Local Discrimination of Three Generalized Bell States Guojing Tian1 2 ∗

Local discrimination of three generalized Bell states Guojing Tian1 2 ∗

1 CAS Key Lab of Network Data Science and Technology, Institute of Computing Technology, Chinese Academy of Sciences, 100190, Beijing, China 2 University of Chinese Academy of Sciences, Beijing, 100049, China Abstract. In this paper, we present a very simple proof for the local discrimination of any three generalized Bell states, that is, any three orthogonal generalized Bell states can be distinguished by local operations and classical communication as long as the dimension of the corresponding quantum system is bigger than two. Keywords: quantum nonlocality, quantum entanglement, local discrimination, generalized Bell state

1 Introduction in the same paper [21]. What interests us much naturally is the local discrim- As an effective and efficient method, local discrimination of three GBSs in every quantum system. That ination of quantum states has always been playing a is, we desire to know whether there exist three locally significant role in the study of the relationship be- indistinguishable GBSs in some quantum system or not. tween quantum nonlocality and quantum entanglement In this paper, we will give a negative answer to this ques- [1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. The tion, i.e., arbitrary three GBSs in any quantm system can so-called “local” here means the spatially separated par- be distinguished by LOCC. ties who shared a secret multi-partite quantum state cho- sen from a known orthogonal states set are only allowed 2 Theorems and Proofs to employ local operations and communicate classically (LOCC). In 1999, Bennett et al. came up with the phe- Consider a bipartite quantum system A ⊗ B of the nomenon of “nonlocality without entanglement”, which dimension d ⊗ d, that is, either subsystem A or B can would also be discovered in many cases [2, 3, 4, 5]. Oth- be regarded as a qudit with d levels labeled by Zd = erwise, any two quantum states, entangled or not, can {0, 1, ··· , d − 1}. Let {|ai|a ∈ Zd} be the computational be discriminated perfectly by LOCC [6], which implies basis, and the standard bipartite maximally entangled quantum entanglement is neither sufficient nor necessary state in this system is expressed as |Φi = √1 P |jji. d j∈Zd condition for quantum nonlocality. We can also define the bit flip and phase flip operators Actually, besides the theoretical meaning the local dis- to be crimination of quantum states has been employed to de- X X a sign quantum secret sharing protocols. In Ref. [7], the Xd = |a ⊕d 1iha|, Zd = ωd |aiha| (1) authors have presented a (2,n)-threshold quantum secret a∈Zd a∈Zd sharing protocol, where any two cooperating players from 2πi disjoint groups can always reconstruct the secret, based where ωd = e d and the subscript denotes the dimension on the local discrimination of their specical GHZ-states of the corresponding quantum system. When the dimen- set. Therefore, we focus on the local discrimination of sion is 2, the above two operators will become the Pauli maximally entangled states, especially the generalized matrices X and Z, respectively. Thus the matrices Bell states (GBSs), which will be introduced in Sec. 2. U = X mZn (2) Until now, there have been series of results on the local nm d d discrimination of GBSs. In 2002, the local discrimination generated by Xd and Zd in Eq. (1) have been called gen- of Bell states has been well studied [15]. Any two Bell eralized Pauli matrices, GPMs for short. states are locally distinguishable, while three or four Bell A generalized Bell state (GBS) is a maximally entan- states cannot be distinguished by LOCC with certainty. gled state obtained by applying a GPM of one party For GBSs, the authors in [16] have presented a sufficient to the standard maximally entangled state. Since only condition for the local discrimination of GBSs and built GBSs are considered here, we shall denote the set of all four 1-LOCC indistinguishable GBSs in 4 ⊗ 4 quantum GBSs by the corresponding GPMs, i.e., system. Another four 4 ⊗ 4 GBSs have been referred in [17] based on a necessary condition they had given beforehand. Both of the two examples have been nicely U = {(Id ⊗ Unm)|Φi|n, m ∈ Zd} = {Unm|n, m ∈ Zd}. certified locally indistinguishable in [18]. Moreover, we have derived that there are three locally indistinguishable Next, we will discuss the local distinguishability of mu- and seven locally distinguishable quadruples in the 4 ⊗ 4 tually orthogonal maximally entangled states in the most quantum system with the help of local invariants referred generalized quantum system, and show that any three

∗ GBSs can be distinguished by 1-LOCC. Before that, it is [email protected] necessary to review the undermentioned result which can be found in Ref. [16], as it gives a specific criterion for Consider now the most general set {I,Unm,Uνµ}, i.e. determining the local discrimination of orthogonal max- {I, X mZn, X µZν } with n, m, µ, ν = 0, 1, ··· , d − 1. imally entangled states. Case i. m, µ, µ − m 6= 0: we choose |xi = |0i. Case ii. µ = m = 0: we have n, ν, n−ν 6= 0 and choose d−1 j(j+1) Lemma 1 (Ghosh et al. [16]) Single copies of k(k ≤ P 2 |xi = j=0 ω |ji. d) number of pairwise orthogonal maximally entangled Case iii. µ = m 6= 0: we have n 6= v and (d) d−1 j(j+1) states |Ψnimi i (for i = 1, 2, ··· , k), taken from the P 2 a). n, ν 6= 0: we choose |xi = j=0 ω |ji. set given in Eq. (2), can be reliably discriminated by b). ν = 0: the set {I, X mZn, X m} is locally equiv- (d) LOCC if there exists at least one state |x i for which alent to {X −m, Zn,I}. (d) (d) (d) (d) (d) (d) m m ν the states Un1mi |x i,Un2m2 |x i, ··· ,Unkmk |x i are c). n = 0: the set {I, X , X Z } is locally equiv- (d) −m ν pairwise orthogonal, where Unms are given by Eq. (2). alent to {X ,I, Z }. Case iv. µ 6= m and µ = 0: we have ν 6= 0 and Although a necessary and sufficient condition has been a). n = 0: we have the set {I, X m, Zν }. referred in [20] for 1-LOCC distinguishability, it is enough b). n 6= 0 and n = ν: the set {I, X mZn, Zn} is for our objective here just applying this Lemma. Next, locally equivalent to {Z−n, X m,I}. we can directly explain the local discrimination of three c). n 6= 0 and n 6= ν: we choose |xi = d−1 j(j+1) GBSs through finding out the particular |xis as in the P 2 j=0 ω |ji. following Theorem 2. Case v. µ 6= m and m = 0: we have n 6= 0 and a). ν = 0: we have the set {I, Zn, X µ}. Theorem 2 In a bipartite quantum system of d ⊗ d with b). ν 6= 0 and ν = n: the set {I, Zn, X µZn} is d ≥ 3, any three GBSs in the form of Eq. (??) can be locally equivalent to {Z−n,I, X µ}. perfectly distinguished by 1-LOCC. c). ν 6= 0 and ν 6= n: we choose |xi = d−1 j(j+1) P 2 Proof. We shall at first prove that the three GBSs j=0 ω |ji. n µ with local unitaries {I,Un0,U0µ}, i.e., {I, Z , X } is one- Until now, we have presented effective |xis for all pos- way LOCC distinguishable for arbitrary nonzero n, µ = sible cases one by one. Therefore, any three orthogonal 0, 1, ··· , d − 1 and d ≥ 3. For this purpose, it is sufficient d ⊗ d GBSs are 1-LOCC distinguishable using every cor- to find a state |xi such that the set {|xi, Zn|xi, X µ|xi} responding |xi as is shown above. is orthogonal for given n, µ, d according to the above Lemma. d 3 Summary In case of even d and n = µ = 2 := h, we denote In this paper, we have considered the local discrimi-  h−1 d j e  P 2 nation of three mutually orthogonal GBSs in the quan- h−1  ωh |h + ji if h is odd, X  j=0 tum system of d ⊗ d, where d is any integer larger than |xi = |ji + h−1  P 2. Employing the sufficient condition of local discrimi- j=0  i|h + ji if h is even,  j=0 nation, we give a very simple proof that there exists no locally indistinguishable set including three GBSs. 2πi Despite of all the above research progress, we still have where ωh = e h , and dj/2e is the smallest integer that is not smaller than j/2. It is straightforward to check that no idea about the local (in)discrimination of more gen- the set {|xi, Zh|xi, X h|xi} is mutually orthogonal. Let eral maximally entangled states. However, for the reason that all the GBSs construct a specific basis, we believe us denote xj = hj|xi for j = 0, 1, ··· , d − 1 and |xj| = 1 by definition, showing that hx|Zh|xi = 0. By noting the result we have derived here will assist us to go ahead −d j e stably for more findings as well as further understanding x x = ω 2 , −i for j = 0, 1, ··· , h − 1 in case of odd h+j j h the properties of local operations and classical communi- m or even m respectively, we have hx|X hZµh|xi = 0 for cation. µ = 0, 1. Otherwise, in case of odd d or even d with n 6= h or References µ 6= h, we choose [1] C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. d−1 X −rj(j+1) Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. |xi = ω 2 |ji, d K. Wootters, Quantum nonlocality without entangle- j=0 ment, Phys. Rev. A 59, 1070 (1999).

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