Entanglement and algebraic independence in fermion systems
aa r X i v : . [ qu a n t - ph ] M a y June 13, 2018 16:53 WSPC/INSTRUCTION FILE paper
International Journal of Quantum Informationc (cid:13)
World Scientific Publishing Company
ENTANGLEMENT AND ALGEBRAIC INDEPENDENCE INFERMION SYSTEMS
FABIO BENATTI
Dipartimento di Fisica, Universit`a di Trieste, 34151 Trieste, ItalyIstituto Nazionale di Fisica Nucleare, Sezione di Trieste, 34151 Trieste, [email protected]
ROBERTO FLOREANINI
Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, 34151 Trieste, Italyfl[email protected]
Received Day Month YearRevised Day Month YearIn the case of systems composed of identical particles, a typical instance in quantum sta-tistical mechanics, the standard approach to separability and entanglement ought to bereformulated and rephrased in terms of correlations between operators from subalgebraslocalized in spatially disjoint regions. While this algebraic approach is straightforwardfor bosons, in the case of fermions it is subtler since one has to distinguish betweenmicro-causality, that is the anti-commutativity of the basic creation and annihilationoperators, and algebraic independence that is the commutativity of local observables.We argue that a consistent algebraic formulation of separability and entanglement shouldbe compatible with micro-causality rather than with algebraic independence.
Keywords : Entanglement; Identical particles; Fermionic Systems
1. Introduction
In trying to apply the standard definitions of separability and entanglement to sys-tems of identical particles, one immediately faces a problem: the indistinguishabilityof the system constituents conflicts with Hilbert space tensor product structure onwhich these notions are based. The point is that the particles are identical andtherefore they can not be singly addressed, nor can their individual properties mea-sured: only collective, global system operators are in fact admissible, experimentallyaccessible observables a , a Entanglement in many-body systems has been widely discussed in the recent literature, e.g. see3-23; however, for the reasons just pointed out, only a limited part of those results are reallyapplicable to the case of identical particle systems.1 une 13, 2018 16:53 WSPC/INSTRUCTION FILE paper F. Benatti, R. Floreanini quantization language proper to quantum many-body theory where the primaryobjects are the algebras of operators rather than states in a Hilbert space 24. Thenew point of view towards separability and entanglement has been advocated be-fore 27-30, but formalized only recently 31-36 with particular attention on bipartiteentanglement, aiming at specific applications to quantum metrology.In the following, we shall consider the second quantized (algebraic) approachto the notions of separability and entanglement in the case of systems composedof fermions whose elementary creation and annihilation operators anti-commute.We shall show that the canonical anti-commutation relations in connection withthe properties of locality and commutativity of the system observables make thetheory of fermion entanglement even richer than in the case of bosonic systems.Indeed, while anti-commutativity of creation and annihilation operators of orthog-onal fermionic states corresponds to the axiom of micro-causality in axiomaticquantum field theory 24 , ,
26, locality has to do with observables localized withinregions that forbid the possibility of interference between their respective measure-ments: these observables must then commute, a property known in the literatureas algebraic independence
37. While for quantum systems consisting of bosonslocality is compatible with micro-causality as the creation and annihilation opera-tors of single particle states obey the canonical commutation relations, in the caseof fermions is not so and this fact clearly emerges when one wants to extend to suchsystems the standard notions of entanglement and separability. In the following wedefine entanglement and separability in terms of micro-causality rather than bas-ing on algebraic independence and argue that this a more consistent choice from aphysical point of view.
2. Entanglement in Fermi systems
We shall consider a many-body system consisting of a fixed number N of fermionseach of which can be found in M different modes, i = 1 , , . . . , M ≥ N : the choice ofmodes is highly non-unique as they correspond to the the orthogonal componentsof any orthonormal basis in the single particle ( M -dimensional) Hilbert space, M being possibly infinite. The second quantization description of such a system as-sociates to each mode i creation and annihilation operators, a † i , a i
24 obeying theCanonical Anti-commutation relations (CAR) { a i , a † j } ≡ a i a † j + a † j a i = δ ij , { a i , a j } = { a † i , a † j } = 0 . (1)The most natural Hilbert space H associated to this system is the Fock spacespanned by the states obtained by applying creation operators to the vacuum vector | i ( a i | i = 0): | n , n , . . . , n M i = ( a † ) n ( a † ) n · · · ( a † M ) n M | i , (2)the integers n , n , . . . , n M representing the occupation numbers of the differentmodes; due to (1), they can take only the two values 0 or 1. The set of polynomialsune 13, 2018 16:53 WSPC/INSTRUCTION FILE paper Entangled Fermions in all creation and annihilation operators, { a † i , a i | i = 1 , , . . . , M } , form an algebra A of bounded operators acting on H ; the observables of the systems are part of thisalgebra.In this setting the notions of separability and entanglement cannot just be ex-trapolated from the case of distinguishable particles. In the case of two non-identicalstandard qubits these notions are connected with the natural Hilbert space productstructure H = C ⊗ C and the corresponding algebraic product structure for thespace of the associated observables A = M ( C ) ⊗ M ( C ), with M ( C ) the set of2 × A ⊗ B = ( A ⊗
1) (1 ⊗ B ) , (3)where A is an observable of the first qubit, while B that for the second one. In otherterms, local observables for two-qubit systems are characterized by being tensorproducts of observables pertaining each to one of the two parties: they commuteand, following 37, we term them as algebraically independent .Consider instead a system composed by two fermions that can occupy twomodes, and thus described by the set of operators ( a , a † , a , a † ): the single particleHilbert space is still C ; however, the CAR in (1) make the total Hilbert space H consist of the vector a † a † | i only and the Fermi algebra A linearly generated by theidentity together with at most second order monomials in a , a † and a , a † . Clearly,the particle Hilbert space tensor product structure is lost as well as the locality ofobservables expressed by the tensor product structure as in (3).The way out is provided by identifying local observables with products of com-muting observables that is with observables that can be simultaneously and inde-pendently measured without the need of attaching them to any particular particleIn quantum many-body theory, a most natural identification of local observablesis in terms of self-adjoint operators supported within disjoint volumes, say a finitevolume V and its complement V = R \ V (disjoint apart from their commonborder). Then, one considers the two subalgebras A , generated by creation andannihilation operators a ( f ) , a † ( f ) and a ( f ) , a † ( f ) of normalized single particlestates f , f supported within the two volumes: a † ( f , ) | i = | f , i .In the case of bosons, the Canonical Commutation Relations (CCR) yield[ a ( f ) , a ( f )] = [ a † ( f ) , a † ( f )] = 0 and[ a ( f ) , a † ( f )] = h f | f i = 0 , ∀ f , : supp( f ) ⊆ V , supp( f ) ⊆ V . (4)The vanishing commutators provide a non-relativistic expression of the so-calledbosonic micro-causality ; in relativistic quantum field theory, micro-causalitymeans that bosonic fields in causally-disjoint space-time regions cannot influenceeach other and must then commute 25 , { a ( f ) , a † ( f ) } to vanish. On the other hand, theune 13, 2018 16:53 WSPC/INSTRUCTION FILE paper F. Benatti, R. Floreanini algebraic independence of operators is based on their vanishing commutators; inorder to check that, one may use the algebraic relation[
AB , C ] = A { B , C } − {
A , C } B . (5)It then follows that two operators supported in disjoint volumes commute when atleast one of them is constructed by means polynomial involving only even powersof creation and annihilation operators. Therefore, given two sub-algebras A , ofthe Fermi algebra A , localized within disjoint volumes, on one hand one has themicro-causality condition expressed by the anti-commutativity of the basic creationand annihilation operators, { a ( f ) , a ( f ) } = 0 ∀ f , : supp( f ) ⊆ V , supp( f ) ⊆ V , (6)where a stands for a or a † . On the other hand, from the point of view of thealgebraic independence of fermionic observables one ought to distinguish the so-called even and odd components of A , . Definition 1.
Let Θ be the automorphism on the Fermi algebra A defined byΘ( a i ) = − a i , Θ( a † i ) = − a † i for all a i , a † i ∈ A .The even component A e of A is thesubset of elements A e ∈ A such that Θ( A e ) = A e , while the odd component A o of A consists of those elements A o ∈ A such that Θ( A o ) = − A o . Remark 1.
The even component A e is generated by the norm closure of evenpolynomials in creation and annihilation operators and is a subalgebra of A , whilethe odd component A o is only a linear space since the product of two odd elementsis even. Even if self-adjoint, odd elements like A o = a ( f ) + a † ( f ) , A o = a ( f ) + a † ( f ) , (7)with a ( f , ) as in (6) are not considered to be observable as they are not compat-ible with superselection rules 40 , ,
42: for instance, they do not leave the numberoperators invariant. Since they do not commute, were they observable, their respec-tive measurements would interfere with each other despite the disjointness of theirsupports. In axiomatic relativistic quantum field theory, they are known as unob-servable fields: however, with their even powers one constructs operators like energyand currents. These are not only observable, but, if supported within causally sep-arated regions, they also commute and are thus algebraically independent. (cid:3)
The splitting of the whole algebra A into a bipartition consisting of two subalge-bras supported within disjoint volumes can be generalized by means of annihilationand creation operators corresponding to different modes. Indeed, a bipartition ofthe algebra A of a system of N fermions each one capable of M ≥ N modes canbe given by splitting the collection of creation and annihilation operators into twodisjoint sets, { a † i , a i | i = 1 , . . . , m } and { a † j , a j , | j = m + 1 , m + 2 , . . . , M } ; it isthus uniquely determined by the choice of the integer m , with 0 ≤ m ≤ M .une 13, 2018 16:53 WSPC/INSTRUCTION FILE paper Entangled Fermions In order to discuss the consequences of the second-quantization (algebraic) ap-proach we start with the following definitions:
Definition 2. (1) Two subalgebras A , A of the Fermi lagebra A will be called disjoint if theyare generated by the norm-closure of polynomials in annihilation and creationoperators of modes belonging to disjoint subsets I and I .(2) An algebraic bipartition of the Fermi algebra A is any pair ( A , A ) ⊂ A of disjoint subalgebras, with only the identity operator in common, such that A ∪ A = A .(3) An operator of A is said to be ( A , A )- local , i.e. local with respect to a givenbipartition ( A , A ), if it is the product A A of an element A of A andanother A in A .In general, a state ω over the Fermi algebra A is any normalized, positive, linear(expectation) functional ω : A 7→ C , such that the average value of any observable O can be expressed as the value taken by ω on it, hOi = ω ( O ), the standardexample being hOi = Tr( ρ O ), namely an expectation functional given by the traceoperation with respect to a density matrices ρ .From the notion of operator locality, a natural definition of state separability(absence of non-local correlations) and entanglement (presence of non-local corre-lations) follows 31: Definition 3.
A state ω on the algebra A will be called separable with respectto the bipartition ( A , A ) if the expectation ω ( A A ) of any local operator A A can be decomposed into a linear convex combination of products of expectations: ω ( A A ) = X k λ k ω (1) k ( A ) ω (2) k ( A ) , λ k ≥ , X k λ k = 1 , (8)where ω (1) k and ω (2) k are given states on A ; otherwise the state ω is said to be entangled with respect the bipartition ( A , A ). Remark 2.
It clearly appears from the previous definition that separability orits absence are properties of states of systems of identical particles which stronglydepend on the chosen bipartition. Indeed, as already remarked in the introduction,there is no a-priori given algebraic split into system 1 and system 2 as in the caseof the tensor product of the algebras of two distinguishable particles 27-36; thisgeneral observation, often overlooked, is at the origin of much confusion in therecent literature. (cid:3)
3. Separable and entangled fermionic states
For bosonic states 31, the two subalgebras A , A commute. As already observed,the condition [ A , A ] = 0 for all A i ∈ A i , i = 1 ,
2, encodes at the algebraic level theune 13, 2018 16:53 WSPC/INSTRUCTION FILE paper F. Benatti, R. Floreanini intuition that entanglement should be connected with the presence of non-classicalcorrelations among commuting , that is algebraically independent, observables. b .However, in the case of fermions, the operators A , are only required to satisfy thecondition of fermionic micro-causality, namely that they must belong to subalgebrasconstructed by anti-commuting annihilation and creation operators. In the followingwe shall clarify the reasons for this choice.Given the algebraic bipartition ( A , A ), one can define the even A ei and odd A oi components of the two subalgebras A i , i = 1 ,
2. Only the operators of thefirst partition belonging to the even component A e commute with any operator ofthe second partition and, similarly, only the even operators of the second partitioncommute with the whole subalgebra A . Structure of separable fermionic states
A crucial observation is that the decomposition in (8) makes sense only when atleast one of the state entering each of the products at the right hand side vanisheson odd elements. This fact follows from a result 39 whose simple proof we reportas it sheds light upon the constraints posed by anti-commutativity.
Lemma 1.
Consider a bipartition ( A , A ) of the fermion algebra A and twostates ω , ω on A . Then, the linear functional ω on A defined by ω ( A A ) = ω ( A ) ω ( A ) for all A ∈ A and A ∈ A is a state on A only if at least one ω i vanishes on the odd component of A i . Proof. If ω , do not vanish on the odd components A o , , there exist self-adjoint A oi ∈ A oi , such that ω i ( A oi ) = 0, i = 1 ,
2. Then, the anti-commutativity of the oddelements A oi yields a contradiction as ω ( A o A o ) = ω ( A o A o ) = − ω ( A o A o ) = ω ( A o ) ω ( A o ) = 0 . It thus turns out that, given a bipartition ( A , A ) of the fermion algebra A , i.e. a decomposition of A in the subalgebra A generated by the first m modes andthe subalgebra A , generated by the remaining M − m ones, the decomposition (8)is meaningful only for local operators A A for which [ A , A ] = 0, so that, alsofor fermions, separable states yield linear convex combination of products of meanvalues on all products of commuting observables. Structure of entangled fermionic states
As a consequence of the previous Lemma, we also have that if a state ω on the Fermialgebra A does not vanish on a local operator A o A o , with the two components b For this reasons, in dealing with fermion systems, the discussion is often restricted just to thecommuting subalgebras A e , A e of even operators 17 une 13, 2018 16:53 WSPC/INSTRUCTION FILE paper Entangled Fermions A o ∈ A o , A o ∈ A o both belonging to the odd part of the two subalgebras, then itis entangled. Indeed, in such a case it cannot be split as in Definition 3.Given a bipartition ( A , A ) where the number of modes is M = 2 N , withodd number of fermions N and A , respectively A , is constructed with creationand annihilation operators of the first N , respectively the second N modes, a verysimple instance of a state with the above characteristics is given by a pure stateconsisting of the balanced superposition of N fermions in the first N modes andnone in the other ones, with no fermions in the first N modes and N in the secondones: | Ψ i = 1 √ (cid:16) | N ; 0 i + | N i (cid:17) = 1 √ (cid:16) a † a † · · · a † N + a † N +1 a † N +2 · · · a † N (cid:17) | i . (9)Consider the product of odd elements A A , where A o = a a · · · a N ∈ A and A o = a † N +1 a † N +2 · · · a † N ∈ A ; then A A | N i = 0 , A A | N ; 0 i ∝ | N i = ⇒ h Ψ | A o A o | Ψ i = 12 . (10)Thus, in full agreement with its evident entangled structure, the pure state | Ψ i isnot separable according to the Definition 3.However, if state separability had been defined by asking the factorization in (8)only relatively to even (and therefore commuting) operators, it would have followedthat, on the algebra generated by A e , , the pure state | Ψ i coincides with the sepa-rable density matrix ρ sep = 12 | N ; 0 ih N ; 0 | + 12 | N ih N | . Indeed, N odd implies h N ; 0 | A e | N i = h N ; 0 | A e | N i = 0 for all A e , , whence h Ψ | A e A e | Ψ i = 12 (cid:16) h N ; 0 | A e | N ; 0 i h N ; 0 | A e | N ; 0 i + h N | A e | N i h N | A e | N i (cid:17) = Tr (cid:16) ρ sep A e A e (cid:17) , ∀ A e ∈ A e , A e ∈ A e . (11)In relativistic quantum field theory a still open problem is the relation betweenthe locality of observables (identified with them being algebraically independent)and their statistical independence which is related to a reference state 37 ,
38. Moreconcretely, the issue at stake is to derive the commutativity of two sub-algebras A , from whether or not the sufficiently many states factorize over product ofobservables A ∈ A and A ∈ A . In this context one has the following definitionof uncorrelated states 38. Definition 4.
Given two (not necessarily disjoint) subalgebras A , of the Fermialgebra A , a state ω on A is ( A , A )-uncorrelated if ω ( P ∧ P ) = ω ( P ) ω ( P ) , (12)une 13, 2018 16:53 WSPC/INSTRUCTION FILE paper F. Benatti, R. Floreanini for every pair of projections P ∈ A and P ∈ A , where P ∧ P = lim n → + ∞ ( P P ) n = lim n → + ∞ ( P P ) n = lim n → + ∞ P ( P P P ) n P = lim n → + ∞ P ( P P P ) n P (13)denotes the largest projector Q ∈ A such that Q ≤ P , Q ≤ P .The above definition refers to any pair of subalgebra, that is not necessarily form-ing an algebraic bipartition in the sense of Definition 3. It is thus interesting to relatethe entangled structure of | Ψ i to the above characterization of uncorrelated states:it turns out that, given a bipartition ( A , A ), the pure ( A , A )-entangled state | Ψ i in (9) is also ( A , A )-correlated. This enforces the need of defining fermionicentanglement basing on micro-causality rather than on algebraic independence.This can be seen by considering the projections P = 1 + a + a † ∈ A , P = 1 + a N +1 + a † N +1 ∈ A , constructed with the creation and annihilation operators of the first and N + 1-thmode that belong to the different subalgebras of the disjoint pair ( A , A ). It provesconvenient to work within the spin-representation provided by the Jordan-Wignertransformation a i = (cid:16) i − O j =1 σ z (cid:17) ⊗ σ − ⊗ (cid:16) N O i +1 (cid:17) , σ − = σ x − iσ y P = 1 + σ x ⊗ (cid:16) N O j =2 (cid:17) (15) P = 12 (cid:16) N O j =1 (cid:17) + (cid:16) N O j =1 σ z (cid:17) ⊗ σ x ⊗ (cid:16) N O j = N +2 (cid:17)! . (16)Then, one computes P P P = 1 + σ x ⊗ (cid:16) N O j =2 (cid:17) . The non-zero eigenvalue of P P P is 1 /
2: thus P ∧ P = 0, h Ψ | P ∧ P | Ψ i = 0,while h Ψ | P | Ψ i = h Ψ | P | Ψ i = 14 , (17)so that | Ψ i cannot fulfill the condition (12) and is thus ( A , A )-correlated.une 13, 2018 16:53 WSPC/INSTRUCTION FILE paper Entangled Fermions
4. Conclusions
In order to properly extend the notions of separability and entanglement to quantumsystems consisting of identical particles, an optimal framework is provided by thesecond quantization formalism. This allows one to resort to the more general modepicture and to abandon the particle based tensor product structure of Hilbert spacesand algebras of observables, valid only for distinguishable particle.Unlike in the case of bosonic systems, in the case of fermions the fundamen-tal anti-commutativity of creation and annihilation operators of orthogonal modes,known as micro-causality, conflicts with the notion of algebraic independence of lo-cal observables, that is with the fact that they must commute in order to be simulta-neously measurable without interferences. In order to reconcile micro-causality withalgebraic independence one usually restricts oneself to considering even fermionicsubalgebras, namely the closures of even polynomials in fermionic creation and an-nihilation operators. Then, the elements of these subalgebras, even if constructedwith anti-commuting creation and annihilation operators of orthogonal modes, docommuteWe have shown that, if separability is defined by restricting to commuting evensubalgebras, that is to algebraic independence, then an apparently entangled statewould be termed separable, whereas the same state is perfectly entangled withrespect to a definition of absence of correlations based on micro-causality, namelywith reference to disjoint full fermionic subalgebras, that is not only to their evencomponents.
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