Optimal Unambiguous Discrimination And Quantum Nonlocality Without Entanglement: Locking And Unlocking By Post-Measurement Information

The phenomenon of nonlocality without entanglement(NLWE) arises in discriminating multi-party quantum separable states. Recently, it has been found that the post-measurement information about the prepared subensemble can lock or unlock NLWE in minimum-error discrimination of non-orthogonal separable states. Thus it is natrual to ask whether the availability of the post-measurement information can inﬂuence on the occurrence of NLWE even in other state-discrimination stratigies. Here, we show that the post-measurement information can be used to lock as well as unlock the occurence of NLWE in terms of optimal nambiguous discrimination. Our results can provide a useful application for hiding or sharing information based on non-orthogonal separable states.


Introduction
Quantum nonlocality is of central importance in multi-party quantum systems. A typical phenomenon of quantum nonlocality is quantum entanglement which is a useful resource for multi-party quantum communication 1 . Quantum entanglement is the correlation that cannot be shared among multiple parties using only local operations and classical communication(LOCC) [1][2][3] . However, it is also known that some nonlocal phenomena in multi-party quantum systems are still possible even in the absence of quantum entanglement.
Nonlocality without entanglement(NLWE) is another nonlocal phenomenon that arises in discriminating non-entangled states of multi-party quantum systems 4,5 . NLWE occurs when what can be achieved with global measurement in discriminating non-entangled states cannot be achieved only by LOCC. In the case of discriminating orthogonal non-entangled states, NLWE occurs when the perfect discrimination cannot be implemented by LOCC [5][6][7][8][9] . On the other hand, in the case of discriminating non-orthogonal non-entangled states, NLWE occurs when the globally optimal discriminations such as minimum-error discrimination(ME) [10][11][12][13] or optimal unambiguous discrimination(OUD) [14][15][16][17] cannot be implemented by LOCC [18][19][20][21] . We also note that some non-local phenomena without entanglement can occur in the generalized probabilistic theories beyond quantum theory 22 .
In quantum state discrimination [23][24][25][26] , orthogonal states can be perfectly discriminated, whereas non-orthogonal states cannot. However, some non-orthogonal states can be perfectly discriminated when the post-measurement information(PI) about the prepared subensemble is available 27 . Nevertheless, some non-orthogonal states cannot be perfectly discriminated even the PI about the prepared subensemble is provided [28][29][30] . Therefore, in optimal discriminations with the PI about the prepared subensemble, the NLWE phenomenon arises when the globally optimal discrimination cannot be implemented by LOCC with the help of PI. Recently, it was shown that the availability of PI can lock or unlock NLWE in terms of ME 31 , therefore it is natural to ask whether the PI affects the occurrence of NLWE in terms of state-discrimination strategies other than ME.
Here, we show that even in OUD, the availability of the PI about the prepared subensemble can affect the occurrence of NLWE. We first provide an ensemble of two-qubit product states having NLWE in terms of OUD, and show that the availability of PI about the prepared subensemble vanishes the occurrence of NLWE, therefore locking NLWE in terms of OUD by PI. We further provide another ensemble of two-qubit product states that does not have NLWE in terms of OUD, and show that NLWE in the OUD can be released when the PI about the prepared subensemble is provided. Thus unlocking NLWE in terms of OUD by PI.
This paper is organized as follows. First, we present the form of two-qubit product state ensemble to be considered. In the "Methods" Section, we review the definitions and properties with respect to OUD without and with PI and provide some useful lemmas in optimal local discrimination. As a main result of this paper, we provide a quantum state ensemble consisting of four two-qubit product states and show the occurrence of NLWE in terms of OUD. With the same ensemble, we further show that NLWE does not occur in the OUD with the PI about the prepared subensemble is available. As another main result of this paper, we provide another quantum state ensemble consisting of four two-qubit product states and show the non-occurrence of NLWE in terms of OUD. With the same ensemble, we further show that NLWE occurs in the OUD with the PI about the prepared subensemble.

Results
Throughout this paper, we only consider the situation of unambiguously discriminating four states from the quantum state ensemble, where ρ i is a 2 ⊗ 2 non-entangled pure state, and {|ϕ i } i∈Λ is a product basis of H . Each η i is the probability that the state ρ i is prepared.
The ensemble E can be seen as an ensemble consisting of two subensembles, where E 0 and E 1 are prepared with probabilities ∑ j∈A 0 η j and ∑ j∈A 1 η j , respectively. The definitions and properties related to OUD of E without and with PI are provided in the "Methods" Section.

Locking NLWE by PI in OUD
In this section, we consider a situation where PI about the prepared subensemble E b locks NLWE in terms of OUD. We first provide a specific example of a state ensemble E and show that NLWE in terms of OUD occurs. With the same ensemble, we further show that the occurrence of NLWE in terms of OUD can be vanished when PI is provided, thus locking NLWE by PI.
To show the occurrence of NLWE in terms of OUD about the ensemble E in Example 1, we first evaluate the optimal success probability p G (E ) defined in Eq. (42) of the "Methods" Section. The reciprocal vectors {|φ i } i∈Λ corresponding to {|ϕ i } i∈Λ defined in Eq. (4) are where

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We can easily verify that the following {M i } i∈Λ is an unambiguous POVM satisfying the error-free condition in Eq. (41): Also, it is optimal because Condition (43) holds for this unambiguous POVM along with a positive-semidefinite operator Thus, the optimality of the POVM {M i } i∈Λ in Eq. (8) and the definition of p G (E ) lead us to In order to obtain the maximum success probability p L (E ) defined in Eq. (45) of the "Methods" Section, we consider lower and upper bounds of p L (E ). A lower bound of p L (E ) can be obtained from the following unambiguous measurement {M i } i∈Λ , which can be implemented by finite-round LOCC because it can be realized by performing local measurements {|0 0|, |1 1|} and {|+ +|, |− −|} on first and second subsystems, respectively. As we can easily verify that the success probability for the unambiguous LOCC measurement in Eq. (11) is 1 2(1+γ) , the success probability is obviously a lower bound of p L (E ), To obtain an upper bound of p L (E ), let us consider a positive-semidefinite operator with Lemma 1 in the "Methods" Section leads us to Inequalities (12) and (15) imply From Eqs. (10) and (16), we note that there exists a nonzero gap between p G (E ) and p L (E ), thus NLWE occurs in terms of OUD in discriminating the states of the ensemble E in Example 1. Now, we show that the availability of PI about the prepared subensemble vanishes the occurrence of NLWE in Inequality (17). To show it, we use the fact that the states of E in Example 1 can be unambiguously discriminated without inconclusive results using LOCC when the PI about the prepared subensemble is available 31 , or equivalently, From the definitions of p PI L (E ) and p PI G (E ), we note that As both p PI G (E ) and p PI L (E ) are bound above by 1, we have Thus, NLWE does not occur in terms OUD in discriminating the states of the ensemble E in Example 1 when the PI about the prepared subensemble is available. Inequality (17) shows that NLWE occurs in terms of OUD about the ensemble E in Example 1, whereas Eq. (20) shows that NLWE does not occur when PI is available. Figure 1
To show the non-occurrence of NLWE in terms of OUD about the ensemble E in Example 2, we first evaluate the optimal success probability p G (E ) defined in Eq. (42) of the "Methods" Section. Since the reciprocal vectors {|φ i } i∈Λ corresponding to {|ϕ i } i∈Λ defined in Eq. (21) are the following POVM {M i } i∈Λ satisfies the error-free condition in Eq. (41),

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Moreover, the unambiguous measurement is optimal because Condition (43) holds for this unambiguous POVM along with the following positive-semidefinite operator Thus, the optimality of the POVM {M i } i∈Λ in Eq. (24) and the definition of p G (E ) lead us to The POVM given in Eq. (24) can be perforemed using finite-round LOCC; two local measurements {|+ +|, |− −|} and {|0 0|, |1 1|} are perforemed on first and second subsystems, respectively. Thus, the success probability for the unambiguous LOCC measurement in Eq. (24) is a lower bound of p L (E ) defined in Eq. (45), therefore Moreover, from the definition of p G (E ) and p L (E ) in Eqs. (42) and (45), respectively, we have Inequalities (27) and (28) lead us to Thus, NLWE does not occur in terms of OUD in discriminating the states of the ensemble E in Example 2. Now, we show that NLWE in terms of OUD occurs when the PI about the prepared subensemble is available. To show it, we use the fact that the states of E in Example 2 can be unambiguously discriminated without inconclusive results when the PI about the prepared subensemble is available 31 , or equivalently, As p PI G (E ) is bound above by 1, we have where The POVM given in Eq. (32) is unambiguous because it satisfies the error-free condition in Eq. (50). Moreover, this measurement can be performed with finite-round LOCC; we first measure {|ν + ν + |, |ν − ν − |} on first subsystem, and then measure {|+ +|, |− −|} or {|0 0|, |1 1|} on second subsystem depending on the first measurement result |ν + ν + | or |ν − ν − |. As we can verify from a straightforward calculation that the success probability for the unambiguous LOCC measurement in Eq. (32) is thus the definition of p PI L (E ) lead us to We also note that the measurement in Eq.
In order to obtain an upper bound of p PI L (E ), let us consider the assumption of Lemma 2 in the "Methods" Section. For each (ω 0 , ω 1 ) ∈ A 0 × A 1 , there does not exist any nonzero product vector |v = |a ⊗ |b satisfying Condition (57); otherwise, |a is not orthogonal to both |0 and |+ . At the same time, |b is orthogonal to the |k 's with k ∈ Λ \ {ω 0 , ω 1 }, which leads us a contradiction. Thus, the guessing probability of E is also an upper bound of p PI L (E ) due to Lemma 2 in the "Methods" Section, that is, Inequalities (35) and (37) imply From Eqs. (31) and (38), we note that there exists a nonzero gap between p PI G (E ) and p PI L (E ), Thus, NLWE occurs in terms of OUD when the PI about the prepared subensemble is available. Equation (29) shows that NLWE in terms of OUD does not occur in discriminating the states of the ensemble E in Example 2, whereas Inequality (39) shows that NLWE occurs when PI is available. Figure 2 illustrates the relative order of p G (E ), p L (E ), p PI G (E ), and p PI L (E ) for the range of 1 3 η 0 < 1 2 . Theorem 2. For OUD of the ensemble E in Example 2, the PI about the prepared subensemble unlocks NLWE.

Discussion
We have shown that the PI about the prepared subensemble can lock or unlock NLWE in terms of OUD. We have provided a quantum state ensemble consisting of four 2 ⊗ 2 non-entangled pure states (Example 1) and shown the occurrence of NLWE in 6/11 terms of OUD with respect to the ensemble. With the same state ensemble, we have further shown that the availability of PI about the prepared subensemble vanishes the occurrence of NLWE, thus locking NLWE in terms of OUD by PI (Theorem 1). Moreover, we have provided another quantum state ensemble consisting of four 2 ⊗ 2 non-entangled pure states (Example 2) and shown the non-occurrence of NLWE in terms of OUD with respect to the ensemble. With the same state ensemble, we have further shown the occurrence of NLWE in the OUD with the PI about the prepared subensemble, thus unlocking NLWE in terms of OUD by PI (Theorem 2).
We remark that the two state ensembles of this paper can also be used to demonstrate locking and unlocking NLWE in terms of ME 31 . Thus, it is a natural future work to investigate locking and unlocking NLWE even in generalized state discrimination strategies such as an optimal discrimination with a fixed rate of inconclusive results [32][33][34][35][36] .
Our results can also provide us with a useful application in quantum cryptography. Whereas the existing quantum data hiding and secret sharing schemes are based on orthogonal states 37-41 , our results can extend those schemes to improved ones using non-orthogonal states. In Example 1, the availability of the PI about the prepared subensemble makes the globally hidden information accessible locally. On the other hand, in Example 2, the PI makes locally accessible information hidden locally but accessible globally. Finally, it is an interesting task to investigate if locking or unlocking NLWE by the PI about the prepared subensemble can depend on nonzero prior probabilities.

Methods
In two-qubit (or 2 ⊗ 2) systems, a state and a measurement are expressed by a density operator and a positive operator valued measure(POVM), respectively, acting on a two-party complex Hilbert space C 2 ⊗ C 2 . A density operator ρ is a positivesemidefinite operator ρ 0 with unit trace Trρ = 1 and a POVM {M i } i is a set of positive-semidefinite operators M i 0 satisfying ∑ i M i = ✶, where ✶ is the identity operator on C 2 ⊗ C 2 . The probability of obtaining the measurement outcome corresponding to M i is Tr(ρM i ) when {M i } i is performed on a quantum system prepared with ρ.
A positive-semidefinite operator is called separable(or non-entangled) if it is a sum of positive-semidefinite product operators; otherwise, it is said to be entangled. Also, a POVM is called separable if all elements are separable. In particular, a LOCC measurement that can be realized by LOCC is a separable measurement 2 .

Optimal unambiguous discrimination
Let us consider the unambiguous discrimination of the states in E of Eq. (1) using a measurement {M i } i∈Λ , where For each i ∈ Λ, M i is to detect ρ i , and M ? gives inconclusive results: "I don't know what state is prepared." The measurement {M i } i∈Λ can be expressed as where {s i } i∈Λ is a non-negative number set and {|φ i } i∈Λ is the set of reciprocal vectors corresponding to {|ϕ i } i∈Λ in Eq. (2) such that ϕ i |φ j = δ i j 42 . We say a POVM {M i } i∈Λ is unambiguous if it satisfies the error-free condition in Eq. (41). OUD of E is to minimize the probability of obtaining inconclusive results. Equivalently, OUD of E is to maximize the average probability of unambiguously discriminating states in E ; where the maximum is taken over all possible unambiguous POVMs satisfying the error-free condition in Eq. (41). It is known that an unambiguous POVM {M i } i∈Λ is optimal if and only if there is a positive-semidefinite operator K satisfying the following condition 21,[43][44][45] , In this case, we have if an unambiguous POVM {M i } i∈Λ and a positive-semidefinite operator K satisfy Condition (43) In the following lemma, we provide an upper bound of p L (E ).

Lemma 1. If H is a positive-semidefinite operator satisfying
for all reciprocal vectors |φ i that is a product vector, then TrH is an upper bound of p L (E ).
Proof. Let us suppose that {M i } i∈Λ is an unambiguous LOCC measurement and χ is the set of all i ∈ Λ such that |φ i is a product vector. Since every LOCC measurement is separable, M i is separable for all i ∈ Λ. For all i ∈ Λ with i / ∈ χ, M i = 0 because M i is proportional to entangled |φ i φ i |. Thus, the success probability is where the inequality is due to the assumption of Inequality (47) For each (ω 0 , ω 1 ) ∈ Ω, M (ω 0 ,ω 1 ) detects a state in E unambiguously or gives inconclusive results depending on PI b ∈ {0, 1}. If ω b =?, the state ρ ω b is detected unambiguously, that is, the POVM {M ω } ω∈Ω satisfies However, if ω b =?, inconclusive results are obtained. We say that a POVM {M ω } ω∈Ω is unambiguous if it satisfies the error-free condition in Eq. (50).
OUD of E with PI is to minimize the probability of obtaining inconclusive results. Equivalently, OUD of E with PI is to maximize the average probability of unambiguously discriminating states where the optimal success probability is defined as

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We note that p PI L (E ) in Eq. (52) can also be rewritten as whereη Since the states of E are non-entangled, NLWE occurs in terms of OUD with PI if and only if OUD of E with PI cannot be achieved using LOCC, that is, For an upper bound of p PI L (E ), let us consider the following quantity, which is the maximum average probability of correct guessing the prepared state when the available measurements are limited to LOCC measurements without inconclusive results [10][11][12][13] . The following lemma shows that p guess (E ) can be used as an upper bound of p PI L (E ). Lemma 2. For each (ω 0 , ω 1 ) ∈ A 0 × A 1 , if there is no nonzero product vector |v satisfying then p guess (E ) is an upper bound of p PI L (E ). Proof. The assumption in (57) implies that for each ω = (ω 0 , ω 1 )∈A 0 ×A 1 , there does not exist any nonzero separable M ω 0 that unambiguously detects the state ρ ω 0 or ρ ω 1 depending on PI b = 0 or 1, respectively. Then, the term ∑ ω∈A 0 ×A 1η ω Tr(ρ ω M ω ) in Eq. (52) disappears. Thus, we have where the inequality is from the fact that p guess (E ) is the maximum obtained from measurements without any constraint, whereas p PI L (E ) is the maximum obtained from unambiguous LOCC measurements.