One-body entanglement as a quantum resource in fermionic systems
OOne-body entanglement as a quantum resource in fermionic systems
N. Gigena, M. Di Tullio, and R. Rossignoli
1, 2 IFLP/CONICET and Departamento de F´ısica, Universidad Nacional de La Plata, C.C. 67, La Plata (1900), Argentina Comisi´on de Investigaciones Cient´ıficas (CIC), La Plata (1900), Argentina
We show that one-body entanglement, which is a measure of the deviation of a pure fermionicstate from a Slater determinant (SD) and is determined by the mixedness of the single-particledensity matrix (SPDM), can be considered as a quantum resource. The associated theory has SDsand their convex hull as free states, and number conserving fermion linear optics operations (FLO),which include one-body unitary transformations and measurements of the occupancy of single-particle modes, as the basic free operations. We first provide a bipartitelike formulation of one-bodyentanglement, based on a Schmidt-like decomposition of a pure N -fermion state, from which theSPDM [together with the ( N − I. INTRODUCTION
Quantum entanglement and identical particles are twofundamental concepts in quantum mechanics. Entangle-ment in systems of distinguishable components is par-ticularly valuable in the field of quantum informationtheory [1] because it can be considered as a resourcewithin the Local Operations and Classical Communica-tion (LOCC) paradigm [1, 2]. Extending the notion of en-tanglement to the realm of indistinguishable particles is,however, not straightforward because the constituents ofthe system cannot be individually accessed. Different ap-proaches have been considered, like mode entanglement[3–5], where subsystems correspond to a set of single-particle (SP) states in a given basis, extensions basedon correlations between observables [6–10] and entangle-ment beyond symmetrization [11–21], which is indepen-dent of the choice of SP basis. Several studies on the re-lation between these types of entanglement [5, 16, 20, 22–28] and on whether exchange correlations can be associ-ated with entanglement [29–33] have been recently made.There is also a growing interest in quantum chemistrysimulations based on optical lattices [34, 35], which wouldbenefit from a detailed characterization of fermionic cor-relations. In this paper we will focus on entanglementbeyond antisymmetrization in fermionic systems and an-alyze its consideration as a quantum resource.Quantum resource theories [36, 37] have recently be-come a topic of great interest since they essentially de-scribe quantum information processing under a restrictedset of operations. Standard entanglement theory in sys-tems of distinguishable components is just one of thesetheories, amongst which we may include others like quan-tum thermodynamics [38, 39], coherence [40, 41], nonlo-cality [42] and non-Gaussianity [43].In the usual entanglement theory a multipartite quan-tum system shared by distant parties is considered.These parties can operate each on their own subsystemand are allowed to communicate via classical channels [2]. From these restrictions the LOCC set arises natu-rally as the set of free operations of the resource theory,and the set of free (separable) states is then derived. Inour case ignoring antisymmetrization correlations definesSlater determinants (SDs) and their convex hull as theset of “free” states S and we are looking for a set of freeoperations O consistent with this set.With this aim, we first define a partial order relationon the Fock space F of the system, based on the mixed-ness of the corresponding single-particle density matrix (SPDM) ρ (1) [also denoted as the one-particle or one-body density matrix (DM)], which determines whethera given pure fermionic state can be considered more en-tangled than another state. A bipartite like formulationfor this one-body entanglement , involving ρ (1) and the( N − N fermions) is also provided. Next we define a class ofoperations consistent with S and the previous partial or-der, through a majorization relation to be fulfilled by theinitial and final SPDMs, which ensures that one-body en-tanglement will not be increased by such operations. Wethen show that number conserving Fermion linear optics(FLO) operations [44–46], which include one-body uni-tary transformations and measurement of the occupancyof a SP state, are indeed within this class. One-bodyentanglement then plays the role of a resource in a the-ory where S is the convex hull of SDs and O is that ofFLO operations. Possible extensions of the set of free op-erations and connection of this resource with a quantumcomputation model and with mode entanglement are alsodiscussed. II. FORMALISMA. One-body entanglement
We consider a SP space H of finite dimension n anda set of fermion creation and annihilation operators c † k a r X i v : . [ qu a n t - ph ] O c t and c k associated with an orthogonal basis of H , sat-isfying the anticommutation relations { c k , c † k (cid:48) } = δ kk (cid:48) , { c k , c k (cid:48) } = { c † k , c † k (cid:48) } = 0. The elements of the SPDM ρ (1) in a general fermionic state ρ are given by ρ (1) kk (cid:48) = (cid:104) c † k (cid:48) c k (cid:105) = Tr ρ c † k (cid:48) c k . (1)They form a Hermitian matrix with eigenvalues λ ν = (cid:104) c † ν c ν (cid:105) ∈ [0 , c † ν = (cid:80) k U kν c † k creates a fermionin one of the “natural” SP orbitals diagonalizing ρ (1) ( (cid:104) c † ν (cid:48) c ν (cid:105) = λ ν δ νν (cid:48) ). For pure states ρ = | Ψ (cid:105)(cid:104) Ψ | , the“mixedness” of ρ (1) then reflects the deviation of | Ψ (cid:105) froma SD [ (cid:81) ν ( c † ν ) n ν ] | (cid:105) , since for the latter λ ν = n ν = 0 or 1 ∀ ν and hence ( ρ (1) ) = ρ (1) .Such mixedness can be rigorously characterizedthrough majorization [47–50]. For states | Ψ (cid:105) and | Φ (cid:105) with the same fermion number N = Tr ρ (1)Ψ = Tr ρ (1)Φ , wewill say that | Ψ (cid:105) is not less one-body entangled than | Φ (cid:105) if ρ (1)Ψ is more (or equally) mixed than ρ (1)Φ , i.e. if theireigenvalues λ = ( λ , . . . , λ n ), sorted in decreasing order,satisfy the majorization relation λ ( ρ (1)Ψ ) ≺ λ ( ρ (1)Φ ) , (2)which means m (cid:88) ν =1 λ ν ( ρ (1)Ψ ) ≤ m (cid:88) ν =1 λ ν ( ρ (1)Φ ) (3)for m = 1 , . . . , n −
1, with identity for m = n . Thus SDsare the least one-body entangled states, as their SPDMmajorizes any other ρ (1) with the same trace. Relation(2) is analogous to that imposed by LOCC operationson reduced states of systems of distinguishable compo-nents, which in the bipartite case lead to the celebratedNielsen’s theorem: | Ψ AB (cid:105) can be converted by LOCC to | Φ AB (cid:105) (and hence is not less entangled than | Φ AB (cid:105) ) if andonly if their reduced states satisfy λ ( ρ A ( B )Ψ ) ≺ λ ( ρ A ( B )Φ )[48, 51]. Local measurements reduce the ignorance aboutthe state of the measured subsystem, decreasing themixedness of reduced states and hence bipartite entan-glement. Similarly, we will show that one-body entan-glement will decrease under operations which reduce theignorance about the SPDM.
1. The associated Schmidt decomposition
We first remark that one-body entanglement also ad-mits a bipartite like formulation: A pure state | Ψ (cid:105) of N fermions ( (cid:80) k c † k c k | Ψ (cid:105) = N | Ψ (cid:105) ) can be expanded as | Ψ (cid:105) = 1 N (cid:88) k,l Λ kl c † k C † l | (cid:105) (4)where C † l = c † l . . . c † l N − , l = 1 , . . . , ( nN − ), are operatorscreating N − l , satisfying (cid:104) | C l C † l (cid:48) | (cid:105) = δ ll (cid:48) , while the coefficients Λ kl form an n × ( nN − ) matrix Λ satisfying Tr ΛΛ † = N . Thus,each term in the sum (4) is a SD which is repeated N times, such that c k | Ψ (cid:105) = (cid:88) l Λ kl C † l | (cid:105) (5)is the (unnormalized) state of remaining fermions whenSP state k is occupied, while C l | Ψ (cid:105) = ( − N − (cid:88) k Λ kl c † k | (cid:105) (6)is that of remaining fermion when the N − l are occupied. In this way, (cid:104) Ψ | Ψ (cid:105) = N Tr ΛΛ † = 1.Moreover, Eqs. (5)–(6) allow us to express the elementsof both the SPDM ρ (1) and the ( N − -body DM ρ ( N − in terms of Λ as ρ (1) kk (cid:48) = (cid:104) Ψ | c † k (cid:48) c k | Ψ (cid:105) = (ΛΛ † ) kk (cid:48) , (7) ρ ( N − ll (cid:48) = (cid:104) Ψ | C † l (cid:48) C l | Ψ (cid:105) = (Λ T Λ ∗ ) ll (cid:48) . (8)Eqs. (7)–(8) are analogous to those for the reduced states ρ A ( B ) of distinguishable subsystems in a standard purebipartite state | Ψ AB (cid:105) = (cid:80) i,j C ij | i A , j B (cid:105) , where ρ Aii (cid:48) = (cid:104)| i (cid:48) A (cid:105)(cid:104) i A |(cid:105) = ( CC † ) ii (cid:48) , ρ Bjj (cid:48) = (cid:104)| j (cid:48) B (cid:105)(cid:104) j B |(cid:105) = ( C T C ∗ ) jj (cid:48) [1]. The only difference is that Tr ρ A ( B ) = Tr CC † = 1whereas Tr ρ (1) = Tr ρ ( N − = N .Eqs. (7)–(8) imply that ρ (1) and ρ ( N − have the samenonzero eigenvalues λ ν , which are just the square of thesingular values of Λ. Moreover, by means of the singularvalue decomposition Λ = U DV † , with D νν (cid:48) = √ λ ν δ νν (cid:48) and U and V unitary matrices (of n × n and ( nN − ) × ( nN − )respectively), we may now obtain from (4) the 1–( N − N -fermion state: | Ψ (cid:105) = 1 N (cid:88) ν (cid:112) λ ν c † ν C † ν | (cid:105) , (9)where c † ν = (cid:88) k U kν c † k , C † ν = (cid:88) l V ∗ lν C † l (10)are the “natural” one- and N −
1- fermion creation op-erators satisfying (cid:104) | c ν c † ν (cid:48) | (cid:105) = δ νν (cid:48) = (cid:104) | C ν C † ν (cid:48) | (cid:105) . (11) (cid:104) Ψ | c † ν c ν (cid:48) | Ψ (cid:105) = λ ν δ νν (cid:48) = (cid:104) Ψ | C † ν C ν (cid:48) | Ψ (cid:105) . (12)Thus, c ν | Ψ (cid:105) = (cid:112) λ ν C † ν | (cid:105) , C ν | Ψ (cid:105) = ( − N − (cid:112) λ ν c † ν | (cid:105) , (13)i.e. the orthogonal natural N − C † ν | (cid:105) arethose of remaining fermions when the natural SP orbital ν is occupied, while c † ν | (cid:105) are the orthogonal states of theremaining fermion when the natural N − C † ν | (cid:105) (which in general is no longer a SD) is occupied.Therefore, in an N -fermion state one-body entanglementis actually the 1–( N −
1) -body entanglement, associatedwith the correlations between one- and N − | Ψ (cid:105) = ( (cid:81) Nν =1 c † ν ) | (cid:105) , λ ν = 1 (0) for ν ≤ N ( > N ), with C † ν ∝ (cid:81) Nν (cid:48) (cid:54) = ν c † ν (cid:48) such that c † ν C † ν | (cid:105) = | Ψ (cid:105) for ν ≤ N . On the other hand, for N = 2 Eq. (9)becomes the Slater decomposition of a two-fermion state[11–13], | Ψ (cid:105) = (cid:88) ν (cid:112) λ ν c † ν c † ¯ ν | (cid:105) = 12 (cid:88) ν (cid:112) λ ν ( c ν C † ν + c ¯ ν C † ¯ ν ) | (cid:105) (14)where C † ν = c † ¯ ν , C † ¯ ν = − c † ν . In this case one-body entan-glement is directly related to that between the set of nor-mal ν and ¯ ν modes, which contain each just one-fermion(see sec. II C).
2. One-body entanglement entropies
We may now define a general one-body entanglemententropy E ( | Ψ (cid:105) ) ≡ E (1) ( | Ψ (cid:105) ) as E ( | Ψ (cid:105) ) = S ( ρ (1)Ψ ) = S ( ρ ( N − ) , (15)where S ( ρ (1) ) is a Schur-concave function [49, 50] of ρ (1) .These entropies will all satisfy E ( | Ψ (cid:105) ) ≥ E ( | Φ (cid:105) ) , (16)whenever the majorization relation of Eq. (2) is fulfilled.For instance, trace-form entropies S ( ρ (1) ) = Tr f ( ρ (1) ) = (cid:88) ν f ( λ ν ) (17)where f : [0 , → R is concave and satisfies f (0) = f (1) = 0 [52], will fulfill (16), with E ( | Ψ (cid:105) ) ≥ ∀ | Ψ (cid:105) and E ( | Ψ (cid:105) ) = 0 if and only if | Ψ (cid:105) is a SD. Such E ( | Ψ (cid:105) )will then be one-body entanglement monotones. Exam-ples are the von Neumann entropy of ρ (1) , S ( ρ (1) ) = − (cid:80) ν λ ν log λ ν , a quantity of interest in various fields[53–57], and the one-body entropy [20, 25] S ( ρ (1) ) = − (cid:88) ν λ ν log λ ν + (1 − λ ν ) log (1 − λ ν ) , (18)which represents, for | Ψ (cid:105) of definite fermion number N ,the minimum relative entropy (in the grand canonical en-semble) between ρ = | Ψ (cid:105)(cid:104) Ψ | and any fermionic Gaussianstate ρ g : S ( ρ (1)Ψ ) = Min ρ g S ( ρ || ρ g ) [25], for S ( ρ || ρ (cid:48) ) = − Tr ρ (log ρ (cid:48) − log ρ ) and ρ g ∝ exp[ − (cid:80) k,k (cid:48) α kk (cid:48) c † k c k (cid:48) ](pair creation and annihilation terms in ρ g are not re-quired for such | Ψ (cid:105) [25]). It is also the minimum overall SP bases of the sum of all single mode entropies [20] − p k log p k − (1 − p k ) log (1 − p k ), where p k = (cid:104) c † k c k (cid:105) . Weremark that the SPDM and hence any measure (15) are in principle experimentally accessible. Measurement ofthe fermionic SPDM in optical lattices has been recentlyreported [58].All measures (15) can be extended to mixed states ρ = (cid:88) α p α | Ψ α (cid:105)(cid:104) Ψ α | (19)of definite N through their convex roof extension E ( ρ ) =Min (cid:80) α p α E ( | Ψ α (cid:105) ), where the minimum is over all repre-sentations { p α ≥ , | Ψ α (cid:105)} of ρ [20]. Such E ( ρ ) representsa one-body entanglement of formation, vanishing if andonly if ρ is a convex mixture of SDs. B. One-body entanglement nongeneratingoperations
1. Definition and basic properties
We now define a class of operations which do not gen-erate one-body entanglement, i.e., which do not increase,on average, the mixedness of the SPDM.
Definition 1.
Let ε ( ρ ) = (cid:80) j K j ρ K † j be a quantumoperation on a fermion state ρ , with {K j , (cid:80) j K † j K j = } a set of Kraus operators, assumed number conserving.Let ρ (1) and ρ (1) j be the SPDMs determined by ρ and ρ j = K j ρ K † j /p j , with p j = Tr[ ρ K † j K j ] . We say that ε is one-body entanglement nongenerating (ONG) if it admits aset of Kraus operators {K j } satisfying ∀ ρ the relation λ ( ρ (1) ) ≺ (cid:88) j p j λ ( ρ (1) j ) , (20) where eigenvalues λ ( ρ (1) j ) are sorted in decreasing order. This majorization relation is analogous to that satis-fied by reduced local states under local operations in thestandard entanglement theory [48] and implies S ( ρ (1) ) ≥ S (cid:88) j p j λ ( ρ (1) j ) ≥ (cid:88) j p j S ( ρ (1) j ) , (21)for any concave entropy S ( ρ (1) ), such as those of Eq. (17).For pure states ρ = | Ψ (cid:105)(cid:104) Ψ | , ρ j = | Φ j (cid:105)(cid:104) Φ j | is also pure ∀ j , with | Φ j (cid:105) ∝ K j | Ψ (cid:105) , and Eqs. (15), (21) imply E ( | Ψ (cid:105) ) ≥ (cid:88) j p j E ( | Φ j (cid:105) ) ≥ E ( ε ( | Ψ (cid:105)(cid:104) Ψ | )) , (22)showing that any one-body entanglement monotone (15)will not increase, on average, after ONG operations. Inparticular, if | Ψ (cid:105) is a SD, E ( | Ψ (cid:105) ) = 0 and Eq. (22) impliesthat all states | Φ j (cid:105) ∝ K j | Ψ (cid:105) must be SDs or zero, i.e. all Kraus operators fulfilling (20) should map free statesonto free states . And for the one-body entanglement offormation of general mixed states ρ , Eq. (22) implies E ( ρ ) ≥ E ( ε ( ρ )) (23)since by using the minimizing representation, E ( ρ ) = (cid:80) α p α E ( | Ψ α (cid:105) ) ≥ (cid:80) α,j p α p αj E ( | Φ αj (cid:105) ) ≥ E ( ε ( ρ )).It also follows from (20) that the set of ONG operationsis convex and closed under composition , i.e., λ ( ρ (1) ) ≺ (cid:80) i,j p ij λ ( ρ (1) ij ) for K ij = K bi K aj and ε ( ρ ) = ε b [ ε a ( ρ )].This property ensures that ONG operations can be ap-plied any number of times in any order. Proposition 1.
The conversion of a pure state | Ψ (cid:105) ∈F into another pure state | Φ (cid:105) ∈ F by means of ONGoperations is possible only if the majorization relation (2) is satisfied by the corresponding SPDMs.Proof. The state conversion will consist in some sequenceof ONG operations, which can be resumed in just oneONG operation due to the closedness under composition.Let {K j } be a set of associated Kraus operators satisfying(20). After this operation is performed, we should have K j | Ψ (cid:105) = √ p j | Φ (cid:105) ∀ j , with p j = (cid:104) Ψ |K † j K j | Ψ (cid:105) , implying ρ (1) j = ρ (1)Φ ∀ j and hence Eq. (2) when (20) is fulfilled.Then, maximally one-body entangled states are thosepure states whose SPDM is majorized by that of anyother state. Due to Eq. (16) they will also maximize E ( | Ψ (cid:105) ) for any choice of S . At fixed fermion number N ≥ n = mN they are states leading to ρ (1) = n /m , (24)for which any SP basis is natural. For m integer such ρ (1) emerges, for instance, from Greenberger-Horne-Zeilinger(GHZ)-like states involving superpositions of SDs in or-thogonal subspaces: | Ψ (cid:105) = 1 √ m m − (cid:88) l =0 c † Nl +1 . . . c † Nl + N | (cid:105) = 1 N √ m n (cid:88) ν =1 c † ν C † ν | (cid:105) , (25)which lead to (cid:104) c † ν c ν (cid:48) (cid:105) = δ νν (cid:48) /m .
2. Fermion linear optics operations as ONG
We now show that number conserving FLO operations[44–46], which include one-body unitary transformationsand measurement of the occupancy of a SP mode, areincluded in the ONG set. First, any number conservingone-body unitary transformation U = exp[ − i (cid:88) k,k (cid:48) H k (cid:48) k c † k (cid:48) c k ] , (26)with U † U = ( H † = H ) is obviously ONG: Since U c † k U † = (cid:80) k (cid:48) U k (cid:48) k c † k (cid:48) , with U = e − iH , it will map theSPDM as ρ (1) → U † ρ (1) U , leaving its eigenvalues un-changed (and hence transforming SDs into SDs). It canbe implemented through composition of phaseshiftingand beamsplitters unitaries [44–46] U p ( φ ) = e − iϕ c † k c k , U b ( θ ) = e − iθ ( c † k c k (cid:48) + c † k (cid:48) c k ) , which are the basic unitary el-ements of the FLO set.FLO operations also include measurements of the oc-cupancy of single-particle modes, described by projectors P k = c † k c k , P ¯ k = c k c † k , (27)which satisfy P k + P ¯ k = . We now show explicitly thefollowing fundamental result. Theorem 1.
The measurement of the occupancy of asingle-particle state | k (cid:105) = c † k | (cid:105) ∈ H , described by theoperators (27) , is a ONG operation.Proof. Consider a general pure fermionic state | Ψ (cid:105) withSPDM ρ (1) . Let ρ (1) k and ρ (1)¯ k be the SPDMs after SPmode | k (cid:105) is found to be occupied or empty, respectively,determined by the states | Ψ k (cid:105) = P k | Ψ (cid:105) / √ p k , | Ψ ¯ k (cid:105) = P ¯ k | Ψ (cid:105) / √ p ¯ k , (28)with p k = (cid:104) Ψ |P k | Ψ (cid:105) = 1 − p ¯ k . Then | Ψ (cid:105) = √ p k | Ψ k (cid:105) + √ p ¯ k | Ψ ¯ k (cid:105) . (29)We will prove relation (20), i.e. (Fig 1), λ ( ρ (1) ) ≺ p k λ ( ρ (1) k ) + p ¯ k λ ( ρ (1)¯ k ) . (30)If the measured state | k (cid:105) is a natural orbital, such that (cid:104) c † k c k (cid:48) (cid:105) = p k δ kk (cid:48) with p k = λ k an eigenvalue of ρ (1) , Eq.(30) is straightforward: In this case (29) leads to ρ (1) = p k ρ (1) k + p ¯ k ρ (1)¯ k , (31) FIG. 1. Measurement of the occupancy of a single fermionmode k . It reduces (or does not increase), on average, themixedness of the SP density matrix ρ (1) ( λ denotes its spec-trum) and hence the one-body entanglement. as (cid:104) Ψ k | c † k (cid:48)(cid:48) c k (cid:48) | Ψ ¯ k (cid:105) = δ k (cid:48)(cid:48) k (1 − δ k (cid:48) k ) (cid:104) c † k c k (cid:48) (cid:105) = 0 ∀ k (cid:48) , k (cid:48)(cid:48) .Eq. (31) implies (30) since λ ( A + B ) ≺ λ ( A ) + λ ( B ) forany two Hermitian n × n matrices A and B [48] (this caseincludes the trivial situation p k = 1 or 0, where | Ψ (cid:105) = | Ψ k (cid:105) or | Ψ ¯ k (cid:105) ; in the following we consider p k ∈ (0 , (cid:104) Ψ k | c † k (cid:48)(cid:48) c k (cid:48) | Ψ ¯ k (cid:105) = 0 for any two SP states | k (cid:48) (cid:105) and | k (cid:48)(cid:48) (cid:105) orthogonal to | k (cid:105) , Eq. (29) implies, for any SP subspace S ⊥ ⊂ H orthogonal to the measured state | k (cid:105) , ρ (1) S ⊥ = p k ρ (1) k S ⊥ + p ¯ k ρ (1)¯ k S ⊥ , (32)where ρ (1) S ⊥ = P S ⊥ ρ (1) P S ⊥ and ρ (1) k (¯ k ) S ⊥ are the restrictionsof ρ (1) and ρ (1) k (¯ k ) to S ⊥ , and P S ⊥ is the associated pro-jector. This result is expected since the measurementis “external” to S ⊥ (if [ K j , O ] = 0 ∀ j ⇒ Tr [ ρO ] =Tr (cid:80) j K j ρ K † j O = (cid:80) j p j Tr[ ρ j O ]; for K j = P k (¯ k ) and O = c † k (cid:48)(cid:48) c k (cid:48) with k (cid:48) and k (cid:48)(cid:48) orthogonal to k , this resultimplies Eq. (32)). And for any S ⊂ H containing thestate | k (cid:105) , we have, as (cid:104) Ψ k | c † k (cid:48) c k (cid:48) | Ψ ¯ k (cid:105) = 0 for k (cid:48) = k or k (cid:48) orthogonal to k ,Tr ρ (1) S = Tr [ p k ρ (1) k S + p ¯ k ρ (1)¯ k S ] . (33)We can now prove the m th inequality in (30), m (cid:88) ν =1 λ ν ( ρ (1) ) ≤ m (cid:88) ν =1 (cid:16) p k λ ν ( ρ (1) k ) + p ¯ k λ ν ( ρ (1)¯ k ) (cid:17) . (34)Let S m ⊂ H be the subspace spanned by the first m eigenstates of ρ (1) , such that λ ν ( ρ (1) S m ) = λ ν ( ρ (1) ) for ν ≤ m and hence Tr ρ (1) S m = (cid:80) mν =1 λ ν ( ρ (1) ). If S m is eitherorthogonal to | k (cid:105) or fully contains | k (cid:105) , Eq. (32) or Eq.(33) holds for S = S m , implying (34) since Tr ρ (1) k (¯ k ) S m ≤ (cid:80) mν =1 λ ν ( ρ (1) k (¯ k ) ) by the Ky Fan maximum principle [48](the m largest eigenvalues λ ν of a Hermitian matrix O satisfy (cid:80) mν =1 λ ν ≥ Tr P (cid:48) m O = (cid:80) mν =1 λ (cid:48) ν for any rank m orthogonal projector P (cid:48) m , with λ (cid:48) ν the sorted eigenvaluesof P (cid:48) m OP (cid:48) m ).Otherwise we add to S m the component | k ⊥ (cid:105) of | k (cid:105) orthogonal to S m , obtaining an m + 1 dimensional SPsubspace S (cid:48) m where Eq. (33) holds and still λ ν ( ρ (1) S (cid:48) m ) = λ ν ( ρ (1) ) for ν ≤ m . It is proved in the Appendix A thatthe remaining smallest eigenvalue satisfies λ m +1 ( ρ (1) S (cid:48) m ) ≥ p k λ m +1 ( ρ (1) k S (cid:48) m ) + p ¯ k λ m +1 ( ρ (1)¯ k S (cid:48) m ) . (35)Hence (cid:80) mν =1 λ ν ( ρ (1) ) ≤ (cid:80) mν =1 p k λ ν ( ρ (1) k S (cid:48) m ) + p ¯ k λ ν ( ρ (1)¯ k S (cid:48) m )due to the trace conservation (33) for S = S (cid:48) m , whichimplies Eq. (34) due to previous Ky Fan inequality. Thiscompletes the proof for pure states. It is easily verified(see Appendix B) that these measurements map SDs ontoSDs, as implied by Eq. (30).The previous proof actually shows the general relation λ ( ρ (1) S ) ≺ p k λ ( ρ (1) k S ) + p ¯ k λ ( ρ (1)¯ k S ) , (36)valid for the restriction of ρ (1) to any subspace S ⊂ H either containing or orthogonal to the measured state | k (cid:105) ([ P S , | k (cid:105)(cid:104) k | ] = 0). Moreover ρ (1) S will be determinedby a mixed reduced state ρ S = Tr S ⊥ | Ψ (cid:105)(cid:104) Ψ | satisfying (cid:104) Ψ | O S | Ψ (cid:105) = Tr ρ S O S for any operator O S involving justcreation and annihilation of SP states ∈ S (see AppendixA). Eq. (36) then shows that (30) holds for general mixedfermionic states ρ (assumed to commute with the fermionnumber ˆ N = (cid:80) k c † k c k or the number parity e iπ ˆ N ) sincethey can be purified and seen as a reduced state ρ S ofa pure fermionic state | Ψ (cid:105) in an enlarged SP space (Eq.(A5) in Appendix A).
3. More general ONG measurements and operations
By composing the basic measurements (27), more com-plex operations satisfying (20) are obtained. In particu-lar, a measurement in a basis of SDs, which is obviouslyONG, results from the composition of all measurements { P k , P ¯ k } in a given SP basis. Extension of the set of freeoperations beyond the standard FLO set can also be con-sidered. The proof of theorem 1 can be extended to moregeneral single-mode measurements: Corollary 1.
A general measurement on single-particlemode k described by the operators M k = α P k + β P ¯ k , M ¯ k = γ P k + δ P ¯ k , (37) where M † k M k + M † ¯ k M ¯ k = ( | α | + | γ | = | β | + | δ | = 1 )is also a ONG operation. The proof is given in Appendix A and implies that Eq.(30) will be satisfied for ρ (1) k (¯ k ) the SPDMs obtained from | Ψ (cid:48) k (¯ k ) (cid:105) ∝ M k (¯ k ) | Ψ (cid:105) and p k → p (cid:48) k = (cid:104) Ψ |M † k M k | Ψ (cid:105) =1 − p (cid:48) ¯ k . This result entails that any pair of Kraus operators M k and M ¯ k for the occupation measurement operation ε ( ρ ) = P k ρ P k + P ¯ k ρ P ¯ k = M k ρ M † k + M ¯ k ρ M † ¯ k (38)will also be ONG operations, since they are a special caseof (37) [ αβ ∗ + γδ ∗ = 0, i.e. ( α βγ δ ) unitary].It is also possible to consider in the present contextoperations which do not conserve the fermion number N but still generate states with definite particle numberwhen applied to such states. In this case it becomesnecessary to extend the partial order (2) to states withdifferent particle number. We then notice that Eq. (2)implies a similar majorization relation (see Appendix C) λ ( D (1)Ψ ) ≺ λ ( D (1)Φ ) (39)for the sorted eigenvalues of the extended 2 n × n SPDM D (1) = ρ (1) ⊕ ( − ρ (1) T ) = (cid:18) ρ (1) − ρ (1) T (cid:19) , (40)with spectrum ( λ ( ρ (1) ) , − λ ( ρ (1) )) and elements (cid:104) c † k (cid:48) c k (cid:105) and (cid:104) c k (cid:48) c † k (cid:105) , the trace Tr D (1) = n of which is fixed by theSP space dimension and is N - independent . For generalstates we then say that | Ψ (cid:105) is not less one-body entangledthan | Φ (cid:105) if Eq. (39) holds. Note that all SDs lead tothe same sorted spectrum λ ( D (1) ) regardless of N , allbeing then least entangled states, with ( D (1) ) = D (1) ifand only if | Ψ (cid:105) is a SD. Similarly, Eq. (20) for numberconserving ONG operations implies λ ( D (1) ) ≺ (cid:88) j p j λ ( D (1) j ) (41)for the extended densities. We then say that an opera-tion not conserving fermion number is ONG if it admitsa set of Kraus operators K j such that (41) is satisfied.Prop. 1 remains then valid for general ONG operationsreplacing (2) by (39). All previous properties satisfied bythe entropies (15) extend to entropies E ( | Ψ (cid:105) ) = S ( D (1) ) , (42)with Tr f ( D (1) ) = Tr f ( ρ (1) ) + Tr f ( − ρ (1) ). In partic-ular, Eq. (18) becomes just the von Neumann entropyof D (1) . Maximally one-body entangled states are nowthose leading to D (1) = n / , (43)which will maximize all entropies (42). Examples are pre-vious GHZ-like states (25) in half-filled SP spaces ( m = 2, N = n/ ≥ c k and c † k as operators: Corollary 2.
A measurement on single-particle mode k described by the operators c k and c † k , which satisfy c † k c k + c k c † k = , is a ONG operation: λ ( D (1) ) ≺ p k λ ( D (1) k ) + p ¯ k λ ( D (1)¯ k ) . (44)Here p k = (cid:104) c † k c k (cid:105) = 1 − p ¯ k , and D (1) and D (1) k (¯ k ) arethe extended SPDMs determined by ρ , ρ k = c k ρc † k /p k and ρ ¯ k = c † k ρc k /p ¯ k . Since these extended densities D (1) k (¯ k ) have clearly the same spectrum as those obtained from P k ρ P k /p k and P ¯ k ρ P ¯ k /p ¯ k , Eq. (44) directly follows fromTheorem 1 and Eq. (C1). On the other hand, previousbasic occupation measurement (27) is just the composi-tion of this measurement with itself (see Appendix C).This extension then enables us to consider the additionof free ancillas (SDs of arbitrary N ) as a free operation,as it will not alter the spectrum of the extended SPDM D (1) in the full SP space. We remark, however, thatgeneral “active” FLO operations which do not conservethe fermion number (for instance, a Bogoliubov trans-formation) may increase the one-body entanglement de-termined by ρ (1) . While we will not discuss these oper-ations here, we mention that if they are also regardedas free one should consider instead the generalized one-body entanglement, determined by the mixedness of thefull quasiparticle DM [20], as the associated resource (seealso Appendix C). C. One-body entanglement as a resource
The identification of number conserving FLO opera-tions as ONG implies that they map SDs onto SDs, asverified in App. B. It has been noted [59, 60] that this factultimately explains why the pure state FLO computationmodel can be efficiently simulated classically, as matrixelements of free unitaries and outcome probabilities of free measurements can be reduced to overlaps (cid:104) Ψ | Φ (cid:105) be-tween SDs, which can be computed in polynomial timethrough a determinant [44].In contrast, the simultaneous measurement of the oc-cupancy of two SP modes k and k (cid:48) , described by opera-tors {M = P ¯ k P ¯ k (cid:48) , M = P ¯ k P k (cid:48) + P k P ¯ k (cid:48) , M = P k P k (cid:48) } ,is not free since M can map a SD onto a state withSlater number 2 [59], i.e. a one-body entangled state. Asimilar measurement with m such outcomes may returna state with an exponentially large (2 m ) Slater number[59], the expectation values of which would be hard toevaluate classically. In [61] this operation is identifiedwith a charge detection measurement in a system of freeelectrons, showing that it is possible to build a controlled-NOT gate with just beamsplitters, spin rotations andcharge detectors. The extended set of FLO plus chargedetection operations then enables quantum computation.If the computational power of this model is to be linkedto the presence of a quantum resource, the ensuing freestates and operations would be S and O , respectively,and the results derived here entail that one-body entan-glement would be an associated resource.One-body entanglement can also be considered as aresource for mode entanglement. In particular, mode en-tanglement with definite particle number N or definitenumber parity e iπN at each component requires one-bodyentanglement . A first example was seen with the nor-mal form (14) for a general two-fermion state [11–13],where the entanglement between the modes k ( A ) and¯ k ( B ), containing each one fermion, is directly linkedto one-body entanglement: The entanglement entropy E ( A, B ) = S ( ρ A ) = S ( ρ B ) of this partition is just E ( A, B ) = (cid:88) ν f ( λ ν ) = E ( | Ψ (cid:105) ) (45)for any entropy S ( ρ ) = Tr f ( ρ ), where E ( | Ψ (cid:105) ) = S ( ρ (1) )is the corresponding one-body entanglement entropy (15)(as (cid:104) c † ν c ν (cid:48) (cid:105) = (cid:104) c † ¯ ν c ¯ ν (cid:48) (cid:105) = λ ν δ νν (cid:48) , (cid:104) c † ν c ¯ ν (cid:48) (cid:105) = 0). In particularany one-body entangled state of two fermions in a SPspace of dimension 4 can be seen as an entangled state oftwo distinguishable qubits, allowing then the realizationof tasks like quantum teleportation [23]. In this casethe one-body entanglement entropy also provides a lowerbound to any bipartite mode entanglement entropy [23].For a general pure fermionic state | Ψ (cid:105) (with definiteparticle number or number parity) we now show that one-body entanglement is always required in order to havebipartite mode entanglement entropy E ( A, B ) > A and B : In such a case, and assum-ing sides A and B correspond to orthogonal subspaces H A and H B of the SP space H = H A ⊕ H B , the ensuingSPDM takes the block-diagonal form ρ (1) = ρ (1) A ⊕ ρ (1) B ,i.e., ρ (1) = (cid:32) ρ (1) A ρ (1) B (cid:33) (46)since for any k A ∈ H A , k B ∈ H B , c † k A c k B connectsstates with different number parity at each side and hence (cid:104) c † k A c k B (cid:105) = 0 in such a state. Then, if | Ψ (cid:105) is a SD,( ρ (1) ) = ρ (1) , implying ( ρ (1) A ( B ) ) = ρ (1) A ( B ) , i.e. the stateat each side must be a SD (a pure state) and no A – B entanglement is directly present. For instance, a singlefermion state √ ( c † k A + c † k B ) | (cid:105) implies entanglement be-tween A and B but at the expense of involving zero andone fermion at each side, i.e., no definite local numberparity. In contrast, | Ψ (cid:105) = √ ( c † k A c † k B + c † k (cid:48) A c † k (cid:48) B ) | (cid:105) leadsto entanglement between A and B with definite fermionnumber (and hence number parity) at each side, but it isnot a SD, i.e., it has nonzero one-body entanglement.Expanding the state in a SD basis as | Ψ (cid:105) = (cid:80) µ,ν C µν A † µ B † ν | (cid:105) , where A † µ = (cid:81) k ( c † k A ) n kµ and B † ν = (cid:81) k ( c † k B ) n kν involve creation operators just on H A and H B respectively (with n k µ ( ν ) = 0 , µ and ν la-beling all possible sets of occupation numbers, suchthat (cid:104) | A µ A † µ (cid:48) | (cid:105) = δ µµ (cid:48) , (cid:104) | B ν B † ν (cid:48) | (cid:105) = δ νν (cid:48) ), stateswith definite number parity at each side correspondto ( − (cid:80) k n kµ and ( − (cid:80) k n kν fixed for all µ and ν with C µν (cid:54) = 0. The reduced DM of side A is ρ A = (cid:80) µ,µ (cid:48) ( CC † ) µµ (cid:48) A † µ | (cid:105)(cid:104) | A µ (cid:48) (and similarly for ρ B ; see Ap-pendix A), and there is entanglement between A and B whenever ρ A has rank ≥
2, i.e., rank( C ) ≥
2. In such acase, previous argument shows that such | Ψ (cid:105) cannot bea SD if the fermion number or number parity is fixed ateach side, i.e. N A or e iπN A fixed.Due to the fermionic number parity superselection rule[5, 62], fixed number parity at each side is required inorder to be able to form arbitrary superpositions, i.e.,arbitrary unitary transformations of the local states, andhence to have entanglement fully equivalent to the dis-tinguishable case. Fixed particle number at each sidemay be also required if the particle number or chargesuperselection rule applies for the fermions considered.The extension to multipartite mode entanglement isstraightforward: For a decomposition H = (cid:76) i H i of theSP space into orthogonal subspaces H i , and for compo-nent i associated to subspace H i , all elements of ρ (1) con-necting different components i and j will vanish if eachcomponent is to have definite fermion number or numberparity in a state | Ψ (cid:105) : (cid:104) c † k i c k j (cid:105) = 0 ∀ k i , k j if i (cid:54) = j . Thus, ρ (1) = (cid:77) i ρ (1) i . (47)If | Ψ (cid:105) were a SD, each ρ (1) i should then satisfy ( ρ (1) i ) = ρ (1) i and hence each subsystem would be in a pure SDstate, implying no entanglement between them. A one-body entangled state is then required.And when all previous subsystems contain just onefermion, one-body entanglement is directly linked tostandard multipartite entanglement. Consider an N -partite system with Hilbert space L = (cid:78) Ni =1 L i , where L i is the Hilbert space of the i -th (distinguishable) con- stituent. Consider also an N -fermion system with SPspace H = (cid:76) Ni =1 H i , such that dim H i = dim L i . Thisenables the definition of an isomorphism Θ i : L i → H i between these two spaces: Any pure separable state in L , | S (cid:105) L = (cid:78) Ni =1 | φ i (cid:105) , with | φ i (cid:105) ∈ L i , can be mapped toa SD | Ψ (cid:105) = [ N (cid:89) i =1 c † i,φ ] | (cid:105) , = Θ( | S (cid:105) L ) , (48)where c † i,φ creates a fermion in the state Θ i ( | φ i (cid:105) ) ∈ H i .The map Θ : L → F is an isomorphism between L and the subspace of F determined by the fermion stateshaving occupation number N i = 1 in H i ⊂ H . Hence anypure state | Φ (cid:105) L of the multipartite system is mappedto a state | Ψ (cid:105) = Θ( | Φ (cid:105) L ) in F . Since there is a fixedfermion number (1) in each constituent, the SPDM ρ (1) determined by | Ψ (cid:105) will satisfy Eq. (47), implying here ρ (1) = (cid:77) i ρ i , (49)where the elements of the matrix ρ i are those ofthe reduced state of the i -th subsystem associated to | Φ (cid:105) L . Thus, one-body entanglement monotones become E ( | Ψ (cid:105) ) = Tr f ( ρ (1) ) = (cid:80) i Tr f ( ρ i ), being then equiva-lent to the multipartite version of the linear entropy ofentanglement [6, 63–65] and constituting monotones forthe multipartite entanglement of the tensor product rep-resentation.This link between one-body entanglement and multi-partite entanglement is not a surprise if we recall thatperforming local operations on the multipartite systemcannot increase, on average, the mixedness of the localeigenvalues. Relation (49) then implies that these opera-tions cannot increase the mixedness of the SPDM associ-ated to the fermionic representation, in agreement withEq. (2). And any local unitary in the tensor product rep-resentation can be implemented as a one-body unitaryin the fermion system, while local projective measure-ments can be performed as occupation measurements,both FLO operations which are one-body entanglementnongenerating. The overlap between one-body entangle-ment and multipartite entanglement described here rein-forces the idea that the first could be the resource be-hind the computational power of the quantum computa-tion model described in [61], since it consists of a map-ping from qubits to fermions just like the map Θ definedabove. III. CONCLUSIONS
We have shown that one-body entanglement, a mea-sure of the deviation of a pure fermionic state from aSD determined by the mixedness of the SPDM ρ (1) , canbe considered as a quantum resource. We have firstprovided a basis-independent bipartite like formulationof one-body entanglement in general N -fermion states,which relates it with the correlation between one and( N − N − ACKNOWLEDGMENTS
The authors acknowledge support from CONICET(N.G. and M.DT.) and CIC (R.R.) of Argentina. Thiswork was supported by CONICET PIP Grant No.11220150100732.
Appendix A: Proof of inequality (35) and Corollary 1
Proof.
We first prove Eq. (35) for the lowest eigenvalueof ρ (1) S (cid:48) m , where S (cid:48) m is the subspace containing the mea-sured SP state | k (cid:105) and the first m eigenstates of ρ (1) . Wewrite the lowest eigenstate of ρ (1) S (cid:48) m as α | k (cid:105) + β | k (cid:48) (cid:105) , with | k (cid:48) (cid:105) ∈ S (cid:48) m orthogonal to | k (cid:105) , such that λ m +1 ( ρ (1) S (cid:48) m ) is thesmallest eigenvalue λ − of the 2 × ρ (1) kk (cid:48) = (cid:18) (cid:104) c † k c k (cid:105) (cid:104) c † k (cid:48) c k (cid:105)(cid:104) c † k c k (cid:48) (cid:105) (cid:104) c † k (cid:48) c k (cid:48) (cid:105) (cid:19) . (A1)Setting (cid:104) c † k c k (cid:105) = p k , its eigenvalues are λ ± = p k + p k (cid:48) ± (cid:113) ( p k − p k (cid:48) ) + |(cid:104) c † k (cid:48) c k (cid:105)| . (A2)Writing again P k = c † k c k , P ¯ k = c k c † k = − P k , a generalstate | Ψ (cid:105) can be expanded as | Ψ (cid:105) = (cid:80) µ,µ (cid:48) P µ P µ (cid:48) | Ψ (cid:105) = (cid:80) µ,µ (cid:48) √ p µµ (cid:48) | Ψ µµ (cid:48) (cid:105) , (A3) where µ = k, ¯ k ; µ (cid:48) = k (cid:48) , ¯ k (cid:48) (so that (cid:80) µ,µ (cid:48) P µ P µ (cid:48) = ) and | Ψ µµ (cid:48) (cid:105) = P µ P µ (cid:48) | Ψ (cid:105) / √ p µµ (cid:48) , with p µµ (cid:48) = (cid:104) Ψ |P µ P µ (cid:48) | Ψ (cid:105) ,are states with definite occupation of SP states | k (cid:105) and | k (cid:48) (cid:105) . We then have p k = p kk (cid:48) + p k ¯ k (cid:48) , p k (cid:48) = p kk (cid:48) + p ¯ kk (cid:48) and (cid:104) c † k (cid:48) c k (cid:105) = r √ p ¯ kk (cid:48) p k ¯ k (cid:48) , with | r | ≤
1. Thus, |(cid:104) c † k (cid:48) c k (cid:105)| ≤ p ¯ kk (cid:48) p k ¯ k (cid:48) , and (A2) implies λ − ≥ p k + p k (cid:48) − p k ¯ k (cid:48) + p ¯ kk (cid:48) , i.e., λ − ≥ p kk (cid:48) ≥ p k λ m +1 ( ρ (1) k S (cid:48) m ) + p ¯ k λ m +1 ( ρ (1)¯ k S (cid:48) m ) , (A4)since p k λ m +1 ( ρ (1) k S (cid:48) m ) ≤ p kk (cid:48) and λ m +1 ( ρ (1)¯ k S (cid:48) m ) = 0 (as thestate k is empty in ρ (1)¯ k S (cid:48) m ). This implies Eq. (35) since λ m +1 ( ρ (1) S (cid:48) m ) = λ − .We also mention that by considering the largest eigen-value λ + in (A2) of a similar 2 × ρ (1) is spanned by | k (cid:105) and | k (cid:48) (cid:105) , it is verified, using again |(cid:104) c † k (cid:48) c k (cid:105)| ≤ p ¯ kk (cid:48) p k ¯ k (cid:48) ,that λ ( ρ (1) ) = λ + ≤ p k + p ¯ kk (cid:48) ≤ p k λ ( ρ (1) k ) + p ¯ k λ ( ρ (1)¯ k ),since λ ( ρ (1) k ) = 1 and p ¯ kk (cid:48) ≤ p ¯ k λ ( ρ (1)¯ k ), which is thefirst ( m = 1) inequality in Eq. (34).The present proof of Theorem 1 also holds for generalmixed fermionic states ρ (assumed to commute with thefermion number ˆ N = (cid:80) k c † k c k or in general the num-ber parity e iπ ˆ N ) since they can always be purified, i.e.considered as reduced states ρ S of a pure fermionic state | Ψ (cid:105) = (cid:88) µ,ν C µν A † µ B † ν | (cid:105) , (A5)of definite number parity. Here A † µ and B † ν contain cre-ation operators in S and in an orthogonal SP space S ⊥ re-spectively, satisfying (cid:104) | A µ (cid:48) A † µ | (cid:105) = δ µµ (cid:48) , (cid:104) | B ν (cid:48) B † ν | (cid:105) = δ νν (cid:48) , (cid:104) | B ν (cid:48) A † ν | (cid:105) = 0. We may assume, for instance,that { A † µ B † ν | (cid:105)} is a complete set of orthogonal SDs.Then ρ S = Tr S ⊥ | Ψ (cid:105)(cid:104) Ψ | = (cid:80) µ,µ (cid:48) ( CC † ) µµ (cid:48) | µ (cid:105)(cid:104) µ (cid:48) | , with | µ (cid:105) = A † µ | (cid:105) a SD in S , satisfies (cid:104) Ψ | O S | Ψ (cid:105) = Tr ρ S O S ∀ operators O S containing creation and annihilation oper-ators of SP states ∈ S . Given then an arbitrary mixedfermionic state ρ S Eq. (A5) is a purification of ρ S forany matrix C satisfying ( CC † ) µµ (cid:48) = (cid:104) µ | ρ S | µ (cid:48) (cid:105) (requiresdim S ⊥ ≥ dim S ).Let us now prove Corollary 1, i.e. the extension of The-orem 1 to the more general occupancy measurement oper-ators of Eq. (37). In terms of the states (28), the ensuingpostmeasurement states | Ψ (cid:48) k (¯ k ) (cid:105) ∝ M k (¯ k ) | Ψ (cid:105) are | Ψ (cid:48) k (cid:105) = (cid:0) α √ p k | Ψ k (cid:105) + β √ p ¯ k | Ψ ¯ k (cid:105) (cid:1) / (cid:113) p (cid:48) k , (A6) | Ψ (cid:48) ¯ k (cid:105) = (cid:0) γ √ p k | Ψ k (cid:105) + δ √ p ¯ k | Ψ ¯ k (cid:105) (cid:1) / (cid:113) p (cid:48) ¯ k , (A7)where | α | + | γ | = 1, | β | + | δ | = 1 and p (cid:48) k = p k | α | + p ¯ k | β | , p (cid:48) ¯ k = p k | γ | + p ¯ k | δ | , (A8)with p (cid:48) k + p (cid:48) ¯ k = 1. We have to prove λ ( ρ (1) ) ≺ p (cid:48) k λ ( ρ (cid:48) (1) k ) + p (cid:48) ¯ k λ ( ρ (cid:48) (1)¯ k ) , (A9)where ρ (cid:48) (1) k (¯ k ) are now the SPDM’s determined by thestates (A6)–(A7). The generalization of Eq. (33),Tr ρ (1) S = Tr [ p (cid:48) k ρ (cid:48) (1) k S + p (cid:48) ¯ k ρ (cid:48) (1)¯ k S ] , (A10)still holds for any subspace S either orthogonal to orcontaining the SP state | k (cid:105) , as [ M k (¯ k ) , c † k (cid:48) c k (cid:48) ] = 0 forboth k (cid:48) = k or k (cid:48) orthogonal to k [see comment belowEq. (32)]. Proceeding in the same way and using previousnotation, we see that p (cid:48) k λ m +1 ( ρ (cid:48) (1) k S (cid:48) m ) is less than or equalto the smallest eigenvalue λ k − of the 2 × (cid:18) | α | p k αβ ∗ (cid:104) c † k (cid:48) c k (cid:105) α ∗ β (cid:104) c † k c k (cid:48) (cid:105) | α | p kk (cid:48) + | β | p ¯ kk (cid:48) (cid:19) , (A11)while p (cid:48) ¯ k λ m +1 ( ρ (cid:48) (1)¯ k S (cid:48) m ) is less than or equal to the smallesteigenvalue λ ¯ k − of a similar matrix with α → γ , β → δ . Itis then straightforward to prove, using Eq. (A2) for λ − ,that λ m +1 ( ρ (1) S (cid:48) m ) = λ − ≥ λ k − + λ ¯ k − , since λ − − λ k − − λ ¯ k − = (cid:113) ( | α | p k ¯ k (cid:48) −| β | p ¯ kk (cid:48) ) + | αβ (cid:104) c † k (cid:48) c k (cid:105)| + (cid:113) ( | γ | p k ¯ k (cid:48) −| δ | p ¯ kk (cid:48) ) + | γδ (cid:104) c † k (cid:48) c k (cid:105)| − (cid:113) ( p k ¯ k (cid:48) − p ¯ kk (cid:48) ) + |(cid:104) c † k (cid:48) c k (cid:105)| ≥ , (A12)with equality for | r | = 1 ( |(cid:104) c † k (cid:48) c k (cid:105)| = √ p ¯ kk (cid:48) p k ¯ k (cid:48) ) or | α | = | β | . Then the m th inequality in (A9), m (cid:88) ν =1 λ ν ( ρ (1) ) ≤ m (cid:88) ν =1 p (cid:48) k λ ν ( ρ (cid:48) (1) k ) + p (cid:48) ¯ k λ ν ( ρ (cid:48) (1)¯ k ) , (A13)follows due to (A10) and the previously used Ky Faninequality. Eq. (A9) also holds within any subspace S containing (or orthogonal to) the SP state | k (cid:105) . Appendix B: Occupation measurements on freestates
For a one-body entanglement nongenerating operation,Eq. (20) implies that the Kraus operators K j satisfyingit should convert free states onto free states, i.e. SDsonto SDs. For the occupation measurements of Eq. (27)(Theorem 1), Eq. (37) (Corollary 1), and Corollary 2,this property can be easily verified. Let | Ψ (cid:105) = ( N (cid:89) ν =1 c † ν ) | (cid:105) , (B1)be a general SD for N fermions, with { c ν , c † ν (cid:48) } = δ νν (cid:48) .A general fermion creation operator c † k = (cid:80) nν =1 α ν c † ν ,with α ν = { c ν , c † k } and { c k , c † k } = (cid:80) ν | α ν | = 1, can bewritten as c † k = √ p k c † k (cid:107) + √ p ¯ k c † k ⊥ , (B2) where √ p k c † k (cid:107) = (cid:80) ν ≤ N α ν c † ν is the component in thesubspace occupied in | Ψ (cid:105) , with p k = (cid:80) ν ≤ N | α ν | = (cid:104) Ψ | c † k c k | Ψ (cid:105) the occupation probability of SP state k and c † k (cid:107) | Ψ (cid:105) = 0, while √ p ¯ k c † k ⊥ = (cid:80) ν>N α ν c † ν is the orthog-onal complement, with p ¯ k = (cid:80) ν>N | α ν | = 1 − p k and c k ⊥ | Ψ (cid:105) = 0. If p k >
0, through a unitary transforma-tion of the c † ν for ν ≤ N , they can be chosen such that c † k (cid:107) = c † ν = N . Hence, for the measurement operators ofcorollary 2, we see that c k | Ψ (cid:105) = √ p k c k (cid:107) | Ψ (cid:105) , c † k | Ψ (cid:105) = √ p ¯ k c † k ⊥ | Ψ (cid:105) , (B3)are clearly orthogonal SDs. For the measurement opera-tors of Theorem 1, Eq. (B3) implies c † k c k | Ψ (cid:105) = √ p k c † k c k (cid:107) | Ψ (cid:105) , (B4) c k c † k | Ψ (cid:105) = √ p ¯ k c k c † k ⊥ | Ψ (cid:105) = √ p ¯ k c † k (cid:48) c k (cid:107) | Ψ (cid:105) , (B5)where c † k (cid:48) = √ p ¯ k c † k (cid:107) − √ p k c † k ⊥ , which are also orthogonalSD’s ( { c k , c † k (cid:48) } = 0). And in the case of the generalizedmeasurement based on the operators (37), we see from(B4)–(B5) that M k | Ψ (cid:105) = ( α √ p k c † k + β √ p ¯ k c † k (cid:48) ) c k (cid:107) | Ψ (cid:105) (B6) M ¯ k | Ψ (cid:105) = ( γ √ p k c † k + δ √ p ¯ k c † k (cid:48) ) c k (cid:107) | Ψ (cid:105) (B7)are as well SDs, not necessarily orthogonal. Appendix C: Comparing one-body entanglement ofstates with different particle number
Given two pure fermionic states | Ψ (cid:105) and | Φ (cid:105) withthe same fermion number N , such that their associatedSPDMs have the same trace Tr ρ (1)Ψ = Tr ρ (1)Φ = N , | Ψ (cid:105) is considered not less entangled than | Φ (cid:105) if Eq. (2)( λ ( ρ (1)Ψ ) ≺ λ ( ρ (1)Φ )) is satisfied. Here λ ( ρ (1) ) denotes thespectrum of ρ (1) sorted in decreasing order. It can beshown that λ ( ρ (1)Ψ ) ≺ λ ( ρ (1)Φ ) = ⇒ λ ( D (1)Ψ ) ≺ λ ( D (1)Φ ) , (C1)for the extended DM defined in Eq. (40), the spectrumof which is ( λ , − λ ). Eq. (C1) follows from the straight-forward properties (see for instance [49, 50])i) λ ≺ λ (cid:48) = ⇒ − λ ≺ − λ (cid:48) ,ii) λ ≺ λ (cid:48) and µ ≺ µ (cid:48) = ⇒ ( λ , µ ) ≺ ( λ (cid:48) , µ (cid:48) ) , where λ , λ (cid:48) , µ , µ (cid:48) ∈ R n are sorted in decreasing order and( λ , µ ) ∈ R n denotes the sorted vector resulting from theunion of λ and µ . The converse relation in (C1) does nothold. These properties also entail that the majorizationrelation (20) implies λ ( D (1) ) ≺ (cid:88) j p j λ ( D (1) j ) , (C2)for the corresponding extended densities.0The advantage of Eqs. (C1)–(C2) is that within a fixedSP space the extended vectors can always be comparedthrough majorization, regardless of the particle number N , since Tr D (1) = n = dim H is fixed by the SP spacedimension. For two states | Ψ (cid:105) and | Φ (cid:105) with definite butnot necessarily coincident fermion number, we then saythat | Ψ (cid:105) is not less entangled than | Φ (cid:105) if Eq. (C1) holds.In particular, it is clear that —up to a permutation—the same eigenvalue vector λ ( D (1) ) is assigned to all SDstates in the Fock space of the system, irrespective of N ,so that they are all least entangled states.The extension of Definition 1 to general ONG oper-ations, not necessarily conserving the particle number,is now straightforward: a quantum operation is ONG ifit admits a set of Kraus operators {K j } satisfying Eq.(C2) ∀ ρ , with D (1) and D (1) j the extended SPDMs de-termined by ρ and ρ j respectively. This extension allowsus to consider operations such as that of Corollary 2, withKraus operators c k and c † k . The extended SPDMs D (1) of the postmeasurement states √ p k c k | Ψ (cid:105) and √ p ¯ k c † k | Ψ (cid:105) have clearly the same spectrum as those obtained from √ p k c † k c k | Ψ (cid:105) and √ p ¯ k c k c † k | Ψ (cid:105) —up to a permutation—,with the same probabilities, such that theorem 1 directlyimplies corollary 2. In fact, the number conserving occu-pation measurement is just a composition of the formerwith itself, as ( c k , c † k ) ◦ ( c k , c † k ) = ( c k c † k , c † k c k ).We can therefore embed the fermion number preserv-ing resource theory within a more general theory in which the set of free states is the convex hull of SD states —ofall possible particle number— and where the free opera-tions are one-body unitaries and operations based on the { c k , c † k } measurement mapping SDs onto SDs. Any SDand hence any free state can be prepared from an arbi-trary state ρ by means of free operations only, i.e., byapplying one-body unitaries and successive { c k , c † k } mea-surements with postselection. Since the starting state isarbitrary, any free state in this theory can be convertedinto any other free state by free operations. Allowingthe particle number to vary implies that appending freestates is also a free operation, since this will not alter thespectrum of the associated D (1) in the full SP space.We have here considered pure states | Ψ (cid:105) with definitefermion number and operators K j which produce states K j | Ψ (cid:105) with definite fermion number when applied to suchstates, suitable for systems where a particle number su-perselection rule applies. The extension to the case wheregeneral fermionic Gaussian states (with no fixed particlenumber but definite number parity) and active FLO op-erations are also considered free is straightforward. Itrequires the consideration of the full extended 2 n × n quasiparticle density matrix containing in addition thepair creation and annihilation contractions (cid:104) c † k c † k (cid:48) (cid:105) and (cid:104) c k (cid:48) c k (cid:105) , the eigenvalues of which remain invariant undergeneral Bogoliubov transformations. 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