arXiv:0705.0795v3 [quant-ph] 8 Oct 2007 naayia eso fti odto sdrvdfrteca the for derived operatio is separable condition a orthogo this by of ( of present distinguishable version set be first analytical finite An to a We for states operations. condition quantum sufficient separable and by necessary states quantum fti eutaete aeul netgtd Remarkably of investigated. class carefully large a then exists are there show result consequ this interesting of of number A consideration. under space eoeorwr nyacasof class a only communi work classical our and operations Before local by realizable being 5 855(05] oal,w bana xlctconstruc dimension explicit of an subspaces obtain we indistinguishable Notably, Re (2005)]. Phys. 080505 [Watrous, 95, subspaces bipartite indistinguishable ubrntls than less not number prtosi n nyi hssaecno easuperpositio a separa be complement by cannot orthogonal state indistinguishable this the is if of only 1 state and (1999)] basis pure if 1070 any operations multipartite 59, A that a Rev. Phys. show of al, also et [Bennett We known was ations opst unu ytmcnitn ftoqtis(resp. qutrits previously dimension the two with than subspace of better bipartite slightly consisting indistinguishable is which system qubits), quantum three composite a iy5mistigu.d.n(uXn,adyingmsh@tsin and Xin), Ying) (Mingsheng (Yu w 100084, also [email protected] [email protected] Beijing is Duan), University, (Runyao Xin Tsinghua I Yu [email protected] Physics, I China. of of and 100084, for Laboratory partment Science Beijing Key Laboratory Computer University, of State National Tsinghua Department (AQIS0 the Technology, Tsinghua and with Science Science Systems, are Information and authors Quantum Technology The of Japan. Conference oto, Asia 2006A No. 2007 Res (Grant Hi-Tech project) the (863 and China 60503001) of and Program 60621062 Development Nos. (Grant China n ookr 1.O h n ad lhuhmn partial operations many LOCC of although structure hand, the one made, been the have On progresses Benne [1]. by coworkers discovered phenomenon weird and entanglement” the without of consequence “nonlocality a of as true, how always quant part, not converse realizable is The locally separable. any necessarily only. that using is known operation (LOCC) subsystems well communication the is it classical of Actually, owners and can- the operations global system by local some composite implemented exist a be there on not as nonlocality. performing interpreted exhibit operations be can quantum can effect systems an quantum Such composite that is rdc Bases. Unexten Product number, Schmidt Orthogonal operations, Separable D or itnusaiiyo unu ttsb Separable by States Quantum of Distinguishability h aeili hspprwspeetdi ata ogtalk long a as part in presented was paper this Founda in Science material Natural The the by supported partly was work This Abstract n ftems rfudfaue fqatmmechanics quantum of features profound most the of One ne Terms Index − 2 1) rhgnlpoutsae,ie,hsa rhgnlSchmidt orthogonal an has i.e., states, product orthogonal uesae,where states, pure esuytedsigihblt fmultipartite of distinguishability the study We — unu olclt,Lcldistinguishability, Local Nonlocality, Quantum — .I I. 3 hsgnrlz h eetwr about work recent the generalize thus , NTRODUCTION D uyoDa,Ya eg uXn n ighn Ying Mingsheng and Xin, Yu Feng, Yuan Duan, Runyao stettldmnino h state the of dimension total the is 3 2 ⊗ ⊗ 3 2 olclsprbeoper- separable nonlocal eaal prtosnot operations separable 7 (or 6 yconsidering by ) c Ya Feng), (Yuan .cn hn.E-mails: China. Operations 8 . A01Z102). Technology, ghua.edu.cn t h De- the ith nformation ac and earch ntelligent .Lett. v. known ) Ky- 7), cation. inof tion inof tion ences tthe at dible ever, eof se we , of n um nal ble ns. t . 7,[] 9,[0;acmlt hrceiainfrteloc the for characterization [6 complete [5], a of states [10]; distinguishability orthogonal [9], dimensional [8], infinite [7], two nonorthogo or two ca distinguishing states result when this case and the perfect [4], to a locally generalized achieved quant [3]; be always partial orthogonal [2], can multipartite states a [1], can two that LOCC for between states by discrimination therein product perfectly orthogonal references [ discriminated of [23], be and sets [22], [12], exist [28] [21], [11], There [20], list. [10], [27], [19], [9], [26], [18], [8], [17], [25], [7], [16], [6], Ref [15], [5], see [4], [14], recent reported, [3], in been [2], have attentions [1], results considerable effectiv interesting dis- received very Many a been tasks, years. operations is has these states quantum and Among quantum one of LOCC. orthogonal by class of realized a crimination be construct is cannot can task that certain effici we local a then and be global if cies, can optimal speaking, different tasks with Roughly processing accomplished LOCC. information by of achieved of kind what is consider operations theory. separable information of quantum in understanding importance particular deep sta quantum of a separability determi the Thus, employing for to by tools ric separable developed class easier well is with the operation the relatively is quantum given hand, is and a other whether restricted It the rather structure. is On mathematical operations understood. separable well of from far is h eut bandb Ghosh by a general obtained using extensively results Hayashi which the [18], and method Owari different by slightly obtained independently was ysprbeoeain sa most at is operations separable by htn ae itnusal ysprbeoeain,and chann operations, quantum of separable class a by construct explicitly to distinguishable proper it employed interesting bases with no subspace that bipartite a found Watrous nltclyecp o h pca ae of cases distinguishabili special local the the sta for decide product to except orthogonal a how analytically know multipartite on don’t of still states set we orthogonal arbitrary Actual task. an of formidable for a set is loca still space finite the state multipartite a determining advances, of significant distinguishability these of Despite yLC 1] e loRf 1] ahno hwdta the that of showed set Nathanson [16]. a of Ref. also number for distinguis see probabilistically condition [15], be LOCC sufficient by to tool and states a quantum necessary as general a operations obtain separable to distinguishabi employed local for Chefles criteria useful particular, obtain respectively. to [28], order Ref. in and [2] Ref. in solved were which eea taeyfrsuyn unu olclt sto is nonlocality quantum studying for strategy general A eyrcnl eaal prtoshv enwdl used widely been have operations separable recently Very d ⊗ d aial nage ttsdistinguishable states entangled maximally 2 ⊗ 2 ttsas a enotie [11]. obtained been has also states tal. et d 1]adtesm result same the and [17] 1]adb a [20]. Fan by and [19] 2 ⊗ n and y even ly, iy In lity. 3 hable [13], ⊗ be n ized 24], tes. en- um not nal tes els ne ty 3 ty al s. ], h e 1 l , 2 with suboptimal environment-assisted capacity [21], which condition for the distinguishability of a set of orthogonal settled an open problem suggested by Hayden and King [22]. quantum states on by separable operations (Theorem 1). Hayashi et al. obtained a connection between the number of Many immediate butH interesting consequences of this condition distinguishable states by separable operations and the average are discussed in details. Sections IV and V devote to the entanglement degree of the quantum states to be discriminated following specific problem: Let Φ be a pure state on | i H [23]. In Ref. [24] we studied the local distinguishability of and let = Ψ1 , , ΨD−1 be an orthonormal basis an arbitrary basis of a multipartite state space and provided of Φ B⊥(i.e.,{| the orthogonali ··· | complementi} of Φ ), determine a universal tight lower bound on the number of locally the{| distinguishabilityi} of by separable operations.| i We show unambiguously distinguishable members in an arbitrary basis. this problem can be solvedB analytically. Two special cases are All these results suggest the following question:“What kind of most notable. The first case is concerned with two qubits, i.e., quantum states can be perfectly distinguishable by separable K = 2 and d1 = d2 = 2. We find a necessary condition operations?” The answer to this question will not only lead toa for the distinguishability of Ψ1 , Ψ2 , Ψ3 by separable {| i | i | i} better understanding of the nature of separable operations, but operations is that the summation of the concurrences of Ψk also accelerate the present research of quantum nonlocality. is equal to that of Φ (Theorem 2). When Φ is maximally| i The purpose of this paper is to study the strength and the entangled, such a| conditioni is also sufficient| i (Corollary 2). limitation of separable operations by considering the distin- The second case of interest is that any basis of Φ ⊥ guishability of quantum states under separable operations. We is indistinguishable by separable operations ifB and{| onlyi} if present a necessary and sufficient condition for the distin- the orthogonal Schmidt number of Φ (the least number of guishability of a set of general multipartite quantum states. orthogonal product states that can linearly| i express Φ ) is not Assisting with this condition, we completely solve the problem less than 3 (Theorem 6). Section VI is of independent| i inter- of distinguishing (D 1) multipartite pure states by separable est. We give an explicit construction of an indistinguishable operations, where D −is the dimension of the multipartite state subspace of dimension 7 (or 6) when considering a composite space under consideration. Two consequences are of particular quantum system consisting of two qutrits (resp. three qubits). interests. First we show that there exists a large class of We conclude the paper in Section VII, and put some additional separable operations performing on a 2 2 quantum system proofs in Appendix. cannot be realized by LOCC. This is⊗ somewhat surprising as it indicates that the capability of separable operations is much more powerful than that of LOCC operations even II. PRELIMINARIES when only the simplest composite quantum system is under consideration, which settles an open problem recently posed A general quantum state ρ is characterized by its density by Gheorghiu and Griffiths [29] and highlights the nonlocal operator, which is a positive semi-definite operator with trace nature of separable operations. It is also worth noting that one on . The density operator of a pure state ψ is simply the projectorH ψ ψ . The support of ρ, denoted by| suppi (ρ), is before our work only nonlocal separable operations acting on | ih | 3 3 systems are known [1]. the vector space spanned by the eigenvectors of ρ with positive ⊗Second we show that any basis of the orthogonal com- eigenvalues. Alternatively, suppose ρ can be decomposed into a convex combination of ψk ψk , say, plement of a multipartite pure state is indistinguishable by | ih | separable operations if and only if this state cannot be a n superposition of 1 or 2 orthogonal product states, i.e., has ρ = pk ψk ψk , (1) | ih | an orthogonal Schmidt number not less than 3, thus provide a k=1 generalization of the result by Watrous [21], who proved that X n any basis of the orthogonal complement of a d d maximally where 0 < pk 1 and k=1 pk = 1. Then supp(ρ) = ⊗ ≤ entangled state is indistinguishable by separable operations span ψk : 1 k n . This fact will be extensively used {| i ≤ ≤ } P when d 3. Furthermore, we also present an explicit construc- without being explicitly stated. In particular, ρ is said to be ≥ tion of indistinguishable bipartite (or tripartite) subspaces of separable if ψk can be chosen as product states. | i dimension 7 (resp. 6), which reduces the minimal dimension of A nonzero positive semi-definite operator E on is said H indistinguishable bipartite subspace from 8 (given by Watrous to be separable if the normalized form E/tr(E) is a separable in Ref. [21]) to 7. quantum mixed state. A separable positive operator-valued Throughout this paper we consider a multipartite quantum measure (POVM) is a collection of semi-definite positive n system consisting of K parts, say A1, , AK . We assume operators Π = Π1, , Πn such that =1 Πk = IH and ··· { ··· } k part Ak has a state space k with dimension dk. The Πk is separable for each 1 k n, where IH is the identity H K ≤ ≤ P whole state space is given by = k with total operator on . Now we are ready to introduce the notion H ⊗k=1H H dimension D = d1 dK . Sometimes we use the notation of separable operation. Intuitively, a separable operation is a ··· d1 dK for . With these assumptions, the rest of trace-preserving completely positive linear map which sends the⊗···⊗ paper is organizedH as follows. In Section II we give separable states to separable ones. For completeness, we state some necessary preliminaries, including a brief review of the a formal definition as follows. notion of separable operations, the concept of generalized Definition 1: A separable operation is a quantum operation Schmidt number of a quantum state, and some technical with product Kraus operators. More precisely, a separable K lemmas. In Section III we present a necessary and sufficient operation acting on a multipartite state space = j E H ⊗j=1H 3 should be of the following form: So E ψ = ψ or E ψ ψ = ψ ψ . Noticing that P = n | i | i | ih | | ih | ψk ψk such that ψk is an orthonormal basis for N k=1 | ih | {| i} † supp(ρ), then EP = P and P E = P . Let Q = I P . Then (ρ)= EkρE , (2) E k P − k=1 X E = (P + Q)E(P + Q)= P + QEQ, K N † where Ek = =1Ekj satisfies E Ek = I, and Ekj is ⊗j k=1 k which implies E P = QEQ 0.  linear operator on j . − ≥ The set of separableH operationsP is a rather restricted class of We also need the following concept of Schmidt number of quantum operations. It is not difficult to show that any LOCC a quantum state. operation is separable, see Ref. [1] for a detailed discussion. Definition 2: The Schmidt number of a multipartite quan- However, the converse part is not true as there exists some tum state ρ, denoted by Sch(ρ), is the minimal number of separable operation acting on 3 3 cannot be implemented product pure states whose linear span contains supp(ρ). The locally [1]. ⊗ orthogonal Schmidt number of ρ, denoted by Sch⊥(ρ), can be defined similarly with the additional requirement that these Let = ρ1, ,ρn be a collection of n quantum states. WeS say{ that ···is perfectly} distinguishable by separable product pure states should be mutually orthogonal. measurements if thereS is a separable POVM Π such that For pure state Ψ the (orthogonal) Schmidt number is exactly reduced to| thei minimal number of (orthogonal) pure tr(Πkρj )= δk,j for any 1 k, j n. Interestingly, if a set of ≤ ≤ product states needed to express Ψ as a superposition and quantum states ρ1, ,ρn can be perfectly distinguishable | i by a separable{ POVM··· Π, then} we can transform this set of this is just the definition given by Eisert and Briegel [30]. quantum states into another set of LOCC distinguishable states It is difficult to determine the Schmidt number of a general by applying a suitable separable operation on the quantum multipartite state. Only for bipartite pure states and some spe- E cial multi-qubit states the Schmidt number can be analytically system. More precisely, let σ1, , σn be any set of sep- arable quantum states that{ can··· be perfectly} distinguishable determined [30], [31]. We should point out that the definition by LOCC, then the mentioned separable operation can be of Schmidt number given here does not coincide with the one constructed as follows: given by Terhal and Horodecki [32]. n By definition, we have the following useful fact: (ρ)= tr(Πkρ)σk. (3) E Sch(ρ) max Sch(Ψ) : Ψ supp(ρ) . (4) k=1 X ≥ { | i ∈ } One can easily verify that (ρk)= σk. The function of can This inequality can be used to provide a lower bound for be intuitively understoodE as performing a separable POVME Sch(ρ). Πk on ρ and then preparing a separable state σk according It is interesting that the Schmidt number provides a simple {to the} outcome k, finally forgetting the classical information. criterion for the separability. In fact, when ρ is separable, Clearly, is a separable quantum operation and can be used supp(ρ) can be spanned by a set of product states. Thus we to perfectlyE discriminate the given states. Due to the above have: reason, when a set of states are perfectly distinguishable Lemma 2: For any quantum state ρ, R(ρ) Sch(ρ), where ≤ by a separable POVM, we directly say they are perfectly R(ρ) is the rank of ρ. The equality holds if ρ is separable. distinguishable by separable operations. A typical use of Lemma 2 is to show a quantum state ρ is The rest of this section is to present some useful notations entangled. Usually this can be achieved by finding a vector and technical lemmas. Ψ in supp(ρ) such that Sch(Ψ) > R(ρ). | i Lemma 1: Let E be a positive semi-definite operator such The Schmidt decomposition of a bipartite pure state is not that 0 E IH, and let ρ be a density matrix on . Then unique. However, when multipartite pure states with Schmidt tr(Eρ)=1≤ ≤if and only if E P 0, where P is the projectorH number 2 are under consideration, we do have a unique − ≥ K K on the support of ρ. decomposition. Let a = k=1 ak and b = k=1 bk | i ⊗ K | i | i ⊗ | i Proof. We first show that if E P 0 and 0 E I, then be product vectors on = k. Then a and b are − ≥ ≤ ≤ H ⊗k=1H | i | i tr(Eρ)=1. To see this, let ψ be any normalized vector in said to be different at the k-th entry if ak = z bk for supp(ρ). Then ψ P ψ =1.| Iti follows from E P 0 that any complex number z. Let h( a , b ) be| thei total 6 | numberi ψ E ψ 1. Onh the| | otheri hand, E I implies −ψ E≥ψ 1. of different entries between a | andi | ib . Then we have the Soh | for| anyi≥ ψ supp(ρ) we have ≤ h | | i≤ following interesting result. | i | i | i ∈ Lemma 3: Let Φ = a + b be an entangled state tr(E ψ ψ )= ψ E ψ =1. | i | i | i | ih | h | | i of such that h( a , b ) 3. Then there cannot exist n productH vectors c |andi | id such≥ that Φ = c + d and Noticing that ρ can be decomposed into k=1 pk ψk ψk n | ih | a , b = c| ,i d . In| i other words,| i Φ |hasi a| uniquei such that ψk supp(ρ) and k=1 pk =1, we have | i ∈ P {|Schmidti | i} decomposition. 6 {| i | i} | i n P Proof. Without loss of generality, assume h( a , b )=3. By tr(Eρ)= pktr(E ψk ψk )=1. | ih | contradiction, suppose there exist another two| producti | i vectors k=1 X c and d which also yield a decomposition of Φ . That is, Conversely, tr(Eρ) = 1 and ψ E ψ 1 imply that | i | i | i h | | i ≤ tr(E ψ ψ )=1 for any normalized vector ψ supp(ρ). a1a2a3 + b1b2b3 = c1c2c3 + d1d2d3 , (5) | ih | | i ∈ | i | i | i | i 4

⊥ iθ where ak is not equal to bk for each 1 k 3. Let a1 the phase factor e into b and resetting β and δ to β and ⊥ ≤ ≤ ⊥ | i ∈ | i | | span a1 , b1 such that a1 a1 = 0 and a1 b1 = 0. δ respectively, we know that condition (iii) is satisfied.  Then{| multiplyingi | i} a⊥ on theh both| sidesi of the aboveh | equationi 6 −| | h 1 | we have that b2b3 is contained in S = span c2c3 , d2d3 . III. CONDITIONSFORTHEDISTINGUISHABILITYOF | i {| i | i} Similarly we can also show a2a2 S. On the other hand, QUANTUM STATES BY SEPARABLE OPERATIONS | i ∈ there are only two unique product vectors c2c3 and d2d3 in It would be desirable to know when a collection of quantum | i | i S (up to some factors). Then it follows that a2a3 , b2b3 is states is perfectly distinguishable by separable operations. {| i | i} essentially c2c3 , d2d3 . Without loss of generality, assume Orthogonality is generally not sufficient for the existence of a {| i | i} a2a3 = α c2c3 and b2b3 = β d2d3 for some complex separable POVM for discrimination. A rather simple but useful | i | i | i | i numbers α and β, then necessary and sufficient condition is as follows. Theorem 1: Let = ρ1, ,ρn be a collection of or- (α a1 c1 ) c2c3 + (β b1 d1 ) d2d3 =0, (6) S { ··· } | i − | i | i | i − | i | i thogonal quantum states of . Then is perfectly distinguish- −1 H S which is possible if and only if a1 = α c1 and b1 = able by separable operations if and only if there exist n positive −1 | i | i | i β d1 . That also means a = c and b = d . A semi-definite operators E1, , En such that Pk + Ek is | i | i | i | i | i { ··· }n contradiction with our assumption.  separable for each 1 k n, and k=1 Ek = P0, where Pk ≤ ≤ n We also need the following result concerning with the is the projector on supp(ρk), and P0 = IH =1 Pk. P − k separability of rank 2 positive semi-definite operators. Proof. Sufficiency is obvious. Suppose that (ii) holds, let us P Lemma 4: Let Ψ and Φ be pure states on and λ 0. define a POVM Π= Π1, , Πn as follows: Πk = Pk +Ek { ··· } Then ρ(λ)= Ψ |Ψ i+λ Φ| iΦ is separable if andH only if≥ one for each 1 k n. It is easy to verify that Π is a separable ≤ ≤ of the following| ih cases| holds:| ih | measurement that perfectly discriminates . S (i) Both Ψ and Φ are product states and λ 0; Now we turn to show the necessity. Suppose is | i | i ≥ S (ii) Ψ is a product state and λ =0; perfectly distinguishable by some separable POVM, say | i (iii) There are two product states a and b and positive Π1, , Πn . Take Ek = Πk Pk for each 1 k n. | i | i { ··· n } − ≤ ≤ real numbers α, β, γ, δ such that Ψ = α a + β b , Φ = Then k=1 Ek = P0. To complete the proof, it suffices to | i | i | i | i γ a δ b , and λ = αβ/γδ. (There may be some global phase show Ek 0. By the assumption, we have tr(Πkρk)=1. | i− | i P ≥  factors before Ψ and Φ .) Then the positivity of Ek follows directly from Lemma 1. Proof. By a| directi calculation| i one can readily verify that (i), (ii) and (iii) are sufficient for the separability of ρ(λ). We In some sense Theorem 1 is only a reformulation of our only need to show that they are also necessary. The case that discrimination problem. One may doubt it can provide any both states are unentangled is trivial, corresponding to (i). We valuable results. However, this seemingly trivial reformulation shall consider the following two cases only: does give us much more operational forms of the measure- (1) One of Ψ and Φ is entangled. First assume that Φ ment operators for discrimination and connects them to the is entangled. Then| i for| anyi λ> 0. ρ(λ) is just a mixture of| ani separability problem, which is one of the central topics in entangled state and a product state, and should be entangled quantum information theory and has been extensively studied for any λ> 0, as shown by Horodecki et al. [34]. So the only since the last two decades. So many well developed tools for possible case is λ = 0, i.e., (ii) holds. Second, when Ψ is the separability problem may be applicable in the context entangled we can show by a similar argument that ρ(λ) cannot| i of discrimination. As a result, Theorem 1 is unexpectedly be separable for any λ 0. powerful. (2) Both states are entangled.≥ We shall show that condition Some special but interesting cases of Theorem 1 deserve careful investigations. When the supports of the states in (iii) is necessary. Suppose that ρ(λ) is separable, then by n S Lemma 2, it follows that Sch(ρ(λ)) = R(ρ(λ))=2. In other together span the whole state space, i.e., supp( k=1 ρk)= , is perfectly distinguishable by separable operations if anHd words, the support of ρ(λ) can be spanned by two product S P only if Pk is separable for each 1 k n. In particular, states a and b . Noticing that both Ψ and Φ are contained ≤ ≤ in the| supporti | ofi ρ(λ), we can write| i | i an orthonormal basis of is perfectly distinguishable by separable operations if andH only if it is a product basis, which Ψ = α a + β b , (7) is a well known result first obtained by Horodecki et al. using | i | i | i Φ = γ a + δ b (8) a rather different method [25], see also [15] for another simple | i | i | i proof. for nonzero complex numbers α,β,γ,δ. Furthermore, a and A slightly more general case is when there exists k such are also the only product states in supp . So| i b (ρ(λ)) ρ(λ) P0 + Pk and Pj are all separable for any j = k. In this | i 6 should be a mixture of a a and b b . Substituting Ψ and case we say that Pj : j = k is completable [33]. It has Φ into ρ(λ) and setting| ih the| coefficients| ih | of a b and| ib a { 6 } | i | ih | | ih | been shown in Ref. [2] that a sufficient condition for a set to be zero we have of orthogonal product states to be completable is that they αβ∗ + λγδ∗ =0. (9) are exactly distinguishable by LOCC. Combining this with Theorem 1, we have the following interesting corollary: By adding suitable phase factors before Ψ and Φ , we can Corollary 1: A set of LOCC distinguishable product states make α and γ positive. The condition λ>| 0i implies| i that β = together with any state in its orthogonal complement is always β eiθ and δ = δ eiθ for some real number θ. Absorbing perfectly distinguishable by separable operations. | | −| | 5

We notice that DiVincenzo and co-workers have shown Πk is separable if and only if λk =0. Otherwise by Lemma −1 in Ref. [3] that any three (or four) orthogonal multipartite 5 we have that Πk is separable if and only if ΨkΦ has (resp. bipartite) product states are distinguishable by LOCC. two anti-parallel eigenvalues and λk = C(Ψk)/C(Φ). So by Hence we conclude immediately by the above corollary that λ1 +λ2 +λ3 =1 we have C(Ψ1)+C(Ψ2)+C(Ψ3)= C(Φ). any four (or five) orthogonal multipartite (resp. bipartite) That completes the proof.  states including three (resp. four) product states are perfectly The most interesting part in the above theorem is Condition distinguishable by separable operations. The case when (ii). It reveals a precise relation between the concurrences and is a set of product states is of particular interest and hasS the distinguishability by separable operations. By a careful been studied carefully in Ref. [3] using a rather different observation, we notice that a similar condition has been method. Specifically, let = ψ1 , , ψn . Then it has previously obtained by Hayashi et al. in Ref. [23], where a S {| i ··· | i} been shown that if Sk = S ψk is completable for each very general relation between distinguishability by separable 1 k n, then S is perfectly− {| distinguishablei} by separable operations and average entanglement degree of the states to ≤ ≤ operations. Interesting examples include any orthogonal UPB be discriminated was presented. When only 2 2 states consisting of four 2 2 2 (or five 3 3) product states. were involved, this relation can be reduced to an⊗ inequality Clearly we have refined⊗ the⊗ results by DiVincenzo⊗ et al. similar to condition (ii). The key difference is that here what As another important consequence of Theorem 1, we shall we obtained is an exact identity that (almost) completely show that the problem of distinguishing (D 1) orthogonal characterizes the distinguishability by separable operations. − pure states by separable operations can be completely solved. Suppose now Φ is maximally entangled. Then Φ= U for Notice that if Φ is a product state, then the only form of some 2 2 unitary| i matrix. Noticing that | i B × that can be distinguishable by separable operations is a product −1 † basis. For simplicity, in what follows we shall assume Φ is tr(ΨkΦ ) = tr(U Ψk)=2 Φ Ψk =0 | i h | i an entangled state. We shall consider two cases Sch⊥(Φ) = 2 −1 for k = 1, 2, 3, we conclude that ΨkΦ should have two and Sch⊥(Φ) 3 respectively in the next two sections. ≥ anti-parallel eigenvalues (assume that Ψk is entangled). Fur- thermore, the concurrence of a 2 |2 maximallyi entangled ⊗ IV. DISTINGUISHABILITY OF ARBITRARY ORTHONORMAL state is one. Collecting all these facts together we have the ⊥ BASIS OF Φ WITH Sch⊥(Φ) = 2 following interesting corollary: {| i} Corollary 2: Let Φ be a 2 2 maximally entangled state, We start with the simplest 2 2 case and provide an | i ⊗ analytical solution. It is well known⊗ that any 2 2 state Ψ and let = Ψ1 , Ψ2 , Ψ3 be an orthonormal basis ⊗ | i for Φ B⊥. Then{| iis| perfectlyi | i} distinguishable by separable can be uniquely represented by a 2 2 matrix Ψ as follows: {| i} B × operations if and only if C(Ψ1)+ C(Ψ2)+ C(Ψ3)=1. Ψ = (I Ψ) Φ+ , (10) Theorem 2 implies that there exists a large class of 2 | i ⊗ | i 2 separable operations that cannot be implemented locally.⊗ where √ is a maximally entangled Φ+ = 1/ 2( 00 + 11 ) We shall exhibit an explicit construction of such separable state. The| concurrencei |of i |is giveni by the absolute value of Ψ operations. We achieve this goal by identifying a set of states the determinant of , i.e.,| i This rewriting Ψ C(Ψ) = det(Ψ) . that is distinguishable by separable operations but is not LOCC form coincides with the original definition| given| by Wootters distinguishable. [35]. So 0 C(Ψ) 1 for any 2 2 state Ψ . In Let Ψ(θ) = cos θ 01 +sin θ 10 and Φ(θ) = cos θ 00 + particular, C(Ψ)≤ vanishes≤ for product states⊗ while attains| i one sin θ 11| foriθ . Then| i for any| 0 i< α | β iπ/4, let |Φ i= for maximally entangled states. We say complex numbers z1 | i ∈ R ≤ ≤ |π i Φ(β) , and let Ψ1 = Ψ(α) , Ψ2 = cos γ Ψ(α 2 ) + and z2 are anti-parallel if there exists a negative real number | i π | i | i | i π | − iπ sin γ Φ(β 2 ) , Ψ3 = sin γ Ψ(α 2 ) cos γ Φ(β 2 ) a such that z1 = az2. First we need the following simplified be an| orthonormal− i | basisi of Φ| ⊥.− Choosei− γ such| that− i version of Lemma 4 in the 2 2 case. {| i} ⊗ Lemma 5: Let Ψ and Φ be two 2 2 entangled pure −1 sin 2α −1 sin 2β states, and let λ | i0. Then| iρ(λ) = Ψ⊗ Ψ + λ Φ Φ is tan γ tan . (11) sin 2β ≤ ≤ sin 2α separable if and only≥ if ΨΦ−1 has two anti-parallel| ih | eigenvalues| ih | s r and λ = C(Ψ)/C(Φ). Note that Φ is entangled implies that By direct calculations we can easily verify that the assump- | i Φ is invertible. tions of Theorem 2 are fulfilled, thus = Ψ1 , Ψ2 , Ψ3 Now the distinguishability of three 2 2 orthogonal states is perfectly distinguishable using separableB {| operations.i | i |How-i} ⊗ π by separable operations is characterized as follows. ever, it is clear whenever 0 <α<β , both Ψ1 and Ψ2 ≤ 4 | i | i Theorem 2: Let Φ be a 2 2 pure state, and let = are entangled, thus is indistinguishable by LOCC, as shown | i ⊗ ⊥ B B Ψ1 , Ψ2 , Ψ3 be an orthonormal basis for Φ . Then by Walgate and Hardy [11]. Interestingly, Ψ3 is reduced to {| isi perfectly| i | distinguishablei} by separable operations{| i} if and a product state when γ takes one of the extreme| i values in Eq. B −1 only if (i) ΨkΦ has two anti-parallel eigenvalues for each (11). entangled state Ψk; and (ii) C(Ψ1)+C(Ψ2)+C(Ψ3)= C(Φ). The above example also implies that the distinguishability Proof. By Theorem 1, the POVM element that can perfectly by separable operations has some continuous property. We identify Ψk should have the form Πk = Ψk Ψk + formalize this intuitive observation as follows: | i | ih | λk Φ Φ , where 0 λk 1 and λ1 + λ2 + λ3 = 1. If Theorem 3: Let c1,c2,c3 be nonnegative real numbers such | ih | ≤ ≤ Ψk is a product state, then it follows from Lemma 4 that that 0 c1 +c2 +c3 1. Then there exists a set of three 2 2 | i ≤ ≤ ⊗ 6

states Ψ1 , Ψ2 , Ψ3 satisfying (i) C(Ψk) = ck for k = rable. According to the assumptions on Φ and employing {| i | i | i} | i 1, 2, 3; and (ii) Ψ1 , Ψ2 , Ψ3 is perfectly distinguishable Lemma 4, we can easily see that Ψ should be of the form {| i | i | i} ′ | i by separable operations. b1 bK−2 Ψ . Second, we have a1 aK−2 should | ··· i ⊗ | i | ··· i There is a nice geometrical way to illustrate the distin- be identical to b1 bK−2 up to some factors. Otherwise guishability of quantum states by different class of operations. span Ψ , Φ |cannot··· containi product states. So we need To see this, we identify a triple of orthogonal quantum states Ψ′ {|Ψ′ i+|λi}Φ′ Φ′ being separable. Now the problem is 3 | ih | | ih | ( ψ1 , ψ2 , ψ3 ) by a point (C(ψ1), C(ψ2), C(ψ3)) of R . reduced to the 2 2 case and the result follows directly from We| sayi | suchi | a pointi is distinguishable by global (or separable, Theorem 2. ⊗  LOCC) operations if the related states are distinguishable by The case that Φ is at least entangled among three parties is the corresponding operations. By a careful analysis, we can much more simple.| i Any distinguishable basis should contain show any point in the regular tetrahedron in Fig. 1 corresponds a unique entangled state with a predetermined form. ⊥ to some orthonormal basis of Φ+ . A detailed proof for Theorem 5: Let Φ = cos θ a + sin θ b be an entangled this interesting fact is given in the{| Appendixi} A. So the points in pure state of such| thati a b |=0i and h|( ia , b ) 3. Then the regular tetrahedron represent a region that can be perfectly can be perfectlyH distinguishableh | i by separable| i | i operations≥ if distinguishable by global quantum operations. The yellow Band only if there is a unique entangled state Ψ in with | i B triangle BCD is exactly the set of points distinguishable by the form Ψ = sin θ a cos θ b , and the other states should | i | i− | i separable operations. Finally, three vertices B, C, and D are form an orthogonal product basis of a , b ⊥. the only points that are distinguishable by LOCC. From such Proof. Sufficiency is obvious. We only{| consideri | i} the necessity. a representation we can clearly see that the class of 2 2 Suppose that is distinguishable by separable operations. B separable operations strictly lies between the classes of LOCC⊗ Noticing that Φ is entangled, we conclude that should | i B and global operations. contain at least one entangled state. By Lemma 3, one can easily show that any entangled state Ψ such that Ψ Ψ + A(1,1,1) λ Φ Φ is separable should be contained| i in span| aih, b| . Noticing| ih | that Ψ is orthogonal to Φ , we can deduce{| i | thati} Ψ is uniquely| determinedi and is given| i by sin θ a cos θ b . | i | i− | i B(0,0,1) D(1,0,0) V. Φ ⊥ HASNOBASESDISTINGUISHABLEBY {| i} SEPARABLE OPERATIONS WHEN Sch⊥(Φ) 3 ≥ Now we consider the case when Sch⊥(Φ) 3. Our result is ≥ C(0,1,0) a generalization of Watrous’s result. It is remarkable that the orthogonal Schmidt number of Φ completely characterizes ⊥ | i Fig. 1. Illustration of the distinguishability of the basis of {|Φ+i} , where the distinguishability of the subspace Φ ⊥. |Φ+i is a 2 ⊗ 2 maximally entangled states and we identify an orthonormal {| i} 3 Theorem 6: Let Φ be an entangled pure state on . Then basis (|ψ1i, |ψ2i, |ψ3i) by a point (C(ψ1), C(ψ2), C(ψ3)) in R . Any | i H point in the regular tetrahedron ABCD corresponds a legal orthonormal basis Φ ⊥ having no orthonormal basis perfectly distinguishable ⊥ of {|Φ+i} , thus is distinguishable by global operations. Vertices B, C, and D {| i} by separable operations if and only if Sch⊥(Φ) > 2. In partic- are the only points corresponding to LOCC distinguishable basis. Interestingly, the triangle △BCD represents bases that are distinguishable by separable ular, when Sch⊥(Φ) = 2, there always exists an orthonormal operations. basis of Φ ⊥ that is perfectly distinguishable by LOCC. B {| i} Proof. Necessity: Suppose Sch⊥(Φ) = 2. Then the exis- Now we consider the multipartite setting. There are two tence of a distinguishable basis by separable operations follows possible cases: Φ is reduced to a bipartite entangled state directly from Theorem 4. Here we further construct a basis | i only between two parties; or Φ is an entangled state at for Φ ⊥ that is distinguishable by LOCC. Notice that Φ | i {| i} | i least between three parties. The following two theorems deal can be written into the form Φ = cos θ Φ0 + sin θ Φ1 , | i | i | i with these two cases separately. Surprisingly, the first case is where Φ0 and Φ1 are orthogonal product states on and | i | i H essentially reduced to 2 2 case. 0 < θ < π/2. Then we can extend Φ0 and Φ1 into a ⊗ ′ | i | i Theorem 4: Let Φ = a1 aK−2 Φ be a pure state on complete orthogonal product basis Φ0 , Φ1 , , ΦD−1 ′ | i | ··· i| i {| i | i ··· | i} such that Φ is an entangled state between parties AK−1 of . We can further assume this basis is distinguishable H | i H and AK . Then is perfectly distinguishable by separable by local projective measurements [2]. Replacing Φ0 and B | i operations if and only if each entangled state Ψ in can Φ1 with Ψ = sin θ Φ0 cos θ Φ1 we obtain a basis for | ′i B | i ⊥ | i | i− | i be written into the form Ψ = a1 aK−2 Ψ for some Φ that is distinguishable by the same local projective |′ i | ··· i| i {| i} bipartite entangled state Ψ between AK−1 and AK such that measurements. ′ ′ | i (i) all Ψ and Φ can be embeded into a 2 2 subspace Sufficiency: In this case we have Sch⊥(Φ) 3. If ′ | i | i ′ ′−1 ⊗ ≥ of K−1 K ; and (ii) Ψ Φ has two anti-parallel Sch(Φ) 3. Then for any λk > 0 we know from Lemma 4 H H ⊗ H ′ ′ ′ ′ ≥ eigenvalues and Ψ′ C(Ψ ) = C(Φ ), where Ψ and Φ are that Π(λk)= Ψk Ψk +λk Φ Φ should be entangled. That 2 2 matrices that are the representations of Ψ′ and Φ′ on implies any basis| ih of Φ| ⊥ |isih indistinguishable| by separable ⊗′ in the sense ofP Eq. (10), respectively. | i | i operations. {| i} H Proof. Let Ψ be any entangled state of . Then there Suppose now that Sch⊥(Φ) 3 and Sch(Φ) = 2. By is some 0 <| λ i 1 such that Ψ Ψ + λ ΦB Φ is sepa- contradiction, assume there exists≥ a basis distinguishable ≤ | ih | | ih | B 7

2 2 by separable operations. Then applying Lemma 3, we know where α =0 and α + β =1. Substituting Φ1 and Φ2 there are two unique product vectors a and b such that into the above6 equation,| | | we| have | i | i | i | i Φ = a + b . So any state Ψk with λk > 0 should also be a 1 superposition| i | i | i of a and b .| Asi there are only two orthogonal ψ = ( 0 (α 0 + √3β 1 )+ α 11 + α 22 ), (14) | i √3 | i | i | i | i | i states in span a| i, b ,| wei conclude the only possibility is {| i | i} which is clearly an entangled state with Schmidt number for that there is a unique state Ψk with λk =1 and all the other 3 | i any α =0. states are product states. On the other hand, let Ψ be the 6 entangled state in span a , b such that Ψ a =0| i. Then Now we turn to show the validity of P2. By contradiction, {| i | i} h | i suppose for some ρ such that supp(ρ)= S we have Sch(ρ)= Ψk Ψk + Φ Φ = a a + Ψ Ψ (12) 3. Then there should exist three unnormalized product states | ih | | ih | | ih | | ih | a1b1 , a2b2 , and a3b3 such that is separable. But the right hand side of the above equation is a | i | i | i summation of an entangled state and a product state, it should Φ1 = a1b1 + a2b2 + a3b3 , (15)  | i | i | i | i be entangled [34]. A contradiction. Φ2 = α1 a1b1 + α2 a2b2 + α3 a3b3 . (16) Let us check some interesting examples. Take Φ to be a | i | i | i | i W -type state, Φ = a 001 + b 010 + c 100 , where| i a 2 + Without loss of generality, we may assume α1 =0. Then it is 2 2 | i | i | i | i | | 6 b + c =1, and abc =0. Then Sch⊥(Φ) = Sch(Φ) = 3. It clear that | | | | 6 follows from Theorem 6 that any orthonormal basis of Φ ⊥ −1 α2 α3 {| i} Φ1 α1 Φ2 = (1 ) a2b2 + (1 ) a3b3 . (17) cannot be perfectly distinguishable by separable operations. | i− | i − α1 | i − α1 | i 3 This yields an indistinguishable subspace with dimension 2 By P1, the left hand side of the above equation has Schmidt − 1=7, which is a slight improvement over the bipartite case, number 3, while the right hand side of the above equation has where a 3 3 indistinguishable subspace of dimension 8 was Schmidt number at most . That is a contradiction. ⊗ 2 given by Watrous [21]. But if we choose the GHZ-type state Now let = Ψk : 1 k 7 be any orthogonal basis π ⊥ ⊥ B {| i ≤ ≤ } Φ = cos θ 000 +sin θ 111 , where 0 θ 2 . Then Φ of S . By Theorem 1, the POVM element identifying Ψk | i | i | i ≤ ≤ {| i} 7 | i does have an orthonormal basis that is perfectly distinguishable is of the form Πk = Ψk Ψk + Ek, where Ek = | ih | k=1 by LOCC. Interestingly, if we take Φ = α 000 + β +++ , Φ1 Φ1 + Φ2 Φ2 and Ek 0. | √i | i | i | ih | | ih | ≥ P where αβ =0 and + = ( 0 + 1 )/ 2. Then we can easily So we can choose Ek such that Ek = 0 and Ek = 6 | i | i | i 6 6 see that Sch(Φ) = 2 but Sch⊥(Φ) 3, as Φ cannot be 01 01 (up to some factor). We shall show that Πk should ≥ | i | ih | written into a superposition of two orthogonal product states. be entangled. There are two cases. If R(Ek)=1 then by ⊥ Thus it follows from the above theorem that Φ does P1 we have Sch(Ek)=3, which follows that Sch(Πk) {| i} ≥ not have any orthonormal basis distinguishable by separable 3 > R(Πk). Similarly, if R(Ek)=2 then by P2 we have operations. Sch(Πk) Sch(Ek)=4 > R(Πk). In both cases Πk is entangled.≥ That completes the proof.  VI.INDISTINGUISHABLE SUBSPACES WITH SMALLER Using the very same method, we can construct a 2 2 2 in- ⊗ ⊗ DIMENSIONS distinguishable subspace with dimension 6. For instance, take Φ1 = ( 001 + 010 + 100 )/√3 and Φ2 = 000 . Then The dimension of indistinguishable subspaces can be further | i | ⊥i | i | i | i | i Φ1 , Φ2 is indistinguishable by separable operations. reduced. We shall show that there exist a 3 3 indistinguishable {| i | i} subspace with dimension 7 and a 2 2 ⊗2 indistinguishable VII. CONCLUSIONS subspace with dimension 6. We first⊗ consider⊗ bipartite case. Let Φ1 = ( 00 + 11 + 22 )/√3 and Φ2 = 01 . Let In summary, we provided a necessary and sufficient condi- | i | i | i | i ⊥ | i | i S = span Φ1 , Φ2 . It is clear that S is a bipartite sub- tion for the distinguishability of a set of multipartite quantum space with{| dimensioni | i}7. Surprisingly, we have the following states by separable operations. A set of three 2 2 pure states ⊗ interesting result: that is perfectly distinguishable by separable operations but is Theorem 7: S⊥ is indistinguishable by separable opera- indistinguishable by LOCC then was explicitly constructed. As tions. a consequence, there exists a large class of nonlocal separable Proof. We need the following easily verifiable properties of operations even for the simplest composite quantum system S to complete the proof: consisting of two qubits. We also showed that the orthogonal P0: Φ2 is the unique product vector (up to a scalar factor) complement of a pure state has no bases distinguishable by in S; | i separable operations if and only if this state has an orthogonal P1. Sch(Ψ) = 3 for any entangled state Ψ S; Schmidt number not less than 3. We believe these results | i ∈ P2. Any positive operator ρ such that supp(ρ)= S satisfies would be useful in clarifying the relation between separable Sch(ρ)=4. operations and LOCC. The validity of P0 can be directly verified. We only consider There are still a number of unsolved problems that are P1 and P2. We first show the validity of P1. Actually, any worthwhile for further study. We have mentioned some in entangled state ψ from S can be written into a superposition the previous context. Here we would like to stress two of | i them. The first one is concerning with the distinguishability of of Φ1 and Φ2 . That is, | i | i orthogonal product pure states. It a simple fact that any set of ψ = α Φ1 + β Φ2 , (13) orthogonal product pure states can be perfectly distinguishable | i | i | i 8

3 by some positive-partial-transpose preserving (PPT) operation. 2 be the set of x satisfying the following equations: Is this also true for separable operations? We have seen that P ∈ R x1 + x2 + x3 1, (25) separable operations are powerful enough to distinguish any ≥ set of orthogonal product pure states in 3 3 and 2 2 2.If the x1 + x2 x3 1, (26) answer is affirmative in general, then how⊗ to prove?⊗ Otherwis⊗ e − ≤ x2 + x3 x1 1, (27) a counterexample would be highly desirable. With little effort − ≤ x3 + x1 x2 1, (28) we can easily show that it is sufficient to verify whether any − ≤ 0 x1, x2, x3 1. (29) UPB is perfectly distinguishable by separable operations. The ≤ ≤ difficulty we met is that the structure of UPB on a multipartite state space remains unknown except for some special cases. Obviously, 2 is just a unit regular tetrahedron. We shall show that is containedP by . That means any point of the regular Another problem is to find more applications of locally 2 1 tetrahedronP correspondsP to some orthonormal basis of Φ ⊥. indistinguishable subspaces (LIS). The work of Watrous sug- {| i} gests that bipartite LIS can be used to construct quantum chan- Theorem 8: 2 1. nels with sub-optimal environment-assisted capacity. It would P ⊆P Proof. For any point x 2, we shall construct a 3 3 be of great interest to employ LIS as a tool to give tighter ∈ P × upper of the capacity. In the asymptotic setting similar problem unitary operation U such that Eq. (22) holds. Without loss of generality, we may assume x1 x2 x3 0. So Eq. (25) is has been thoroughly studied by Winter [36]. Hopefully, these ≥ ≥ ≥ efforts would reveal some deep properties of LIS and testify reduced to the following equations: the richness of the mathematical structure of this notion. x1 + x2 + x3 1, (30) ≥ x1 + x2 x3 1, (31) − ≤ APPENDIX: GEOMETRIC REPRESENTATION OF THE 0 x3 x2 x1 1. (32) CONCURRENCES OF ORTHOGONAL BASES OF Φ ⊥ ≤ ≤ ≤ ≤ {| i} Construct a unitary matrix U as follows: First we recall a useful representation of the concurrence of a two-qubit pure state. Let Φk :1 k 4 be the magic u11 u12 u13 1 0 0 {| i ≤ ≤ } iθ basis [35], and let ψ be any pure state such that ψ = U = u21 u22 u23 . 0 e 0 , 4 | i | i    iθ  λk Φk . Then the concurrence of ψ is given by the u31 u32 u33 0 0 e k=1 | i | i following formula:     P where [ukl]3×3 is a real orthogonal matrix, and 0 θ π. ≤ ≤ C(ψ)= λ2 + λ2 + λ2 + λ2 . (18) By a simple calculation, we have | 1 2 3 4| iθ iθ u11 u12e u13e Suppose now we choose Φ to be one of Φk :1 k 4 , iθ iθ | i ⊥{| i ≤ ≤ } U = u21 u22e u23e . say Φ4 . Then any vector from Φ should be a linear  iθ iθ  | i {| i} u31 u32e u33e combination of Φ1 , Φ2 , Φ3 . As a direct consequence, {| i | i | i}   we have the following interesting lemma which connects the Our purpose is to choose suitable real numbers u11, u21, u31, ⊥ concurrences of orthonormal bases of Φ to 3 3 unitary and θ such that matrices. {| i} × 2 2 2iθ Lemma 6: Let ψ1 , ψ2 , ψ3 be an orthogonal basis for u11 + (1 u11)e = x1, (33) ⊥ {| i | i | i} 2 − 2 2 Φ , then there exists a 3 3 unitary matrices U = [ukl]3×3 iθ (34) {| i} × u21 + (1 u21)e = x2, such that 2 − 2 2iθ u + (1 u )e = x3, (35) 31 − 31 2 2 2 2 2 2 C(ψ1)= u11 + u12 + u13 , (19) u11 + u21 + u13 = 1. (36) | 2 2 2 | C(ψ2)= u21 + u22 + u23 , (20) From the first three equations, we have | 2 2 2 | C(ψ3)= u31 + u32 + u33 . (21) | | 2 2 2 sin θ xk cos θ u1k = ± − , k =1, 2, 3. (37) Conversely, for any unitary matrix U = [ukl]3×3, there exists 2 sin θ ⊥ p an orthogonal basis for Φ , say ψ1 , ψ2 , ψ3 such We shall choose a suitable θ such that the last normalized that Eq. (19) holds. {| i} {| i | i | i} 3 relation holds. That is, choose θ such that Motivated by the above lemma, let 1 be the set of x P ∈ R such that there exists a unitary matrix such that 2 2 2 U = [ukj ]3×3 sin θ x cos2 θ x cos2 θ x cos2 θ =0, ± 1 − ± 2 − ± 3 − 2 2 2 q q q x1 = u11 + u12 + u13 , (22) −1 | | where cos (x3) θ π/2. The difficulty here is how to 2 2 2 ≤ ≤ x2 = u21 + u22 + u23 , (23) choose suitable signs for u1k. Let | 2 2 2 | x3 = u31 + u32 + u33 . (24) | | f(θ) = sin θ x2 cos2 θ x2 cos2 θ x2 cos2 θ, − 1 − − 2 − − 3 − Then it is clear that 1 is exactly the set of points correspond- P ⊥ q 2 2 q 2 2 q 2 2 ing to the concurrences of the orthogonal bases of Φ . Let g(θ) = sin θ x1 cos θ x2 cos θ + x3 cos θ. {| i} − − − − − q q q 9

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