Direct calculation of mutual information of distant regions

We consider the (Rényi) mutual information, InAB=SAn+SBn−SA∪Bn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {I}^{(n)}\left(A,B\right)={S}_A^{(n)}+{S}_B^{(n)}-{S}_{A\cup B}^{(n)} $$\end{document}, of distant compact spatial regions A and B in the vacuum state of a free scalar field. The distance r between A and B is much greater than their sizes RA,B. It is known that InAB∼CABn0ϕrϕ002\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {I}^{(n)}\left(A,B\right)\sim {C}_{AB}^{(n)}{\left\langle 0\left|\phi (r)\phi (0)0\right|\right\rangle}^2 $$\end{document}. We obtain the direct expression of CABn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {C}_{AB}^{(n)} $$\end{document} for arbitrary regions A and B. We perform the analytical continuation of n and obtain the mutual information. The direct expression is useful for the numerical computation. By using the direct expression, we can compute directly I(A, B) without computing SA, SB and SA∪B respectively, so it reduces significantly the amount of computation.

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 Ω is defined as (1. 2) The limit n → 1 coincides with the entanglement entropy lim n=1 S (n) Ω = S Ω .In this paper, we consider the (Rényi) mutual information, A∪B , of distant compact spatial regions A and B in the vacuum state of a free scalar field.The distance r between A and B is much greater than their sizes R A,B .It is known that [26], when r R A,B , the (Rényi) mutual information behaves as where C (n) AB depends on the shapes of the regions A and B. When both A and B are the spheres and the scalar field is massless, the coefficient C (n) AB was calculated analytically by Cardy [26].However, it is difficult to calculate C (n) AB analytically when both A and B are not the spheres or the scalar field is not massless.In this paper, we obtain the direct expression of C (n) AB for arbitrary regions A and B in the vacuum state of a scalar field which has a general dispersion relation.We perform the analytical continuation of n and obtain the mutual information I(A, B) = lim n→1 I (n) (A, B).The direct expression is useful for the numerical computation.By using the direct expression, we can compute directly I(A, B) without computing S A , S B and S A∪B respectively, so it reduces significantly the amount of computation.
We comment on the advantages of this direct expression over the conventional numerical computation by the real time formalism.Entanglement entropy in free scalar fields can be calculated numerically by the real time formalism [20,21].In order to calculate the coefficient C (n=1) AB by the real time formalism, we have to plot the mutual information I(A, B) as a function of r and extract the coefficient [25].So we have to calculate numerically S A∪B many times to plot I(A, B) as a function of r.On the other hand, in our method, we separate the r dependence of I(A, B) analytically and obtain the direct expression of C (n=1) AB .So, it reduces significantly the amount of computation.
To obtain the direct expression of C (n) AB , we use the operator method to compute the Rényi entropy developed in [27].This operator 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 [26], Calabrese et al. [28] and Headrick [29].This idea was generalized to an arbitrary density matrix ρ and the local operator was explicitly constructed in [27].
The present paper is organized as follows.In section 2, we review the operator method to compute the Rényi entropy developed in [27].In section 3, we expand the glueing operator which plays the important role in the operator method to compute the (Rényi) mutual information.In section 4, we compute the (Rényi) mutual information and obtain the direct expression of 2 The review of the operator method to compute the Rényi entropy We review the operator method to compute the Rényi entropy developed in [27].We consider n copies of the scalar fields in (d+1) dimensional spacetime 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, where H is the Hilbert space of one scalar field.We define the density matrix ρ where ρ is an arbitrary density matrix in H.We can express Trρ n Ω as where 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 where The useful property of the glueing operator for calculating the mutual information is the locality.When Ω = A ∪ B and A ∩ B = ∅, From the locality (2.6), the mutual Rényi information in the vacuum state can be expressed as the correlation function of the glueing operators, (2.7) We consider (d + 1) dimensional free scalar field theory.For free scalar fields, it is useful to represent the glueing operator E Ω in (2.3) as the normal ordered operator.We decompose φ and π into the creation and annihilation parts, where here E p is the energy and The commutators of these operators are where we have defined the matrices W and W −1 which has continuous indices x, y in (2.10) and W −1 is the inverse of W . W and W −1 are positive definite symmetric matrices.By using (2.10) and the Baker- where : O : means the normal ordered operator of O. From (2.11) we can rewrite E Ω in (2.3) as the normal ordered operator, where J (n+1) = J (1) and (2.13)

The expansion of the glueing operator
We consider a complex scalar field φ because it is useful for later calculation.The mutual information of a real free scalar field can be obtained by dividing the mutual information of the complex free scalar field by 2.Then, the glueing operator becomes where For the free scalar field, it is useful to use the following Fourier transformation, where f (l) is an arbitrary n dimensional vector and f (k) is its Fourier transformation, i.e. (3.3) is the definition of the Fourier transformation.The Fourier transformation diagonalizes the glueing operator, where In order to expand : exp[iQ (k) ] : in E Ω , we define . . .as where . . . is an arbitrary function of J(k) and K(k) .When Ω is a compact spatial region, we express Ω as a sum of the local operators at a conventionally chosen point x 0 inside Ω.Thus, we expand In order to represent the Gauss integrals of K(k) and J(k) , we will use the following matrix notation, where x Ω(Ω c ) and y Ω(Ω c ) are the coordinates in Ω(Ω c ), where Ω c is the complement of Ω.
In order to calculate J(k) (x) J(k) * (y) , we perform the K(k) integral first, (3.12) From (3.12), we obtain In order to separate the n dependence of J(k) (x) J(k) * (y) , we rewrite it as where Thus we obtain where Z = OX and we discretized the space coordinates in order to regularize the scalar field.In the appendix A, we show that the range of the eigenvalues λ i is Finally, when Ω is a compact spatial region, we obtain the expansion of E Ω as where

The (Rényi) mutual information of distant regions
We apply above results to the mutual Rényi information I (n) (A, B) of disjoint compact spatial regions A and B in the vacuum states of the free scalar field.From (2.7), (3.4) and (3.19), we obtain where x A and x B are some conventionally chosen points inside A and B, r = |x A − x B |, and From (4.1), we obtain the mutual Rényi information as Ω in (3.20) into (4.3) and obtain where We can perform explicitly the summation in (4.6) and obtain (see Appendix B) From (4.7), for n = 1, 2, 3 and 4, we obtain (1 + q 2 ) (1 + q) 2 (4.10) (4.12) When n = 2, 3, F (n, a, b) is a product of the function of a and b and C (n) AB becomes, where

C(n)
A(B) is a function which is determined by the shape of A(B).So, when n = 2, 3, C (n) AB is not entangled, i.e. it is a simple product of functions each of which is determined by the shape of A(B).In general, F (n, a, b) is not a product of the function of a and b and C ) is an elementary function of n, its analytical continuation is trivial.So we can take n → 1 limit in AB in (4.5).From (4.7) and (4.9), we obtain where

Conclusion and discussions
In this paper, we considered the (Rényi) mutual information, A∪B , of distant compact spatial regions A and B in the vacuum state of a free scalar field.The distance r between A and B is much greater than their sizes R A,B and the (Rényi) mutual information behaves as AB for arbitrary regions A and B. We performed the analytical continuation of n and obtain the mutual information is not entangled, i.e. it is a simple product of functions each of which is determined by the shape of A(B).For general n, C (n) AB is not a simple product of functions each of which is determined by the shape of A(B) and The direct expression is useful for the numerical computation.By using the direct expression, we can compute directly I(A, B) without computing S A , S B and S A∪B respectively, so it reduces significantly the amount of computation.
It is an interesting future problem to apply our direct expression to study the shape dependence of C (n) AB .For example, the corner contribution to mutual information in (2+1) dimension is an interesting problem.The corner contributions to entanglement entropy in (2+1) dimension are universal and have important information of the QFT [30][31][32][33], however, the corner contribution to mutual information has not been studied well.Our method is useful for studying the corner contributions of mutual information.It is also an interesting future problem to generalize our method to the entanglement negativity [34,35].
Because A −1 is a positive definite matrix and (A.3), 1 − Y is a positive definite matrix and we obtain λ i < 1.Therefore, we have shown 0 ≤ λ i < 1.
B The calculation of F (n, a, b) in (4.6) We calculate the summation F (n, a, b) in (4.6) for 0 ≤ a < 1, 0 ≤ b < 1.We expand 1 − cos 2πk n 2 in (4.6) and rewrite F (n, a, b) as In order to calculate the summations in (B.1), we use the following expansion, where we have used We substitute (B.7) into (B.5) and obtain In the same way as above, by using the expansion in (B.2), we obtain ρ(b) q+q e i(q−q ) 2πk n . (B.10) From (B.6), we can rewrite the summations in (B.10) as We substitute (B.11) into (B.10) and obtain The last term in (B.12) can be evaluated as The third term in (B.12) can be evaluated as The fourth term in (B.12) is obtained by interchanging a and b in the third term in (B.12).We perform the p and q summations in the second term in (B.12) and obtain Thus, we substitute (B.16) into (B.15) and obtain the second term in (B.12) (B.17) We perform the p and q summations in the first term in (B.12) and obtain

2 2 3 4 4 7 5and discussions 8 A 9 B
The review of the operator method to compute the Rényi entropy The expansion of the glueing operator The (Rényi) mutual information of distant regions Conclusion Derivation of 0 ≤ λ i < 1 The calculation of F (n, a, b) in (4.6) 10 B.1 The calculation of n− is entangled.The calculation of the matrix Z and the eigenvalues λ i is simple matrix computation.So, we can compute C (n=1) AB numerically.