Quantum entanglement of locally excited states in Maxwell theory

In 4 dimensional Maxwell gauge theory, we study the changes of (Rényi) entanglement entropy which are defined by subtracting the entropy for the ground state from the one for the locally excited states, generated by acting with gauge invariant local operators on the state. The changes for the operators which we consider in this paper reflect the electric-magnetic duality. The late-time value of changes can be interpreted in terms of electromagnetic quasi-particles. When the operator constructed of both electric and magnetic fields acts on the ground state, it shows that the operator acts on the late-time structure of quantum entanglement differently from free scalar fields.

However the definition of (Rényi) entanglement entropy in gauge theories has subtleties [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. In gauge theories, physical states have to be gauge invariant. They obey constraints which guarantee their gauge invariance. They make it difficult to divide the Hilbert space into subsystems A and B because the physical degrees of freedom in A depends on the freedom in B due to the constraints. Their boundary is ∂A. Then the definition of (Rényi) entanglement entropy needs a precise method of dividing Hilbert space JHEP12(2016)069 and defining the reduced density matrix ρ A which is given by tracing out the degrees of freedom in B, ρ A = tr B ρ. (1.1) On the other hand, the entropy in QFTs depends on a UV cutoff (ultraviolet cutoff) δ because by definition it has the UV divergence. It is given by a series expansion in conformal field theories. The physical degrees of freedom around ∂A have the significant effect on the terms which depend on δ. The method of dividing the Hilbert space is expect to affect the degrees of freedom around ∂A in the direct fashion. In the present paper, we study the changes of (Rényi) entanglement entropy ∆S

(n)
A which is defined by subtracting the entropy for the ground state from the one for the locally excited state, which is defined by acting with a local operator on the ground state. Here we assume that the operator is located far from ∂A. We will explain it more in the next section. As in [42][43][44][45][46][47][48], their changes do not possess the UV divergence. More precisely, they measure how the local operator changes the structure of quantum entanglement. Therefore, they are expected to avoid the subtleties which (Rényi) entanglement entropy has.
In this paper we study ∆S

(n)
A in 4d Maxwell gauge theory, which is a free CFT [49]. The previous works [44][45][46] show the time evolution of ∆S (n) A can be interpreted in terms of relativistic propagation of entangled quasi-particles which are created by local operators. In the free theories, the late-time value of ∆S (n) A is given by a constant, which depends on the operators. It comes from the quantum entanglement between quasi-particles. As in [46], the late-time entanglement structure depends on the kind of quasi-particles. The authors in [47] show that in the specific 2d CFTs, it is related to the quantum dimension of the operator which acts on the ground state. The authors in [48,50] have shown that in holographic theories the late time value of ∆S (n) A logarithmically increases similarly to the behavior of entanglement entropy for the local quenches [51,52]. ∆S

(n)
A in the finite temperature system was investigated by the authors in [53]. Many works have been done to study the fundamental properties of ∆S A depend on theories and the quasi-particles created by the local operator. Then we study how the structure of quantum entanglement is changed by gauge invariant local operators such as electric and magnetic fields. In particular, we study how the late-time structure of quantum entanglement depends on them. More precisely, we study the effect of quasi-particles on the structure. We also study whether ∆S

(n)
A for a gauge invariant locally excited state reflects electric-magnetic duality.
Summary. Here, we briefly summarize our results in this work. We study how the various local gauge invariant operators change the structure of quantum entanglement by measuring the time evolution of ∆S (n) A for various gauge invariant local operators. We also study whether ∆S Electric-magnetic duality. As it will be explained later, ∆S  A reflects the electric-magnetic duality, the entropy for B i is equal to that for E i , which is the electric field along the same direction as that of magnetic field. ∆S

(n)
A for the electric and magnetic fields along the direction vertical to the entangling surface increases slower than those for fields along the directions parallel to the surface. However there are no differences between the effects of electromagnetic field and that of scalar one on the entanglement structure at the late time. 1 Composite operators. If the composite operator such as B 2 acts on the ground state, they lead to the late-time structure of quantum entanglement in the same manner as a specific scalar operator. Then the late-time value of ∆S (n) A for that can be interpreted in terms of quasi-particles created by the scalar operator. However ∆S (n) A for some specific operators (e.g, E 2 B 3 ) constructed of both electric and magnetic fields can not be interpreted in terms of the scalar quasi-particles but of electromagnetic ones, which is explained in section 4. Here B 3 (E 3 ) and B 2 (E 2 ) are the magnetic (electric) fields along the direction perpendicular to ∂A as we will explain it later.
Late-time algebra. We interpret the late-time values of ∆S (n) A in terms of electromagnetic quasi-particles created by an electromagnetic field, and derive a late-time algebra which they obey. There are commutation relations between the particles of the same kinds of fields . As we will show later, there are also additional relations between E 2 (E 3 ) and B 3 (B 2 ), which are parallel to the entangling surface. They make the effect of electromagnetic fields different from that of scalar fields on the late-time structure of quantum entanglement.
Organization. This paper is organized as follows. In section 2, we will explain locally excited states and how to compute ∆S A in terms of entangled quasi-particles in section 4. We study how they have the effect on the late-time structure of quantum entanglement. We finish with the conclusion, future problems and the detail of propagators is included in appendices.
2 How to compute excesses of (Rényi) entanglement entropy By measuring the excess of (Rényi) entanglement entropies ∆S (n) A , we study how local gauge-invariant operators changes the structure of quantum entanglement in the 4d Maxwell gauge theory: where F µν = ∂ µ A ν − ∂ ν A µ and g µν = diag (−1, 1, 1, 1). In this section, we explain the definition of locally excited state and how to compute ∆S The definition of locally excited states. The locally excited state is defined by acting with a gauge invariant local operator O such as F µν on the ground state:

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where N is a normalization constant and |0 is a gauge invariant state. As in figure 1, O is located at t = −t, x 1 = −l and x = (x 2 , x 3 ).

(n)
A approaches to a constant, which comes from quantum entanglement between entangled quasi-particles. In this paper we would like to study how the constant depends on gauge invariant operators. Therefore the region in figure 1 is chosen as A.
Excesses of (Rényi) entanglement entropy. Here we explain more about the definition of ∆S (n) A . (Rényi) entanglement entropy for the ground state is a static quantity, which does not depend on time. Then we define the excesses of (Rényi) entanglement entropy ∆S , are (Rényi) entanglement entropies for the excited states in (2.2) and the ground state |0 , respectively. In the sense that ∆S

(n)
A does not depend on δ, it is a "renormalized" (Rényi) entanglement entropy.

The replica trick
We would like to study the time evolution of ∆S (n) A in 4d Minkowski spacetime. However in this paper we do not directly study the changes of entanglement structure in the spacetime. Instead, we compute ∆S (n) A in Euclidean space by the replica trick. After that we perform JHEP12(2016)069 the analytic continuation, which we will explain later. Then we compute the real time evolution of ∆S (n) A . As in [44][45][46][47], a reduced density matrix in Euclidean space is given by where τ is Euclidean time. By introducing a polar coordinate, (τ l,e , −l) is mapped to (r 1,2 , θ 1,2 ) as in figure 2.
The reduced density matrix of ρ introduced above, which is our interest now, can be written in the form of path integral where the indices i, µ is i = 1, 2, 3, µ = 0, 1, 2, 3, and x i runs over the region A, which is in this case x 1 > 0 and x 2 , x 3 ∈ R. The Φ 1 = Φ 1 (x i ) and Φ 2 = Φ 2 (x i ) introduced in the path integral inside the δ-function with a positive value ∆ 1 (which is taken to the zero limit in the end), represent the boundary condition at region A. Z EX 1 is defined as (2.6) The n-th power of the reduced density matrix is written as Note that since the path integral is performed over an n-sheeted Riemann surface, the action is now represented as S n [Φ]. x l 1 , x l 2 represent the operator insersion points on the l-th sheet, and the boundary condition Φ 1 , Φ 2 is now at the 1-st and n-th Riemann sheet, respectively.
Thus, in the replica trick (Rényi) entanglement entropies for (2.4) and the ground state are respectively given by taking the trace log of (2.7), 2 where θ k 1,2 = θ 1,2 + 2(k − 1)π. The actions S n and S 1 are defined on the n-sheeted geometry Σ n (see figure 3) and the flat space Σ 1 , respectively. By substituting (Rényi) entanglement entropies in (2.8)  We only need to compute propagators on Σ n in order to compute ∆S . If we choose a specific gauge, 3 their Green's functions obey the same equation of motion as that for 4d free massless scalar field theory, The solution of the equation is given by where t 0 is defined by (2.12) (2.11) has been obtained by the authors in [44-46, 48, 60, 61].
Analytic continuation. After computing Green's functions on Σ n in Euclidean space, we perform the following analytic continuation, where is a smearing parameter which is introduced to keep the norm of the excited state finite. Analytic-continued Green's functions depend on . We are interested in the behavior of ∆S  A is the difference of the Rényi EE of ground state and excited state. More precisely, when we decompose the Hilbert space into two components, as discussed in [26], we need to specify a precise boundary condition. Here, we assume that this boundary condition can be imposed by an insertion of a corresponding local operator at the boundary. On the other hand, as is discussed in [45], the operator at the boundary does not contribute to the result since l 1. Therefore, through out the computation of ∆S

(n)
A , the result does not depend on the boundary condition. Moreover, we are allowed to choose an arbitrary boundary condition. Thus, we can set a gauge invariant boundary condition for the Green's functions of gauge invariant operators, so that the result will be gauge invariant.

Excesses of (Rényi) entanglement entropy
In this section, we study the time evolution and the late time value of ∆S A does not depend on the operator. It can be interpreted in terms of the quasi-particle created by a scalar operator φ.
(ii) Composite operators which act on the ground state are constructed of only electric or magnetic fields such as E 2 and B 2 . ∆S

(n)
A for E 2 is equivalent to the entropy for B 2 . Thus ∆S (iii) Local operators are constructed of both electric and magnetic fields such as E 2 A shows that there is a significant difference between the effect of E 1 (B 1 ) and E 2,3 (B 2,3 ) on the late-time entanglement structure. Here E 1 (B 1 ) is the electric (magnetic) field along the direction vertical to the entangling surface. On the other hand, E 2,3 (B 2,3 ) is the electric (magnetic) field along the direction parallel to the entangling surface. As it will be explained in the next section, the difference comes from the commutation relation between electromagnetic quasi-particles created by E 2 (E 3 ) and the particles created by B 3 (B 2 ).
Here locally excited states are defined by acting with only E i or B i on the ground state. ∆S

(n)
A is given by (2.9) in the replica method with Euclidean signature. After performing the analytic continuation in (2.13) and taking the limit → 0, their time evolution is given as follows. ∆S

(n)
A vanishes before t = l(> 0), but after t = l, it increases. The detail of their time evolution is summarized in table 1. After taking the late time limit (0 < l t), they are given by ∆S Their late time value is the same as that for φ in free massless scalar field theories with any spacetime dimensions. It can be interpreted as (Rényi) entanglement entropy for maximally entangled state in 2-qubit system. Therefore they do not show the difference between the effect of electromagnetic fields and that of free scalar one on the late-time structure of quantum entanglement. However, the time evolution of ∆S where · · · are contributions from the higher order O t−l · · · are contributions from the higher order O t−l l 3 . Their time evolution shows that quasi-particles created by E 2,3 (B 2,3 ) enter the region A faster than those generated by E 1 (B 1 ). These behaviors seem to be natural since particles created by E 1 (B 1 ) do not propagate along the direction parallel to x 1 . ∆S

(n)
A in table 1 shows that they are invariant under the transformation, A for E 1 (B 1 ) and E 2,3 (B 2,3 ). The horizontal and vertical axes correspond to time t/l and ∆S (2) A , respectively. The red and blue lines correspond to ∆S where µνρσ is an anti-symmetric tensor. Under the transformation in (3.4), the local operator E i (B i ) changes to −B i (E i ). Therefore, this duality changes a locally excited state to a different one.

Composite operators constructed of only electric or magnetic fields
The excited states which we consider here are generated by acting with the following operators: (a) E i E j or B i B j , (b) E 2 or B 2 . We study the time evolution and the late-time value of ∆S (n) A for them.

First we consider ∆S (n)
A for the excited states generated by acting with E i E j or B i B j on the ground state. When i = j, the late time-value of ∆S It is the same as that of ∆S

(n)
A for φ 2 in the massless free scalar field theories as in [44,45]. Therefore, the late-time value of ∆S  Table 2. ∆S (n)

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for O in the region 0 < l < t. A for E 1 E 1 and E 2 E 2 . The horizontal and vertical axes correspond to time t/l and the excess of ∆S (2) A , respectively. The blue line represents the evolution for E 1 E 1 , and the red line for E 2 E 2 . In the limit t → ∞, they are all log 8 3 .
When i = j, ∆S  1, 1, 1). It is the same as ∆S

(n)
A for the excited state given by acting with the operator φ a φ b on the ground state. Here a, b denote the kind of scalar fields, and a = b. There are two kinds of massless free scalar fields. The time evolution of ∆S

(n)
A for them is summarized in table 2. Table 2 shows ∆S   A for E 1 E 2 , E 2 E 3 (E 2 B 2 ) and E 1 B 1 . The horizontal and vertical axes correspond to time t/l and the excess of ∆S (2) A , respectively. The red line represents the evolution for E 1 E 2 , the blue line for E 2 E 3 (E 2 B 2 ) and the green line for E 1 B 1 . In the limit t → ∞, they are all 2 log 2.

O = E 2 or B 2
In order to study whether E 1 acts on the late-time structure of quantum entanglement differently from E 2,3 , 5 we study the late-time value of ∆S

(n)
A for the given locally excited state: Before studying its late-time value, we comment on its time evolution. Before t = l, ∆S A is given by where D, N i and P i are defined in table 3. If we take the late time limit (0 < l t), the ratios of P i and N i to D reduce to constant numbers [62], where we ignore the higher order contribution O l t . Amazingly, the sum of them is 1. Therefore if the effective reduced density matrix is defined by  Table 3. ∆S (n)
The excess of n−th (Rényi) entanglement entropy, entanglement entropy and Min entropy are respectively given by which can be interpreted in terms of quasi-particles created by φ 1 2 + φ 2 2 + φ 3 2 , which is constructed of three kinds of free scalar fields. Therefore, there are no differences between the effect of E 1 and that of E 2,3 on the late-time structure of quantum entanglement. As in the table 2, ∆S

(n)
A for E 2 is equivalent to that for B 2 . Therefore, they are electric-magnetic duality invariants.

Composite operators constructed of both electric and magnetic fields
In the previous two subsections we study how the entanglement structure changes at the late time if either electric or magnetic fields act on the ground state. However, we do not uncover how it changes at the late time when both of them act on the ground state. Here we study ∆S

(n)
A for (a) E 2 1 + B 2 1 , (b) E i B j and (c) F µν F µν and B · E, which can show that E i and B i act on the late-time structure of quantum entanglement differently from scalar fields such as φ a .

E
Here in order to study whether there are differences between the effects of electric and magnetic fields on the late-time structure of quantum entanglement, we study ∆S

(n)
A for the following excited state: A reduces to the (Rényi) entanglement entropy whose effective reduced density matrix is given by 1, 1, 1) . (3.14) Its entropies are given by ∆S (n) It shows there are no differences between the effects of electric and magnetic fields on the late-time structure.

E i B j
Here let's find out how the operators constructed of both electric and magnetic fields, E i B j , affect the late-time structure of quantum entanglement. The late-time values of ∆S

(n)
A for E i B j except for E 2 B 3 and E 3 B 2 are the same as (3.6). Their time evolution is summarized in table 2.
On the other hand, after t = l the time evolution of ∆S (n) A for E 2 B 3 or E 3 B 2 is summarized in table 3. We can see that it has the electric-magnetic duality from the table 3. The late-time value of ∆S (n) A is given by (Rényi) entanglement entropy whose reduced density matrix is given by diag (25,7,7,25).  It shows how different the effect of E 1 (B 1 ) is from that of E 2,3 (B 2,3 ) on the structure. The value can not be interpreted in terms of quasi-particles created by scalar fields such as φ a φ b . As we will explain later, in the entangled quasi-particle interpretation, there is a commutation relation between the quasi-particle created by E 2 (B 2 ) and that by B 3 (E 3 ).

B · E and
We finally study ∆S

(n)
A for more complicated operators, B · E, F µν F µν and B 2 E 3 − B 3 E 2 . Before t = l, ∆S (n) A for them vanishes, but after t = l, it increases. Their details are summarized in table 3. 6 It shows that ∆S

(n)
A for B · E is the same as that for F µν F µν .

∆S (n)
A is commuted by the Green's functions in appendix B.

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The effective reduced density matrices for B · E (F µν F µν ) and B 2 E 3 − B 3 E 2 are given by diag (30,30,16,16,49,49,1,1) , 128 diag (50,50,7,7,7,7) ,  which are for O = B 2 E 3 − B 3 E 2 . As we will explain in the next section, they can be reproduced by using a late-time algebra which electromagnetic quasi-particles obey. The time dependence of E 2 , F µν F µν and B 2 E 3 − B 3 E 2 are shown in figure 7. We can see that their ∆S (2) A increases diffently and reach different finite values in the limit t → ∞. As we see in the following section, we find significant differences in the late time algebra between E 2 and the others.

A late-time algebra
We interpret the late-time value of ∆S (n) A in terms of quasi-particles. More precisely, let's interpret the effective reduced density matrix in (3.10) in terms of quasi-particles. The effective reduced density matrix for the excited state generated by a composite operator whereN is a normalization constant. The operator O is assumed to be constructed of electric and magnetic fields. 7 As in [44][45][46]48], these fields can be decomposed into left 7 Here ρ e A is not the same as the reduced density matrix for the locally excited state. It is for a "effective" stateN O |0 . It is different form the "original" locally excited state.
A respectively. The red line represents the evolution for E 2 , the blue line for F µν F µν and the green line for moving and right moving electromagnetic quasi-particles as follows, where since we take x 1 ≥ 0 as A in this paper, left-moving and right-moving quasi-particles correspond to particles included in B and A at late time, respectively. The ground state for them is defined by The late-time algebra which quasi-particles obey is given by which is obtained so that the results by the replica trick are reproduced. Here C is a real number. 8 In the gauge theory in addition to (4.2), we need the following commutation relation for different particles: where X R,L and Y R,L are given by (4.6) 8 The redefinition of quasi-particles can absorb the constant C.

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Here X R,L and Y R,L are real numbers. 9 The commutation relation between electric (magnetic) quasi-particles is determined so that the effective density matrices computed by (4.4) are consistent with (3.11) respectively. The relation for the quasi-particles of E 1 should be the same as that for B 1 so that the effective density matrix in (4.1) reproduces ∆S

(n)
A for the matrix in (3.14). That relation between quasi-particles generated by E 2 (E 3 ) and those by B 3 (B 2 ) reproduces the matrix in (3.16). We also check that ∆S are reproduced by using the commutation relation in (4.4) and (4.5).
The relation in (4.5) shows that the effect of fields along the direction vertical to ∂A is significantly different from that along the direction parallel to ∂A on the late-time structure. It makes the effects of electromagnetic fields different from that of free scalar fields on the late-time structure of quantum entanglement.

Conclusion and future problems
We have studied how gauge invariant operators such as E i , B i and the composite operators constructed of them changes the structure of quantum entanglement by studying ∆S (n) A . We studied whether ∆S

(n)
A for locally excited states created by gauge invariant local operators reflects the electric-magnetic duality. ∆S

(n)
A , which we studied in this paper, is invariant under the duality transformation. If only E i or B i acts on the ground state, without taking the late time limit, the time evolution of ∆S (n) A depends on them. Due to the duality, ∆S A for E 2,3 (B 2,3 ) increases slower than that for E 1 (B 1 ). However they can not show the difference between the effects of electromagnetic fields and that of scalar fields on the late-time structure because the late-time values of ∆S

(n)
A for them can be interpreted in terms of quasi-particle created by scalar fields.
On the other hand, the late-time values of ∆S (n) A for the specific operators constructed of both electric and magnetic fields can not be interpreted in terms of quasi-particles by scalar fields. They show that there are differences between the effects of electromagnetic fields and that of scalar fields on the late-time structure of quantum entanglement. If their late-time values are interpreted in terms of electromagnetic quasi-particles in (4.2), there are commutation relations between E 2 (E 3 ) and B 3 (B 2 ), which make the effect of electromagnetic field significantly different from that of scalar fields on the late-time structure. The effect of E 1 and B 1 on the late-time structure is different from that of E 2,3 and B 2,3 .
Future problems. We finish with comments on a few future problems: • In this paper we only consider 4d Maxwell gauge theory which has conformal symmetry. D( = 4) dimensional Maxwell gauge theory is not a CFT. Therefore, it is interesting to generalize the analysis in 4d Maxwell theory to that in the theories with general dimensions. 9 We find the correspondence between propagators and commutation relations. The commutations can be defined by the late time limit of propagators. We will discuss the detail of the correspondence in [62]. When we use this correspondence, XL = − 3 4 C.

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• We expect that the structure of the late-time algebra depends on the spacetime dimension D. Then, it is also interesting to study it in general dimensions.

Acknowledgments
MN thanks Tadashi Takayanagi for useful discussions and comments on this paper. MN and NW thank Pawel Caputa, Tokiro Numasawa, Shunji Matsuura and Akinori Tanaka for useful comments on this work.

A Green's functions
The relation between E i , B i and field strengths which are defined in Euclidean space is given by The analytic continued Green's functions are defined by If the limit → 0 is taken, their leading terms for n = 1 are given by That for the arbitrary n in 0 < t < l are the same as given in (A.3).

(A.4)
The contribution of the other propagators is much smaller than those in (A.4).