Variation of Entanglement Entropy in Scattering Process
aa r X i v : . [ h e p - t h ] J a n Prepared for submission to JHEP
Variation of Entanglement Entropy in ScatteringProcess
I.Y. Park, a Shigenori Seki b and Sang-Jin Sin c a Department of Applied Mathematics, Philander Smith CollegeLittle Rock, AR 72223, USA b Research Institute for Natural Science, Hanyang UniversitySeoul 133-791, Republic of Korea c Department of Physics, Hanyang University,Seoul 133-791, Republic of Korea
E-mail: [email protected] , [email protected] , [email protected] Abstract:
In a scattering process, the final state is determined by an initial state andan S-matrix. We focus on two-particle scattering processes and consider the entanglementbetween these particles. For two types initial states; i.e. , an unentangled state and anentangled one, we calculate perturbatively the change of entanglement entropy from theinitial state to the final one. Then we show a few examples in a field theory and in quantummechanics. ontents φ -like interaction 83.2 Time-dependent interaction in quantum mechanics 10 Entanglement is a characteristic feature in a quantum theory. The entanglement in quan-tum field theories has been studied extensively in the past decade. When one considersa sub-system A and its complement A , the entanglement entropy between A and A isdefined by the von Neumann entropy S E = − tr A ρ A log ρ A with the reduced density ma-trix ρ A . Calabrese and Cardy have systematically studied it in a conformal field theorywith the use of a replica trick [1]. The other remarkable recent progress is the holographicderivation of entanglement entropy suggested by Ryu and Takayanagi [2, 3]. Following it,one can obtain an entanglement entropy by calculating S E = A / (4 G N ) , where A is thearea of a minimal surface whose boundary is the boundary of the sub-system A. In otherwords, the holographic entanglement entropy provides us with a geometric understandingof entanglement.Then there is the other geometric interpretation of entanglement entropy conjecturedrecently by Maldacena and Susskind [4]. Its original purpose was to resolve the firewallparadox [5] . This conjecture states that an Einstein-Rosen-Podolski pair, i.e. , a pair ofentangled objects, is connected by an Einstein-Rosen bridge (or a wormhole). Therefore theconjecture is symbolically called the ER=EPR conjecture. From the point of view of theAdS/CFT correspondence, some examples supporting the ER=EPR conjecture have beenshown. An entangled pair of accelerating quark and anti-quark was studied in Ref. [7]. In-vestigating the causal structure on the world-sheet minimal surface that is the holographicbulk dual of such a quark and anti-quark on the AdS boundary, Ref. [7] has found that thereexists a wormhole on the minimal surface and that any open strings connecting the quark See Ref. [6] for an earlier work. It has predicted an energetic curtain, which is similar to the firewall,on the assumptions different from Ref. [5]. – 1 –nd anti-quark must go through the wormhole. Therefore the entanglement of the accel-erating quark and anti-quark coincides with the existence of the wormhole. Furthermore,Ref. [8] considered Schwinger pair creation of a quark and an anti-quark and confirmedthat there is a wormhole on the string world-sheet of their bulk dual. Ref. [9] focused ona pair of scattering gluons as an EPR pair. Since Ref. [10] had shown the minimal surfacesolution corresponding to the gluon scattering, Ref. [9] calculated the induced metric on theminimal surface and found a wormhole connecting the gluon pair. One can then naturallyguess that, in a scattering process , an interaction induces the variation of entanglementfrom an incoming state to an outgoing one. We know these states are associated with eachother by an S-matrix. So the question is how the variation of entanglement entropy andthe S-matrix are related. In this paper we attack this problem by a perturbative analysis ina weak coupling λ . In order to evaluate the entanglement entropy, it is useful to calculateRényi entropy by the replica trick when one can calculate it exactly. For instance, Ref. [12]explicitly calculated the time evolution of the entanglement entropy between two free scalarfield theories with a specific interaction. However, this method is often unavailable for aperturbative analysis. Therefore we apply the method developed by Ref. [13, 14], in whichthe entanglement between two divided momentum spaces was studied perturbatively.In Section 2, we consider the variations of entanglement entropy from two kinds ofinitial states; one is an unentangled initial state and the other is an entangled one. InSection 3, we evaluate the variation of entanglement entropy in the field theory with a φ -like interaction. We also consider the time-dependent interaction in quantum mechanics.Section 4 is devoted to conclusion and discussion. Since we are interested in a scattering process of two particles, A and B, and their entan-glement, let us consider the Hamiltonian with an interaction: H = H + λH int , H = H A ⊗ + ⊗ H B . (2.1)It is usually difficult to divide the total Hilbert space H to H A ⊗ H B due to the interaction.However an initial state far in the past and a final state far in the future in a scatteringprocess can be regarded as states generated by an asymptotically free Hamiltonian. Further-more, although a field theory in general includes arbitrary multi-particle states in its Hilbertspace, we concentrate only on an elastic scattering of two particles such as A + B → A + B with a weak coupling. That is to say, we restrict the Hilbert space to the (1+1)-particleFock space, in which the initial and final states are. Since such a restriction usually violatesunitarity for local interaction terms, we assume in this paper specific theories that do notproduce states of more than 1+1 particles at lower orders of perturbation (see an examplein Section 3.1). Then the unitarity is approximately recovered at a weak coupling. Underthis assumption, we can divide the Hilbert space of the initial and final states to H A ⊗ H B , Ref. [11] has studied the entanglement entropy in a decay process in terms of the Wigner-Weisskopfmethod. – 2 –nd these states are denoted by a (1+1)-particle state generated by the free Hamiltonian H , namely, a state of a particle A and B with momentum p and q : | p, q i := | p i A ⊗ | q i B . (2.2)One can express the infinite time evolution from the initial state to the final one in termsof S-matrix by definition, lim t →∞ h fin | e − iHt | ini i = h fin | S | ini i , S := + i T . (2.3) T is a transition matrix in O ( λ ) which is induced by the interaction. Then the final stateis described as | fin i = Z dkdl | k, l ih k, l | S | ini i , (2.4)in which we used the completeness relation of (1+1)-particles’ states, i.e. , ( ) (1+1)-particles = R dkdl | k, l ih k, l | , and an inner product of states, i.e. , h k, l | p, q i = δ ( k − p ) δ ( l − q ) . Althoughthe norm h p, q | p, q i =: V has an infinite volume, we shall fix a normalization at the stageof a reduced density matrix. Here we comment that one can easily formulate the case ofdiscrete spectra by replacing R dkdl with P k,l . As an example we shall show in Section 3.2the theory with a time-dependent interaction in non-relativistic quantum mechanics.The total density matrix of the final state is ρ (fin) = | fin ih fin | , and we obtain thereduced density matrix ρ (fin) A by taking trace of ρ (fin) with respect to the particle B, i.e. , ρ (fin) A = tr B ρ (fin) up to normalization. In the case of (2.4) we can write down the reduceddensity matrix as ρ (fin) A = 1 N Z dkdk ′ (cid:18)Z dl h k, l | S | ini ih ini | S † | k ′ , l i (cid:19) | k ih k ′ | , (2.5)where N is a normalization constant determined by tr A ρ (fin) A = 1 , namely, N = Z dkdl |h k, l | S | ini i| . (2.6)Then the entanglement entropy between A and B in the final state is S (fin) E = − tr ρ (fin) A log ρ (fin) A , (2.7)and the variation of entanglement entropy from the initial state to the final one is ∆ S E = S (fin) E − S (ini) E , (2.8)where S (ini) E is the entanglement entropy of the initial state. We shall calculate these en-tanglement entropies perturbatively.The replica trick allows us to calculate a Rényi entropy, S ( n ) = − n log tr A ρ nA . The en-tanglement entropy is given by the n → limit of Rényi entropy, namely, S E = lim n → S ( n ) = − lim n → ∂∂n tr A ρ nA . Therefore the method to derive an entanglement entropy via a Rényi– 3 –ntropy is often useful. However, we are confronted with a problem when we analyze aquantum theory with a coupling λ in terms of perturbation. When one obtains a pertur-bative expansion of tr A ρ nA , the term of order λ n relevantly contributes to the entanglemententropy because the operation lim n → ∂∂n acts on λ n and yields a term of λ log λ order. Inother words, the higher order terms in the Rényi entropy are responsible for the convergenceof the entanglement entropy under the n → limit. Hence any λ n -order terms in tr A ρ nA arenecessary in order to obtain a meaningful entanglement entropy. In this paper, instead ofthe replica trick, we apply the perturbative method developed by Ref. [13] for calculatingan entanglement entropy. Let us consider the simplest single state with fixed momenta p and q for the initial stateof particle A and B, | ini i ∼ | p , q i . (2.9)The normalization of states will be properly fixed later in normalizing a density matrix sothat tr A ρ (fin) A = 1 . This initial state is obviously unentangled, i.e. , S (ini) E = 0 . Then we candescribe the final state (2.4) as | fin i = Z dkdl | k, l iS kl ; p q = S p q ; p q V | p , q i + iλ Z k = p dk T kq ; p q V | k, q i + iλ Z l = q dl T p l ; p q V | p , l i + iλ Z k = p l = q dkdl T kl ; p q | k, l i , (2.10)where we introduced an infinite spacial volume V := R dx e ix · = δ (0) due to the divergenceof norms, i.e. , h p | p i A = h q | q i B = δ (0) . The integral R k = p dk means R dk (1 − V − δ ( k − p )) . S kl ; pq and T kl ; pq denote S and T-matrix elements, S kl ; pq := h k, l | S | p, q i , T kl ; pq := 1 λ h k, l | T | p, q i . (2.11) S includes an identity , while T is given by an interaction with coupling λ . Therefore thepossible lowest orders of (2.11) with respect to λ are S pq ; pq ∼ O ( λ ) , S p ′ q ′ ; pq | ( p ′ ,q ′ ) =( p,q ) ∼ O ( λ ) , T kl ; pq ∼ O ( λ ) . (2.12)We employ the method developed by Ref. [13] in order to perturbatively calculate theentanglement entropy. Since Eq. (2.10) is rewritten as | fin i = S p q ; p q V | ˜ p i A ⊗ | ˜ q i B + Z k = p l = q dkdl (cid:18) λ T kq ; p q T p l ; p q S p q ; p q + iλ T kl ; p q (cid:19) | k, l i , (2.13)with | ˜ p i A = | p i A + iλV Z k = p dk T kq ; p q S p q ; p q | k i A , | ˜ q i B = | q i B + iλV Z l = q dl T p l ; p q S p q ; p q | l i B , (2.14)– 4 –e can calculate the reduced density matrix (2.5) as ρ (fin) A = 1 N (cid:18) |S p q ; p q | V | ˜ p ih ˜ p | + λ V Z k,k ′ = p dkdk ′ M kk ′ | k ih k ′ | (cid:19) ,M kk ′ = 1 V Z l = q dl (cid:18) λ T kq ; p q T p l ; p q S p q ; p q + i T kl ; p q (cid:19)(cid:18) λ T k ′ q ; p q T p l ; p q S p q ; p q + i T k ′ l ; p q (cid:19) ∗ . (2.15) N is the normalization factor which is fixed by tr A ρ (fin) A = 1 , namely, N = |S p q ; p q | V + λ V Z k = p dk M kk . (2.16)Here we recall (2.12) and it leads to M kk ′ ∼ O (1) . After a perturbative expansion, thereduced density matrix (2.15) becomes ρ (fin) A = (cid:18) − λ Z k = p dk M kk (cid:19) V | ˜ p ih ˜ p | + λ Z k,k ′ = p dkdk ′ M kk ′ | k ih k ′ | + O ( λ ) , (2.17) M kk ′ = 1 V Z l = q dl T kl ; p q T ∗ k ′ l ; p q + O ( λ ) , ( k, k ′ = p ) (2.18)When the eigenvalues of M kk ′ at leading order are denoted by m k , we obtain Z k = p dk M kk = tr A M kk ′ = Z k = p dk m k , (2.19)up to O ( λ ) . Therefore the entanglement entropy of final state (2.7) becomes S (fin) E = − (cid:18) − λ Z k = p dk m k (cid:19) log (cid:18) − λ Z k = p dk m k (cid:19) − Z k = p dk ( λ m k ) log( λ m k ) + O ( λ )= − λ log λ Z k = p dk m k + λ Z k = p dk m k (1 − log m k ) + O ( λ ) . (2.20)Only the T-matrix elements T kl ; p q with k = p and l = q contribute to the entanglemententropy of the final state at leading order. Of course, since the entanglement entropy of theunentangled initial state (2.9) vanishes, the variation of entanglement entropy (2.8), ∆ S E ,is equal to S (fin) E itself in (2.20). Let us consider an entangled initial state, | ini i ∼ u | p , q i + u | p , q i , (2.21)with p = q , p = q , u + u = 1 , u , = 0 and u ≥ u . The entanglement entropy ofthis state is S (ini) E = X j =1 | u j | log | u j | . (2.22)– 5 –e can write down the final state in terms of the S-matrix (or T-matrix), | fin i = S p q V | p , q i + S p q V | p , q i + iλ T p q V | p , q i + iλ T p q V | p , q i + iλ Z l = q ,q dl X j =1 T p j l V | p j , l i + iλ Z k = p ,p dk X j =1 T kq j V | k, q j i + iλ Z k = p ,p l = q ,q dkdl T kl | k, l i , (2.23)where S kl := u S kl ; p q + u S kl ; p q , T kl := u T kl ; p q + u T kl ; p q . (2.24)Note that S p q = u V + iλ T p q and S p q = u V + iλ T p q . Firstly we diagonalize thefirst line in (2.23) by the use of Q = S p q iλ T p q iλ T p q S p q ! , W = iλ T p q S p q − ζ S p q − ζ iλ T p q ! , W Q W − = ζ ζ ! , (2.25)where ζ + ζ = S p q + S p q , ζ − ζ = q ( S p q − S p q ) − λ T p q T p q . (2.26)Following this diagonalization, the basis is transformed as | p i| p i ! = W t | ˆ p i| ˆ p i ! , | q i| q i ! = W − | ˆ q i| ˆ q i ! . (2.27)Then we can rewrite the final state (2.23) as | fin i = X j =1 ζ j V | ˆ p j , ˆ q j i + iλ Z l = q ,q dl X j =1 A j ( l ) V | ˆ p j , l i + iλ Z k = p ,p dk X j =1 B j ( k ) V | k, ˆ q j i + iλ Z k = p ,p l = q ,q dkdl T kl | k, l i , (2.28)where A ( l ) = iλ T p l T p q + T p l ( S p q − ζ ) , A ( l ) = T p l ( S p q − ζ ) + iλ T p l T p q ,B ( k ) = iλ T kq T p q − T kq ( S p q − ζ )det W , B ( k ) = −T kq ( S p q − ζ ) + iλ T kq T p q det W . (2.29)Furthermore we can rearrange the basis so that | fin i = X j =1 ζ j V | ˜ p j i A ⊗ | ˜ q j i B + Z k = p ,p l = q ,q dkdl λ X j =1 A j ( l ) B j ( k ) ζ j + iλ T kl ! | k, l i , (2.30)– 6 –here | ˜ p j i A = | ˆ p j i A + iλ Vζ j Z k = p ,p dk B j ( k ) | k i A , | ˜ q j i B = | ˆ q j i B + iλ Vζ j Z l = q ,q dk A j ( k ) | k i B . ( j = 1 , (2.31)As a result, we obtain the reduced density matrix (2.5) after a similarity transformation, ρ (fin) A = 1 N X j =1 | ζ j | V | ˜ p j ih ˜ p j | + λ V Z k,k ′ = p ,p dkdk ′ R kk ′ | k ih k ′ | ! , (2.32) R kk ′ = 1 V Z l = q ,q dl λ X j =1 A j ( l ) B j ( k ) ζ j + i T kl ! λ X j =1 A j ( l ) B j ( k ′ ) ζ j + i T k ′ l ! ∗ . (2.33)The leading term of R kk ′ does not depend on A j and B j but on T kl ( k = p , p , l = q , q ).Using the normalization tr A ρ (fin) A = 1 , N is computed as N = X i =1 | ζ j | V + λ V Z k = p ,p dk R kk . (2.34)Then one can write down the reduced density matrix in perturbative expansion, ρ (fin) A = (cid:18) u + λf + λ g − λ u Z k = p ,p dk R kk (cid:19) V | ˜ p ih ˜ p | + (cid:18) u − λf − λ g − λ u Z k = p ,p dk R kk (cid:19) V | ˜ p ih ˜ p | + λ Z k,k ′ = p ,p dkdk ′ R kk ′ | k ih k ′ | + O ( λ ) , (2.35) R kk ′ = 1 V Z l = q ,q dl T kl T ∗ k ′ l + O ( λ ) , (2.36)with V f = 2 u u ( u Im T p q − u Im T p q ) , (2.37) V g = 4 u u ( u Im T p q + u Im T p q )( u Im T p q − u Im T p q )+ u |T p q | − u |T p q | − u u ( u + u ) u − u Re( T p q T p q ) . (2.38)Note that f and g are anti-symmetric with respect to the indices 1 and 2. Since (2.35)implies a reduced density matrix after a similarity transformation, we can calculate S (fin) E ,the entanglement entropy of the final state. Here we introduce r k ( k = p , p ) which denotesthe eigenvalues of R kk ′ . Subtracting the initial entanglement entropy (2.22) from S (fin) E , weobtain the variation of entanglement entropy as ∆ S E = − λ log λ Z k = p ,p dk r k − λf log u u + λ (cid:18)Z k = p ,p dk r k (1 − S (ini) E − log r k ) − f u u − g log u u (cid:19) + O ( λ ) . (2.39)– 7 –he leading term is of order λ log λ and is similar to the case of the unentangled initialstate (2.20). While the sub-leading term in the case of the unentangled initial state is oforder λ , the sub-leading term in the case of the entangled initial state appears at order λ . This order λ contribution comes from the mutual transition between the states, | p , q i and | p , q i . When the particles A and B at the initial state are maximally entangled, i.e. , u = u = 1 / √ , the term of order λ vanishes.In the same way, one can consider an n coherent state as an initial state, namely, | ini i ∼ P nj =1 u j | p j , q j i , P nj =1 u j = 1 . Since the final state includes S p i q j V − | p i q j i , we firstlydiagonalize the matrix Q = ( S p i q j ) ( i, j = 1 , . . . , n ) so that W Q W − = diag( ζ , . . . , ζ n ) ,and replace | p i , q i i with | ˆ p i , ˆ q i i like (2.27). Then, by a procedure similar to (2.31), we canobtain a simplified reduced density matrix like (2.33). Therefore the leading contributionto the variation of entanglement entropy is λ log λ R k = p ,...,p n dk R l = q ,...,q n dl V − T kl T ∗ kl , inwhich T kl = P nj =1 u j T kl ; p j q j . φ -like interaction We consider two real scalar fields, φ A and φ B , of which action with a φ -like interaction is S = − Z d d +1 x (cid:18) ∂ µ φ A ∂ µ φ A + 12 ∂ µ φ B ∂ µ φ B + 12 m ( φ A + φ B ) + λ φ A φ B (cid:19) . (3.1)We focus on a scattering process of two incoming particles and two outgoing particles suchas A + B → A + B . Since we can assume that the incoming and outgoing particles arefree on-shell particles in the far past and future, one can describe a Fock space of such(1+1)-particle states as | ~p , ~q i = a ~p † | i A ⊗ b ~q † | i B . (3.2) a ~p † and b ~q † are the creation operators of particles A and B and are defined by the followingmode expansion for free scalar fields: φ A ( x ) = Z d d ~p (2 π ) d E ~p ( a ~p e − ip · x + a ~p † e ip · x ) , φ B ( x ) = Z d d ~q (2 π ) d E ~q ( b ~q e − iq · x + b ~q † e iq · x ) , (3.3)where p = E ~p = p ~p + m . The factor d d ~p/ ((2 π ) d E ~p ) is a Lorentz invariant integrationmeasure. The creation and annihilation operators obey the commutation relations: [ a ~p , a ~k † ] = 2 E ~p (2 π ) d δ ( d ) ( ~p − ~k ) , [ b ~q , b ~l † ] = 2 E ~q (2 π ) d δ ( d ) ( ~q − ~l ) . (3.4)Now let us study the case that the initial state is | ini i = | ~p , ~q i . Since the identityoperator on the (1+1)-particle Hilbert space is ( ) (1+1)-particle = Z d d ~p (2 π ) d E ~p d d ~q (2 π ) d E ~q | ~p, ~q ih ~p , ~q | , (3.5)– 8 –he final state (2.4) is described as | fin i = S | ini i = Z d d ~k (2 π ) d E ~k d d ~l (2 π ) d E ~l | ~k,~l ih ~k,~l | S | ~p , ~q i = 12 E ~p E ~q L d | ~p , ~q ih ~p , ~q | S | ~p , ~q i + Z ~k = ~p ~l = ~q d d ~k (2 π ) d E ~k d d ~l (2 π ) d E ~l | ~k,~l ih ~k,~l | i T | ~p , ~q i , (3.6)where L originates from the spacial volume of phase space, L d = (2 π ) d δ ( d ) (0) = R d d ~x e i~x · ~ .The final state (3.6) does not contain the states proportional to | ~k ( = ~p ) , ~q i and | ~p ,~l ( = ~q ) i , which appear in the second line of (2.10), because such states vanish due to the factorof momentum conservation in the S-matrix element, namely, h ~k,~l | S | ~p , ~q i ∼ δ ( d +1) ( k + l − p − q ) .As we have studied in Section 2, the variation of entanglement entropy in a scatteringprocess is determined by the transition matrix T . From the action (3.1), we perturbativelycalculate the S-matrix element, h ~k,~l | S | ~p , ~q i = 2 E ~p E ~q (2 π ) d δ ( d ) ( ~k − ~p )(2 π ) d δ ( d ) ( ~l − ~q ) − iλ (2 π ) d +1 δ ( d +1) ( k + l − p − q ) + O ( λ ) . (3.7)Substituting this S-matrix element into (3.6), we obtain the final state. Then the reduceddensity matrix automatically becomes block-diagonal, ρ (fin) A = (cid:18) − λ Z ~k = ~p d d ~k (2 π ) d E ~k M ~k~k (cid:19) E ~p L d | ~p ih ~p | + λ Z ~k = ~p d d ~k (2 π ) d E ~k M ~k~k E ~k L d | ~k ih ~k | + O ( λ ) , (3.8) M ~k~k = 12 E ~p E ~q E ~p + ~q − ~k L d (cid:8) πδ ( E ~k + E ~p + ~q − ~k − E ~p − E ~q ) (cid:9) . (3.9)Notice that we have normalized this density matrix so that tr A ρ (fin) A = 1 . Then the variationof entanglement entropy (2.20) is computed as ∆ S E = − λ log λ Z ~k = ~p d d ~k (2 π ) d E ~k M ~k~k + λ Z ~k = ~p d d ~k (2 π ) d E ~k M ~k~k (cid:18) − log M ~k~k E ~k L d (cid:19) + O ( λ ) . (3.10)We shall calculate it further by employing a center of mass frame, that is, ~p = − ~q =: ~p cm and p ~p cm + m =: E cm . Of course the momenta of outgoing particles obey ~k = − ~l dueto the momentum conservation. Then the d -dimensional integration can be replaced witha spherical integration as d d ~k = dkd Ω d − k d − , because the integration kernel in (3.10)– 9 –epends only on the norm of ~k . Therefore we finally obtain ∆ S E = − λ log λ π − d d +3 Γ( d ) L d − | ~p cm | d − E cm + λ π − d d +3 Γ( d ) L d − | ~p cm | d − E cm (cid:0) E cm L d − ) (cid:1) + O ( λ ) . (3.11)When the number of the spacial dimension d is equal to three, the leading term of thevariation of entanglement entropy is proportional to | ~p cm | /E cm . This is consistent withthe cross section, which is ( dσ/d Ω) cm = λ π | ~p cm | /E cm , because both the variation ofentanglement entropy and the cross section originate from a square of the absolute value ofthe scattering amplitude. Notice that the remaining factor L in the entanglement entropy isan artifact caused by choosing the single-mode initial state whose norm has delta-functionaldivergence. The volume dependence of entanglement entropy in field theories was discussedalso in Ref. [13, 14], where the momentum-space entanglement entropy is proportional toa spacial volume. The difference between the volume dependence of Ref. [13, 14] and oursis mostly caused by the absence of integration with respect to the initial state momenta inour calculation. In this subsection we turn to quantum mechanics with a time-dependent interaction, λH int ( t ) .We set the initial state so that | ini i = | p , q i at t = 0 . Then the time evolution of thisinitial state is described as | Ψ( t ) i = | p , q i + λ X k = p C kq ; p q ( t ) e − iE k t | k, q i + λ X l = q C p l ; p q ( t ) e − iE l t | p , l i + λ X k = p ,l = q C kl ; p q ( t ) e − iE kl t | k, l i , (3.12)up to normalization. E p , E q and E pq are energy eigenvalues which are defined in terms ofthe non-interacting part of the Hamiltonian (see (2.1)), H A | p i A = E p | p i A , H B | q i B = E q | q i B , H | p, q i = E pq | p, q i . (3.13)The interacting Hamiltonian λH int ( t ) yields C kl ; pq ( t ) . By the use of the well-known time-dependent perturbation theory, we can calculate C kl ; p q ( t ) = − i Z t dt ′ e iω kl ; p q t ′ T kl ; p q ( t ′ ) , T kl ; p q ( t ) := h k, l | H int ( t ) | p , q i . (3.14)where ω kl ; p q := E kl − E pq . Since the time-dependent density matrix of | Ψ( t ) i is givenby ρ ( t ) = | Ψ( t ) ih Ψ( t ) | , we can calculate the reduced density matrix ρ A ( t ) = N − tr B ρ ( t ) together with the normalization by tr A ρ A = 1 . After the same procedure as Ref. [13] orSection 2.1, we obtain the entanglement entropy, S E ( t ) = − λ log λ X k = p ,l = q Z t dt ′ Z t dt ′′ e iω kl ; p q ( t ′ − t ′′ ) T kl ; p q ( t ′ ) T ∗ kl ; p q ( t ′′ ) + O ( λ ) . (3.15)Notice that one can regard λT kl ; p q ( t = ∞ ) as a kind of transition matrix.– 10 – Conclusion and discussion
We have studied the variation of entanglement entropy from an initial state to a finalstate in a scattering process. We concentrated on the scattering of → particles andperturbatively calculated the entanglement entropy of final states for the two kinds ofsimple initial states: the unentangled state (2.9) and the entangled state (2.21). In bothcases the leading terms of the variation of entanglement entropy, (2.20) and (2.39), areof order λ log λ and are proportional to the trace of a square of the absolute value ofT-matrix elements, which are, in other words, the scattering amplitudes. The next leadingterm in the case of the unentangled initial state is of order λ . On the other hand the nextleading term in the case of the entangled initial state appears at order λ , because there isa mutual transition between the states | p , q i and | p , q i .We have considered the model of two real scalar fields with the φ -like interaction asan example in a field theory. The variation of entanglement entropy has been computedperturbatively. If we employ the center of mass frame, the leading term (at order λ log λ )in the variation of entanglement entropy depends on the momenta of initial particles as | ~p cm | d − /E cm . Notice that this factor becomes | ~p cm | /E cm , when the space dimension isequal to three, i.e. , the coupling λ is dimensionless. The same factor also appears in thecross section, because it originally comes from the scattering amplitude. Therefore, as weexpected, the variation of entanglement entropy is proportional to the cross section.We have also mentioned the time-dependent interaction as an example in quantummechanics. The time-evolution of entanglement entropy from the simple initial state | p , q i can be written in terms of the transition matrix at the leading order λ log λ .With the AdS/CFT correspondence, one can identify the scattering amplitude in a fieldtheory of strong coupling with exp( −A ) , where A is an area of minimal surface in a bulkgravity theory, while the holographic entanglement entropy [2, 3] is given by A ′ / (4 G N ) ,where A ′ is an area of another minimal surface. That is to say, both of the scatteringamplitude and entanglement entropy in a strongly coupled field theory are associated withminimal surfaces from the point of view of the AdS/CFT correspondence. In this paper wehave shown the relation between the scattering and the variation of entanglement entropyby the perturbative calculations in a weak coupling. It is then in order to ask whether wecan clarify such a relation from a field theory in a strong coupling. For this purpose weneed to test it in an exactly calculable model. Moreover, the holographic understanding ofsuch a relation, or a relation between those minimal surfaces, is another problem for thefuture. Acknowledgments
This work was supported by Mid-career Researcher Program through the National ResearchFoundation of Korea (NRF) grant No. NRF-2013R1A2A2A05004846. IP thanks Sang-JinSin for his hospitality during the IP’s visit to Hanyang University through its foreign scholarinvitation program. SS was also supported in part by Basic Science Research Programthrough NRF grant No. NRF-2013R1A1A2059434.– 11 – eferences [1] P. Calabrese and J. L. Cardy, “Entanglement entropy and quantum field theory,” J. Stat.Mech. (2004) P06002 [hep-th/0405152].[2] S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy fromAdS/CFT,” Phys. Rev. Lett. (2006) 181602 [hep-th/0603001].[3] S. Ryu and T. Takayanagi, “Aspects of Holographic Entanglement Entropy,” JHEP (2006) 045 [hep-th/0605073].[4] J. Maldacena and L. Susskind, “Cool horizons for entangled black holes,” Fortsch. Phys. (2013) 781 [arXiv:1306.0533 [hep-th]].[5] A. Almheiri, D. Marolf, J. Polchinski and J. Sully, “Black Holes: Complementarity orFirewalls?,” JHEP (2013) 062 [arXiv:1207.3123 [hep-th]].[6] S. L. Braunstein, “Black hole entropy as entropy of entanglement, or it’s curtains for theequivalence principle,” [arXiv:0907.1190v1 [quant-ph]] published as S. L. Braunstein,S. Pirandola and K. Życzkowski, “Better Late than Never: Information Retrieval from BlackHoles,” Phys. Rev. Lett. (2013) 101301 [arXiv:0907.1190 [quant-ph]].[7] K. Jensen and A. Karch, Phys. Rev. Lett. (2013) 21, 211602 [arXiv:1307.1132 [hep-th]].[8] J. Sonner, “Holographic Schwinger Effect and the Geometry of Entanglement,” Phys. Rev.Lett. (2013) 21, 211603 [arXiv:1307.6850 [hep-th]].[9] S. Seki and S. -J. Sin, “EPR = ER and Scattering Amplitude as Entanglement EntropyChange,” Phys. Lett. B (2014) 272 [arXiv:1404.0794 [hep-th]].[10] L. F. Alday and J. M. Maldacena, JHEP (2007) 064 [arXiv:0705.0303 [hep-th]].[11] L. Lello, D. Boyanovsky and R. Holman, “Entanglement entropy in particle decay,” JHEP (2013) 116 [arXiv:1304.6110 [hep-th]].[12] A. Mollabashi, N. Shiba and T. Takayanagi, “Entanglement between Two Interacting CFTsand Generalized Holographic Entanglement Entropy,” JHEP (2014) 185[arXiv:1403.1393 [hep-th]].[13] V. Balasubramanian, M. B. McDermott and M. Van Raamsdonk, “Momentum-spaceentanglement and renormalization in quantum field theory,” Phys. Rev. D (2012) 045014[arXiv:1108.3568 [hep-th]].[14] T. C. L. Hsu, M. B. McDermott and M. Van Raamsdonk, “Momentum-space entanglementfor interacting fermions at finite density,” JHEP (2013) 121 [arXiv:1210.0054 [hep-th]].(2013) 121 [arXiv:1210.0054 [hep-th]].