Separability of N-particle Fermionic States for Arbitrary Partitions
aa r X i v : . [ qu a n t - ph ] J a n Separability of N -particle Fermionic States for Arbitrary Partitions Tsubasa Ichikawa , Toshihiko Sasaki and Izumi Tsutsui Research Center for Quantum Computing, Interdisciplinary Graduate School of Science and Engineering,Kinki University, 3-4-1 Kowakae, Higashi-Osaka, Osaka 577-8502, Japan Theory Center, Institute of Particle and Nuclear Studies,High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
We present a criterion of separability for arbitrary s partitions of N -particle fermionic pure states.We show that, despite the superficial non-factorizability due to the antisymmetry required by theindistinguishability of the particles, the states which meet our criterion have factorizable correlationsfor a class of observables which are specified consistently with the states. The separable states andthe associated class of observables share an orthogonal structure, whose non-uniqueness is found tobe intrinsic to the multi-partite separability and leads to the non-transitivity in the factorizabilityin general. Our result generalizes the previous result obtained by Ghirardi et. al. [J. Stat. Phys. (2002) 49] for the s = 2 and s = N case. PACS numbers: 03.65.-w, 03.65.Ta, 03.65.Ud.
I. INTRODUCTION
One of the distinguished features of quantum theory is the existence of indistinguishable particles. The indistin-guishability of particles, which leads to the symmetry or antisymmetry of states depending on the particle statistics,demystifies miscellaneous physical phenomena and properties, including black-body radiation, the electronic shellstructure of atoms, Bose-Einstein condensation, superconductivity, and Hanbury Brown and Twiss effect. Mean-while, the indistinguishability mystifies us as well, especially in its perplexing consequence in quantum correlationor entanglement. For example, the cluster decomposition property [1–3] of quantum field theory ensures that cor-relation functions of local observables are factorizable for indistinguishable particles, despite that the states of theparticles are necessarily intertwined with each other due to the (anti-)symmetry prescribed. The puzzling aspect ofindistinguishability of particles in quantum theory derives from the fact that the entire Hilbert space is not just thetensor product of the constituent Hilbert spaces of individual particles; rather it is the (anti-)symmetrized subspaceof it. Consequently, the conventional criterion of entanglement for distinguishable particles is no longer valid forindistinguishable particles.A definite criterion of entanglement for indistinguishable particles is indispensable also for quantum informationscience, where characterization of entangled multi-partite states forms a key element to achieve quantum computationand communication effectively, or any other operations designed. In view of this, in recent years the criterion ofentanglement for indistinguishable particles has been studied intensively by several groups [4–11]. Among them isan algorithmic approach [4–6], where (anti-)symmetric N -particle states separable into N single-particle states areexamined based on the Slater rank which is defined from the orthonormal decomposition of the states, generalizingthe Schmidt numbers considered for distinguishable particles [2, 12]. A related analysis in terms of the von Neumannentropy has also been made for the bipartition separability criterion [7–9]. Although this approach has an advantagein the operational directness, enabling one to construct useful operators such as the entanglement witness, it leaves theextension to the general case of separability into arbitrary subsystems highly non-trivial. Another approach, which hasbeen proposed by Ghirardi et. al in [9–11], characterizes separable states based on the possession of ‘complete set of theproperties’ associated with the subsystems into which they are separable. Their criterion of bipartition separability(an N -particle system separable into two subsystems) and full separability (an N -particle system separable into N subsystems), given by conditions using projection operators, are satisfied by (anti-)symmetrized direct product states.For fermionic systems, the separable (non-entangled) states determined by these two approaches coincide completelyin the case of N = 2 bipartition separability [9], which suggests that the two approaches are consistent at least forthe restricted cases of fermionic states, even though their manipulations are quite different.In this article, we present a general framework of separability for the case of indistinguishable fermionic particlesand furnish a criterion capable of treating arbitrary separability, i.e. , an N -particle system separable into an arbitrarynumber s of subsystems. The states which meet our criterion are found to be anti-symmetrized states of productstates belonging to subsystems whose state spaces are mutually orthogonal, and for the two extreme cases, s = 2and s = N , they coincide with those found in [10]. It is confirmed that these states have factorizable correlations(as well as weak values [13–15]) for a certain class of observables in a way which ensures the cluster decompositionproperty when the fermions are localized remotely from each other. Our argument highlights the important fact thatseparability of states is defined only with reference to a class of observables admitting an orthogonal structure. In theactual experimental setups, such a class may correspond to measurements which allow for unambiguous specificationof the subsystems in the separation.This paper is organized as follows. Sec. II illustrates our motivation by the simple N = 2 case from physicalgrounds, where we discuss why indistinguishability of particles calls for an appropriate revision on the criterion ofseparability of states. The novel aspects pertinent to fermionic states and their consequences in the required revisionwill be mentioned for our direct generalization of later sections. In Sec. III, we introduce a number of basic toolsthat we use for the system of general N particles partitioned into arbitrary subsystems. Two types of observablesappropriate for fermionic states, one considered in [10] and the other considered here, are shown to be equivalentat the level of expectation values. Sec. IV contains our main argument on arbitrary separability. After introducingan orthogonal structure with which a particular set of states and a class of observables are defined, we present ourcriterion of arbitrary separability and prove that these sets of states indeed meet the criterion and possess factorizablecorrelations for the corresponding observables. Some of the notable points associated with the orthogonal structure,as well as the observation on the cluster decomposition, are mentioned in Sec. V. Sec. VI is devoted to the conclusionsand discussions. II. PROLEGOMENON
In this section, we outline our motivation of the present work by discussing the N = 2 case referring to the relationbetween separable states, characterized by factorization of correlations, and the local realism signified, e.g. , by theBell inequality [16–19]. We shall see that for indistinguishable particles the condition of separability must be revisedproperly in order to maintain the relation of the two, and this revision will be generalized in later sections when wedeal with systems of N ≥ | Ψ i ∈ H ⊗ H be a vector describing the state of thecomposite system of two particles i = 1 , H i for individual states. Given the observables O i for the particles i = 1 ,
2, the combined observable for the composite system reads O ⊗ O , which admits the trivialidentity, O ⊗ O = ( O ⊗ ) ( ⊗ O ) (2.1)with i being the identity operator in H i . Corresponding to this, one may find that, for some | Ψ i and O i , theexpectation value also factorizes: h Ψ | O ⊗ O | Ψ i = h Ψ | O ⊗ | Ψ ih Ψ | ⊗ O | Ψ i . (2.2)As shown in [10], for pure states of distinguishable particles, the factorization (2.2) occurs for any O i if and only ifthe state | Ψ i is a product state | Ψ i = | ψ i | ψ i , (2.3)where | ψ i i i ∈ H i are arbitrary states of the individual particles. This implies that we can use the factorizationof correlations (2.2) as a criterion to examine the non-entanglement of states of composite systems, if the latter ischaracterized by the form of product states.To see the connection between separability and local realism, let us consider the case where O , O correspondto dichotomic observables A, B (= ±
1) in the constituent particle systems which are remotely separated from eachother (see Fig.1). Now, in local realistic theory (LRT) [16], it is assumed that the outcomes of the measurement isdetermined by hidden variables collectively denoted by λ , and that locality of the two particles and the measurementsetup ensures that the outcomes are independent of the choice of the local measurement parameters a , b and also ofthe outcomes themselves, i.e. , they are determined as A ( a , λ ) and B ( b , λ ), respectively. From these, the correlationsof the measurement outcomes are given by C L ( a , b ) = Z d λ ρ ( λ ) A ( a , λ ) B ( b , λ ) , (2.4)under a given probability distribution ρ ( λ ) of hidden variables specifying the state of the system. We then have thecelebrated Bell (CHSH) inequality [17], | C L ( a , b ) + C L ( a , b ′ ) + C L ( a ′ , b ) − C L ( a ′ , b ′ ) | ≤ , (2.5)which holds for any choice of parameters a , a ′ , b and b ′ . We mention that the Bell inequality (2.5) holds even forstochastic cases [20] where the dichotomic observables A, B are replaced by ‘averages’ ¯ A, ¯ B for which we have | ¯ A | ≤ | ¯ B | ≤ FIG. 1: Schematic diagram of experiments to test the Bell inequality. In a system of distinguishable particles 1 and 2, dichotomicobservables
A, B are measured along the axes a , b represented by the operators σ ( a ), σ ( b ), respectively. Accompanied tothese, apparatuses to measure the positions L, R of the particles represented by the observables | L i h L | , | R i h R | are alsoinstalled. If the particles are indistinguishable, the entire observables must be made invariant under the exchange of theparticles in accordance with states which respect the (anti-)symmetry required by the statistics of the particles. On the other hand, in quantum mechanics, the dichotomic observable can be represented in the state space H i = C by the spin operators O i = σ i ( a ) := σ i · a where σ i = ( σ xi , σ yi , σ zi ) are Pauli matrices. Here, the outcomes of the spinmeasurement in the direction a are given by the dichotomic values ± C Q ( a , b ) = h Ψ | O ⊗ O | Ψ i = h Ψ | σ ( a ) ⊗ σ ( b ) | Ψ i (2.6)for which we have the Cirel’son inequality [21], | C Q ( a , b ) + C Q ( a , b ′ ) + C Q ( a ′ , b ) − C Q ( a ′ , b ′ ) | ≤ √ . (2.7)The upper bound of the Cirel’son inequality (2.7) is attained by the singlet state | Ψ i = | Singlet i given by | Singlet i := 1 √ | + i |−i − |−i | + i ) (2.8)with the eigenstates σ zi |±i i = ±|±i i under a suitable choice of the parameters (configurations of the measurementaxes). This shows that some quantum states, though not all, do realize correlations that cannot be achieved by LRT.If, in particular, the state | Ψ i is a product state (2.3), the factorization (2.2) yields C Q ( a , b ) = h Ψ | σ ( a ) ⊗ | Ψ ih Ψ | ⊗ σ ( b ) | Ψ i = h ψ | σ ( a ) | ψ i h ψ | σ ( b ) | ψ i . (2.9)We then observe that, if we put ¯ A ( a ) = h ψ | σ ( a ) | ψ i , ¯ B ( b ) = h ψ | σ ( b ) | ψ i we have | ¯ A ( a ) | ≤ | ¯ A ( a ) | ≤ C Q ( a , b ) in (2.9) coincides formally with the stochastic version of C L ( a , b ) in (2.4) whenthe observables are independent of the hidden variables. Thus, for those factorized states, the upper bound in theinequality (2.7) is found to be 2, which implies that the correlations of factorizable states can always be emulatedby LRT. To sum up, we find that for distinguishable particles in pure states, the three criteria (factorization ofcorrelations, direct-productness of state, and emulation by LRT) are all consistent.At this point we note that in the actual measurement we identify the measurement outcomes A, B with thoseassociated with the particles i = 1 ,
2, respectively. For this, it is customary to consider the spatial degrees of freedomof the particles in addition to the spins, that is, we introduce in each of the constituent systems a state | L i i localizedaround the measurement apparatus on the left and similarly a state | R i i localized around the measurement apparatuson the right, where the locality implies i h L | R i i = 0. With these, the observable and the singlet state of the compositesystem are given, more precisely, by σ ( a ) | L i h L | ⊗ σ ( b ) | R i h R | and | Singlet i | L i | R i . (2.10)Fortunately, thanks to the direct product structure of the states and the observables (2.10), the introduction of thespatial degrees of freedom does not affect the outcomes of correlations, and our argument goes through withoutinvoking them for distinguishable particles.However, for indistinguishable particles, this is no longer the case. To see this, we first note that due to theindistinguishability the observable for the composite system must also be made invariant under the exchange ofparticles. Thus, the observable corresponding to (2.10) should be replaced by σ ( a ) | L i h L | ⊗ σ ( b ) | R i h R | → σ ( a ) | L i h L | ⊗ σ ( b ) | R i h R | + σ ( b ) | R ih R | ⊗ σ ( a ) | L ih L | . (2.11) Distinguishable particles Fermions O = O ⊗ O O = A ( O ⊗ O ) A Observables O (1) = O ⊗ O (1) = A ( O ⊗ I ) AO (2) = ⊗ O O (2) = A ( I ⊗ O ) A Factorization of observables O = O (1) O (2) O = O (1) O (2) Factorization of correlations h Ψ | O | Ψ i = h Ψ | O (1) | Ψ ih Ψ | O (2) | Ψ i h Ψ | O | Ψ i = h Ψ | O (1) | Ψ ih Ψ | O (2) | Ψ i Separable states | Ψ i = | ψ i | ψ i | Ψ i = A | ψ i | ψ i TABLE I: Comparison of observables and separable states between distinguishable particles and fermions.
In the case of fermions, one can show (see Proposition 2) that (2.11) is equivalent to σ ( a ) | L i h L | ⊗ σ ( b ) | R i h R | → A ( σ ( a ) | L i h L | ⊗ σ ( b ) | R i h R | ) A, (2.12)where A is the (rescaled) anti-symmetrizer A = 1 √ ⊗ − π ) (2.13)given from the permutation operator π defined by π | ψ i | ψ i = | ψ i | ψ i . (2.14)For notational conciseness, hereafter we use the second form (2.12) for observables of fermionic systems.In order to address the question of separation of correlations, we need to provide beforehand a possible form ofseparation of operators pertaining to each of the fermions analogously to (2.1) in the distinguishable case. Clearly,the problem is that, despite the indistinguishable nature of the particles, we need to somehow label the particles bythe observables in the measurement in order to define the correlations, and one possible approach for this is to utilizethe locality which is also presupposed for the distinguishable case. In the present situation, we have one observablemeasured by an apparatus on the left and, remotely separated from it, we have another observable measured by anapparatus on the right. These are represented by O := σ ( a ) | L i h L | , O := σ ( b ) | R i h R | , (2.15)with which the observable of the composite system in (2.12) becomes simply A ( O ⊗ O ) A . One can show (seeCorollary 2) that it admits the separation with respect to the local observables: A ( O ⊗ O ) A = A ( O ⊗ I ) A · A ( I ⊗ O ) A. (2.16)Here we have introduced the projectors, I := I | L i h L | , I := I | R i h R | , (2.17)with I i being the identity matrix in the spin space of particle i . Operationally, each of the projectors (2.17) may beinterpreted as the observable which confirms merely the presence of the particles at the apparatus without measuringthe spin.Now that we have introduced the factorization (2.16) for observables of fermions analogously to the factorization(2.1) for observables of distinguishable particles, we may also define the factorization of correlations by h Ψ | A ( O ⊗ O ) A | Ψ i = h Ψ | A ( O ⊗ I ) A | Ψ ih Ψ | A ( I ⊗ O ) A | Ψ i . (2.18)One can show (see Proposition 3) that, in analogy with (2.2) and (2.3) in the case of distingushable particles, thefactorization (2.18) occurs if the state | Ψ i is given by | Ψ i = A | ψ i | ψ i , | ψ i = | α i | L i , | ψ i = | α ′ i | R i , (2.19)where | α i , | α ′ i are arbitrary spin states. We note that, as stated before, such states (2.19) possess a complete set ofproperties mentioned in Sec. IV (see Definition 1 and Theorem 2).The correspondence of observables and separable states between distinguishable particles and fermions is summa-rized in Table I. One finds that the transition rule from distinguishable particles to fermions is rather simple, i.e. ,it is accomplished essentially by attaching the antisymmterizer A to the states and also to the observables in theconjugate manner. In fact, this rule is applicable even to finding states which saturates the Cirel’son inequality withthe observables given by (2.12), where one learns that the upper bound of the inequality is attained by the state A ( | Singlet i ⊗ | L i | R i ) = | Singlet i ⊗ √ | L i | R i + | R i | L i ) . (2.20)An important point to note in the transition rule is that the assignment of observables entails introducing an‘orthogonal structure’ there, that is, due to the locality property i h L | R i i = 0 the operators O and O are made to actin orthogonal subspaces when the individual spaces of the states are identified. More generally, the states | ψ i , | ψ i of the subsystems appearing in (2.20) must be orthogonal in the same sense. On physical grounds, one may argue thatthis orthogonal structure is required for indistinguishable particles to make a distinction between the measurementoutcomes obtained for the two observables, while such a structure is unnecessary for distinguishable particles onaccount of its intrinsic capability of distinction. As we shall see in the following sections, these characteristics ofseparability for indistinguishable (fermionic) states will be seen also for the system of general N particles under anarbitrary partition into s subsystems. Unlike the present N = 2 case, however, the general N case may admit morethan one orthogonal structure, and we shall also discuss the meaning of this ambiguity with respect to the separabilitycriterion there. III. ANTISYMMETRIC SPACES AND PARTITIONED OBSERVABLES
To begin our discussions on arbitrary separability of N -particle fermionic states, we first provide a basic accountof identical N -particle systems along with necessary tools to treat the separation into arbitrary subsystems.Our total system is formed out of N constituent systems representing the N particles. All of the constituent systemsare identical to each other, and the Hilbert space H i associated with the constituent system i = 1 , . . . , N is assumedto be d -dimensional, that is, H i = C d for all i . We assume d ≥ N to treat antisymmetric states. If we let | a i i i , a i = 0 , , · · · , d −
1, be a complete orthonormal basis in H i , then the set of the direct product states N Ni =1 | a i i i forms acomplete orthonormal basis in the Hilbert space H of the total system defined by the tensor product H tot = N Ni =1 H i .For the case of identical particles, a subsystem is a set of constituent systems in arbitrary number. To specify howthe total system is decomposed into such subsystems, it is convenient to introduce a partition Γ of the set of labels { , , · · · , N } of the constituents, i.e. ,Γ = ( Γ , . . . , Γ s (cid:12)(cid:12)(cid:12)(cid:12) s [ k =1 Γ k = { , , · · · , N } , Γ k = ∅ , Γ j ∩ Γ k = ∅ , j = k ) . (3.1)Given a partition Γ, a subset is specified by those constituent systems whose labels comprise Γ k in Γ for some k , andwe use Γ k to denote the subsystem as well. The Hilbert space of subsystem Γ k is then given by the tensor product H (Γ k ) = O i ∈ Γ k H i , (3.2)which is equipped with a complete orthonormal basis N i ∈ Γ k | a i i i in H (Γ k ). If we let O k be an observable in H (Γ k ),it admits the spectral decomposition, O k = X µ λ µ | µ i Γ k Γ k h µ | , λ µ ∈ R , | µ i Γ k ∈ H (Γ k ) , (3.3)fulfilling Γ k h µ | ν i Γ k = δ µν , | µ i Γ k ∈ H (Γ k ) , µ, ν = 0 , , . . . , d | Γ k | − . (3.4)Here, δ µν is the Kronecker delta and | Γ k | is the cardinality of the set Γ k , which in our case is the number of constituentscontained in the subsystem Γ k .In order to deal with identical fermionic particles, we first introduce an isomorphic map among the constituentHilbert spaces H i by means of the correspondence between the basis vectors | a i i i ∈ H i and | a i i j ∈ H j for all i, j = 1 , . . . , N . The isomorphism is highly non-unique because of the arbitrariness in the choice of bases, but thefollowing arguments hold under any fixed choice of bases. One then introduces permutations σ ∈ S N of the symmetricgroup S N on the states of constituents, which are implemented by self-adjoint operators π σ on H tot defined by π σ N O i =1 | a i i i ! = N O i =1 | a i i σ ( i ) . (3.5)Note that π σ acts on the bra vectors as N O i =1 i h a i | ! π σ = N O i =1 σ ( i ) h a i | , (3.6)since π σ are self-adjoint. The action of permutation σ on the labels of the constituents induces a map Γ into anotherpartition Γ σ = { Γ σk } sk =1 with Γ σk = { σ ( i ) | i ∈ Γ k } . Accordingly, we may classify permutations into two classes; (i)those leaving the partition Γ unchanged, i.e. , Γ σ = Γ, and (ii) those changing Γ, i.e. , Γ σ = Γ. We denote the set ofpermutations belonging to the former class by I (Γ) and the set of the latter by ¯ I (Γ). The symmetric group S N isthus a disjoint union of these, S N = I (Γ) ∪ ¯ I (Γ), and we may further consider the quotient set S N / I (Γ) of S N withrespect to I (Γ).The anti-symmetrizer A , which plays a key role in the subsequent discussions of fermionic states, is constructedfrom the permutation operators as A := 1 N ! X σ ∈ S N sgn( σ ) π σ , (3.7)where sgn( σ ) = 1 for even permutations, and sgn( σ ) = − A is aprojection operator, satisfying A = A , A † = A and π σ A = A π σ = sgn( σ ) A (3.8)for all σ ∈ S N .The space of fermionic states H tot A is the antisymmetric linear subspace of H tot obtained by collecting all antisym-metric states in H tot , which is written as H tot A = Asym H tot := (cid:8) | Ψ i (cid:12)(cid:12) A| Ψ i = | Ψ i , | Ψ i ∈ H tot (cid:9) . (3.9)Likewise, we may construct the antisymmetric subspace of a given multi-particle space K by considering only those π σ corresponding to the permutations σ ∈ S | Γ k | which map vectors from K to K . Utilizing these permutation operators,we can define the anti-symmetrizer A on K by (3.7) upon restriction to the permutations appropriate for K (weuse the same symbol A for simplicity), and thereby define the antisymmetric subspace Asym K analogously to (3.9)from K . For example, if this construction is applied to the space H (Γ k ) in (3.2), we then have the Hilbert space H A (Γ k ) := Asym H (Γ k ) of the subsystem Γ k consisting of only antisymmetric states.Having furnished necessary tools for our discussion, we now discuss observables in H tot A constructed out of observableson subsystems specified by the partition Γ. Given an observable O k in H A (Γ k ) for each k = 1 , . . . , s , we may consider,by taking account of the anti-symmetric property of the states required, the operator in H tot A , O (Γ) := A (Γ) s O k =1 O k ! A (Γ) , (3.10)where we have introduced, for our technical convenience, the rescaled anti-symmetrizer, A (Γ) := p M (Γ) A (3.11)with the multinomial coefficient associated with the partition Γ, M (Γ) := N ! Q sk =1 | Γ k | ! . (3.12)We denote the class of operators associated with the partition Γ of the form (3.10) by C (Γ). A salient property of theclass C (Γ) is that it is invariant under permutation, C (Γ σ ) = C (Γ) for any σ ∈ S N . This can be seen from Proposition 1
To an observable O (Γ) ∈ C (Γ) defined in (3.10), one can always find an observable O (Γ σ ) ∈ C (Γ σ ) for any σ ∈ S N such that O (Γ) = O (Γ σ ) . (3.13) Proof.
Using (3.8) and the spectral decomposition of O k with the label µ written here as µ k to distinguish thesubsystems the states belong to, we find O (Γ) = A (Γ) s O k =1 X µ k λ µ k | µ k i Γ k Γ k h µ k | ! A (Γ)= X µ , ··· ,µ s s Y k =1 λ µ k ! A (Γ) s O k =1 | µ k i Γ k ! s O k =1 Γ k h µ k | ! A (Γ)= X µ , ··· ,µ s s Y k =1 λ µ k ! A (Γ) π σ s O k =1 | µ k i Γ k ! s O k =1 Γ k h µ k | ! π σ A (Γ)= X µ , ··· ,µ s s Y k =1 λ µ k ! A (Γ σ ) s O k =1 | µ k i Γ σk ! s O k =1 Γ σk h µ k | ! A (Γ σ )= A (Γ σ ) s O k =1 X µ k λ µ k | µ k i Γ σk Γ σk h µ k | ! A (Γ σ ) (3.14)where we have used A (Γ) = A (Γ σ ). The last line gives an observable belonging to C (Γ σ ), and hence if we just defineit to be our O (Γ σ ), we have O (Γ) = O (Γ σ ).The above proposition shows that, as far as the observables belonging to the class C (Γ) are concerned, any partitionswhich are related under permutations lead to the same outcomes in the total system H tot A . On the other hand, onemay instead consider a different class ˜ C (Γ) of observables constructed from the same set of observables O k in H A (Γ k )by collecting all possible permutations of the product,˜ O (Γ) := X σ ∈ S N / I (Γ) π σ s O k =1 O k ! π σ , (3.15)where we take the summation over the quotient set S N / I (Γ) to eliminate possible degeneracies due to the invariantpermutations in I (Γ). The two observables, O (Γ) in (3.10) and ˜ O (Γ) in (3.15), enjoy a simple relationship as seen in Lemma 1
The observables O (Γ) and ˜ O (Γ) defined in (3.10) and (3.15), respectively, are related by A ˜ O (Γ) A = O (Γ) . (3.16) Proof.
Using (3.8) and noting the cardinality of S N / I (Γ) being M (Γ), we have A ˜ O (Γ) A = X σ ∈ S N / I (Γ) A π σ s O k =1 O k ! π σ A = X σ ∈ S N / I (Γ) O (Γ) M (Γ) = O (Γ) . (3.17)Now, it is straightforward to show Proposition 2
Two observables O (Γ) ∈ C (Γ) and ˜ O (Γ) ∈ ˜ C (Γ) which are constructed from a given set of observ-ables { O k } sk =1 by the prescriptions (3.10) and (3.15), respectively, share the same expectation values under arbitraryantisymmetric states | Ψ i ∈ H tot A , that is, h Ψ | O (Γ) | Ψ i = h Ψ | ˜ O (Γ) | Ψ i , ∀| Ψ i ∈ H tot A . (3.18) Proof.
Using Lemma 1 and A| Ψ i = | Ψ i for | Ψ i ∈ H tot A , we obtain (3.18).At this point we briefly consider the problem of constructing observables of N -particle systems from those ofsingle-particle constituent systems. For simplicity, we use the abbreviated notation A := A (Γ) hereafter. To examinethe physical meaning of the observable O (Γ) defined in (3.10), let us consider the N = 2 case Γ = { Γ , Γ } withΓ = { } and Γ = { } , where we have observables O , O ′ in H and O , O ′ in H . We assume that O and O (andsimilarly O ′ and O ′ ) are operators sharing the same spectrum decomposition with the same eigenvectors in each ofthe constituent Hilbert space. In other words, we just have two sets of physically distinct operators acting on thespaces, H and H , and accordingly we may drop the indices 1 and 2 which indicate the labels of the subsystemsthat the operators act on and write them simply as O and O ′ . The observable O (Γ) acting on the total system H tot A constructed from the two observables O and O ′ would then be O (Γ) = A ( O ⊗ O ′ ) A. (3.19)We are thus tempted to interpret O (Γ) as an operator implementing a ‘simultaneous’ measurement of O and O ′ onthe constituent systems. On the other hand, we may expect that the individual measurements of the two observableson the N = 2 system are represented by Q (Γ) = A ( O ⊗ ) A, and Q ′ (Γ) = A ( O ′ ⊗ ) A, (3.20)respectively. We observe that, assuming that the two observables commute [ O, O ′ ] = 0, the product of the twooperators in (3.20) becomes a self-adjoint operator and reads Q (Γ) Q ′ (Γ) = O (Γ) + A ( OO ′ ⊗ ) A, (3.21)where O (Γ) is the one given in (3.19). Imposing further that the two operators be orthogonal (or their supports bemutually disjoint) OO ′ = 0, we obtain Q (Γ) Q ′ (Γ) = O (Γ) . (3.22)When this is done, the final form admits the interpretation that the product of two individual measurements yieldsthe simultaneous measurement given by the product form (3.19).Our argument can be easily extended to the case of infinite dimensional Hilbert spaces such as H i = C d ⊗ L ( R ),where one may consider two spatially localized observables possessing no overlaps. Such an example has been men-tioned in [10] in considering separability for bipartition of N -particle states, where observables of the class ˜ C (Γ) whichfulfill the orthogonality OO ′ = 0 are adopted. Our discussion to be expounded below will be based on the observablesof the class C (Γ), but the consistency between the two is ensured by Proposition 2 which asserts that at the level ofexpectation values for | Ψ i ∈ H tot A , the choice of the class, C (Γ) or ˜ C (Γ), is immaterial. More on this point will bediscussed in Sec. V where we consider the physical implications of separability. IV. ARBITRARY SEPARABILITY FOR N -PARTICLE ANTISYMMETRIC STATES We now move on to present our condition for separability of N -particle states into arbitrary subsystems andprovide states which fulfill the condition. To this end, following the procedure employed in [10], and also hinted bythe argument given at the end of the previous section, we furnish a set of orthogonal subspaces within C d and therebyobtain operators which are mutually orthogonal and admit their simultaneous measurement. This will allow us tofind a set of states in H tot A which are factorizable with respect to a given partition Γ of the original system under theorthogonal structure. A. Orthogonal Structure in States and Observables
Let us first choose a set V of mutually orthogonal linear subspaces V j , j = 1 , . . . , m , in C d , i.e. , V = ( V , . . . , V m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m M k =1 V k ⊆ C d , V k ⊥ V l , k = l ) . (4.1)Such a set V can be used to furnish an orthogonal structure in each of the constituent Hilbert spaces, by consideringthe subspaces V k within H i = C d and denote them as V k [ H i ] ⊂ H i for k = 1 , . . . , m . For our purposes, we choose V in (4.1) such that m = s , where s is the number of subsystems in Γ = { Γ k } sk =1 , and that dim V k ≥ | Γ k | forall k . We associate a common subspace V k to all the constituents belonging to Γ k , and then define the subspace W (Γ k , V k ) ⊂ H A (Γ k ) by W (Γ k , V k ) := Asym O i ∈ Γ k V k [ H i ] ! . (4.2)In more concrete terms, we choose those states in H (Γ k ) which are confined to the subspace V k in each of theconstituent H i for i ∈ Γ k and then anti-symmetrize them with the procedure analogous to (3.9) and the remarks thatfollow. The resultant states form the space W (Γ k , V k ) in (4.2).Since W (Γ k , V k ) is itself a Hilbert space and has a complete orthonormal basis, we may consider observables O k with its support only in W (Γ k , V k ) ⊂ H A (Γ k ). In other words, O k admits the spectral decomposition, O k = X µ λ µ | µ i Γ k Γ k h µ | , λ µ ∈ R , | µ i Γ k ∈ W (Γ k , V k ) . (4.3)Note that O k is still a self-adjoint operator in H A (Γ k ), and out of these observables in the subsystems we can constructoperators on the form (3.10) as before. More explicitly, we consider the class of operators in H tot A given by C (Γ , V ) := ( O (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) O = A s O k =1 O k ! A ) , (4.4)with the requirement (4.3). Note that C (Γ , V ) is uniquely defined to a given pair (Γ , V ). Note also that, by con-struction, C (Γ , V ) is not closed under addition, that is, O + O ′
6∈ C (Γ , V ) for O , O ′ ∈ C (Γ , V ) in general. Further, anoperator O may belong to more than one such classes defined from different orthogonal sets V .Similarly, we also consider the set of states which are obtained by anti-symmetrizing fully factorized states withrespect to the partition Γ S (Γ , V ) := ( | Ψ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | Ψ i = A s O k =1 | ψ k i , | ψ k i ∈ W (Γ k , V k ) ) . (4.5)Note that S (Γ , V ) ⊆ H tot A and that S (Γ , V ) is not a linear space. Note also that, in general, two sets S (Γ , V ) and S (Γ , V ′ ) for different V and V ′ are not mutually exclusive in H tot A . This implies that a single state may belong bothto S (Γ , V ) and S (Γ , V ′ ) or possibly more, as we see explicitly in Sect. V. Hereafter we verify that the states in S (Γ , V )correspond to the separable states of distinguishable particles from the two view points: Decomposition of correlationsand complete set of the properties. B. Factorization of Correlations
The next lemma is useful in our following arguments.
Lemma 2
The inner product of any two states | Ψ i , | Φ i ∈ S (Γ , V ) is factorizable according to the partition Γ , h Φ | Ψ i = s Y k =1 h φ k | ψ k i . (4.6) In other words, for | ψ k i , | φ k i ∈ W (Γ k , V k ) , k = 1 , . . . , s , we have s O k =1 h φ k | A ! A s O k =1 | ψ k i ! = s Y k =1 h φ k | ψ k i . (4.7) Proof.
In terms of the set I (Γ) of invariant permutations and its complement ¯ I (Γ), the anti-symmetrizer A can bedecomposed into A = F + G, with F := 1 N ! X σ ∈I (Γ) sgn( σ ) π σ , G := 1 N ! X σ ∈ ¯ I (Γ) sgn( σ ) π σ . (4.8)0For permutations σ ∈ I (Γ), we obtain π σ s O k =1 | ψ k i = sgn( σ ) s O k =1 | ψ k i , (4.9)which yields F s O k =1 | ψ k i = 1 M s O k =1 | ψ k i , (4.10)with M = M (Γ) in (3.12), because the cardinality of I (Γ) is equal to Q sk =1 | Γ k | !. Also, we have s O k =1 h φ k | ! G s O k =1 | ψ k i = 0 . (4.11)This can be seen by noticing that any permutation σ ∈ ¯ I (Γ) contained in G maps all the states in V k [ H i ] for some i ∈ Γ k to the corresponding states in V k [ H σ ( i ) ] with σ ( i ) Γ k . As a result, the l.h.s. of (4.11) vanishes due to theorthogonality V k [ H σ ( i ) ] ⊥ V l [ H σ ( i ) ] for l for which Γ l ∋ σ ( i ). Using (4.10) and (4.11), we then see h Φ | Ψ i = s O k =1 h φ k | A ! A s O k =1 | ψ k i ! = M s O k =1 h φ k | ! A s O k =1 | ψ k i = M s O k =1 h φ k | ! F s O k =1 | ψ k i + M s O k =1 h φ k | ! G s O k =1 | ψ k i = s Y k =1 h φ k | ψ k i . (4.12)Lemma 2 is also useful to show that the set S (Γ , V ) is closed under the action of the observables in C (Γ , V ), thatis, Lemma 3
For any | Ψ i ∈ S (Γ , V ) and O ∈ C (Γ , V ) , we have O | Ψ i ∈ S (Γ , V ) . More explicitly, if | Ψ i and O are givenby (4.5) and (4.4), respectively, then O | Ψ i = A s O k =1 O k | ψ k i ∈ S (Γ , V ) . (4.13) Proof.
Employing the spectral decomposition (4.3), we find O | Ψ i = A s O l =1 O l ! A A s O m =1 | ψ m i ! = A s O l =1 X µ l λ µ l | µ l ih µ l | ! A A s O m =1 | ψ m i ! = A " X µ , ··· ,µ s s Y l =1 λ µ l ! s O l =1 | µ l i ! s O l =1 h µ l | ! A A s O m =1 | ψ m i ! . (4.14)1With the help of (4.7), and using the spectral decomposition again and changing the dummy indices in summand, weobtain O | Ψ i = A X µ , ··· ,µ s s Y l =1 λ µ l ! s O k =1 | µ k i ! " s O k =1 h µ k | A ! A s O k =1 | ψ k i ! = A X µ , ··· ,µ s s Y l =1 λ µ l ! s O k =1 | µ k i h µ k | ψ k i = A s O k =1 X µ k λ µ k | µ k i h µ k | ψ k i = A s O k =1 O k | ψ k i . (4.15)Since O k has the support only in W (Γ k , V k ), we observe O k | ψ k i ∈ W (Γ k , V k ) for all k , and hence O | Ψ i ∈ S (Γ , V ).Factorizability of weak values can then be established by the help of Lemma 2 and Lemma 3 as Theorem 1
The weak values of O ∈ C (Γ , V ) under | Ψ i , | Φ i ∈ S (Γ , V ) is factorizable according to the partition Γ , h Φ | O | Ψ ih Φ | Ψ i = s Y k =1 h φ k | O k | ψ k ih φ k | ψ k i . (4.16) Proof.
The factorization of the denominator of the l.h.s. follows from Lemma 2 for | Ψ i , | Φ i ∈ S (Γ , V ). Since O | Ψ i also belongs to S (Γ , V ) by Lemma 3, again from Lemma 2 the factorization of the numerator of the l.h.s. follows,too. Combining these we find (4.16).As a trivial corollary of the theorem, we note here that the expectation values of observables O ∈ C (Γ , V ) arefactorizable for states in S (Γ , V ): Corollary 1
The expectation values of O ∈ C (Γ , V ) under a normalized state | Ψ i ∈ S (Γ , V ) is factorizable accordingto the partition Γ , h Ψ | O | Ψ i = s Y k =1 h ψ k | O k | ψ k i . (4.17)The factorizability of expectation values according to (arbitrary) partitions Γ discussed above generalizes the previ-ous result [10] obtained for the case of bipartitions s = 2. The set C (Γ , V ) of observables for which the factorizabilityis seen is also an extension of the set used in [10].It is worth mentioning that the space C (Γ , V ) is closed under multiplication provided that the self-adjointness ofthe operator is preserved. Namely, we have Lemma 4
Given two elements O , O ′ ∈ C (Γ , V ) with O = A s O k =1 O k ! A, O ′ = A s O k =1 O ′ k ! A, (4.18) for which the product O k O ′ k of the two self-adjoint operators O k and O ′ k is again self-adjoint for all k , we have OO ′ = A s O k =1 O k O ′ k ! A ∈ C (Γ , V ) . (4.19) Proof.
With the use of the spectral decomposition, we find OO ′ = A s O k =1 O k ! A · A s O l =1 O ′ l ! A = A s O k =1 X µ k λ µ k | µ k ih µ k | ! A · A s O k =1 X ν k λ ′ ν k | ν k ih ν k | ! A = X µ , ··· ,µ s X ν , ··· ,ν s s Y k =1 λ µ k λ ′ ν k ! A s O k =1 | µ k i ! " s O k =1 h µ k | A ! A s O k =1 | ν k i ! s O k =1 h ν k | ! A. (4.20)2From (4.7) and using the spectral decomposition again, we obtain OO ′ = X µ , ··· ,µ s X ν , ··· ,ν s s Y k =1 λ µ k λ ′ ν k ! A s O k =1 | µ k i ! s Y k =1 h µ k | ν k i ! s O k =1 h ν k | ! A = X µ , ··· ,µ s X ν , ··· ,ν s s Y k =1 λ µ k λ ′ ν k ! A s O k =1 | µ k i ! s O k =1 h µ k | ! s O k =1 | ν k i ! s O k =1 h ν k | ! A = A s O k =1 X µ k λ µ k | µ k ih µ k | ! s O k =1 X ν k λ ′ ν k | ν k ih ν k | ! A = A s O k =1 O k O ′ k ! A. (4.21)Since O k and O ′ k are in the form (4.3), so is the product O k O ′ k . Besides, since the product O k O ′ k is assumed to beself-adjoint for all k , we obtain (4.19).Let I k be the projection operator onto W (Γ k , V k ) from H (Γ k ), i.e. , the operator (4.3) with λ µ = 1 for all µ , whichacts as the identity operator when restricted to W (Γ k , V k ). Given an O ∈ C (Γ , V ) in (4.4) with O k , k = 1 , . . . , s , weconsider the corresponding observables in H tot A given by O ( k ) (Γ , V ) := A k − O i =1 I i ! ⊗ O k s O i = k +1 I i ! A. (4.22)Note that, since I i are all self-adjoint, we have O ( k ) (Γ , V ) ∈ C (Γ , V ). We then find from Lemma 4 that Corollary 2
An observable O ∈ C (Γ , V ) given in (4.4) with O k , k = 1 , . . . , s , admits the factorization, O = s Y k =1 O ( k ) (Γ , V ) , (4.23) in terms of the observables O ( k ) (Γ , V ) for k = 1 , . . . , s defined in (4.22). It is readily seen that for | Ψ i ∈ S (Γ , V ) the factorization corresponding to (4.23) also holds at the level of theexpectation value: Proposition 3
For a normalized state | Ψ i ∈ S (Γ , V ) , the expectation value of an observable O ∈ C (Γ , V ) in (4.4)factorizes as h Ψ | O | Ψ i = s Y k =1 h Ψ | O ( k ) (Γ , V ) | Ψ i , (4.24) where O ( k ) (Γ , V ) , k = 1 , . . . , s , are observables given in (4.22). Proof.
Application of Corollary 1 to the both sides of (4.24) yields the same result given by the r.h.s. of (4.17).
C. Criterion of Separability and the Complete Set of Properties
We now present our criterion of separability of N -particle fermionic systems into arbitrary subsystems: Definition 1
A normalized N -particle antisymmetric pure state | Ψ i ∈ H tot A is separable with respect to (Γ , V ) ifthere exist one-dimensional projection operators P k , k = 1 , · · · , s , projecting in W (Γ k , V k ) from H A (Γ k ) such that h Ψ | P ( k ) (Γ , V ) | Ψ i = 1 , (4.25) with P ( k ) (Γ , V ) := A k − O i =1 I i ! ⊗ P k ⊗ s O i = k +1 I i ! A. (4.26)3This definition of separability with respect to (Γ , V ) is motivated by the fact that it gives a necessary and sufficientcondition for states to have factorizable inner products among themselves according to the partition Γ as well asfactorizable expectation values for observables belonging to the class C (Γ , V ). On account of the uniqueness of C (Γ , V )assigned to a given pair (Γ , V ), one can also regard that our criterion refers to separability of states with respect to C (Γ , V ). This may be more natural from the physical viewpoint that correlations are evaluated only with respectto the set of observables under consideration/preparation in advance. Now, the aforementioned fact motivating thedefinition is shown by Theorem 2
A normalized N -particle antisymmetric pure state | Ψ i ∈ H tot A is separable with respect to (Γ , V ) if andonly if | Ψ i ∈ S (Γ , V ) . Proof.
Suppose | Ψ i ∈ S (Γ , V ). Applying (4.17) of Corollary 1 to P ( k ) = P ( k ) (Γ , V ) in (4.26), we obtain h Ψ | P ( k ) | Ψ i = h ψ k | P k | ψ k i . (4.27)Setting P k = | ψ k ih ψ k | for all k , we see h Ψ | P ( k ) | Ψ i = 1 , k = 1 , . . . , s. (4.28)Conversely, suppose that there exists a state | Φ i ∈ H tot A fulfilling (4.25). We then have P ( k ) | Φ i = | Φ i + | Ω k i (4.29)for some state | Ω k i ∈ H tot A with h Φ | Ω k i = 0 . (4.30)Since P ( k ) is a projector, using (4.29) and (4.30), we obtain1 = h Φ | P ( k ) | Φ i = h Φ | ( P ( k ) ) | Φ i = ( h Φ | + h Ω k | ) ( | Φ i + | Ω k i ) = 1 + h Ω k | Ω k i , (4.31)from which we find | Ω k i = 0. This implies P ( k ) | Φ i = | Φ i for all k , and thus if we consider the product of theprojectors, P := s Y k =1 P ( k ) = A s O k =1 P k ! A, (4.32)we find P | Φ i = s Y k =1 P ( k ) | Φ i = | Φ i . (4.33)Note that since all P ( k ) belong to C (Γ , V ) defined in (4.4), so does the product P ∈ C (Γ , V ) by Corollary 2. On theother hand, since the operator P k in P ( k ) is a one-dimensional projector in W (Γ k , V k ), one can put P k = | ξ k ih ξ k | forsome normalized | ξ k i ∈ W (Γ k , V k ). Then, recalling (4.32), we have P = A s O k =1 | ξ k ih ξ k | ! A = | Ξ ih Ξ | , (4.34)where | Ξ i := A s O k =1 | ξ k i ∈ S (Γ , V ) . (4.35)Combining (4.34) with (4.33), we arrive at | Φ i = h Ξ | Φ i | Ξ i . (4.36)which shows | Φ i ∈ S (Γ , V ).4We learn from (4.36) that | Φ i is equivalent to | Ξ i in (4.35) up to the global phase h Ξ | Φ i . It then follows that thereexists a one-to-one correspondence between a projector P and a state in S (Γ , V ) , and if | Φ i and P are in such acorrespondence, we have h Φ | P | Φ i = 1. Concerning the ontological aspect of our separability criterion, Theorem 2implies that separable states under the partition Γ consists of states | ξ k i ∈ W (Γ k , V k ), k = 1 , . . . , s , each possessing the‘complete set of properties’ in the subsystem Γ k as mentioned in [10]. In fact, the measurement outcome h Φ | P | Φ i = 1(or (4.25)) ensures that the possession of the properties specified by P k can be confirmed with certainty to the setof number | Γ k | of particles, not to the individual particles in each of the subsystems Γ k , in accordance with theindistinguishability of the particles. Note that this holds simultaneously for all k , since P ( k ) (Γ , V ) commute with eachother among themselves.We point out that it is possible to regard our criterion (Definition 1) as a generalization of the criterion given in[10] for the bipartition s = 2 case in which only one projection (rather than s = 2 projections) is used. Indeed, we seethat for the arbitrary s -partite case our criterion can actually be replaced by a slightly more economical one whichuses ( s −
1) projections, that is,
Proposition 4
A normalized N -particle antisymmetric pure state | Ψ i ∈ H tot A is separable with respect to (Γ , V ) ifand only if there exist ( s − one-dimensional projection operators P k in W (Γ k , V k ) from H A (Γ k ) for k = 1 , · · · , t − , t + 1 , · · · , s with an integer t ∈ [1 , s ] , such that (4.25) are fulfilled with (4.26). Proof.
Consider the product of ( s −
1) projectors, P ′ := t − Y k =1 P ( k ) ! s Y k = t +1 P ( k ) ! = A t − O k =1 P k ! ⊗ I t ⊗ s O k = t +1 P k !! A. (4.37) P ′ is a projection operator because, from Lemma 4, we have P ′ = A t − O k =1 P k ! ⊗ I t ⊗ s O k = t +1 P k !! A = A t − O k =1 P k ! ⊗ I t ⊗ s O k = t +1 P k !! A = P ′ . (4.38)The same argument used to show (4.33) then implies that, if a state | Ψ i ∈ H tot A fulfills (4.25) for k = t , we have P ′ | Ψ i = | Ψ i . (4.39)Since P k is a one-dimensional projection operator in W (Γ k , V k ), one can put P k = | ξ k ih ξ k | for some normalized | ξ k i ∈ W (Γ k , V k ). Using any normalized basis {| µ t i} in W (Γ t , V t ), we also have I t = P µ t | µ t ih µ t | . Then P ′ can bewritten as P ′ = A t − O k =1 | ξ k ih ξ k | ! ⊗ X µ t | µ t ih µ t | ! ⊗ s O k = t +1 | ξ k ih ξ k | !! A = X µ t | Ξ µ t ih Ξ µ t | , (4.40)where | Ξ µ t i = A t − O k =1 | ξ k i ! ⊗ | µ t i ⊗ s O k = t +1 | ξ k i !! . (4.41)It follows that | Ψ i = P ′ | Ψ i = X µ t h Ξ µ t | Ψ i | Ξ µ t i = A t − O k =1 | ξ k i ! ⊗ X µ t h Ξ µ t | Ψ i | µ t i ! ⊗ s O k = t +1 | ξ k i !! ∈ S (Γ , V ) . (4.42)The converse is trivial.The criterion for the full separation, where we have s = N and Γ k = { k } for k = 1 , , . . . , N , is also given in [10]with N projectors (see (4.43) below) in a way slightly different from the bipartition s = 2 case. However, again, theequivalence to our criterion in that case can be established by5 | Ψ i | Ψ ′ i | Ψ ′′ i S (Γ , V ) S (Γ , V ′ ) FIG. 2: A state may be separable with respect to more than one orthogonal structure under an identical partition Γ and, as aresult, the property of separability may not be transitive. Here, both of the states | Ψ i and | Ψ ′ i belong to S (Γ , V ) and hencethey are mutually separable ( e.g. , their inner product is factorizable). Similarly, | Ψ ′ i and | Ψ ′′ i belong to S (Γ , V ′ ), and theyare also mutually separable. However, the two states | Ψ i and | Ψ ′′ i are not necessarily separable, unless they share a commonorthogonal structure. Proposition 5 An N -particle antisymmetric pure state | Ψ i ∈ H tot A is fully separable if and only if there exist one-dimensional projection operators P i , i = 1 , , · · · , N , which are commonly defined in each constituent Hilbert space C d and are mutually orthogonal P i P j = 0 for i = j such that h Ψ | E ( i ) | Ψ i = 1 , (4.43) with E ( i ) = ⊗ N − ( − P i ) ⊗ N . (4.44) Proof.
We first expand E ( i ) as E ( i ) = − X σ ∈ S N N X k =1 ( − k N ! (cid:18) Nk (cid:19) π σ (cid:16) P ⊗ ki ⊗ ⊗ ( N − k ) (cid:17) π σ , (4.45)and thereby consider their product Q Ni =1 E ( i ) . Since each term in E ( i ) has at least one projector P i in the constituents,and since they are mutually orthogonal P i P j = 0 for i = j , we find E := N Y i =1 E ( i ) = X σ ∈ S N π σ N O i =1 P i ! π σ . (4.46)On account of (3.8), we obtain A E A = X σ ∈ S N A π σ N O i =1 P i ! π σ A = X σ ∈ S N A N O i =1 P i ! A = N ! A N O i =1 P i ! A = P, (4.47)where P is the product defined in (4.32) whose multinomial coefficient (3.12) is now M (Γ) = N !.Suppose that a state | Ψ i ∈ H tot A fulfills (4.43) for all i . Since E ( i ) are all projectors, employing again the sameargument to reach (4.33) we find E | Ψ i = | Ψ i . (4.48)It then follows from (4.47) and A| Ψ i = | Ψ i that P | Ψ i = A E A| Ψ i = A E | Ψ i = | Ψ i . (4.49)This is just the condition (4.33), and hence we arrive at the same conclusion | Ψ i ∈ S (Γ , V ) as before. The converseis trivial.A notable feature of our criterion is that it renders explicit the dependence of separability on the orthogonalstructure V used. An important consequence of this is that such a structure may not be assigned to a state uniquely,that is, a single state | Ψ i may admit more than two orthogonal structures under a given partition Γ. Moreover, thevery separable nature of states is not necessarily transitive, that is, even if | Ψ i and | Ψ ′ i , and also | Ψ ′ i and | Ψ ′′ i , areseparable between them sharing the same orthogonal structures, respectively, | Ψ i and | Ψ ′′ i may not be separable,unless they share the same orthogonal structure (see Figure 2). These properties will be seen more explicitly byexamples in the next section.Prior to this, however, we add6 Definition 2 An N -particle antisymmetric pure state | Ψ i ∈ H tot A is separable if it is separable with respect to (Γ , V ) for some partition Γ and set of orthogonal spaces V . This form of criterion will be more convenient when one is working in situations where the dependence of V isimmaterial and only the possibility of separability of states is of concern. V. IMPLICATIONS OF THE ORTHOGONAL STRUCTURES: EXAMPLES
In this section, related to the orthogonal structures encountered in our criterion, we elaborate on two technical pointswhich require particular attention in the discussion of separability. These are non-uniqueness of the space S (Γ , V ) thatcan be assigned to a factorizable state, and non-transitivity of factorizability among individual factorizable states.We also touch upon the physical role of the orthogonal structures concerning the cluster decomposition property. Fordefiniteness, in this section we illustrate these points by examples using only normalized states both for the totalsystem and the subsystems. A. Non-uniqueness
In Sect. III we mentioned that a state may belong to S (Γ , V ) for more than one V . To see this, consider a threeparticle N = 3 state with d = 4, | Ψ i = A (cid:20) √ | i | i − | i | i ) | i (cid:21) , (5.1)where we employ abbreviated notations such as | i | i in place of | i ⊗ | i . Clearly, the state | Ψ i belongs to S (Γ , V ) with Γ = { Γ , Γ } , Γ = { , } , Γ = { } (5.2)and V = { V , V } , V = span {| i , | i} , V = span {| i , | i} . (5.3)On the other hand, | Ψ i also belongs to S (Γ , V ′ ) with the same Γ in (5.2) and V ′ = { V ′ , V ′ } , V ′ = span {| i , | i , | i} , V ′ = span {| i} . (5.4)Consequently, the state | Ψ i is separable with respect to both (Γ , V ) and (Γ , V ′ ).Note that in the above example the two orthogonal structures V and V ′ are considered under the same basis vectors {| a i i } . However, this is not necessary as can be seen in the familiar case of the singlet state, | Ψ i = A ( | i | i ) = 1 √ | i | i − | i | i ) (5.5)in a two particle N = 2 system with d = 2. Indeed, we see immediately that | Ψ i ∈ S (Γ , V ) forΓ = { Γ , Γ } , Γ = { } , Γ = { } (5.6)and infinitely many sets of spaces V ( θ, φ ) = { V ( θ, φ ) , V ( θ, φ ) } with V ( θ, φ ) = span { cos θ | i + e i φ sin θ | i} , V ( θ, φ ) = span { sin θ | i − e − i φ cos θ | i} (5.7)parametrized by angles 0 ≤ θ ≤ π and 0 ≤ φ < π . Thus, the singlet state | Ψ i is separable with respect to allsuch (Γ , V ( θ, φ )). However, this ambiguity is deceptive because the singlet state | Ψ i is the only element spanningthe total Hilbert space H tot A in this case, and hence the resultant space S (Γ , V ( θ, φ )) is actually unique. Accordingly,all observables in C (Γ , ( θ, φ )), or even those in C (Γ), are proportional to the one dimensional projector | Ψ ih Ψ | . Forexample, from (4.4) one finds that any observable in C (Γ , V (0 , O = A ( k | i h | ⊗ k | i h | ) A = k k | Ψ ih Ψ | , k , k ∈ R , (5.8)7which is obviously the case for all C (Γ , V ( θ, φ )).At this point, we remark on the outcome of joint spin measurements of two particles obtained by the spin operator σ ( a , b ) := σ ( a ) ⊗ σ ( b ) where σ ( a ) = a · σ and σ ( b ) = b · σ are defined from normalized real 3-vectors a , b and Pauli matrices σ i = ( σ xi , σ yi , σ zi ) for i = 1 ,
2. Since σ ( a , b ) acts on H tot = H ⊗ H but not on H tot A , for thefermionic system we may consider the corresponding operator A σ ( a , b ) A which is in the form (3.10) and belongs to C (Γ). However, it is important to notice that, if we are to regard O = σ ( a ) and O = σ ( b ) as a set of operators forindividual spin measurements of the particles, then σ ( a , b ) does not belong to C (Γ , V ), because the spin operators,when considered in an identical subsystem Γ i , have a non-vanishing support σ i ( a ) σ i ( b ) = 0 for i = 1 , A σ ( a , b ) A = − ( a · b ) | Ψ ih Ψ | (5.9)that it is impossible to find an orthogonal set of operators O and O for A σ ( a , b ) A to form an observable O ∈ C (Γ , V )as (4.4) such that they depend on each of the measurement parameters a and b separately, O = O ( a ), O = O ( b ),because if so the resultant operator would then be of the form (5.8) with the factorized eigenvalues k = k ( a ) and k = k ( b ) in contradiction with (5.9). This indicates that, if we restrict ourselves only to the spin degrees of freedom,the formal non-factorizability of the correlation h Ψ | σ ( a , b ) | Ψ i = − a · b (from which the Cirelson operator is formed[17, 21]) for the singlet state [16], despite it belongs to the separable set S (Γ , V ), can arise from the fact that theoperator A σ ( a , b ) A does not belong to C (Γ , V ) under the identification O = σ ( a ) , O = σ ( b ) based on whichthe spin correlation is considered. A clearer account of spin correlation for fermions, with the orthogonal structure V equipped naturally to the actual experimental setup, can be gained by introducing the spatial degrees of freedompossessed by the particles, as we shall see in the last examples in this section.To sum up, the lesson one learns here is that, even if a state is separable with respect to some (Γ , V ), the expectationvalue may not be separable for observables unless they belong to the corresponding class C (Γ , V ) in which individualmeasurements of subsystems can be defined properly. B. Non-transitivity
In Sect. III we noted that, if two states | Ψ i and | Ψ ′ i belong to one S (Γ , V ) ⊆ H tot A , their inner product is factorizable.It is important to notice that this property is not transitive, that is, even if the inner product is factorizable between | Ψ i and | Ψ ′ i , and also between | Ψ ′ i and | Ψ ′′ i , it does not necessarily imply that it is so between | Ψ i and | Ψ ′′ i , unlessall the three belong to the same S (Γ , V ) (see Figure 2). To illustrate this, let us consider an N = 2 system with d = 6.We have Γ in (5.6) and consider the two sets of space, V = { V , V } with V = span {| a i | a = 0 , , } , V = span {| a i | a = 3 , , } , (5.10)and V ′ = { V ′ , V ′ } with V ′ = span {| a i | a = 0 , , } , V ′ = span {| a i | a = 2 , , } . (5.11)In terms of normalized vectors α = ( α , α , α ) ∈ C , the state vectors in V i [ H i ] and V ′ i [ H i ] can be written as | ψ ( α ) i = α | i + α | i + α | i ∈ V [ H ] , | ψ ( α ) i = α | i + α | i + α | i ∈ V [ H ] , (5.12)and | ψ ′ ( α ) i = α | i + α | i + α | i ∈ V ′ [ H ] , | ψ ′ ( α ) i = α | i + α | i + α | i ∈ V ′ [ H ] , (5.13)respectively. The states in S (Γ , V ) and S (Γ , V ′ ) can thus be obtained as | Ψ( α, α ′ ) i = A | ψ ( α ) i| ψ ( α ′ ) i ∈ S (Γ , V ) , | Ψ ′ ( α, α ′ ) i = A | ψ ′ ( α ) i| ψ ′ ( α ′ ) i ∈ S (Γ , V ′ ) . (5.14)Observe that, although S (Γ , V ) and S (Γ , V ′ ) are different, they are not exclusive. Indeed, choosing β = ( β , β ,
0) and β ′ = (0 , β ′ , β ′ ) , (5.15)we find | Ψ( β, β ′ ) i = | Ψ ′ ( β, β ′ ) i ∈ S (Γ , V ) ∩ S (Γ , V ′ ) . (5.16)8Let now ¯ S (Γ , V ) be the complement of S (Γ , V ) in H tot A satisfying S (Γ , V ) ∩ ¯ S (Γ , V ) = ∅ and S (Γ , V ) ∪ ¯ S (Γ , V ) = H tot A . (5.17)If we choose two states | Ψ( α, α ′ ) i and | Ψ ′ ( γ, γ ′ ) i such that | Ψ( α, α ′ ) i ∈ S (Γ , V ) ∩ ¯ S (Γ , V ′ ) and | Ψ ′ ( γ, γ ′ ) i ∈ ¯ S (Γ , V ) ∩ S (Γ , V ′ ) , (5.18)then from Lemma 2 we find that their inner products with | Ψ( β, β ′ ) i in (5.16) are factorizable, h Ψ( α, α ′ ) | Ψ( β, β ′ ) i = h ψ ( α ) | ψ ( β ) i h ψ ( α ′ ) | ψ ( β ′ ) i , h Ψ ′ ( β, β ′ ) | Ψ ′ ( γ, γ ′ ) i = h ψ ′ ( β ) | ψ ′ ( γ ) i h ψ ′ ( β ′ ) | ψ ′ ( γ ′ ) i . (5.19)However, it is readily confirmed that h Ψ( α, α ′ ) | Ψ ′ ( γ, γ ′ ) i 6 = h ψ ( α ) | ψ ′ ( γ ) i h ψ ( α ′ ) | ψ ′ ( γ ′ ) i (5.20)in general, showing that the factorizability of the inner product is not transitive in this case. C. Cluster Decomposition Property
Consider an N = 2 system with d = 4. We regard the constituent Hilbert space C as C ⊗ C , where the first C is spanned by the basis {| + i , |−i} while the second C is spanned by {| L i , | R i} . The names of the basis vectorsare motivated by the idea that set {| + i , |−i} refers to internal spin states, and the set {| L i , | R i} refers to localizedstates on the left/right in space which are separated from each other; for example, | + i| L i implies that the particle islocalized on the left with the spin state | + i .As before, our Γ is given in (5.6), and if we choose the set V as V = { V , V } , V = span {| a i| L i | a = + , −} , V = span {| a i| R i | a = + , −} , (5.21)then we can write the states in S (Γ , V ) as | Ψ( α, α ′ ) i = A ( | α i | L i | α ′ i | R i ) = 1 √ | α i | L i | α ′ i | R i − | α ′ i | R i | α i | L i ) , (5.22)where | α i i and | α ′ i i , i = 1 ,
2, denote the spin states of the i -th particle. Namely, the space S (Γ , V ) consists of statesof two particles one of which is on the left and the other on the right, and the vanishing overlap h L | R i = 0 expressesthe remoteness between them.With the observables σ ( a ) and σ ( b ) in the spin space of the two particles defined earlier, respectively, we considerthe observables O = σ ( a ) | L i h L | , O = σ ( b ) | R i h R | . (5.23)We then construct the observable O ∈ C (Γ , V ) on the two particles according to the prescription (4.4): O = A ( O ⊗ O ) A = A ( σ ( a , b ) ⊗ | L i h L | ⊗ | R i h R | ) A. (5.24)This corresponds clearly to the simultaneous measurement of spins on one particle on the left and on the other onthe right, and for the state | Ψ( α, α ′ ) i ∈ S (Γ , V ) in (5.22) its expectation value factorizes h Ψ( α, α ′ ) | O | Ψ( α, α ′ ) i = h α | σ ( a ) | α i h α ′ | σ ( b ) | α ′ i = h α ′ | σ ( b ) | α ′ i h α | σ ( a ) | α i , (5.25)by Corollary 2. Note that, on account of Lemma 1, the factorization is also seen for the operator ˜ O in (3.15),˜ O = σ ( a , b ) ⊗ | L i h L | ⊗ | R i h R | + σ ( b , a ) ⊗ | R i h R | ⊗ | L i h L | . (5.26)The outcome shows that, even though the states (5.22) are not direct product states because of the (rescaled) anti-symmetrizer A , the correlation of the two local measurements O and O disappears. In other words, if the twofermions are localized and remotely separated from each other, they behave independently as long as their spins areconcerned. This is a prime example to illustrate the cluster decomposition property [1–3] in the fermionic system,and has been mentioned in [10].9However, the factorizability may break down for states which admit intrinsically long range correlations. To seethis, let us consider the antisymmetric state, | Φ( α, α ′ ) i = A (cid:0) | ψ + ( α, α ′ ) i ⊗ | L i | R i (cid:1) = | ψ + ( α, α ′ ) i ⊗ √ | L i | R i − | R i | L i ) , (5.27)where | ψ + ( α, α ′ ) i = N ( α, α ′ ) ( | α i | α ′ i + | α ′ i | α i ) , N ( α, α ′ ) = 1 p | h α | α ′ i | ) , (5.28)where we used h α ′ | α i = h α ′ | α i derived from the isomorphism between the constituents. Clearly, since the state | Φ( α, α ′ ) i is not an anti-symmetrized state of a direct product of constituent states, we have | Φ( α, α ′ ) i / ∈ S (Γ , V ) for S (Γ , V ) given in (5.21). Such a state with orthogonality in the internal degree of freedom h α | α ′ i = 0 can be foundas a 1s2s state of electrons in Helium atom in the first order approximation of perturbation theory, where the statefor the spatial degrees of freedom of | Φ( α, α ′ ) i results from the anti-symmetrization of the product state | L i | R i .Because of the factorization between the spin part and the spatial part, we have the density matrix for the spin partin the pure state form, X i,j = L,R h i | h j | ( | Φ( α, α ′ ) ih Φ( α, α ′ ) | ) | i i | j i = | ψ + ( α, α ′ ) ih ψ + ( α, α ′ ) | . (5.29)The expectation value of the observable O in (5.24) with | Ψ( α, α ′ ) i then becomes h Φ( α, α ′ ) | O | Φ( α, α ′ ) i = N ( α, α ′ ) (cid:16) h α | σ ( a ) | α i h α ′ | σ ( b ) | α ′ i + h α ′ | σ ( a | α ′ i h α | σ ( b ) | α i + h α | σ ( a ) | α ′ i h α ′ | σ ( b ) | α i + h α ′ | σ ( a ) | α i h α | σ ( b ) | α ′ i (cid:17) , (5.30)which is no longer factorized and gives rise to correlations between the constituents in general. Also, utilizing therelations h α | σ ( a ) | α i = h α | σ ( a ) | α i and so on which come from the isomorphism between constituents, one canconfirm Proposition 2 by direct calculation of the expectation value of ˜ O in (5.26) with respect to | Φ( α, α ′ ) i .To extend the above examples to a system with more than three subsystems, let us consider an N = 4 system with d = 6. The constituent Hilbert space C can be thought of as the direct product of C spanned by the basis {| + i , |−i} and C spanned by the basis {| L i , | C i , | R i} . As before, we regard {| + i , |−i} as internal spin states and {| L i , | C i , | R i} as spatial states localized on the left, the center and the right, respectively, which are separated from each other andshare no support in common. We choose the partition Γ and the set of subspaces V asΓ = { Γ , Γ , Γ } , Γ = { } , Γ = { } , Γ = { , } , (5.31)and V = { V , V , V } with V = span {| a i| L i| a = + , −} , V = span {| a i| C i| a = + , −} , V = span {| a i| R i| a = + , −} . (5.32)The states in S (Γ , V ) can be written as | Ψ( α, β, γ ) i = A ( | α i | L i | β i | C i | γ i ) , (5.33)where | γ i = | Singlet i ⊗ | R i | R i ∈ W (Γ , V ) (5.34)is the state of subsystem Γ representing the spin singlet state localized in the right of the space. From the prescriptionin (4.4), the observables in C (Γ , V ) read O = A ( O ⊗ O ⊗ O ) A, (5.35)with O = σ ( a ) | L i h L | , O = σ ( b ) | C i h C | , O = | γ i
34 34 h γ | . (5.36)Under the state | Ψ( α, β, γ ) i ∈ S (Γ , V ), the observable O has the expectation value which is factorized as h Ψ( α, β, γ ) | O | Ψ( α, β, γ ) i = h α | σ ( a ) | α i h β | σ ( b ) | β i h γ | O | γ i . (5.37)0On the other hand, the factorizability does not hold for states which do not belong to S (Γ , V ). For example, wehave the state | Φ i = A (cid:0) | ψ + ( α, α ′ ) i ⊗ | L i | C i ⊗ | + i | R i |−i | R i (cid:1) (5.38)which does not belong to S (Γ , V ) due to the entangled spin part | ψ + ( α, α ′ ) i in (5.28). The expectation value of thestate is given by h Φ | O | Φ i = N ( α, α ′ ) (cid:16) h α | σ ( a ) | α i h α ′ | σ ( b ) | α ′ i + h α ′ | σ ( a ) | α ′ i h α | σ ( b ) | α i + h α | σ ( a ) | α ′ i h α ′ | σ ( b ) | α i + h α ′ | σ ( a ) | α i h α | σ ( b ) | α ′ i (cid:17) · h γ | O | γ i , (5.39)which is not factorized with respect to the partition Γ in (5.31). VI. CONCLUSIONS AND DISCUSSIONS
In this paper, we have presented a general and coherent framework to examine the separability of states for N -particle fermionic systems under arbitrary partitions. The system we have considered consists of N constituentsystems of fermions each possessing the d -dimensional Hilbert space C d , and the decomposition of the system intosubsystems is specified by an arbitrary partition Γ which provides a grouping of the constituents in the system. Withthe help of an orthogonal structure V defined in the constituent space C d , we have found in the entire Hilbert space H tot A a set S (Γ , V ) of states for which the inner products are factorizable according to the partition Γ. Moreover, wehave seen that there exists a class of observables C (Γ , V ) on H tot A whose expectation values (and weak values) arefactorizable for all states in S (Γ , V ) (Theorem 1 and Corollary 1). In the N = 2 example mentioned in Sect. II, weintroduced the additional (spatial) degrees of freedom to realize an orthogonal structure V in order to recover thebehavior of distinguishable particles for fermionic states. However, our argument shows that the same goal is achievedwithout invoking such additional degrees of freedom, if the constituent system has a sufficiently large dimension d tosustain fermionic states within the internal degrees of freedom alone, i.e. , if d ≥ N .Our separability criterion (Definition 1) for arbitrary s partitions, given by a set of projection operators, is anextension (with a slight modification in form) of the criterion proposed earlier in [10] for the case s = 2 and s = N in the context of ‘complete sets of properties’. Our Theorem 2 shows that our criterion is fulfilled by a state if andonly if it belongs to S (Γ , V ) for some V under a given Γ, that is, S (Γ , V ) furnishes the entire set of states which areseparable with respect to the partition Γ and the orthogonal structure V , generalizing the result obtained in [10] forthe fermionic case.A salient feature of our criterion, or of multi-partite separability in general, is the explicit dependence on theorthogonal structure V . Most notably, it implies that a state which is separable under V may no longer be separableunder different V even with the same partition Γ. In a sense this is natural because, on physical grounds, the choiceof V corresponds to the available experimental setups prepared for measurement, where the orthogonal structure V ensures that the measurement yields unambiguous outputs concerning the separation of the subsystems. The class ofobservables C (Γ , V ) for which the state possesses factorizable expectation values will then be determined by the setupsused. Indeed, the example of spin measurement for remotely separated fermions given in Sect. V shows that localizedmeasurement setups have the class C (Γ , V ) of observables for which no correlations arise with the set S (Γ , V ) of anti-symmetrized product states in line with the cluster decomposition property. However, the same class of observablesadmit inseparable states which are entangled in the spin part and the spatial part, separately. In addition, we havementioned the non-uniqueness in the assignment of S (Γ , V ) to a separable state, as well as the non-transitivity of theseparability, as important consequences of the V -dependence of our separability criterion. These considerations on theimplications of the orthogonal structure V suggest that it might be actually more appropriate to ask the separabilityof the system based on the experimental setups qualified as measurement for separation, not on the states as we do.In closing we remark that our framework can in principle be extended to the bosonic (totally symmetric) statesby use of the symmetrizer instead of the anti-symmetrizer, coupled with some extra consideration pertinent to thebosonic systems. It should also be important to study how to quantify the amount of inseparability, namely, to findsuitable entanglement measures for N -particle indistinguishable particles relevant to the separability under arbitrarypartitions. The family of measures proposed recently by us in [22] which are specifically designed to examine thearbitrary separability (for distinguishable particles) may be convenient for that purpose.1 ACKNOWLEDGEMENTS
The authors thank Prof. Akira Shimizu for calling our attentions to the topic. T. I. is supported by ‘Open ResearchCenter’ Project for Private Universities; matching fund subsidy, T. S. is supported by Global COE Program ‘thePhysical Sciences Frontier’, and I. T. is supported by the Grant-in-Aid for Scientific Research (C), No. 20540391-H21,all of MEXT, Japan. [1] E. H. Wichmann and J. H. Crichton, Cluster Decomposition Properties of the S Matrix,
Phys. Rev. (1963) 2788.[2] Asher Peres,
Quantum Theory: Concepts and Methods , Kluwer, Dordrecht, 1993.[3] Steven Weinberg,
Quantum Theory of Fields, Volume I, Foudations , Cambridge university press, Cambridge, 1995.[4] John Schliemann, J. Ignacio Cirac, Marek Ku´s, Maciej Lewenstein and Daniel Loss, Quantum correlations in two-fermionsystems,
Phys. Rev.
A64 (2001) 022303.[5] Y. S. Li, B. Zeng, X. S. Liu, and G. L. Long, Entanglement in a two-identical-particle system,
Phys. Rev.
A64 (2001)054302.[6] K. Eckert, J. Schliemann, D. Bruß and M. Lewenstein, Quantum correlations in systems of indistinguishable particles,
Ann. Phys. (2002) 88-127.[7] R. Paskauskas and L. You, Quantum correlations in two-boson wave functions,
Phys. Rev.
A64 (2001) 042310.[8] H. M. Wiseman and J. A. Vaccaro, Entanglement of Indistinguishable Particles Shared between Two Parties,
Phys. Rev.Lett. (2003) 097902.[9] GianCarlo Ghirardi and Luca Marinatto, General criterion for the entanglement of two indistinguishable particles, Phys.Rev.
A70 (2004) 012109.[10] G.-C. Ghirardi, L. Marinatto and T. Weber, Entanglement and Properties of Composite Quantum Systems: a Conceptualand Mathematical Analysis,
J. Stat. Phys. (2002) 49.[11] GianCarlo Ghirardi and Luca Marinatto, Entanglement and properties,
Fortschr. Phys. (2003) 379.[12] Michael A. Nielsen and Isaac L. Chuang, Quantum Computaion and Quantum Information , Cambridge university press,Cambridge, 2000.[13] Yakir Aharonov, David Z. Albert, and Lev Vaidman, How the Result of a Measurement of a Component of the Spin of aSpin- Particle Can Turn Out to be 100,
Phys. Rev. Lett. (1988) 1351.[14] Yakir Aharonov and Daniel Rohrich, Quantum Paradoxes , WILEY-VCH, Weinheim, 2005.[15] Kazuhiro Yokota, Takashi Yamamoto, Masato Koashi and Nobuyuki Imoto, Direct observation of Hardy’s paradox by jointweak measurement with an entangled photon pair,
New J. Phys. (2009) 033011.[16] J. S. Bell, On the Einstein-Podolsky-Rosen Paradox, Physics (1964) 195.[17] J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Proposed Experiment to Test Local Hidden-Variable Theories, Phys. Rev. Lett. (1969) 880.[18] A. Aspect, J. Dalibard and G. Roger, Experimental Test of Bell’s Inequalities Using Time- Varying Analyzers, Phys. Rev.Lett. (1982) 1804.[19] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter and A. Zeilinger, Violation of Bell’s Inequality under Strict EinsteinLocality Conditions, Phys. Rev. Lett. (1998) 5039.[20] J. S. Bell, Introduction to the Hidden-Variable Question, in Foundation of Quantum Mechanics , Proc. Int. Sch. Phys.‘Enrico Fermi’, ed. B. d’Espagnat, Academic, New York, 1971.[21] B. S. Cirel’son, Quantum generalizations of Bell’s inequality,
Lett. Math. Phys. (1980) 93.[22] Tsubasa Ichikawa, Toshihiko Sasaki and Izumi Tsutsui, Entanglement measures for intermediate separability of quantumstates, Phys. Rev.