LOCC indistinguishable orthogonal product quantum states

We construct two families of orthogonal product quantum states that cannot be exactly distinguished by local operation and classical communication (LOCC) in the quantum system of 2k+i ⊗ 2l+j (i, j ∈ {0, 1} and i ≥ j ) and 3k+i ⊗ 3l+j (i, j ∈ {0, 1, 2}). And we also give the tiling structure of these two families of quantum product states where the quantum states are unextendible in the first family but are extendible in the second family. Our construction in the quantum system of 3k+i ⊗ 3l+j is more generalized than the other construction such as Wang et al.’s construction and Zhang et al.’s construction, because it contains the quantum system of not only 2k ⊗ 2l and 2k+1 ⊗ 2l but also 2k ⊗ 2l+1 and 2k+1 ⊗ 2l+1. We calculate the non-commutativity to quantify the quantumness of a quantum ensemble for judging the local indistinguishability. We give a general method to judge the indistinguishability of orthogonal product states for our two constructions in this paper. We also extend the dimension of the quantum system of 2k ⊗ 2l in Wang et al.’s paper. Our work is a necessary complement to understand the phenomenon of quantum nonlocality without entanglement.

In quantum cryptography, quantum entangled states are easily distinguished by performing global operation if and only if they are orthogonal. Entanglement has good effects in some cases, but it has bad effects in other cases such as entanglement increases the difficulty of distinguishing quantum states when only LOCC is performed 1 . When many global operations cannot be performed, LOCC becomes very useful. The phenomenon of quantum nonlocality without entanglement 2 is that a set of orthogonal states in a composite quantum system cannot be reliably distinguished by LOCC. The study of quantum nonlocality is one of the fundamental problems in quantum information theory. LOCC is usually used to verify whether quantum states are perfectly distinguished  or not. In refs 3-12, they focus on the local distinguishability of quantum states such as multipartite orthogonal product states can be exactly distinguished 10 or how to distinguish two quantum pure states 11,12 . Moreover, locally indistinguishability [13][14][15][16][17][18][19][20][21][22][23] of quantum orthogonal product states plays an important role in exploring quantum nonlocality.
The nonlocality problem is considered in the bipartite setting case that Alice and Bob share a quantum system which is prepared in an known set contained some mutually orthogonal quantum states. Their aim is to distinguish the states only by LOCC. Bennett et al. 2 proposed a set of nine pure orthogonal product states in quantum system of C 3 ⊗ C 3 in 1999, which cannot be exactly distinguished by LOCC. In 2002, Walgate et al. 16 also proved the indistinguishability of the nine states by using a more simple method. Zhang et al. 19 extended the dimension of quantum system in Walgate et al.'s 16 . Yu and Oh 22 give another equivalent method to prove the indistinguishability and this method is used to distinguish orthogonal quantum product states of Zhang et al. 21 . Furthermore, Wang et al. 20 constructed orthogonal product quantum states under three quantum system cases of C 2k ⊗ C 2l , C 2k ⊗ C 2l+1 and C 2k+1 ⊗ C 2l+1 . The smallest dimension of C 2k ⊗ C 2l can be constructed is C 6 ⊗ C 6 in Wang et al.'s paper 20 . However, the smallest dimension of C 2k ⊗ C 2l can be constructed is C 4 ⊗ C 4 in our paper. Ma et al. 24 revealed and established the relationship between the non-commutativity and the indistinguishability. By calculating the non-commutativity, the quantumness of a quantum ensemble can be quantified for judging the indistinguishability of a family of orthogonal product basis quantum states. For the orthogonal product states, we firstly use a method to judge the indistinguishability of the set, the proof is meaningful. In this paper, we calculate the non-commutativity to judge the indistinguishability if and only if there exists a set to satisfy the inequality of Lemma 2.
In this paper, we construct two families of orthogonal product quantum states in quantum systems of C 2k+i ⊗ C 2l+j with i, j ∈ {0, 1} (i ≥ j) and C 3k+i ⊗ C 3l+j with i, j ∈ {0, 1, 2} and the two families of orthogonal product Figure 1. The tiling structure of orthogonal product quantum states in quantum system of (a) C 2k ⊗ C 2l with k, l ≥ 2 and (b) C 2k+1 ⊗ C 2l+1 with k, l ≥ 1.
Scientific RepoRts | 6:28864 | DOI: 10.1038/srep28864   Figure 2. The tiling structure of orthogonal product quantum states in quantum system of (a) C 4 ⊗ C 4 , Scientific RepoRts | 6:28864 | DOI: 10.1038/srep28864 2, 3 and   0, 2  1, 2 , , 2  1,  2  2;   3, 5, , 2  3, 2  1, 2 , 2  1, 2  1,  2  1;   2  1,  0 , 2, 4, ,  {2  2 Proof. We only discuss the case of Alice measures firstly and the same as Bob. We consider the subspace 2 1 } A can be expressed as follows The necessary and sufficiency of Lemma 1 has already been proved by Walgate in ref. 16. Now we apply the necessary and sufficiency of Lemma 1 to verify a 00 = a 11 and a 10 = a 01 = 0 in the subspace {|0〉 , |1〉 } A . Suppose, the form However, there also exist quantum states |0 ± 1〉 A in the subspace {|0〉 , |1〉 } A . The reduction to absurdity is used to verify the correctness of the conclusion. Suppose there exists one POVM element that is not proportional to identity to distinguish these quantum states, the express of the POVM element is as follows holds. It produces contradiction between results and assumption. So it does not exist a non-trivial measurement to distinguish the orthogonal product quantum states. For the other subspaces, we have the same conclusions. After Alice performs a general measurement, the effect of this positive operator upon the following states , where 3and 0, 0, , where 3 and 0, is entirely specified by those elements a 00 , a 11, a 01, a 10 draw from the {|0〉 , |1〉 } A subspace. It means that Alice cannot perform a nontrivial measurement upon the {|0〉 , |1〉 } A subspace. Thus, the corresponding submatrix must be proportional to the identity. Then, we obtain a 00 = a 11 = a, a 01 = a 10 = 0. For the states Scientific RepoRts | 6:28864 | DOI: 10.1038/srep28864 and the subspace {|1〉 , |2〉 } A , we make the same argument. Then we get the result a 11 = a 22 = a, a 12 = a 21 = 0. For the states   where a is a real number. We now consider the states ψ . For the states , therefore a = 0. It produces contradictory with the theorem of Walgate 16 . So, † M M m m is proportional to the identity and the 4kl orthogonal product states are indistinguishable. ☐ Example 1. Now we will give 16 orthogonal product quantum states in quantum system of C 4 ⊗ C 4 (see Fig. 2(a)).
Case 2. Secondly, we construct LOCC indistinguishable orthogonal product quantum states in quantum system of C 2k+1 ⊗ C 2l+1 with k, l ≥ 1 and l ≤ l (see Fig. 1(b)) and also give an example in the smallest dimension (see Fig. 2(b)).
Here we just give the construction for k ≤ l. When k > l, it should be rotated along the clockwise direction for Fig. 1(b) to get the construction.
Example 2. Now we will give 9 orthogonal product quantum states in quantum system of C 3 ⊗ C 3 (see Fig. 2 where Case 3. Thirdly, we consider the indistinguishable orthogonal product states in quantum system C 2k+1 ⊗ C 2l with k ≥ 2, l ≥ 3 (see Fig. 3) and give an example in the smallest dimension (see Fig. 2(c)).   Example 3. Now we will give 30 orthogonal product quantum states in quantum system of C 5 ⊗ C 6 (see Fig. 3(c)). where | ± 〉 = | 〉 ± | 〉 i j i j ( ) 1 2 with 0 ≤ i ≤ 4 and 0 ≤ j ≤ 5.
LOCC indistinguishable orthogonal product quantum states in quantum system of C 3k+i ⊗ C 3k+j with i, j ∈ {0, 1, 2}. We give LOCC indistinguishable orthogonal product quantum states in quantum system v v    Figure 3. The tiling structure of orthogonal product quantum states in quantum system of C 2k+1 ⊗ C 2l with k ≥ 2, l ≥ 3.

Discussion
The orthogonal product quantum states constructed by us are indistinguishable by performing local operation and classical communication, but not separable operations 25 . Now, we discuss whether the separable operations can distinguish these product quantum states or not.
Similar to Zhang et al.'s paper 19 , the multipartite quantum systems can be constructed when m = n = d. Such as in the quantum system 1, 1, 2, , 2 and ϕ | 〉 i AB in Eqs (1, 12) of C 2k ⊗ C 2l and C 2k+1 ⊗ C 2l+1 . However, Wang et al.'s construction 20 cannot be extended into multipartite quantum systems because the set of orthogonal product states is extendible.
LOCC indistinguishable orthogonal product quantum states in quantum system of C 3k+i ⊗ C 3l+j with i, j ∈ {0, 1, 2}. Similar to the first construction, the second construction is extendible and distinguished by separable operations. Firstly, these states in Eqs (17)(18)(19)(20)(21) all can be extended to mn orthogonal product states. Then, the proof of the process is the same as above. Finally, we construct the 3(m + n) − 9 quantum states respectively in Eqs (17)(18)(19)(20)(21) that can be distinguished by the separable operations.

Methods
In ref. 16, Walgate et al. gave a necessary and sufficient condition to prove the local indistinguishability of a set of orthogonal product states. If a quantum system which is a qubit does not exist, a uniform conclusion cannot be drawn yet. In all LOCC protocols, there must be a party to leave. Lemma 1 16 . Alice and Bob share a C 2 ⊗ C n dimensional quantum system: Alice has a qubit, and Bob has an n dimensional system that may be entangled with that qubit. If Alice goes first, a set of orthogonal states {|ϕ i 〉 } is exactly locally distinguishable if and only if there is a basis {|0〉 , |1〉 } A such that in that basis The Lemma 1 is used to prove the indistinguishability of C 2k+i ⊗ C 2l+j quantum system with i, j ∈ {0, 1} (i ≤ j) and C 3k+i ⊗ C 3l+j quantum system with i, j ∈ {0, 1, 2} in Results.
be a set of operators. The total non-commutativity for this set is defined  The quantity non-commutativity is used to quantify the quantumness of a quantum ensemble for judging the indistinguishability.
Here, we use the simply method in Lemma 2 to judge the indistinguishability of orthogonal product states in 24 by calculating the non-commutativity N. The orthogonal product quantum states in Eqs (1,13,15) are complete. Such as the set of complete orthogonal product states in Eq. (1), we give the briefly process. Firstly, we give the sets of ε A and ε B .