Quantum nonlocality without entanglement depending on nonzero prior probabilities in optimal unambiguous discrimination

Nonlocality without entanglement(NLWE) is a nonlocal phenomenon that occurs in quantum state discrimination of multipartite separable states. In the discrimination of orthogonal separable states, the term NLWE is used when the quantum states cannot be discriminated perfectly by local operations and classical communication. In this case, the occurrence of NLWE is independent of nonzero prior probabilities of quantum states being prepared. Recently, it has been found that the occurrence of NLWE can depend on nonzero prior probabilities in minimum-error discrimination of nonorthogonal separable states. Here, we show that even in optimal unambiguous discrimination, the occurrence of NLWE can depend on nonzero prior probabilities. We further show that NLWE can occur regardless of nonzero prior probabilities, even if only one state can be locally discriminated without error. Our results provide new insights into classifying sets of multipartite quantum states in terms of quantum state discrimination.

www.nature.com/scientificreports/ multipartite states is not in general. A typical example is "indistinguishable product basis" presented by Duan et al. 44 , in which all states cannot be unambiguously discriminated using LOCC. We also note that NLWE in terms of optimal UD may occur regardless of nonzero prior probabilities. However, it is still a natural question whether there is a set of separable states that cause NLWE in terms of optimal UD depending on nonzero prior probabilities.
In this work, we consider a complete product basis, in which only one state can be discriminated unambiguously using LOCC, and establish a necessary and sufficient condition that the occurrence of NLWE in terms of optimal UD depends on nonzero prior probabilities (Corollary 2). We provide an example of six 2 ⊗ 3 product states where NLWE in terms of optimal UD occurs depending on nonzero prior probabilities (Example 1). We also provide an example that is not indistinguishable product basis but causes NLWE in terms of optimal UD regardless of nonzero prior probabilities (Example 2). We further investigate the case where optimal UD measurement detects only one state unambiguously (Theorem 1), and obtain a necessary and sufficient condition for optimal UD measurement to detect two or more states unambiguously, regardless of nonzero prior probabilities (Corollary 1).

Results
Optimal unambiguous discrimination. Let {|ψ 1 �, . . . , |ψ n �} be a set of n linearly independent pure states and η i be the corresponding non-zero probability of |ψ i � being prepared for i = 1, . . . , n . The UD of {η i , |ψ i �} n i=1 is to discriminate the prepared state without error using a positive operator-valued measure(POVM) that consists of n + 1 positive semidefinite operators M 0 , M 1 , . . . , M n on H S satisfying n i=0 M i = 1 . Here H S is the subspace spanned by {|ψ 1 �, . . . , |ψ n �} and 1 is the identity operator on H S . M 0 provides inconclusive results for the prepared state and M i (i = 1, . . . , n) gives conclusive results for |ψ i � unambiguously.
The no-error condition for where p i is the probability of success in unambiguously discriminating |ψ i � and |ψ i � is the reciprocal vector defined by where −1 is the inverse matrix of n-by-n Gram matrix with (�) ij = �ψ i |ψ j � . We note that �ψ i |ψ j � = (� −1 ) ij . Because {|ψ 1 �, . . . , |ψ n �} is a set of linearly independent vectors in H S satisfying �ψ i |ψ j � = δ ij , selecting a UD measurement can be understood as selecting a point � p = (p 1 , . . . , p n ) T ∈ R n such that The optimization task of UD is to maximize the average probability of success, whose maximum is denoted by P max . The optimal p that gives P max is unique 50 , which can also be verified from optimality condition including Lagrangian stability; see Lemma 1 in the "Methods" Section. Some components of optimal p may be zero. In other words, the optimal UD measurement of {η i , |ψ i �} n i=1 may not detect all given quantum states unambiguously. If p i is zero, the UD measurement does not detect |ψ i � unambiguously. The following theorem shows a necessary and sufficient condition for the optimal UD measurement of {η i , |ψ i �} n i=1 to detect only |ψ 1 � unambiguously. The proof of Theorem 1 is provided in the "Methods" Section.
, the optimal measurement detects only |ψ 1 � unambiguously if and only if In the case of n = 2 41 , it is easy to see that the condition η 2 /η 1 |�ψ 1 |ψ 2 �| 2 is equivalent that the optimal p 2 is zero. As we can check in the proof of Theorem 1 in "Methods" Section, the choice of |ψ 1 � in Theorem 1 can be arbitrary. That is, any of {|ψ 1 �, . . . , |ψ n �} can be used to play the role of |ψ 1 � in Theorem 1.

Corollary 1 For any set of nonzero prior probabilities {η
, the optimal UD measurement of {η i , |ψ i �} n i=1 detects more than one state unambiguously if and only if, for any j, there is k satisfying �ψ j |ψ k � = 0.
Proof For fixed j, let us first suppose that there exists k satisfying �ψ j |ψ k � = 0 , which implies Scientific Reports | (2021) 11:17695 | https://doi.org/10.1038/s41598-021-97103-y www.nature.com/scientificreports/ because the left-hand side is positive and the right-hand side is zero. Thus, from Theorem 1, the optimal UD measurement of {η i , |ψ i �} n i=1 cannot be achieved by detecting only |ψ j � unambiguously. Since this argument is true for any |ψ j � , the optimal UD measurement must detect more than one state unambiguously. Now, let us suppose that �ψ j |ψ k � � = 0 for all k. We can see that with nonzero prior probabilities defined as Thus, from Theorem 1, the optimal UD measurement of {η i , |ψ i �} n i=1 detects only |ψ j � unambiguously.
For example, let us consider four pure states with the reciprocal vectors Since �ψ 1 |ψ 2 � and �ψ 3 |ψ 4 � are zero, for any i, there exists j satisfying �ψ i |ψ j � = 0 . Therefore, for any set of nonzero prior probabilities, the optimal UD measurement detects more than one state unambiguously.

Multipartite state discrimination.
Let H be a multipartite Hilbert space and {| i �} n i=1 a set of n linearly independent pure states in H , where each | i � is prepared with nonzero prior probability ξ i . We use | i � , |˜ i � , and P Global max instead of |ψ i � , |ψ i � , and P max to highlight the multipartite quantum system to be considered. Also, we use P LOCC max to denote the maximum of average success probability that can be obtained in UD restricted to asymptotic LOCC, whereas P Global max is the maximum of average success probability that can be obtained using global measurement. If there exists a product state | � that satisfies � | 1 � � = 0 and � | i � = 0 ∀i � = 1 , | 1 � can be unambiguously discriminated using finite-round LOCC, otherwise the state cannot be unambiguously discriminated even using SEP [43][44][45] . In particular, when {| i �} n i=1 is a basis of H (that is, H S = H ), the existence of a product state | � satisfying � | 1 � � = 0 and � | i � = 0 ∀i � = 1 is determined by the corresponding reciprocal vector |˜ 1 � 44 . To be precise, | 1 � can be unambiguously discriminated using LOCC if and only if |˜ 1 � is a product vector. Note that when H S = H , even if |˜ 1 � is entangled, a product state that detects only | 1 � may exist in H.
In the case of n = 2 32,46-48 , the globally optimal UD is always possible only with finite-round LOCC no matter what two pure states with arbitrary prior probabilities are given, i.e., P LOCC max = P Global max . However, in the case of n > 2 , P LOCC max can be less than P Global max . An example is the indistinguishable product basis 44 in which no state can be unambiguously discriminated using asymptotic LOCC, i.e., P LOCC max = 0 . It should be noted that P LOCC max = 0 means P LOCC max < P Global max because P Global max is always nonzero for any linearly independent pure states. Moreover, this example illustrates the case that NLWE in terms of optimal UD occurs regardless of nonzero prior probabilities.
The following theorem shows that if {| i �} n i=1 is a basis of H and only | 1 � can be unambiguously discriminated using LOCC, P LOCC max = P Global max is determined by whether the optimal UD measurement requires unambiguous detection other than | 1 � . The proof of Theorem 2 is provided in the "Methods" Section. Theorem 2 When H S = H and only |˜ 1 � is a product vector, the globally optimal UD of {ξ i , |� i �} n i=1 can be achieved using LOCC if and only if Theorem 2 tells us that the possibility of globally optimal UD using LOCC can depend on nonzero prior probabilities. In other words, when |� 1 �, . . . , |� n � are fixed, both P LOCC max = P Global max and P LOCC max < P Global max can occur according to ξ 1 , . . . , ξ n . As we can check in the proof of Theorem 2 in "Methods" Section, the choice of |˜ 1 � in Theorem 2 can be arbitrary. That is, any of {|� 1 �, . . . , |� n �} can be used to play the role of |˜ 1 � in Theorem 2.

Corollary 2
For the optimal UD of n product states |� 1 �, . . . , |� n � such that H S = H and only |˜ 1 � is a product vector, if �˜ i |˜ 1 � is nonzero for all i, NLWE occurs depending on the nonzero prior probabilities. Otherwise NLWE occurs for any nonzero prior probabilities.
Proof If �˜ i |˜ 1 � � = 0 for all i, satisfying Inequality (11) for all i depends on nonzero prior probabilities; Inequality (11) for each i holds by the nonzero prior probabilities defined as It follows from Theorem 2 that the possibility of the globally optimal UD using LOCC depends on nonzero prior probabilities. Thus, NLWE occurs depending on the nonzero prior probabilities.
However, if �˜ j |˜ 1 � = 0 for some j, no set of nonzero prior probabilities satisfies Inequality (11) for all i. It follows from Theorem 2 that for any set of nonzero prior probabilities cannot be achieved using LOCC. Therefore, NLWE occurs regardless of nonzero prior probabilities.
According to Theorem 2, we can achieve the globally optimal UD between six states of (14) using LOCC if and only if Furthermore, since the given six states are product vectors, we can see from Corollary 2 that the occurrence of NLWE in terms of optimal UD depends on nonzero prior probabilities. Figure 1 illustrates two cases classified by the occurrence of NLWE when considering nonzero prior probabilities determined by two variables a ∈ (0, 1) and b ∈ (0, 1) as follows:    www.nature.com/scientificreports/ Also, only |˜ 1 � is a product vector because the reduced nonnegative operator of |˜ i ��˜ i | is not rank one for i = 1 . Since �˜ 1 |˜ 2 � is zero, according to Corollary 2, no matter what nonzero prior probabilities ξ 1 , . . . , ξ 6 are given, the globally optimal UD of {ξ i , |� i �} 6 i=1 cannot be achieved using LOCC. This example that is not an indistinguishable product basis 44 shows that NLWE can occur regardless of nonzero prior probabilities. We also note that, in two qubits, there is no basis consisting of four product states that satisfy the assumption of Theorem 2; see "Methods" Section.

Discussion
We have shown that NLWE in terms of optimal UD can depend on the nonzero prior probabilities of quantum states being prepared. We have first established a necessary and sufficient condition for optimal UD measurement of linearly independent pure states to detect only one specific state unambiguously (Theorem 1). We have also provided a necessary and sufficient condition for optimal UD measurement to detect two or more states unambiguously, regardless of nonzero prior probabilities (Corollary 1). By generalizing Theorem 1 to multipartite state discrimination, we have established a necessary and sufficient condition for globally optimal UD of linearly independent pure states using LOCC provided that only one state can be discriminated unambiguously (Theorem 2). From Theorem 2, we have provided a sufficient condition that NLWE occurs depending on nonzero prior probabilities, as well as a sufficient condition that NLWE occurs regardless of nonzero prior probabilities (Corollary 2). Finally, we have illustrated the occurrence of NLWE in Corollary 2 by providing Examples 1 and 2.
Our result means that sets of linearly independent product states can be classified into three types in terms of optimal UD: Type I where NLWE does not occur regardless of nonzero prior probabilities (e.g., two multipartite pure states 6,32,46-48 ), Type II where NLWE occurs regardless of nonzero prior probabilities (e.g., domino states 5 , indistinguishable product basis 44 , and Example 2), and Type III where NLWE occurs depending on nonzero prior probabilities (e.g. Example 1). Surprisingly, Type III is a new type that only occurs in nonorthogonal states.
The quantum states of Type III can be useful for possible construction of cryptographical schemes such as quantum data hiding [51][52][53][54] or quantum secret sharing 55,56 with nonorthogonal states. In most quantum datahiding schemes, classical bits are hidden by orthogonal quantum states of Type II which cannot be perfectly (20) |� 1 � = 5|00�,  www.nature.com/scientificreports/ discriminated using LOCC regardless of nonzero prior probabilities. In many quantum secret sharing schemes, classical bits are shared among parties using orthogonal quantum states of Type I that can be perfectly discriminated using LOCC regardless of nonzero prior probabilities. By using nonorthogonal quantum states of Type III, an unified cryptographic scheme can be constructed that can be both data-hiding and secret-sharing scheme depending on nonzero prior probabilities. We believe this is an interesting challenge as a future work. Also, we have provided a new example of Type II (Example 2). Interestingly, this example is different from the domino states 5 , which can be unambiguously discriminated using LOCC. Moreover, this example is also different from the indistinguishable product basis 44 , which cannot be unambiguously discriminated using LOCC. In Example 2, one state can be unambiguously discriminated using LOCC, but all other states cannot. This suggests that Type II can be subdivided into more inequivalent classes.

Methods
Semidefinite programming to optimal UD of linearly independent pure states. The optimal UD of {η i , |ψ i �} n i=1 can be expressed as where Equation (21) is called a primal problem. M p is said to be feasible if it safisfies the two constraints of (21), and strictly feasible if both M p and 1 − M � p are positive-definite. When constructing the Lagrangian 57 as We can derive the dual problem 58,59 of (21): where E and K are two Hermitian operators on H S corresponding to the Lagrange multipliers associated with M p 0 and 1 − M � p 0 . As in the primal problem, a pair (E, K) is said to be feasible if they satisfy the dual constraints of (24).
The first and second constraints of (24) are the positivity conditions of Lagrange multipliers and the last condition means the Lagrangian stability, i.e., ∂L /∂M � p = 0 . The first constraint of (24) does not mean that E is positive semidefinite. The first and last constraints of (24) can be expressed as therefore E can be omitted. When E is omitted, K is called feasbile if it is positive-semidefinite and satisfies Inequality (25) for all i. We use the superscript ⋆ to express the optimality of primal and dual variables.
The primal optimal value P max is an lower bound of the dual optimal value TrK ⋆ because for any feasible M p and (E, K). From Inequality (26) The convex optimization problem given in (21) satisfies the well-known Slater's condition 57 (i.e., the existence of strictly feasible M p ) as a sufficient condition for strong duality, so primal and dual problems provide the same optimal value, i.e., TrK ⋆ = P max . This implies that Eq. (27) minimize TrK subject to �ψ i |E|ψ i � 0 ∀i, K 0, and K = ρ + E, www.nature.com/scientificreports/ The approach to finding an optimal solution using KKT condition is called the linear complementarity problem (LCP) approach 4,61 . The characteristic of LCP is that it considers both primal and dual variables, that it is neither the minimum nor the maximum problem [i.e., there is no objective function], and that the constraints are KKT condition (30).
From KKT condition (30), we can see that both 1 − M ⋆ � p and K ⋆ are positive-semidefinite, nonzero, and not full rank. M ⋆ p is also positive-semidefinite and nonzero, but it may be full rank. Note that if (M p , E, K) and (M � p ,Ē,K) satisfy KKT condition, so do (M � p ,Ē,K) and (M � p , E, K) because both M p and M � p are primal optimal and both (E, K) and (Ē,K) are dual optimal. Furthermore, the optimality condition for (M p , E, K) can be converted to one for (M p , K) by replacing the constraints for E with (25) (31), a is determined as η 1 |�ψ 1 |ψ 1 �| −2 , that is, Due to (iv) of Condition (31), the following inequality holds, which proves the necessity of our theorem.
To prove sufficiency, we assume that It is straightforward to verify that the following (M p , K) satisfies Condition (31); and Thus, (M p , K) is optimal, so and p ⋆ 2 = · · · = p ⋆ n = 0 . This completes our proof.
Proof of Theorem 2. In this section, we suppose that only |˜ 1 � is a product vector, which implies only | 1 � can be unambiguously discriminated using LOCC.
To prove the necessity, we assume that the globally optimal UD of {ξ i , |� i �} n i=1 is possible using LOCC. Thus, the globally optimal UD of {ξ i , |� i �} n i=1 is the optimally unambiguous detection of | 1 � . From Theorem 1, Inequality (11) holds for each i, which proves the necessity of our theorem.
To prove sufficiency, suppose Inequality (11) holds for each i. According to Theorem 1, the globally optimal UD measurement of {ξ i , |� i �} n i=1 detects only one state | 1 � unambiguously. Thus, the optimally unambiguous detection of | 1 � means the projective measurement Moreover, this projective measurement can be done using finite-round LOCC. To show this, let H k be each party's Hilbert space, i.e., H = m k=1 H k . Since |˜ 1 � is a product vector, its normalized form can be written as in which |φ k � is a pure state in H k . Each party performs a projective measurement {|φ k ��φ k |, 1 k − |φ k ��φ k |} , and all parties share their local measurement results through classical communication. Here 1 k is the identity operator on H k . If the measurement result is |φ k ��φ k | for all party k, they unambiguously discriminate | 1 � , otherwise it is considered as inconclusive results. Thus, we can detect only | 1 � unambiguously and optimally using only finite-round LOCC. This implies that the globally optimal UD of {ξ i , |� i �} n i=1 can be achieved using LOCC, which proves the sufficiency of our theorem.