TTarget Space Entanglement Entropy
Edward Mazenc & Daniel Ranard
Department of Physics, Stanford University,Stanford, CA 94305-4060, USA
Abstract
We define a notion of target space entanglement entropy. Rather than partitioning the basespace on which the theory is defined, we consider partitions of the target space. This is the physicalcase of interest for first-quantized theories, such as worldsheet string theory. We associate to eachsubregion of the target space a suitably chosen sub-algebra of observables A . The entanglemententropy is calculated as the entropy of the density matrix restricted to A . As an example, weillustrate our framework by computing spatial entanglement in first-quantized many-body quantummechanics. The algebra A is chosen to reproduce the entanglement entropy obtained by embeddingthe state in the fixed particle sub-sector of the second-quantized Hilbert space. We then generalizeour construction to the quantum field-theoretical setting. a r X i v : . [ h e p - t h ] O c t ontents c =1 matrix quantum mechanics . . . . . . . . . . . . . . 28
11 Acknowledgements 2812 Appendix: The algebra for bosons 29 Introduction
Previous studies of entanglement in field theory mostly address entanglement with respect to partitionsof the base space [1, 2]. Recall that in quantum field theory we speak of both the base space and thetarget space. For instance, in standard d + 1 scalar field theory, the field φ ( (cid:126)x ) take values in the targetspace R . The base space R d parametrizes instead which degree of freedom we are speaking of: it labelsthe (cid:126)x ∈ R d of φ ( (cid:126)x ) . Colloquially, we refer to the base space as the space the field “lives on.” Tocalculate the entanglement entropy of a spatial partition, we partition the base space R d ; see Figure 1.Meanwhile, in promising theories of quantum gravity, a “spatial” partition may not be associ-ated with a partition of the base space, but rather a partition of the target space. For example, infirst-quantized string theory, a spacetime subregion corresponds to a restriction of the embedding co-ordinates of the string. It is thus a partition of the target space, not a partition of the base worldsheet.Figure 1: Distinction between partitions of the base space (left) versus target space (right). We illustrate thecase of 2+1d scalar field theory for concreteness. At every point of the base space ( ∼ = R ) labeled by (cid:126)x , the localdegree of freedom takes values in the target space ( ∼ = R ) with coordinate φ . Previous studies of entanglemententropy have focused on subregions of the base space where the values of (cid:126)x are restricted. Instead, we considerrestrictions on the values of the field φ . Likewise, Matrix theory is a dimensional theory that describes, in a particular frame, quantumgravity in an 11-dimensional spacetime [3]. In this case, the “base space” is nothing more than a point.Clearly, it is senseless to partition it. Instead, subregions of the physical spacetime correspond to asubspace of the moduli space of D branes [4, 5].Motivated by these examples, the main goal of this paper is to define reduced density matricesand entanglement entropies of states with respect to subregions in target space. The first challenge is3hat target space partitions do not correspond to tensor factorizations of the Hilbert space, whereasthe usual framework for entanglement entropy hinges upon such a factorization. We therefore leveragethe powerful algebraic framework, which defines a reduced density matrix relative to a subalgebra ofobservables, and treats tensor factorizations as a special case. This algebraic framework is by no meansnew, though it has only recently gained widespread traction in the high energy community via thework of [6–8], which offer excellent introductions to the subject. Our task therefore reduces to findingwhich subalgebra most accurately reflects an agent having access only to observables confined to thespatial subregion of interest. We then define target space entanglement entropy as the entropy of thestate restricted to this subalgebra. H BFSS = 12 p T r N ⇥ N P i P i + [ X i , X j ] + fermions S W S = 14 ⇡↵ Z d p gg µ⌫ @ µ X a ( ) @ ⌫ X b ( ) G ab ( X )
8O 2 A
Figure 2: (Top) For many of our most promising theories of quantum gravity, such as worldsheeet string theoryor the BFSS matrix quantum mechanics, the emergent physical spacetime of interest is encoded in the targetspace of the theory. (Bottom) To each subregion of the target space, we associate a particular subalgebra ofobservables A . The reduced density matrix ρ A is defined via restriction of the state to the subalgebra A . Wedefine the target space entanglement entropy as the entropy of ρ A . A single particle on a line furnishes the simplest toy model. We may think of the position of theparticle x ( t ) as 0+1-dimensional QFT. The base space is a single point, and the target space R is thephysical space the particle is moving in. To define a notion of spatial entanglement, we must partitionthe “target space.” First-quantized many body quantum mechanics provides the ideal testing groundfor our proposed definition because we have a firm grasp on its second quantized formulation - plainold QFT - where we understand entanglement entropy well. Making a similar comparison in the stringtheory context would require the intricacies of string field theory (see [9] for recent work directly inthat context).In the non-relativistic case, our definition of entanglement entropy for the above quantum-mechanicalsystem agrees with the the standard field theory definition. For relativistic quantum field theory, wefind two ostensibly natural notions of locality and discuss their relative merits. An explicit computa-4ion for one-particle excited states shows to what extent the entanglement entropy associated to thesedifferent “spatial” partitionings can be compared.Further, we stress our framework is by no means limited to quantum mechanics. We generalize ourconstruction to partitions of the target spaces of arbitrary sigma models and interacting field theories.We compute the entanglement entropy for a half (target) space partition in the the simplest example,a massive scalar field on two spatial lattice sites.We conclude by discussing the important role played by reparametrization invariance in theoriessuch as worldsheet string theory and point out the limitations of our framework. We then apply thelessons learned in Section 6 to the case of 2d string theory and the “baby cousin” of the AdS/CFTcorrespondence, the holographic c = 1 matrix model. There, we face multiple notions of emergentlocality [5], and we sketch the possible role of different factorizations of the Hilbert space. To handle partitions of target space, we will need the algebraic framework for entanglement entropy.However, let us motivate it further by exploring in more detail the example of particle on a line.The Hilbert space L ( R ) = span {| x (cid:105) : x ∈ R } may also be considered as the Hilbert space of a0+1-dimensional QFT, where the base space is a single point, and L ( R ) = span {| φ (cid:105) : φ ∈ R } is thespace of field values φ at that point. From the latter perspective, we will call R the “target space.”Alternatively, from the perspective of quantum mechanics on the line, R is simply the space on whichthe particle lives.We can partition the target space R into a region A and its complement ¯ A ; for instance, we mightchoose half-spaces A ≡ { x : x ≤ x } and ¯ A ≡ { x : x > x } . This bi-partition induces a decompositionof the Hilbert space into a direct sum, H = L ( A ∪ ¯ A ) = V A ⊕ V ¯ A (1)where V A ≡ span {| x (cid:105) : x ∈ A } , (2) V ¯ A ≡ span {| x (cid:105) : x ∈ ¯ A } . We emphasize that this decomposition is not a tensor factorization. Therefore one might wonderhow to define a subsystem, a partial trace and reduced density matrix, or an entanglement entropy.To proceed, we thus review a more general notion of subsystems, based on sub-algebras rather thantensor factors.
Traditionally, one defines the entanglement of a state | ψ (cid:105) ∈ H relative to some bi-partition of theHilbert space H = H A ⊗ H ¯ A , where we have divided the degrees of freedom into subsystem A and itscomplement ¯ A . We will consider pure states on H , in which case the entanglement can be quantifiedby the the von Neumann entanglement entropy of the reduced state ρ A = Tr ¯ A ( | ψ (cid:105)(cid:104) ψ | ) .5here are many ways to factorize a Hilbert space H as H = H A ⊗ H ¯ A , and different factorizationsmay be appropriate for different purposes. Given a factorization, it is natural to consider the algebra A of operators local to A , i.e. operators of the the form O A ⊗ ¯ A . These operators represent theobservables and operations available to an observer confined to subsystem A .Even without a factorization of the Hilbert space, we can still choose a sub-algebra of “accessible”observables A and use this to define the subsystem A . Recall that an algebra A of operators on aHilbert space H is a subset A ⊂ L ( H ) where L ( H ) denotes the space of all linear operators; here weconsider finite-dimensional “von Neumann algebras,” required to be closed under addition, multiplica-tion, scaling, and Hermitian conjugation. By identifying any subalgebra
A ⊂ L ( H ) with an abstract“subsystem,” we generalize the notion of a subsystem beyond tensor factors.We review a few key facts about algebras of operators. The most important theorem is that givenany algebra A ⊂ L ( H ) , there exists a decomposition of the Hilbert space as a direct sum of tensorproducts, H = (cid:77) i H A,i ⊗ H ¯ A,i (3)such that the operators O A ∈ A are precisely those which take the form O A = (cid:88) i O A,i ⊗ H ¯ A,i (4)for some O A,i ∈ L ( H A,i ) . This follows from a pedestrian version of the Artin-Wedderburn theorem.Schematically, we can write A = (cid:77) i L ( H A,i ) ⊗ H ¯ A,i . (5)In the case that the above sum has only one term, A is called a “factor,” and indeed the Hilbertspace tensor factorizes as H A ⊗ H ¯ A . In infinite dimensions, the existence of this tensor factorizationhinges upon the “Type” of algebra; see Section 9 for more.Given an algebra A ⊂ L ( H ) , an important related algebra is the commutant A (cid:48) ⊂ L ( H ) , the setof operators that commute with all the operators on A . Given the above decomposition (3), Schur’slemma allows us to easily write down the commutant, which takes the form A (cid:48) = (cid:77) i H A,i ⊗ L ( H ¯ A,i ) . (6)We can also define the center Z ( A ) ≡ A∩A (cid:48) , the set of operators on A that commute with all operatorson A . The center may be expressed as Z ( A ) = span { Π i } i (7)where Π i are the projectors onto the direct sum sectors H A,i ⊗ H ¯ A,i . In practice, we often start outwith A , then determine the minimal projectors spanning its center. This in turn allows us to actually Though we often refer to it as an “algebra of observables,” not all of the elements need be Hermitian. Here we willrequire an algebra to include the identity element. There are additional subtleties in infinite dimensions that we will notimmediately discuss, though the results presented above all hold for finite direct sums of Type I factors. See Section 9for more discussion of infinite dimensions. A is a factor, the center contains only multiples of the identity.With these ingredients in hand, we can easily define the reduced density matrix with respect toa sub-algebra. Say we have a state ρ and want to define a reduced density state ρ A with respect to A . First recall that for an ordinary tensor factorization H = H A ⊗ H ¯ A , the reduced state ρ A can bedefined as the unique state on H A such that Tr( ρ A O A ) = Tr( ρ O A ) for all O A ∈ L ( H ) . With thatdefinition, one can show ρ A is given by the familiar partial trace.Analogously, for the case of an algebra, we will define ρ A to be the unique element of A such that Tr( ρ A O A ) = Tr( ρ O A ) (8)for all O A ∈ A . Given the decomposition of Eqn. 3, it turns out one can easily express ρ A by usingpartial traces on each sector. Let p i ≡ Tr(Π i ρ Π i ) , (9) ρ i ≡ p i Π i ρ Π i . (10)Then one can show ρ A must be given by ρ A = (cid:88) i p i Tr ¯ A,i ( ρ i ) ⊗ H ¯ A,i dim ( H ¯ A,i ) , (11)where ρ i is a state living on the i ’th sector H A,i ⊗H ¯ A,i . The partial traces on each sector are well-definedbecause each sector factorizes individually.To define the entanglement entropy of ρ , we further consider the state ˜ ρ A = (cid:88) i p i Tr ¯ A,i ( ρ i ) (12)on the Hilbert space H A ≡ (cid:76) i H A,i , where we have simply stripped off the identity factors. Then wedefine the entanglement entropy of ρ with respect to A as the ordinary von Neumann entropy of thestate ˜ ρ A on the Hilbert space H A . That is, we define S ( ρ, A ) ≡ S (˜ ρ A ) (13) = S (cid:32)(cid:88) i p i Tr ¯ A,i ( ρ i ) (cid:33) (14) = − (cid:88) i p i log( p i ) + (cid:88) i p i S ( ρ i ) (15) ≡ S ( ρ, A ) classical + S ( ρ, A ) quantum . (16)We find that the entanglement entropy breaks into two pieces, a “classical” piece and a “quantum”piece. S ( ρ, A ) classical is the Shannon entropy of the classical probability distribution { p i } over the This definition differs from the naive definition Tr H ( ρ A log ρ A ) by the term ∆ S = (cid:80) i p i log (cid:16) dim ( H ¯ A,i ) (cid:17) . Toreproduce the standard entropy for the case of a factor, we must use the definition outlined in the main text. SeeAppendix A.7.2 of [7] or more broadly [10]. To avoid any confusion: throughout the text, when we refer to the vonNeumann entropy of ρ A , we really mean S ( ρ, A ) , or equivalently S (˜ ρ A ) . S ( ρ, A ) quantum on the other hand is theweighted sum of the quantum von Neumann entropy of the reduced density matrices ρ i within eachblock [6, 11].Another simple way to define S ( ρ, A ) is to embed H into an extended Hilbert space H ext which does have a tensor factorization. In particular, we define H A ≡ (cid:77) i H A,i , (17) H ¯ A ≡ (cid:77) i H ¯ A,i , so that we can define the extended Hilbert space H ext ≡ H A ⊗ H ¯ A = (cid:77) i,j H A,i ⊗ H ¯ A,j ⊃ (cid:77) i H A,i ⊗ H ¯ A,i = H . (18)Therefore we can also view the state ρ on H as a state ρ ext on the extended Hilbert space H ext . Then S ( ρ, A ) is then precisely the “ordinary” entanglement entropy obtained by taking the partial trace of ρ over H ¯ A and then computing the von Neumann entropy of this reduced density matrix , i.e. S ( ρ, A ) = Tr H A ( ρ A log ρ A ) (19)with ρ A ≡ Tr H ¯ A ( ρ ext ) In the case that A corresponds to the set of operators on a tensor factor, the quantity S ( ρ, A ) agrees with the standard von Neumann entanglement entropy. However, beyond agreement with the“ordinary” case, what motivates this definition of S ( ρ, A ) ?We might first ask the motivation for ordinary von Neumann entanglement entropy. Besides prov-ing a useful tool for analyzing field theories and many-body physics, the entanglement entropy affordsseveral operational or information-theoretic interpretations. For instance, the von Neumann entan-glement entropy between subsystems A and ¯ A also equals the “distillable entanglement,” the numberof Bell pairs that can be distilled by observers on A and ¯ A using only local operations on A, ¯ A andclassical communication. The entanglement entropy with respect to an algebra affords an analogous in-terpretation as distillable entanglement, where observers on A and ¯ A are restricted to using operationsassociated to A and A (cid:48) , respectively. However, it turns out the distillable entanglement is equal to thequantum piece S ( ρ, A ) quantum alone [12–14]. See Equation 46 for an elaboration of the operationalinterpretation. Readers familiar with the literature on entanglement entropy in gauge theory might object that the “extended Hilbertspace” and algebraic definitions famously disagree. However, the extended Hilbert space construction in gauge theorydiffers from the one in Eq. 18. Algebraic entanglement entropy for first-quantized systems
We first apply the algebraic definition of entanglement entropy to our example of a particle on aline, spelling out the details on this first pass. We then proceed more generally to the first-quantizedquantum mechanics of N particles. Later in Section 5, we will confirm that our framework gives thesame spatial entanglement entropy had we instead embedded the first-quantized, N -particle Hilbertspace into the N -particle sector of a second-quantized Fock space, and defined the entanglemententropy using the tensor factorization associated to the Fock space. Nonetheless, we develop thealgebraic approach as a general tool, applicable even when no obvious second-quantized theory exists. We return to the particle on a line, introduced in Section 2. The Hilbert space is simply H = L ( R ) ,and we can think of R alternately as the space on which the particle lives, or the target space of a0+1-dimensional QFT. For this section, we primarily adopt the language of the former.Partitioning the line R into a region A ⊂ R and its complement ¯ A , we obtain the decomposition ofEqn. 1, H = V A ⊕ V ¯ A , (20)where V A = span {| x (cid:105) : x ∈ A } and likewise V ¯ A = span {| x (cid:105) : x ∈ ¯ A } .Now we choose an algebra A ⊂ L ( H ) to associate to the region A . We propose the followingalgebra, A = (cid:28) {| x (cid:105) (cid:104) x (cid:48) | : x, x (cid:48) ∈ A } ∪ H (cid:29) . (21)The angular brackets denote “the algebra generated by,” i.e. the algebra of all operators generatedby addition, multiplication, and scaling of the operators within the brackets. To physically motivatethis choice, note the Hermitian operators in A correspond to what observers situated in the region A of the line could measure. Including the identity is crucial. Physically, it corresponds to the fact thatan observer should be able to act trivially on the system.It will also be useful to define the projector Π A = (cid:90) x ∈ A dx | x (cid:105) (cid:104) x | , (22)which acts on the subspace V A as the identity A . We denote the orthogonal complement as Π ¯ A = ¯ A .Written in the position basis, with the basis partitioned into elements in A and ¯ A , all operators O ∈ A take the following form O = (cid:32) O A c ¯ A (cid:33) (23)where c is an arbitrary constant and O A = Π A O Π A = (cid:90) x,x (cid:48) ∈ A dx dx (cid:48) O ( x, x (cid:48) ) | x (cid:105) (cid:104) x (cid:48) | .
9o analyze the structure of this algebra, we compute the commutant A (cid:48) , again the set of operatorsthat commute with all those in A . By Schur’s Lemma, A (cid:48) is given by all operators of the form O (cid:48) = (cid:32) c A O ¯ A (cid:33) (24)with c some other arbitrary constant. Thus we could also denote A (cid:48) as the algebra ¯ A correspondingto the complementary region ¯ A , with analogous definition A (cid:48) = ¯ A ≡ (cid:28) {| x (cid:105) (cid:104) x (cid:48) | : x, x (cid:48) ∈ ¯ A } ∪ H (cid:29) . (25)Hence the center Z = A ∩ A (cid:48) is given by Z = (cid:28) (cid:90) x ∈ A dx | x (cid:105) (cid:104) x | ∪ H (cid:29) = span { Π A , Π ¯ A } . (26)The center being non-trivial simply reflects the fact that A does not induce a simple tensor factorization.As guaranteed by the theorem of Eqn. 3, the algebra A induces a decomposition of the Hilbertspace. The decomposition is apparent from the form of A and A (cid:48) in Eqns. 23, 24 above. We have H = (cid:77) i =0 , H A,i ⊗ H ¯ A,i . (27)where H A, = C (28) H ¯ A, = V ¯ A = span {| x (cid:105) : x ∈ ¯ A } (29) H A, = V A = span {| x (cid:105) : x ∈ A } (30) H ¯ A, = C (31)so that H = ( C ⊗ V ¯ A ) ⊕ ( V A ⊗ C ) (32) = V A ⊕ V ¯ A , (33)recovering Eqn. 1.The decomposition here is slightly trivial, because the Hilbert spaces H A, and H ¯ A, happen tobe the trivial space C . Each sector corresponds to the number of particles in A . For example, H A, ⊗ H ¯ A, is the sector where the particle is within A . It is the tensor product of H A, , the space ofwavefunctions on A , with the trivial space H ¯ A, , whose single ray represents the state of ¯ A with zeroparticles. Likewise, we can think of the sector H A, ⊗ H ¯ A, as the sector where the particle is within ¯ A . Now we compute the reduced density matrix of a state with respect to A . Let ρ = | ψ (cid:105)(cid:104) ψ | be ageneral pure state for | ψ (cid:105) ≡ (cid:90) dx ψ ( x ) | x (cid:105) = (cid:90) x ∈ A dx ψ ( x ) | x (cid:105) + (cid:90) x ∈ ¯ A dx ψ ( x ) | x (cid:105) (34) ≡ | ψ A (cid:105) + | ψ ¯ A (cid:105) . (35)10e compute the reduced density matrix ρ A using Eqn. 11, taking note of the decomposition in Eqn.27. First we project the density matrix ρ into each of the two sectors, yielding | ψ A (cid:105)(cid:104) ψ A | and | ψ ¯ A (cid:105)(cid:104) ψ ¯ A | .Following Eqn. 9, define p ≡ Tr( | ψ A (cid:105)(cid:104) ψ A | ) = (cid:104) ψ A | ψ A (cid:105) , (36) p ≡ Tr( | ψ ¯ A (cid:105)(cid:104) ψ ¯ A | ) = (cid:104) ψ ¯ A | ψ ¯ A (cid:105) , (37)and ρ = 1 p | ψ ¯ A (cid:105)(cid:104) ψ ¯ A | , (38) ρ = 1 p | ψ A (cid:105)(cid:104) ψ A | . (39)Finally, plugging these into Equations 11 and 13, we obtain S ( ρ, A ) = − p log( p ) − p log( p i ) , (40)and we find the entanglement entropy has a contribution only from the classical term. This classicalpiece is the Shannon entropy associated to the probabilities of the single particle appearing in A or ¯ A .In the multi-particle case, we will see that there is generically a quantum piece as well. Nothing in our construction relied on properties of the simple target line R . Indeed, we may consider aparticle moving on some general d -dimensional target space T , with coordinates (cid:126)x . The Hilbert spaceis given by L ( T ) and admits the same decomposition H = V A ⊕ V ¯ A where A ∪ ¯ A = T . We can take A to as complicated a region as we would like. We define the relevant subalgebra as A = (cid:28) {| (cid:126)x (cid:105) (cid:104) (cid:126)x (cid:48) | : (cid:126)x, (cid:126)x (cid:48) ∈ A } ∪ H (cid:29) , (41)which will again have non-trivial center.All subsequent steps follow through straightforwardly. We consider now the general set-up of N particles propagating on a general target space T , for instance T = R d . A large literature exists on the entanglement of identical particles [15], including e.g. analgebraic approach in [16]. However, here we will study the entanglement with respect to partitions of T , not the set of particles. We first study bosons. Denoting the single particle Hilbert space by H = L ( T ) , the physical Hilbertspace is the symmetric quotient 11 N ≡ Sym( H ⊗ N ) ≡ H ⊗ N S N (42)where the S N quotient arises from the indistinguishability of the N particles. That is, H N consists ofpermutation-symmetric wavefunctions ψ ( (cid:126)x , ..., (cid:126)x N ) .Given a partition of the target space T into complementary regions A, ¯ A , we want to associate asub-algebra of observables A ⊂ L ( H N ) to A .We propose the following algebra, A ≡ (cid:28) { P S N ( | (cid:126)x (cid:105) (cid:104) (cid:126)x (cid:48) | ⊗ ⊗ ... ⊗ N ) P S N : (cid:126)x, (cid:126)x (cid:48) ∈ A } ∪ H N (cid:29) , (43)where P S N is the projection onto the symmetric subspace of H ⊗ N , P S N ≡ N ! (cid:88) σ ∈ S N P σ , (44)and where P σ permutes the subsystems according to the permutation σ ∈ S N . The appearance of P S N in Eqn. 43 is crucial for generating all multi-particle operators. The subscripts on the kets areparticle labels, denoting which copy of H within H ⊗ N the operator acts on. For instance, unpackingthe notation for the case of N = 2 , we have P S ( | (cid:126)x (cid:105) (cid:104) (cid:126)x (cid:48) | ⊗ ) P S = 12! ( | (cid:126)x (cid:105) (cid:104) (cid:126)x (cid:48) | ⊗ + ⊗ | (cid:126)x (cid:105) (cid:104) (cid:126)x (cid:48) | ) . (45)To motivate this algebra operationally, note that in ordinary quantum mechanics, if an externalapparatus X situated in region A were coupled to the system of identical particles H N in a way thatrespected permutation symmetry and particle-number conservation, the apparatus X could only becoupled with a Hamiltonian of the form H int = (cid:88) i O Xi ⊗ O Ai (46)for operators O Xi ∈ L ( X ) and O Ai ∈ A . (The fact that interactions must take this form may be moreobvious from the form of the algebra in Eqn. 47.) If observers on A and ¯ A are allowed to performoperations only using such apparatuses, the amount of entanglement distillable through local operationsand classical communication (LOCC) will be equal to the (quantum piece of the) entanglement entropywith respect to A . This operational interpretation follows as a corollary to the discussions in [12–14].To better understand the above algebra, note that we can decompose H N ≡ H ⊗ N S N = ( V A ⊕ V ¯ A ) ⊗ N S N = N (cid:77) k =0 V ⊗ kA S k ⊗ V ⊗ N − kk ¯ A S N − k . (47)where we define V A = V A = C . The sectors indexed by k in the sum correspond to states with k particles in A and N − k particles in ¯ A . It turns out that when our algebra A of Eqn. 43 above12s decomposed in the general way of Eqn. 3, we obtain precisely the above decomposition. That is,schematically, we have A = N (cid:77) k =0 L (cid:32) V ⊗ kA S k (cid:33) ⊗ Sym( V ⊗ N − k ¯ A ) . (48)To justify the above using the definition in Eqn. 43, see Appendix A. The above demonstrates thealgebra decomposes according to the particle number “superselection” sectors reviewed in [15].We may now write down the reduced density matrix. Let Π k be the projector onto the k ’th sectorin the decomposition of Eqn. 47; for an explicit expression, see Appendix A. Following the definitionfor the reduced density matrix in Eqn. 11, we have ρ A = N (cid:88) k =0 p k Tr Sym ( V ⊗ N − k ¯ A ) ρ k ⊗ Sym ( V ⊗ N − k ¯ A ) dim ( Sym ( V ⊗ N − k ¯ A )) (49)where p k ≡ Tr H N (Π k ρ Π k ) (50)and ρ k ≡ p k (Π k ρ Π k ) (51)In particular, for a pure state, we have p k = (cid:18) Nk (cid:19) (cid:90) A dx ...dx k (cid:90) ¯ A dx k +1 ...dx N | ψ ( x , ..., x N ) | , (52)which is the probability of finding k particles in A .We can immediately compute the classical part of the entanglement entropy, (cid:80) k − p k log( p k ) , cor-responding to the Shannon entropy for finding varying numbers of particles in A and ¯ A . Meanwhile,unlike for the case of a single particle, here the blocks ρ k of the density matrix are generically entangledbetween A and ¯ A , so the quantum term of the entanglement entropy is nonzero. All the machinery we have built generalizes quite simply to fermions. In that case, we need to considerthe algebra A F = (cid:28) { P Asym N ( | (cid:126)x (cid:105) (cid:104) (cid:126)x (cid:48) | ⊗ ⊗ ... ⊗ N ) P Asym N : (cid:126)x, (cid:126)x (cid:48) ∈ A } ∪ H N (cid:29) (53)where P Asym N is the projector onto the anti-symmetric subspace of H ⊗ N , defined via P asym N = 1 N ! (cid:88) σ ∈ S N ( − σ P σ . (54)13 Comparison with embedding into second-quantized theory
To define entanglement in a first-quantized theory of many particles, rather than use an algebraicdefinition, we could also translate to the second-quantized picture where a natural tensor factorizationdoes exist. Here, we consider the latter approach and confirm that it agrees with the calculations ofthe previous section.In the second-quantized approach, we consider the N -particle Hilbert space H N ≡ Sym( H ⊗ N ) (55)as a subspace of the Fock space H F ≡ ∞ (cid:77) N =0 Sym( H ⊗ N ) . (56)Let us re-phrase the familiar process of second quantization as the process whereby, given a basis of thesingle-particle Hilbert space H , we induce a tensor factorization of the Fock space H F . For instance,choosing the position basis of H , we write the Fock space as H F = (cid:79) (cid:126)x H (cid:126)x (57)where H (cid:126)x = span {| (cid:105) (cid:126)x , | (cid:105) (cid:126)x , ... } is the countably infinite-dimensional Hilbert space whose basis states | n (cid:105) (cid:126)x indicate n particles occupying position (cid:126)x . For concreteness, using the factorization of Eqn. 57, the zero-particle state in the Fock space lookslike (cid:78) (cid:126)x | (cid:105) (cid:126)x ∈ H F , and the single-particle state | (cid:126)y (cid:105) ∈ H embeds into the Fock space as (cid:16)(cid:78) (cid:126)x (cid:54) = (cid:126)y | (cid:105) (cid:126)x (cid:17) ⊗| (cid:105) (cid:126)y ∈ H F .More generally, by defining raising and lower operators a † (cid:126)x , a (cid:126)x for each factor H (cid:126)x such that | n (cid:105) (cid:126)x = a † (cid:126)x √ n ! | (cid:105) (cid:126)x , we can neatly rewrite the embedding of the state | ψ (cid:105) = (cid:90) d(cid:126)x ...d(cid:126)x N ψ ( (cid:126)x , ...., (cid:126)x N ) | x (cid:105) ⊗ .... ⊗ | x N (cid:105) ∈ H N (58)as a state in the Fock space | ψ (cid:105) F = (cid:90) d(cid:126)x ...d(cid:126)x N ψ ( (cid:126)x , ...., (cid:126)x N ) a † (cid:126)x ...a † (cid:126)x N ( ⊗ (cid:126)x | (cid:105) (cid:126)x ) ∈ H F (59)The “target space” of the first-quantized theory thus becomes the base space of the second-quantizedtheory. The partition of the target space in the first-quantized theory becomes an ordinary partition ofthe base space for the second-quantized theory. Given a region A , we want to check that the algebraicentanglement entropy S ( ρ, A ) of a pure state ρ living on H N matches the ordinary entanglemententropy of ρ when viewed as a state on H F .For a region A , we decompose the single-particle Hilbert space as H = V A ⊕ V ¯ A . (60) The above tensor product is purely formal; it’s a continously indexed tensor product. However, if we chose acountable basis for the single-particle Hilbert space H , rather than the naive basis of position kets, the above tensorproduct would be countably indexed, so that it could be made rigorous. V A , ( V A ) F ≡ ∞ (cid:77) N =0 Sym( V ⊗ NA ) (61) = (cid:79) (cid:126)x ∈ A H (cid:126)x , and likewise for ¯ A .The entire Fock space therefore factorizes as H F = ( V A ) F ⊗ ( V ¯ A ) F (62) = (cid:32) ∞ (cid:77) N =0 Sym( V ⊗ NA ) (cid:33) ⊗ (cid:32) ∞ (cid:77) N =0 Sym( V ⊗ N ¯ A ) (cid:33) (63) = ∞ (cid:77) N,M =0 Sym( V ⊗ NA ) ⊗ Sym( V ⊗ M ¯ A ) (64) ⊃ N (cid:77) k =0 Sym( V ⊗ kA ) ⊗ Sym( V ⊗ N − k ¯ A ) = H N . (65)where the last line uses the decomposition of Eqn. 47. Thus we embed H N ⊂ H F in a way neatlycompatible with the factorization into A , ¯ A .Combining the above embedding with Eqn. 48, which illustrates the structure of the algebra A inthe first-quantized picture, and recalling the definition of S ( ρ, A ) in either Eqn. 13 or 19, we concludethat S ( ρ, A ) = S ( ρ F ) (66)where ρ F indicates the state ρ embedded in the Fock space H F . S ( ρ F ) is the ordinary von Neumannentanglement entropy with respect to the factorization of Eqn. 62, i.e. S ( ρ F ) = Tr A ( ρ A log ρ A ) . (67) In Section 5 we saw that the algebraic entanglement entropy in the first-quantized setting agrees withthe ordinary entanglement entropy in the second-quantized setting. However, we must take care withrelativistic field theories, where we find two competing notions of locality. While one appears quitenatural from the first quantized perspective, the other serves as the standard in most QFT calculationsof entanglement entropy.Consider the free scalar field in d + 1 dimensions. We will discuss two alternative factorizations ofthe Hilbert space, given a spatial partition. Similar discussion appears already in [17]. Afterward, wereturn to the subject of algebraic entanglement entropy.Let us start by reviewing the “ordinary” spatial factorization of a quantum field theory, ignoringsubtleties associated to the continuum [8]. While the content may be familiar, we must be explicit toavoid confusion between the alternative factorizations.15he Hilbert space formally factorizes as H QF T = (cid:79) (cid:126)x ∈ R d P (cid:126)x (68)where P (cid:126)x ≡ span {| φ (cid:105) (cid:126)x : φ ∈ R } ∼ = L ( R ) (69)is the Hilbert space associated to the field degree of freedom living at base point (cid:126)x . This is theordinary tensor factorization of a field theory. When free field theory is viewed as a collection ofcoupled harmonic oscillators, P (cid:126)x is the Hilbert space of the harmonic oscillator ”living" at (cid:126)x .The field operator ˆ φ ( (cid:126)x ) living at a point (cid:126)x is local to the tensor factor P (cid:126)x , and it acts on states φ | φ (cid:105) (cid:126)x ∈ P (cid:126)x as ˆ φ ( (cid:126)x ) | φ (cid:105) (cid:126)x = φ | φ (cid:105) (cid:126)x . (70)Given a field configuration φ : R d → R denoted φ ( x ) , one can then define a field ket | φ (cid:105) ∈ H QF T as the simultaneous eigenstate of the field operators ˆ φ ( (cid:126)x ) with respective eigenvalues φ ( x ) . That is, | φ (cid:105) ≡ (cid:79) (cid:126)x ∈ R d | φ ( x ) (cid:105) (cid:126)x . (71)Finally, the wavefunctional Ψ[ φ ] expands an arbitrary state in H QF T in terms of field kets | φ (cid:105) . Givena region A ⊂ R d and complementary region ¯ A , we obtain a bipartite factorization H QF T = H QF T,A ⊗ H
QF T, ¯ A (72)where H QF T,A = (cid:79) (cid:126)x ∈ A P (cid:126)x (73)and likewise for H QF T, ¯ A .We will call this factorization of the Hilbert space the “ordinary” or “field-based” factorization. Itis the usual factorization used to define entanglement in relativistic field theories, wherein the vacuumexhibits an area law divergence in entanglement entropy. (Again, for a continuum field theory, thisordinary factorization does not actually exist as a tensor product [8].)Meanwhile, we also have a “Fock-based” factorization of the Hilbert space, akin to the factorizationexpressed in Eqn. 57. We utilize the Fock struture of the free theory, H QF T ∼ = H F ≡ ∞ (cid:77) N =0 Sym( H ⊗ N ) , (74)where H is the single-particle Hilbert space. How do we identify the two Hilbert spaces above? Wecan use the momentum basis H = span {| (cid:126)p (cid:105)} for the single-particle space. Let a (cid:126)p , a † (cid:126)p be the ladderoperators that raise/lower the occupancy of the | (cid:126)p (cid:105) state in the Fock space. If we identify | (cid:126)p (cid:105) ∈ H | (cid:126)p (cid:105) ∈ H QF T , then the ladderoperators are related to the field operators in the usual way (taking d + 1 = 3 + 1 for simplicity) ˆ φ ( (cid:126)x ) = (cid:90) d (cid:126)p (2 π ) (cid:112) E (cid:126)p (cid:16) a (cid:126)p e i(cid:126)p · (cid:126)x + a † (cid:126)p e − i(cid:126)p · (cid:126)x (cid:17) , (75) E (cid:126)p ≡ (cid:112) (cid:126)p + m , (76)using the normalization conventions of [18].If we instead choose the position basis for the single-particle Hilbert space H = span {| (cid:126)x (cid:105)} , with themomentum and position basis related by the ordinary Fourier transform, we can define the corespondingladder operators that raise/lower the occupancy of the | (cid:126)x (cid:105) state in the Fock space. These are given by a (cid:126)x ≡ (cid:90) d (cid:126)p (2 π ) e i(cid:126)p · (cid:126)x a (cid:126)p , (77) a † (cid:126)x ≡ (cid:90) d (2 π ) (cid:126)pe − i(cid:126)p · (cid:126)x a † (cid:126)p , (78)and these ladder operators are local to the factors of the tensor factorization in Eqn. 57, H F = (cid:79) (cid:126)x H (cid:126)x , (79)where the local Hilbert spaces H (cid:126)x = span {| (cid:105) (cid:126)x , | (cid:105) (cid:126)x , ... } have basis states | n (cid:105) (cid:126)x that count the numberof particles occupying single-particle state (cid:126)x .This defines the “Fock-based” factorization referred to above. The ladder operators a (cid:126)x † , a (cid:126)x raise andlower the particle number of the free theory. In this factorization, the vacuum is just the zero-particlestate (cid:78) (cid:126)x | (cid:105) (cid:126)x . Note this is a product state! The vacuum is unentangled with respect to the Fockfactorization. Clearly, the Fock-based factorization differs from the ordinary field-based factorization.To sharpen the distinction between the factorizations, let us define ladder operators α (cid:126)x , α † (cid:126)x associ-ated to the “harmonic oscillator” Hilbert space P (cid:126)x , the local degrees of freedom in the ordinary tensorfactorization. That is, take α (cid:126)x = 1 √ φ ( x ) + i ˆ π ( (cid:126)x )) , (80) α † (cid:126)x = 1 √ φ ( x ) − i ˆ π ( (cid:126)x )) were ˆ π ( (cid:126)x ) is the canonical conjugate of the field operator ˆ φ ( (cid:126)x ) , acting as − i δδφ ( (cid:126)x ) on the wavefunctional.Note these are not the same as the ladder operators a (cid:126)x † , a (cid:126)x associated to the Fock-based factorization.Any operator local to (cid:126)x in the ordinary factorization should commute with φ ( (cid:126)y ) for all (cid:126)y (cid:54) = (cid:126)x ,whereas [ a (cid:126)x , ˆ φ ( (cid:126)y )] ∝ K ( (cid:126)x, (cid:126)y ) , (81)where K ( (cid:126)x, (cid:126)y ) is the convolution kernel K ( (cid:126)x, (cid:126)y ) ≡ (cid:90) d (cid:126)p (2 π ) e i(cid:126)p · ( (cid:126)x − (cid:126)y ) (cid:112) E (cid:126)p , (82)17mphasizing that the operators a (cid:126)x † , a (cid:126)x local in the Fock-based factorization are slightly non-local inthe ordinary field-based factorization.In one sense, the two factorizations are “close,” because the kernel K ( x, y ) is peaked near (cid:126)x ∼ (cid:126)y .Thus an operator local to a region A in the Fock-based factorization will be well-approximated by anoperator local to a sufficiently larger B ⊃ A in the ordinary factorization.In another sense, the alternatives yield drastically different entanglement entropies: the vacuum isunentangled in the Fock-based factorization, while it exhibits diverging entanglement in the ordinaryfactorization. It turns out that multi-particle excited states yield a middle ground: if the wavefunc-tions of the particles are sufficiently spread, the two factorizations will yield approximately equalentanglement entropies, up to a correction which is precisely the vacuum entanglement.Which factorization is “correct”? Of course they merely constitute different choices. If we wantto leverage the operational interpretation of entanglement entropy, we must ask which algebra ofobservables is available to an observer who “has access to region A ” ? We will not further pursue thisquestion, but a point in favor of the ordinary factorization is that the Hamiltonian is truly local withrespect to this factorization. Moreover, only in the ordinary factorization is there a strict lightcone,i.e. exact commutation of spacelike-separated Heisenberg operators. Our algebraic setup calculates the entanglement entropy relative to the Fock-based tensor productfactorization of the QFT. In this section, we show there is a sense in which the entanglement entropyof a multi-particle state decomposes into the universal, divergent area-law piece and an additive contri-bution we can associate to the particles’ wavefunction. The extra entanglement due to the excitationshas been called the “excess of entanglement” above the vacuum [19, 20]. We will address the simplecase of single-particle excitations, but the account of finitely multi-particle excitations is similar.Our calculation of the entanglement entropy of single-particle excitations (with respect to theordinary tensor factorization) has precedent in the related calculations of [19, 20]. However, thosearguments only apply to momentum eigenstates, rather than to excited states with more generalwavefunctions. The argument sketched here has a different scope.In the Fock basis, we can describe a single particle state as | ψ (cid:105) = (cid:90) d xψ ( x ) a † (cid:126)x | (cid:105) = (cid:90) d p (2 π ) ˜ ψ ( p ) a † (cid:126)p | (cid:105) (83)The entanglement entropy for a spatial subgion A , can be immediately computed as H ( { p ( a x ) A , − p ( a x ) A } ) = − p ( a x ) A log( p ( a x ) A ) − (1 − p ( a x ) A ) log(1 − p ( a x ) A ) (84)with p ( a x ) A = (cid:90) A d x | ψ ( x ) | (85)18e wish now wish to compare this to the entanglement entropy computed relative to the field-basedfactorization. First, let us rewrite the state | ψ (cid:105) as | ψ (cid:105) = (cid:90) d xf ( x ) ˆ φ ( (cid:126)x ) | (cid:105) = (cid:90) d p (2 π ) ˜ f ( p ) (cid:112) E (cid:126)p a † (cid:126)p | (cid:105) (86)Eqn. 83 therefore identifies ˜ ψ ( p ) = ˜ f ( p ) √ E (cid:126)p , or alternatively, in position space ψ ( (cid:126)x ) = (cid:82) d(cid:126)yK ( (cid:126)x, (cid:126)y ) f ( y ) .Below, we give a proof (on the lattice) that we may well approximate the entanglement entropy as S ( ρ A ) ≈ S + H ( { p A , − p A } ) (87)where S is the entanglement of the vacuum and H ( { p A , − p A } ) = − p A log( p A ) − (1 − p A ) log(1 − p A ) is the Shannon entropy of the classical probability distribution, but now with p A = (cid:82) A d x | f ( (cid:126)x ) | .Before delving into the mechanics of the proof, we stress we may meaningfully compare the Shannonentropies H ( { p A , − p A } ) and H ( { p ( a x ) A , − p ( a x ) A } ) . When the kernel K ( (cid:126)x, (cid:126)y ) is narrowly peaked near (cid:126)x ∼ (cid:126)y , and the regions A are taken sufficiently large, these quantities are in fact close (at least on thelattice). In fact, H ( { p ( a x ) A , − p ( a x ) A } ) → H ( { p A , − p A } ) (88)precisely in the limit described in Section 7.1 below, as the wavefunctions are spread over large regions. Consider a free, massive scalar field on a finite square lattice in d spatial dimensions. The discretizedfield theory Hamiltonian is that of coupled harmonic oscillators, H = (cid:88) x i φ ( x i ) + π ( x i ) + (cid:88) (cid:104) x i ,x j (cid:105) m ( φ ( x i ) − φ ( x j )) , (89)for fields φ ( x i ) at site i and conjugate momenta π ( x i ) . We consider the single-particle excitation | ψ (cid:105) = (cid:88) x i f ( x i ) φ ( x i ) | Ω (cid:105) , (90)not necessarily an energy or momentum eigenstate, where | Ω (cid:105) is the vacuum, and f ( x i ) is some“wavefunction” of the discrete positions x i , normalized so that the overall state is normalized. Partitionthe lattice into complementary, contiguous regions A, ¯ A , and consider the reduced state ρ A . We wantto show that S ( ρ A ) ≈ S + H ( { p A , − p A } ) (91)where S is the entanglement of the vacuum, H ( { p A , − p A } ) = − p A log( p A ) − (1 − p A ) log(1 − p A ) isthe Shannon entropy of the classical probability distribution, and where p A ≡ (cid:88) x i ∈ A | f ( x i ) | (92) However, note the norm of | ψ (cid:105) is not given by (cid:80) i | f ( x i ) | , because the states φ ( x i ) | Ω (cid:105) are not orthogonal for distinct i .
19s essentially the probability of finding the particle in A (at least for large A , due to subtleties aboutmeasuring particle position in this context). Eqn. 91 will hold with small error when the system haslarge volume and the wavefunction f i is not too concentrated around the boundary of A, ¯ A . To bemore precise, let X R ⊂ A be the sub-region of A consisting of sites at a distance larger than R latticeunits from the boundary ∂A , and let B R = A \ X R be the buffer region between X and A . We canquantify the amount of the wavefunction f ( x i ) concentrated in the buffer region as p B R ≡ (cid:88) x i ∈ B R = A \ X R | f ( x i ) | . (93)We will prove that for p B R sufficiently small for a choice of buffer size R sufficiently large, and for totallattice volume sufficiently large, Eqn. 91 holds to arbitrarily good approximation. That is, we showEqn. 91 holds exactly in the limit of a sequence of systems and wavefunctions where | A | , | ¯ A | → ∞ , and p B R → for a choice of buffer sizes R → ∞ . One could also prove the result with more fine-grainederror analysis, but proving the simple limit will serve our illustration.Now we sketch the proof. Proof sketch.
Divide the state into two terms | ψ (cid:105) = (cid:88) x i ∈ A f ( x i ) φ ( x i ) | Ω (cid:105) + (cid:88) x i ∈ ¯ A f ( x i ) φ ( x i ) | Ω (cid:105) (94) ≡ | ψ A (cid:105) + | ψ ¯ A (cid:105) . (95)We can approximate the state instead as | ψ (cid:105) ≈ | (cid:101) ψ (cid:105) ≡ (cid:88) x i ∈ X R f ( x i ) φ ( x i ) | Ω (cid:105) + (cid:88) x i ∈ ¯ A f ( x i ) φ ( x i ) | Ω (cid:105) (96) ≡ | ψ X R (cid:105) + | ψ ¯ A (cid:105) . (97)Then (cid:104) (cid:101) ψ | ψ (cid:105) = 1 − (cid:104) ψ B R | ψ (cid:105) → (98)in the given limit where p B R → , so | (cid:101) ψ (cid:105) approaches | ψ (cid:105) in trace-distance. Then the entanglemententropy of | (cid:101) ψ (cid:105) approaches the entanglement entropy of | ψ (cid:105) , using the continuity of the entanglemententropy with respect to trace distance [21]. The continuity result of [21] requires the same assumptionsas those discussed in Section 9, which the single-particle states here satisfy. Thus we can examine theentanglement entropy of | (cid:101) ψ (cid:105) rather than | ψ (cid:105) . The reduced density matrix has four terms Tr ¯ A ( | (cid:101) ψ (cid:105)(cid:104) (cid:101) ψ | ) = Tr ¯ A ( | ψ B R (cid:105)(cid:104) ψ B R | + | ψ B R (cid:105)(cid:104) ψ ¯ A | + | ψ ¯ A (cid:105)(cid:104) ψ B R | + | ψ ¯ A (cid:105)(cid:104) ψ ¯ A | ) . (99)Let’s start with the fourth term, call it σ A ≡ Tr ¯ A ( | ψ ¯ A (cid:105)(cid:104) ψ ¯ A | ) . Note that the connected correlationfunctions of local operators exponentially decay with distance in this massive free lattice theory. Ac-tually, we use the stronger fact that the mutual information I ( B R : ¯ A ) in the vacuum tends to zeroas the size R of the buffer region increases, which can be shown with the methods of [1]. Then theconnected correlation of any bounded operators on B R and ¯ A must tend to zero for large R , using the In a more detailed argument, some care must be taken with how the continuity bound depends on lattice size. O A on A with operator norm 1, Tr( σO A ) = (cid:104) Ω | O A (cid:88) x i ∈ ¯ A f ( x i ) φ ( x i ) | Ω (cid:105) (100) → (cid:104) Ω | O A | Ω (cid:105)(cid:104) Ω | (cid:88) x i ∈ ¯ A f ( x i ) φ ( x i ) | Ω (cid:105) (101) → (cid:104) Ω | O A | Ω (cid:105) (1 − p A ) , (102)where again all limits are taken as described above. Thus the state σ/ (1 − p A ) approaches the reducedstate of the vacuum.Likewise, we have Tr A ( | ψ B R (cid:105)(cid:104) ψ B R | ) /p A approaching the reduced state of the vacuum as well.(Note the previous expression traces out A rather than ¯ A , but the entanglement entropy will be thesame the same whether we trace out A or ¯ A .)Now we analyze the second and third terms of Eqn. 99. These terms are actually equal by theHermiticiy of φ ; call this term τ ≡ Tr ¯ A ( | ψ ¯ A (cid:105)(cid:104) ψ B R | ) . Similar to the above calculation, we have Tr( τ O A ) → (cid:104) Ω | O A | Ω (cid:105)(cid:104) Ω | (cid:88) x i ∈ ¯ A f ( x i ) φ ( x i ) | Ω (cid:105) = 0 (103)for any fixed norm operator O A local to A , again using the fact that I ( B R : ¯ A ) → in the vacuum as R increases. The RHS is zero above simply because φ has zero vacuum expectation. So τ → , andwe can discard these terms.Finally, the first and fourth terms of Eqn. 99 tend to orthogonal operators, so the entropy of theirsum is the average of their entropies, plus the Shannon entropy associated to their traces. We conclude S ( ρ A ) → S + H ( { p A , − p A } ) (104)in the given limit, as desired. Above we considered the spatial entanglement in first-quantized many-particle systems, alternativelyinterpreted as target space entanglement in a 0+1-dimensional theory. Now we consider target spaceentanglement for a more general d + 1 -dimensional theory, i.e. with a higher-dimensional base space.This quantity is more akin to what might be desired in worldsheet string theory, if one desires topartition the target spacetime. However, worldsheet string theory will offer further complications dueto re-parameterization invariance, as further discussed in Section 10.Consider a d + 1 -dimensional field theory defined on the base space B (for example, B ∼ = R d ), withits field φ taking values in the target space T . Referring again to the ordinary base space factorization Because the operators φ ( x i ) are not bounded, and the mutual information only upper bounds connected correlationsof bounded operators, we cannot directly apply the upper bound as stated. However, it turns out the action of φ ( x i ) onthe vacuum can be sufficiently well-approximated by the action of a bounded operator for our purpose. H QF T = (cid:79) (cid:126)x ∈ B P (cid:126)x P (cid:126)x ≡ span {| φ (cid:105) (cid:126)x : φ ∈ T } ∼ = L ( T ) . Consider a partition of target space T into complementary regions A, ¯ A ⊂ T . We associate the followingalgebra to the target space region A , A ≡ (cid:28) {| φ (cid:105) (cid:126)x (cid:104) φ (cid:48) | (cid:126)x ⊗ (cid:126)y (cid:54) = (cid:126)x P (cid:126)y : φ, φ (cid:48) ∈ A, ∀ (cid:126)x ∈ B } ∪ H QFT . (cid:29) (105)Defining the projector Π (cid:126)xA on local Hilbert space P (cid:126)x at (cid:126)x as Π (cid:126)xA ≡ (cid:90) φ ∈ A dφ | φ (cid:105) (cid:126)x (cid:104) φ | (cid:126)x . (106)we can immediately write down the center of A Z ( A ) = (cid:28) { Π (cid:126)xA ⊗ (cid:126)y (cid:54) = (cid:126)x P (cid:126)y : ∀ (cid:126)x } ∪ H QFT . (cid:29) (107)These algebras resemble those of Section 4.3.1 because we may view our QFT as the first-quantizedtheory of many distinguishable particles, labeled by (cid:126)x , moving in target T with coordinates φ .With this definition in hand, we can calculate the target space entanglement entropy of generalfield theories defined on the lattice. Unfortunately, the calculation does not appear straightforward. We settle for the simplest non-trivial example: a massive scalar “field theory” on two spatial latticepoints, i.e. two coupled harmonic oscillators.Consider two field degrees of freedom φ and φ at lattice points 1 and 2, with Hilbert space H = L ( R ) ⊗ L ( R ) and Hamiltonian H = 12 ( π + π + ( φ − φ ) + m ( φ + φ )) (108)where π , π are the conjugate momentum operators and m the mass. The interaction term ( φ − φ ) comes from the lattice discretized spatial gradient of the field.We choose to compute the entanglement entropy of the ground state, partitioning the target spaceinto the positive and negative half-lines, A ≡ { φ ∈ R : φ > } ⊂ R . In this case, the algebra A of Eqn.105 has four sectors. Defining the projectors Π A , Π A , Π A , Π A as in Eqn. 105, the four sectors of A aresimply the images of the four projectors Π A Π A , Π A Π A , Π A Π A , Π A Π A . The sector projected onto by Π A Π A indicates “the field on both lattice points takes values in A ,” while the sector projected onto by Π A Π A indicates φ ∈ A, φ ∈ ¯ A , and so on. Note there are four rather than three sectors because thelattice sites are distinguishable.The normalized ground state wavefunction for the Hamiltonian (108) is given by ψ ( φ , φ ) = (cid:16) ω + ω − π (cid:17) / e − ( ω + φ + ω − φ − ) (109)22igure 3: Base (Left) vs. Target (Right) Space Partition & Associated Algebras for scalar field on twolattice sites.where φ ± = ( φ ± φ ) / √ , ω + = m , ω − = √ m + 2 ; see for instance the similar example in [23].We consider the state ρ = | ψ (cid:105)(cid:104) ψ | projected separately onto the four sectors described above. In thesector where φ , φ ∈ A , the Hilbert space factorizes in a trivial way, as in the discussion surroundingEqn. 27 for the particle on a line. Hence the projection of the state onto this sector yields a productstate, with no contribution to the quantum piece of the entanglement entropy. The same holds forthe sector associated to φ , φ ∈ ¯ A . The only contribution to the quantum part of the entanglemententropy thus comes from from the two sectors where φ , φ are in different regions of the target space.Since the groundstate is symmetric under the exchange of φ ↔ φ the contribution in each such sectorwill be identical. Thus we need only consider one sector, say the image of Π A Π A .The sector factorizes as V A ⊗ V A , where V A ≡ span {| φ (cid:105) : φ ∈ A } and V A ≡ span {| φ (cid:105) : φ ∈ ¯ A } .We need to take the state projected on this sector, Π A Π A | ψ (cid:105)(cid:104) ψ | Π A Π A , and trace out the second factor V A . We obtain the (non-normalized) density matrix σ on V A given by σ ( x , y ) = (cid:90) x ∈ ¯ A dx ψ ( x , x ) ψ ( y , x ) ∗ . (110)The integral above can be expressed in terms of error functions. To calculate the entanglement entropy,it remains to diagonalize the above density matrix σ . Returning attention to the full reduced state ρ A , the classical part of the entanglement entropy is then given by S ( ρ, A ) classical = H ( { p, p, − p, − p } ) (111)where H ( {·} ) is the classical (Shannon) entropy of the probability distribution, and p = Tr( σ ) , with σ We find this quite reminiscent of the early days of base space entanglement entropy, where analytical progressappeared similarly difficult. Both [22, 23] ultimately resorted to numerical methods to discover the area law in theground state of scalar field theory. What we need is a suitable analog of the powerful path integral replica trick. S ( ρ, A ) quantum = 2 pS ( σ/ Tr( σ )) . (112)In lieu of an analytic method, we discretize the x , y coordinates of Eqn. 110 and numericallydiagonalize the resulting finite matrix. We ensure that the discretization is at sufficient resolution thatthe results converge when decreasing the spacing or increasing the total number of discretized points.Section 9 guarantees convergence, the end result being finite. Ultimately we produce a numericalanswer for the quantum and classical piece of the entanglement entropy of the ground state, as afunction of the mass in the Hamiltonian. The results are depicted in Figure 4. Numerical error due todiscretization appears to be somewhat smaller than − , but we do not include a rigorous analysis.Figure 4: Entanglement entropy with respect to a partition of target space, for two coupled oscillatorsgoverned by the Hamiltonian in Eqn. 108.At high mass, the two harmonic oscillators approximately de-couple. The wavefunction spreadsequally between the four sectors, so that the classical piece of the entanglement entropy gives two bits.Meanwhile, the quantum piece of the entanglement entropy tends to zero, because the only sectors thatcan contribute must have φ and φ in different regions A and ¯ A , and in these sectors, the wavefunctionapproximately factorizes due to the de-coupling of the oscillators.Figure 4 also illustrates that the quantum term of the target space entanglement entropy is notmonotonic with respect to the mass parameter. The non-monotonicity is associated with the fact thatin Eqn. 112, the first factor p increases monotonically with mass, whereas the second factor S ( σ/ Tr( σ )) decreases monotonically. The entanglement entropies discussed in this paper involve infinite-dimensional Hilbert spaces andalgebras. In infinite dimensions, we must take care that density matrices and entropies remain well-defined. Fortunately, we will see that most of the infinities present here are of a relatively tamevariety. 24n this section, we will take more mathematical care, recalling for instance that the “positioneigenstate” | ψ (cid:105) is not a true state in the Hilbert space L ( R ) as traditionally defined.The algebra associated to a region in multi-particle quantum mechanics (like Eqn. 43) is a finitedirect sum of factors, where each factor is an infinite-dimensional “Type I ” factor, according to thetype theory of von Neumann algebras [24]. Type I factors are algebras which are isomorphic to the fullalgebra of bounded operators on some Hilbert space. The Type I property of this algebra is thereforeapparent from the schematic form of the algebra in Eqn. 48.Similarly, an algebra associated to a region in the target space of a lattice field theory – like thealgebra in Eqn. 105 on a finite lattice, or the algebra in Figure 3 – is also a direct sum of Type I sectors, even when the target space is infinite-dimensional.For the general algebra decomposition of Eqn. 3 to make sense as written, it is indeed essential thealgebra is a direct sum of Type I factors. Otherwise the use of the tensor product there is incorrect.Even for these Type I algebras, we must take care with the entanglement entropy. The formulafor the algebraic entanglement entropy in Eqn. requires defining the von Neumann entropy of thepartial trace of a pure state in a bipartite Hilbert space H ⊗ H , where the factors H , H maybe countably infinite-dimensional. (The full algebraic entanglement entropy was then a sum of suchentropies in each sector of the algebra.) We therefore focus on the question of ordinary von Neumannentanglement entropies of pure states in bipartite Hilbert spaces. For any (mathematically legitimate,i.e. normalizable) state | ψ (cid:105) ∈ H ⊗ H , the partial trace ρ = Tr ( | ψ (cid:105)(cid:104) ψ | ) can be taken using any(legitimate) orthonormal basis. The result will be a trace-class Hermitian operator ρ . (To see that ρ istrace class, we can take its trace in any orthonormal basis, and the resulting sum will be convergent bythe normalizability of | ψ (cid:105) .) Recall that a trace-class Hermitian operator ρ has an eigen-decomposition ρ = ∞ (cid:88) i =1 λ i | v i (cid:105)(cid:104) v i | (113)for some countably infinite set of eigenvectors {| v i (cid:105)} and eigenvalues λ i . Thus we are in the positionto define the entanglement entropy S ( ρ ) ≡ ∞ (cid:88) i =1 − λ i log( λ i ) . (114)However, the above sum may be infinite, even though a normalized state | ψ (cid:105) guarantees (cid:80) i λ i = 1 .In fact, the set of states | ψ (cid:105) with infinite entanglement entropy is dense in the total Hilbert space H ⊗ H , so in some sense the divergence is generic.Yet, for states of interest, the sum is often finite. For instance, a finite energy condition may implyfiniteness. The authors of [21] prove that, for any non-interacting Hamiltonian H = H ⊗ + ⊗ H on H ⊗H with discrete spectrum such that Tr( e − βH ) is finite for all β > , any state | ψ (cid:105) ∈ H that hasfinite expected energy (cid:104) ψ | H | ψ (cid:105) < ∞ with respect to this Hamiltonian will have finite entanglemententropy. Note the state | ψ (cid:105) may have nonzero overlap with energy eigenstates of arbitrarily highenergy; as long as the expected energy is finite, the theorem applies.Although the theorem of [21] requires one to find a non-interacting Hamiltonian with respect towhich | ψ (cid:105) has finite energy, this reference Hamiltonian need not bear any relation to the dynamics ofthe system of interest. Rather, the assumption of finite energy with respect to the reference merely25nsures that ρ , which might have infinite nonzero eigenvalues, nonetheless has sufficiently accuratelow-rank approximations. For instance, if we have H , H = L ( R d ) and one chooses H to be theHamiltonian of two independent harmonic oscillators, H = (cid:126)p + (cid:126)p + (cid:126)x + (cid:126)x , (115)then any state | ψ (cid:105) with a smooth spatial wavefunction that decays at spatial infinity at least as fastas /r ( d +3) / will have finite energy with respect to H , and hence the theorem of [21] implies thislarge class of wavefunctions has finite entanglement entropy. (If one tries to weaken this condition toinclude wavefunctions that are not smooth but decay, or decay but are not smooth, counterexamplesexist with infinite entanglement entropy in both cases.) In particular, entanglement entropy of thedensity matrix in Eqn. 110 will be finite, as corroborated by the convergence of the numerics used forFig. 4.Similarly, the algebraic entanglement entropies of Section 5 will be finite for states with smooth,decaying wavefunctions. The finiteness highlights the difference between the two notions of localitydiscussed in Section 6. Our first-quantized algebraic approach uses Type I algebras and gives finiteentanglement entropies, whereas the ordinary “factorization” of field theory gives area-law divergences,associated to the Type III sub-algebras present in field theory.
10 Discussion
This work has highlighted the largely unexplored realm of target space partitions and their relevancein the quantum gravitational context. However, we remain far from our original hope of using analgebraic approach to define target space entanglement entropy in worldsheet string theory.In ordinary field theory, our algebraic definition successfully captured the entanglement entropywith respect to a certain factorization. However, as discussed in Section 6, the field theory admits atleast two seemingly natural factorizations, which we called the “Fock-based” and “field-based” factor-izations. It is the entanglement with respect to the former that is captured by our algebraic definition,whereas only the latter factorization exhibits the divergent area law contribution. On one hand, thecalculation outlined in Section 6 demonstrates that the additional entanglement of particle excitationsatop the vacuum can be meaningfully compared between the two factorizations. On the other hand,the first-quantized algebraic approach remains unable to analyze the area law contribution itself.In fact, it might appear senseless to imagine a first-quantized particle description teaching usanything about the spatial structure of the vacuum. The wordline framing of QFT suggests otherwise.Consider the relativistic free massive scalar field. (Interactions can be incorporated but are not thefocus of the argument.) The logarithm of its partition function can be recast as the path integral of a26oint particle coupled to -dimensional gravity on its worldline [25] : log Z QF T = log (cid:90)
Dφe − (cid:82) d d x √ g ( g µν ∂ µ φ ( x ) ∂ µ φ ( x )+ m φ ( x ) )= −
12 Tr (cid:2) log (cid:0) − g µν ∇ µ ∇ ν + m (cid:1)(cid:3) = (cid:90) d d x (cid:104) x µ | (cid:90) ∞ (cid:15) ds s e − s ( p µ p µ + m ) | x µ (cid:105) = (cid:90) d d x (cid:90) y µ ( s )= x µ y µ (0)= x µ Dy µ ( τ ) De ( τ ) V ol ( Dif f ) e − (cid:82) s dτe ( e ∂ τ y µ ( τ ) ∂ τ y ν ( τ ) g µν ( y ( τ ))+ m ) This worldline approach to field theory is the most immediate field-theoretic analog of worldsheetstring theory. In the worldline setting, we know we can access the area law entanglement pattern ofthe QFT via a replica trick Euclidean path integral. Schematically, we can compute it as S EE = (1 − n∂ n ) log Z QF T [ n ]) (cid:12)(cid:12)(cid:12)(cid:12) n =1 (116)The right hand side of this equation, including the the necessary field-theoretic U V -regulator, maybe completely recast in terms of worldline quantities. The euclidean path integral immediately givesus the entropy. Its Lorentzian interpretation on the other hand, remains elusive.Indeed, while the area law manifests itself as above in the worldline formalism, it is unclear thereexists any partition of the point particle Hilbert space that yields this entropy. An algebraic approachwould require such a partition. However, hope remains. Two salient features deserve further notice.Firstly, we see the spatial arguments of the fields, the x µ in φ ( x µ ) appear as boundary conditions onthe worldline trajectories. This is the familiar statement that, in string theory, D-branes help us probetarget space locality [26, 27]. Note the states | x µ (cid:105) do not belong to the physical subspace of the pointparticle Hilbert space, as they do not satisfy the constraint ˆ p + m | ψ (cid:105) = 0 . In the language of [25],they do not reside in the BRST coholomogy of Q BRST = c ( p + m ) . This simply reflects the fact thatreparametrization invariance breaks down at the endpoints of the worldline. Secondly, from a morealgebraic perspective, we know that any reduced density matrix reproducing all two-point correlationfunctions (cid:104) φ ( x µ ) φ ( y ν ) (cid:105) , (cid:104) φ ( x µ )Π( y ν ) (cid:105) and (cid:104) Π( x µ )Π( y ν ) (cid:105) for x µ , y ν ∈ A will gives us the field theoryentanglement entropy relative to the φ ( x ) tensor product factorization. These correlators can also berewritten purely in terms of worldline variables: (cid:104) φ ( x µ ) φ ( y ν ) (cid:105) = (cid:90) ∞ (cid:15) ds (cid:104) x µ | e − s ( p + m ) | y ν (cid:105) (117)We would therefore need to consider some sort of restriction on the set of allowable “D-branes” forthe worldline. While we have not succeeded in defining an associated reduced density matrix, it is atleast clearly a Lorentzian setup.The single particle Hamiltonian (cid:112) (cid:126)p + m considered in Section 6 arises via gauge fixing therelativistic point particle action. More precisely, it is the canonical Hamiltonian after choosing staticgauge x ( τ ) = τ . We feel there is something important about reparametization invariance we have yetto pinpoint, and hope to explore this avenue in future work. This might be more familiar under the guise of the “Schwinger paramterization” of Feynman diagrams. c =1 matrix quantum mechanics The discussion in Section 6, teasing out the subtle differences in our notion of “spatial locality,” couldappear somewhat artificial. Yet such competing notions of locality might, in fact, be rather genericwithin the emergent spacetime paradigm. The c = 1 matrix quantum mechanics provides a sharpholographic example. As highlighted in [5], there exists at least two seemingly natural emergent spatialdimensions. On one hand, the matrix quantum mechanics in the singlet sector can be recast as a localfermionic field theory on matrix eigenvalue space. On the other hand, the low-energy target spacedynamics, derived from the worldsheet Liouville string theory, is most naturally formulated in terms ofthe string embedding coordinates X (the c = 1 boson) and φ (the Liouville field). Section 11 of [29]shows precisely how bosonization of the matrix model’s fermionic field theory maps onto the closedstring tachyon dynamics in the target spacetime. In momentum space, a simple multiplicative phasefactor relates the two - the celebrated “leg-pole factor” - in close parallel to the φ ( p ) ∼ (2 E p ) − / a † p example discussed in Section 7. In position space, this gives a non-local map. Natsuume and Polchinskiargued all the (admittedly very simple) gravitational dynamics on the 2d target space were encoded inthe matrix model via this non-local map [30]. Reference [31] reproduced the entanglement entropy ofthe bulk 2d tachyon by partitioning the matrix eigenvalue space. It failed, however, in capturing any O (1 /g st ) contribution - the closest 2d relative of Area/4 G . One might blame this on choosing a notionof locality similar to the Fock space factorization discussed above, thereby capturing only excitationsaround the background. Making this precise might help guide future attempts at diagnosing emergentlocality from matrix degrees of freedom.
11 Acknowledgements
It is a pleasure to thank Sean Hartnoll for stressing over the years the importance of defining spatialentanglement entropy in first-quantized (matrix) quantum mechanics. We thank him, Tom Hartmanand Jordan Cotler for many fruitful discussions and early collaborations on this topic. We also grate-fully acknowledge helpful conversations with Eva Silverstein and Ronak Soni. Finally, we would like tothank the Yukawa Institute for Theoretical Physics in Kyoto, where part of this work was completedduring the workshop “Quantum Information and String Theory.” [28] points out important subtleties in viewing the Liouville direction as spatial coordinate. They consider insteadyet another space on which they define a string field theory of loop operators, parametrized by the length of the stringsthey create. Here we provide more detail justifying Eqn. 43 using the definition of the algebra A in Eqn. 48.First off, we can write down the commutant A (cid:48) as A = (cid:28) { P S N ( | (cid:126)x (cid:105) (cid:104) (cid:126)x (cid:48) | ⊗ ⊗ ... ⊗ N ) P S N : (cid:126)x, (cid:126)x (cid:48) ∈ ¯ A } ∪ H N (cid:29) (118)so that the center Z ( A ) is generated by Z = (cid:28) { P S N (cid:18)(cid:90) A d(cid:126)x | (cid:126)x (cid:105) (cid:104) (cid:126)x | ⊗ ⊗ ... ⊗ N (cid:19) P S N } ∪ H N (cid:29) (119)At this point we wish to identify the minimal projectors which span the center Z , as in Eqn. 7.Here there are N + 1 such projectors, which we can write as Π k = (cid:18) Nk (cid:19) P S N Π A ⊗ ... ⊗ Π A (cid:124) (cid:123)(cid:122) (cid:125) k times ⊗ Π ¯ A ⊗ ... ⊗ Π ¯ A (cid:124) (cid:123)(cid:122) (cid:125) ( N − k ) times P S N (120)Physically, Π k is the projector onto the subspace with k particles in A and N − k particles in ¯ A .The algebra Π k A Π k projected onto this subspace takes the form Π k A Π k = (cid:28) P S N ( | (cid:126)x (cid:105) (cid:104) (cid:126)x (cid:48) | ⊗ Π A ... ⊗ Π A ⊗ Π ¯ A ⊗ ... ⊗ Π ¯ A ) P S N : (cid:126)x, (cid:126)x (cid:48) ∈ A (cid:29) (121)has trivial center on Π k H N . To see this, we first write its commutant restricted to the subspace A (cid:48) (cid:12)(cid:12) Π k H N = (cid:28) P S N ( | (cid:126)x (cid:105) (cid:104) (cid:126)x (cid:48) | ⊗ Π ¯ A ... ⊗ Π ¯ A ⊗ Π A ⊗ ... ⊗ Π A ) P S N : (cid:126)x, (cid:126)x (cid:48) ∈ ¯ A (cid:29) (122)so that indeed the center on this subspace is trivial A ∩ A (cid:48) (cid:12)(cid:12) Π k H N = P S N Π A ⊗ ... ⊗ Π A (cid:124) (cid:123)(cid:122) (cid:125) k times Π ¯ A ⊗ ... ⊗ Π ¯ A (cid:124) (cid:123)(cid:122) (cid:125) N − k times P S N = (cid:12)(cid:12) Π k H N (123)Since the algebra restricted to each subspace is a factor, we know there exists a tensor productfactorization in each block such that all O ∈ A take the form O = ⊕ Nk =0 O A k ⊗ ¯ A k (124)What is this tensor product factorization? It is nothing but the decomposition ( H A ⊕ H ¯ A ) ⊗ N S N = ⊕ Nk =0 (cid:32) H ⊗ kA S k ⊗ H ⊗ N − k ¯ A S N − k (cid:33) (125)where we define H A = H A = C . In particular Π k H N = H ⊗ kA S k ⊗ H ⊗ N − k ¯ A S N − k (126)29inally, we stress the symmetric projectors P S N are crucial for A to contain multi-particle operators.For example, in the case of N = 2 , multiplying two basis algebra elements can generate all symmetric2-particle operators: P S ( | (cid:126)x (cid:105) (cid:104) (cid:126)x (cid:48) | ⊗ ) P S P S ( | (cid:126)y (cid:105) (cid:104) (cid:126)y (cid:48) | ⊗ ) P S = 12! P S | (cid:126)x (cid:105) (cid:104) (cid:126)x (cid:48) | ⊗ | (cid:126)y (cid:105) (cid:104) (cid:126)y (cid:48) | P S (127) + 12! δ ( x (cid:48) − y ) P S ( | (cid:126)x (cid:105) (cid:104) (cid:126)y (cid:48) | ⊗ ) P S (128) References [1] Horacio Casini and Marina Huerta. Entanglement entropy in free quantum field theory.
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