28], see also the case for Liouville CFT [30]. The feature of such theory is a logarithmic growth in the intermediate time [26][27][28]. But we expect in the limit t → ∞ the Rényi entropy or entanglement entropy to approach a constant [27]. In rational CFTs this constant is related to the quantum dimension of the inserted operator. However, for large c CFT the quantum dimension is not so well defined as rational CFT. As far as we know, this is still an unsolved problem at the moment.
In any CFT, the sum rule is still true for S L , so one can obtain S L as long as the result of locally excited states created by one primary operator is known. Two local operators can still be expanded as OPE blocks as in (3.2), consequently S R (3.28) can similarly be calculated in large c CFTs, except that the sum over p may be replaced by an integration if the spectrum of the theory is continuous. By the definition of λ p we know it is only associated with conformal blocks. In large c CFT, the details of the conformal blocks are known for few cases [49,50]. One of them is the correlator for the primary operator O L with conformal dimension h L to be fixed in the limit c → ∞.
In this case, the Virasoro blocks reduce to representations of the global conformal group.

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The holomorphic Virasoro blocks [50] are F p (z) = z hp 2 F 1 (h p , h p , 2h p ; z), (5.2) where h p is the conformal dimension of the intermediate operator, which is also assumed fixed in the limit c → ∞. If we consider the locally excited state by two light operators O L (0, 0)O L (−l, 0) |0 , the "probability " λ p , as shown in eq. (2.20) is well defined in this case, since lim z→1 F 1 (h p , h p , 2h p ; z) F 1 (h p , h p , 2h p ; z) = C(h p )/C(h p ), (5.3) where C(h p ), C(h p ) is only a constant depending on the conformal dimension h p and h p [29]. In rational CFTs, we know that the ratio λ p /λ p is associated with the the quantum dimension d p /d p . In the present case the constant C(h p ), C(h p ) may be an alternative of quantum dimension in large c CFT.
To check this claim, we will need to know the result of Rényi entropy for state O L |0 in the limit t → ∞. One more subtle problem is the entanglement entropy S p of the state |p . For rational CFTs we show in section 3.4 that S p is equal to the entanglement entropy of the state O p |0 . It is not straightforward to generalize the result to large-c CFTs, due to the lack of the simple fusion transformation.
Our setup depends on the leading behavior of OPE and success of the replica trick. The identity might break down due to the two facts. Firstly, the constraint should be modified for irrational CFTs, e.g. Liouville field theory. The spectrum of Liouville field theory is continuous and no vacuum exists in the Hilbert space. The OPE involves integration over continuous spectrum instead of discrete summation. Secondly, in the (z,z) → (1, 1) limit, the dominant conformal block to the REE is no longer identity block in this limit. The author [30] have carefully studied the variation of REE of local excited states in late time by same bootstrap equation, showing that the late time of REE is associated with fusion matrix element instead of quantum dimensions.
It is also interesting to study the gravity dual of multiple local excitations, e.g., the bilocal quench can be associated with black hole creation in AdS 3 [51,52].