Entanglement Entropy of Disjoint Regions in Excited States : An Operator Method

We develop the computational method of entanglement entropy based on the idea that $Tr\rho_{\Omega}^n$ is written as the expectation value of the local operator, where $\rho_{\Omega}$ is a density matrix of the subsystem $\Omega$. We apply it to consider the mutual Renyi information $I^{(n)}(A,B)=S^{(n)}_A+S^{(n)}_B-S^{(n)}_{A\cup B}$ of disjoint compact spatial regions $A$ and $B$ in the locally excited states defined by acting the local operators at $A$ and $B$ on the vacuum of a $(d+1)$-dimensional field theory, in the limit when the separation $r$ between $A$ and $B$ is much greater than their sizes $R_{A,B}$. For the general QFT which has a mass gap, we compute $I^{(n)}(A,B)$ explicitly and find that this result is interpreted in terms of an entangled state in quantum mechanics. For a free massless scalar field, we show that for some classes of excited states, $I^{(n)}(A,B)-I^{(n)}(A,B)|_{r \rightarrow \infty} =C^{(n)}_{AB}/r^{\alpha (d-1)}$ where $\alpha=1$ or 2 which is determined by the property of the local operators under the transformation $\phi \rightarrow -\phi$ and $\alpha=2$ for the vacuum state. We give a method to compute $C^{(2)}_{AB}$ systematically.


Introduction
The entanglement entropy in the quantum field theory (QFT) plays important roles in many fields of physics such as the string theory, condensed matter physics, and the physics of the black hole. The entanglement entropy is a useful quantity which characterize quantum properties of given states. For example, the entanglement entropy of ground states follows the area law [1][2][3][4] if we consider a local quantum field theory with a UV fixed point, while non-local field theories [5,6] or QFTs with fermi surfaces [7] at UV cut off scale can violate the area law.
For a given density matrix ρ of the total system, the entanglement entropy of the subsystem Ω is defined as where ρ Ω = Tr Ω c ρ is the reduced density matrix of the subsystem Ω and Ω c is the complement of Ω. The Rényi entropy S (1. 2) The limit n → 1 coincides with the entanglement entropy lim n=1 S (n) Ω = S Ω .
In this paper we develop the computational method of Rényi entanglement entropy based on the idea that Trρ n Ω is written as the expectation value of the local operator at Ω. We apply this method to consider the mutual Rényi information I (n) (A, B) = S (n) of disjoint compact spatial regions A and B in the locally excited states defined by acting the local operators at A and B on the vacuum of a (d + 1)-dimensional field theory, in the limit when the separation r between A and B is much greater than their sizes R A,B .
Our method is based on the idea that Trρ n Ω is written as the expectation value of the local operator at Ω. This idea was originally used to compute I (n) (A, B) in the vacuum state by Cardy [8], Calabrese et al. [9] and Headrick [10]. We generalize this idea to an arbitrary density matrix ρ and construct explicitly the local operator. The density matrix of total system ρ can be a mixed state and an excited state. We consider the general scalar field and do not specify its Hamiltonian. (Our method is applicable to QFT with interaction. ) We summarize our method. We consider n copies of the scalar fields and the j-th copy of the scalar field is denoted by {φ (j) }. Thus the total Hilbert space, H (n) , is the tensor product of the n copies of the Hilbert space, H (n) = H ⊗ H · · · ⊗ H where H is the Hilbert space of one scalar field. We define the density matrix ρ (n) in H (n) as where ρ is an arbitrary density matrix in H. We can express Trρ n Ω as (1. 5) where π(x) is a conjugate momenta of φ(x), [φ(x), π(y)] = iδ d (x − y), and J (j) (x) and K (j) (x) exist only in Ω and J (n+1) = J (1) and we normalize the measure of the functional integral as DJ (j) exp[i d d xJ (j) (x)f (x)] = x∈Ω δ(f (x)) where f (x) is an arbitrary function. Notice that φ and π in (1.5) are operators and the ordering is important. We call this operator E Ω as the glueing operator. When ρ is a pure state, ρ = |Ψ Ψ|, the equation (1.4) becomes Trρ n Ω = Ψ (n) | E Ω |Ψ (n) (1.6) where |Ψ (n) = |Ψ |Ψ . . . |Ψ . (1.7) For a free scalar field, we can rewrite E Ω in (1.5) using the normal ordering. In the case n = 2, we obtain a simple expression of E Ω and reproduce the result that I (2) (A, B) in the vacuum state is proportional to the product of the electrostatic capacitance of each regions obtained by Cardy [8]. Furthermore the simple expression of E Ω is useful for numerical calculation.
The advantages of this method are that we can use ordinary technique in QFT such as OPE and the cluster decomposition property and that we can use the general properties and the explicit expression of the glueing operator to compute systematically the Rényi entropy for an arbitrary state.
We apply this method to the mutual Rényi information I (n) (A, B) in the locally excited states. We consider the following locally excited state, 8) where N is a real normalization constant and O iA and O i A (O jB and O j B ) are operators on A (B) and i and i (j and j ) label a kind of operators. For the general QFT which has a mass gap, we compute I (n) (A, B) explicitly and find that this result is interpreted in terms of an entangled state in a quantum mechanical system which has finite degrees of freedom. For a free massless sacalar field, we show that for some classes of excited states, AB /r α(d−1) where α = 1 or 2 which is determined by the property of the local operators under the transformation φ → −φ and α = 2 for the vacuum state.
The mutual information for a free scalar field in higher dimensions has been studied in only a few papers [8,[11][12][13][14][15]24]. In these papers, the authors considered only the mutual information for the vacuum state. The relation between the mutual information and the physics of the black hole was considered in [13]. Recently, it is proposed [16] that the mutual information is obtained by the quantum correction to the holographic entanglement entropy formula [17]. It would be interesting to use our results to check this proposition.
The entanglement entropy for an excited state defined by acting the local operator on the vacuum was considered in [18][19][20][21]. In [18], the subsystem is a half of the total space and the local operator exists at the complement of the subsystem and the time evolution of the entanglement entropy was considered. It was found that the entanglement entropy at late time is interpreted in terms of an entangled state in quantum mechanics [18]. This result is analogous to our result in the general QFT which has a mass gap.
The present paper is organized as follows. In section 2.1 we develop the computational method of the entanglement entropy. We derive the basic formula (1.4) and construct explicitly the glueing operator E Ω . In section 2.2 we investigate the general properties of E Ω . In section 3 we consider the mutual Rényi information I (n) (A, B) in the locally excited states in the general QFT which has a mass gap. We compute I (n) (A, B) explicitly and find that this result is interpreted in terms of an entangled state in quantum mechanics. In section 4 we consider free scalar fields. In section 4.1 we rewrite E Ω in (1.5) using the normal ordering. In the case n = 2, we obtain a simple expression of E Ω and reproduce the result that I (2) (A, B) in the vacuum state is proportional to the product of the electrostatic capacitance of each regions obtained by Cardy. In section 4.2 we consider a free massless sacalar field. We show that for some classes of excited states where α = 1 or 2 which is determined by the property of the local operators under the transformation φ → −φ and α = 2 for the vacuum state. In section 5 we summarize our conclusion.
2 Operator formalism 2.1 Operator representation of Trρ n Ω We represent the trace of the nth power of the reduced density matrix as the expectation value of the local operator. As a model amenable to unambiguous calculation we deal with the scalar field as a collection of coupled oscillators on a lattice of space points, labeled by capital Latin indices, the displacement at each point giving the value of the scalar field there. The local Hermitian variablesq A andp B (coordinates and the conjugate momentum) obey the canonical commutation relations The density matrix ρ of the total system in coordinate representation is where {q A } denotes the collection of all q A 's. We consider the arbitrary density matrix ρ.
Now consider a subsystem (or subregion) Ω in the space. The oscillators in this region will be specified by lowercase Latin letters, and those in its complement Ω c will be specified by Greek letters. We can obtain a reduced density matrix ρ Ω for Ω by integrating out over q α ∈ R for each of the oscillators in Ω c , and then we have We obtain the trace of the nth power of the reduced density matrix ρ Ω as We consider n copies of the oscillators and the j-th copy of the oscillators is denoted by {q A }. Thus the total Hilbert space, H (n) , is the tensor product of the n copies of the Hilbert space, H (n) = H ⊗ H · · · ⊗ H. We define the following density matrix, i.e.
Then we can rewrite Trρ n Ω as (2.8) We call E Ω as the glueing operator. When ρ is a pure state, ρ = |Ψ Ψ|, the equation (2.7) becomes We represent E Ω as a function ofq andp. We can rewrite E Ω as where we have written the delta functions as the Fourier integrals. Note thatq a is a operator and not a integral variable. The middle term in (2.11) is the tensor product of the following operator, (2.13) We substitute (2.13) into (2.11) and obtain (2.14) From (2.14), for the (d + 1) dimensional scalar field theory, we obtain , and J (j) (x) and K (j) (x) exist only in Ω and J (n+1) = J (1) and we normalize the measure of the functional integral as is an arbitrary function.

General properties of the glueing operator E Ω
We investigate some general properties of E Ω .

Locally excited states in the general QFT which has a mass gap
We consider the mutual Rényi information A∪B of disjoint compact spatial regions A and B in the locally excited states of the general QFT which has the mass gap m in the limit when the separation r between A and B is much greater than their sizes R A,B and 1/m (r R A,B , 1/m). We consider the following locally excited state, (we have omitted the labels is for simplicity.) We impose following orthogonal conditions for simplicity, This state is similar to the EPR state. We compute the mutual Rényi information of this state. In this calculation, the general properties (2), (5) and the cluster decomposition property in the QFT play important roles. From the normalization condition Ψ|Ψ = 1 we obtain where we have used the cluster decomposition property |0 and the orthogonal conditions (3.2). By using (2.9), we obtain Trρ n Ω for ρ = |Ψ Ψ| as where O (l) are operators in the Hilbert space of the l-th copy and the Ω is A, B or A ∪ B.
First, we consider Trρ n A∪B . From (2.20) and (3.4) we obtain where we have used the cluster decomposition property and the conditions r, R A , R B 1/m and ρ 0A(B) is the reduced density matrix of the vacuum state.
Next we consider Trρ n A . We expand the product in (3.4). The terms in the expansion in (3.4) have the following form, The number of the terms which are proportional to .7) is n C l . By using (2.20), we obtain By the same way, we obtain Trρ n B as where We can reproduce these results from the quantum mechanics. Let us consider the following state, 14) where N is the real normalization constant, and |i(i ) A and |j(j ) B are the pure state of the subsystem A and B and From (3.14), (3.15) and the normalization condition Ψ|Ψ = 1, we obtain Because the total system is a pure state, Trρ n qmA∪B = 1. From (3.16) and (3.17) we obtain the mutual Rényi information as (3.11) is the same as (3.18) when we replace the states as follows: Interestingly, the mutual information in the QFT measures only the quantum entanglement in the limit r → ∞ although the mutual information measures generally the total of the quantum entanglement and the classical one [22]. So, in this limit, the mutual information is a good measure of quantum entanglement in this sense.

Explicit calculation of the glueing operator E Ω
We consider (d + 1) dimensional free scalar field theory. For free scalar fields, it is useful to represent the glueing operator E Ω in (2.15) as the normal ordered operator. We decompose φ and π into the creation and annihilation parts, here E p is the energy and [a p , a † p ] = (2π) d δ d (p − p ). The commutators of these operators are where we have defined the matrix W which has continuous indices x, y in (4.3) and W −1 is the inverse of W . By using (4.3) and the Baker-Campbell-Hausdorff (BCH) formula where J (n+1) = J (1) and For the vacuum state ρ 0 = |0 0|, from (4.5) we obtain We can show that (4.7) reproduces the same result as that of the real time approach [1,2,23] which is based on the wave functional calculation (see Appendix A). By expanding the exponential in the normal ordered product in (4.5) and performing the Gauss integral of J and K, we can rewrite the E Ω as a series of operators. Note that the odd powers of the expansion of (4.5) vanish from the symmetric property (2.16).

The case n = 2
In the case n = 2, it is useful to define the following linear combinations, (2) ).
(4.8) From (4.8) we rewrite (4.5) as where we have performed J + and K + integrals. In order to represent the Gauss integrals of K − and J − , we will use the following matrix notation, where x Ω(Ω c ) and y Ω(Ω c ) are the coordinates in Ω(Ω c ). Thus the propagators of J − and K − are and (4.13) Then we can expand the E Ω for n = 2 as (4.14) Next let us apply above results to the mutual Rényi information I (2) (A, B) of disjoint compact spatial regions A and B in the vacuum states of the massless free scalar field. We express E Ω as a sum of the local operators at some conventionally chosen points (r A , r B ) inside A and B. From (4.14), we have where From the local property (2.17), E A∪B = E A E B , and (4.15) we obtain We use the notation E p = |p| and where B d is a numerical constant. Thus the the mutual Rényi information I (2) (A, B) is We can compute C A(B) numerically at least. The expression (4.16) of C A(B) is useful for numerical computation. Furthermore, we can obtain the alternative expression of C A(B) as follows. Let us consider the following generating function, where σ is a c number. We can obtain the coefficients of : φ 2m − (r A ) : in the expansion of E A from G(σ). For example, we have We can rewrite G(σ) by using the (d + 1) dim Euclidean path integral as where where τ is the Euclidean time coordinate and ϕ(τ, x) is a scalar field in (d + 1) dimensional Euclidean space. First, we perform the J − integral and obtain We decompose ϕ into the quantum part ϕ q and the classical part ϕ cl . The ϕ cl is the solution of ∂ 2 ϕ = 0 and satisfy the boundary condition, Thus the region (τ = 0, x ∈ A) acts like a conductor where electrostatic potential is σ/2 and ϕ cl (τ, x) is electrostatic potential at (τ, x). We perform the ϕ q integral and obtain is the electrostatic energy and we have rewritten it by using C A which is electrostatic capacitance of the conductor (τ = 0, x ∈ A). From (4.21) and (4.26) we have

The mutual information of locally excited states
We consider the mutual Rényi information I (n) (A, B) = S (n) A∪B of disjoint compact spatial regions A and B in the locally excited states of the (d+1) dimensional free massless scalar field theory in the limit when the separation r between A and B is much greater than their sizes R A,B .
We consider the following locally excited state, where N is a real normalization constant and O iA and O i A (O jB and O j B ) are operators on A (B) and i and i (j and j ) label a kind of operators. We impose following orthogonal conditions, This state is similar to the EPR state. We compute the mutual Rényi information of this state. When r → ∞, the mutual information I (n) (A, B) is the same as that of the general QFT which have a mass gap in (3.11). In the mass gap case there is a correction which is O(e −mR A(B) ) for (3.11). In the free massless case there is a correction which is O(1/R A(B) ).
(ii)The case |O iA | = |O i A | and |O jB | = |O j B | There are some terms which are odd under the sign changing transformation (4.38). As an example, in the expansion of Trρ n A∪B in (4.32), we consider the following term, In (4.45), the operators at A and at B are both odd under the sign changing transformation (4.38). So the leading r dependent term of it come from the operators φ (l) . Thus the mutual Rényi information is where I (n) (A, B)| r→∞ is the same form as (3.11).

Conclusion and discussions
We developed the computational method of Rényi entanglement entropy based on the idea that Trρ n Ω is written as the expectation value of the local operator at Ω. We expressed Trρ n Ω as the expectation value of the glueing operator E Ω , Trρ n Ω = Tr(ρ (n) E Ω ). We constructed explicitly E Ω and investigated its general properties. For a free scalar field, we rewrote E Ω in (2.15) using the normal ordering. In the case n = 2, we obtained a simple expression of E Ω and reproduced the result that I (2) (A, B) in the vacuum state is proportional to the product of the electrostatic capacitance of each regions obtained by Cardy [8]. The coefficients of the expansion of E Ω is obtained by the propagators of J − and K − in (4.12) and (4.13). We can compute these propagators numerically at least and the expression (4.9) is useful for numerical calculation.
The advantages of this methods are that we can use ordinary technique in QFT such as OPE and the cluster decomposition property and that we can use the general properties and the explicit expression of the glueing operator to compute systematically the Rényi entropy for an arbitrary state.
We applied this method to consider the mutual Rényi information I (n) (A, B) of disjoint compact spatial regions A and B in the locally excited states defined by acting the local operators at A and B on the vacuum of a (d + 1)-dimensional field theory, in the limit when the separation r between A and B is much greater than their sizes R A,B . For the general QFT which has a mass gap, we computed I (n) (A, B) explicitly and find that this result is interpreted in terms of an entangled state in quantum mechanics. Interestingly, the mutual information in the QFT measures only the quantum entanglement in the limit r → ∞ although the mutual information measures generally the total of the quantum entanglement and the classical one [22]. So, in this limit, the mutual information is a good measure of quantum entanglement in this sense. For a free massless sacalar field, we showed that for some classes of excited states I (n) (A, B) − I (n) (A, B)| r→∞ = C (n) AB /r α(d−1) where α = 1 or 2 which is determined by the property of the local operators under the transformation φ → −φ and α = 2 for the vacuum state.
Finally we discuss the generalization of our method. Although we considered only the locally excited states, we can apply our method to more general excited states, for example, many particle states and thermal states. We might be able to generalize our method to fermionic fields. We could apply our method to perturbative calculation in an interacting field theory.
A Rényi entropy for the vacuum state in free scalar field theory In this appendix we show that (4.7) reproduces the same result as that of the real time approach [1,2,23] which is based on the wave functional calculation.
From (4.6) and (4.7), we perform the K integral in (4.7) and obtain  and we have used the matrix notation in (4.10) and (4.11).
The following calculation is analogous to that in [23].