Locally indistinguishable orthogonal product bases in arbitrary bipartite quantum system

As we know, unextendible product basis (UPB) is an incomplete basis whose members cannot be perfectly distinguished by local operations and classical communication. However, very little is known about those incomplete and locally indistinguishable product bases that are not UPBs. In this paper, we first construct a series of orthogonal product bases that are completable but not locally distinguishable in a general m ⊗ n (m ≥ 3 and n ≥ 3) quantum system. In particular, we give so far the smallest number of locally indistinguishable states of a completable orthogonal product basis in arbitrary quantum systems. Furthermore, we construct a series of small and locally indistinguishable orthogonal product bases in m ⊗ n (m ≥ 3 and n ≥ 3). All the results lead to a better understanding of the structures of locally indistinguishable product bases in arbitrary bipartite quantum system.

The following situation is often encountered in quantum cryptography 1 and quantum algorithms 2 . Suppose that Alice and Bob share a bipartite quantum system. They are told that the state they own comes from a set of orthogonal states that are known to each of them, but they are not told that which state their combined system is in. What they need to do is to identify the given state by local operations and classical communication (LOCC) since they are in two different places. It is well known that orthogonal quantum states can always be distinguished by global operations. However, this is not always true if we restrict the set of actions on the bipartite system to LOCC only. Bennett et al. 3 first constructed a set of nine orthogonal product states that cannot be perfectly distinguished by LOCC in a 3 ⊗ 3 quantum system. Their work showed the counterintuitive phenomenon of nonlocality without entanglement, i.e., entanglement is not necessary for the local indistinguishability of orthogonal quantum states. Later, a simple proof for the nonlocality of the nine product states was given by Walgate et al. 4 . Inspired by their work, many scholars are engaged in the research of the local distinguishability of orthogonal product states. With further research, numerous results [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] have been presented up to now.
Unextendible product basis (UPB) is a set of orthogonal product states that spans a subspace whose complementary subspace contains no product state. As an object with rich mathematical structure, it was introduced by Bennett et al. 21 and has been thoroughly studied in the literatures [6][7][8]17,20 . Bennett et al. presented two different UPBs, each of which has five product states in a 3 ⊗ 3 quantum system. Furthermore, they proved that a UPB cannot be perfectly distinguished by LOCC. DiVincenzo et al. 20 gave a complete characterization of unextendible product bases by orthogonality graphs and presented several generalizations of UPBs to arbitrary high dimensions and multipartite systems. Chen et al. 6 made a further research on the minimum size of unextendible product bases. On the other hand, many huge advances have been made on the orthogonal product states that cannot form a UPB. Yu et al. 22 constructed 2d − 1 orthogonal states that are locally indistinguishable in d ⊗ d (d ≥ 3) and conjectured that any set of no more than 2(d − 1) product states is locally distinguishable in a d ⊗ d (d ≥ 3) quantum system. Wang et al. 23 presented a small set with only 3(m + n) − 9 orthogonal product states and proved the local indistinguishability of these states in an m ⊗ n quantum system, where m ≥ 3 and n ≥ 3. Recently, Zhang et al. 24 constructed 3n + m − 4 locally indistinguishable orthogonal product states that do not constitute a UPB and presented a smaller set with 2n − 1 orthogonal product states that cannot be perfectly distinguished by LOCC in m ⊗ n (3 ≤ m ≤ n). All the results show it is a meaningful work to research the structure of the locally indistinguishable product basis and the smallest number of locally indistinguishable orthogonal product states in arbitrary high-dimensional quantum systems. In this paper, we construct a series of completable and locally indistinguishable orthogonal product bases, which have eight members, twelve members, ···, 4 min(m, n) − 4 members, respectively, in a general m ⊗ n (m ≥ 3 and n ≥ 3) quantum system. Our results show that Yu et al.'s conjecture 22 , i.e., any set of no more than 2(d − 1) product states is locally distinguishable in a d ⊗ d (d ≥ 3) quantum system, is not true. In fact, eight is so far the smallest number of locally indistinguishable states of a completable orthogonal product basis [22][23][24] . On the other hand, we construct a series of small and locally indistinguishable orthogonal product bases, which contain five members, seven members, ···, 2 min(m, n) − 1 members respectively, in m ⊗ n (m ≥ 3 and n ≥ 3). It should be pointed out that five is so far the smallest number of locally indistinguishable states of an orthogonal product basis by Refs. 20 and 21. These new results lead to a better understanding of the structures of locally indistinguishable product bases in arbitrary bipartite quantum systems.

Results
it is easy to see that E t is a positive semidefinite operator. We will say that the measurement is a positive operator-valued measure (POVM) and the objects E t are the POVM elements corresponding to each measurement outcome t 4 .
It is easy to see that POVM is a general measurement to a measured quantum state according to the definition 3. As we mentioned in the preceding part, LOCC denote local operations and classical communication. When it comes to identify a given state that is chosen from a known set of orthogonal states by LOCC, the local operations are local POVMs or local unitary operations. That is, if a measured state is a bipartite (or multipartite) quantum system, each party that holds one particle of the bipartite (or multipartite) quantum system can only perform POVM or unitary operations on his (or her) own particle. For simplicity, we usually say a set of orthogonal states is locally distinguishable if it can be distinguished by LOCC. Definition 4 4 . We will say that a POVM is trivial if all the POVM elements are proportional to the identity operator since such a measurement yields no information about the measured state. Any measurement not of this type will be called nontrivial.
Different from Definition 4, we give a new definition about trivial measurement here. It should be noted that we say a measurement is trivial if it satisfies our new definition.

if and only if we cannot get any useful information about the measured state that is arbitrarily selected from the set by the POVM, i.e., for each of the
Definition 6 4 . Alice goes first if Alice is the first person to perform a nontrivial measurement upon the system.
Local indistinguishability of completable orthogonal product basis. Now we construct a completable product basis with 4p − 4 members that cannot be locally distinguished in a general m ⊗ n (m ≥ 3 and n ≥ 3) quantum system and give a proof for its indistinguishability. theorem 1. In an m ⊗ n quantum system, the 4p − 4 orthogonal product states Scientific RepoRts | 6:31048 | DOI: 10.1038/srep31048 cannot be perfectly distinguished by LOCC, where m ≥ 3, n ≥ 3, p is an arbitrary integer from 3 to min(m, n), j = i + 1 when i = 1, ···, p − 2 and j = 1 while i = p − 1.
Proof. Many proof techniques are borrowed from Ref. 22. To distinguish these states, one of the two parties (Alice and Bob) has to start with a nondisturbing measurement, i.e., the postmeasurement states should be mutually orthogonal. Without loss of generality, suppose that Alice goes first with a set of general m × m POVM elements The post measurement states {M t ⊗ I B |ψ i 〉: i = 1, ···, 4p − 4} should be mutually orthogonal. For the states |ψ i 〉 and |ψ j 〉 , where ( according to the completeness of the measurement operators. This means that any state of the 4p − 4 states can lead to the outcome that is corresponding to M t with the same probability a t 00 , i.e., the measurement {M t ⊗ I B | t = 1, 2, ···, l} is trivial to the 4p − 4 states. In other words, Alice cannot get any information about which the measured state will be by the measurement {M t }. Thus Alice cannot go first. In fact, a similar argument can be used to exhibit that Bob faces the same dilemma, i.e., he cannot gain any useful information by a nondisturbing measurement, either. Therefore, they cannot perfectly distinguish these states by LOCC. In other words, the 4p − 4 states cannot be perfectly distinguished by LOCC. This completes the proof.
In general, the local indistinguishability of an incomplete PB are proved by Definition 4 4,22,23 . That is, in order to prove the local indistinguishability of an incomplete PB, we just need to show that all the POVM elements are proportional to the identity operator to keep the orthogonality of the postmeasurement states no matter who performs the first measurement. However, it is not a necessary condition for the local indistinguishability of an incomplete PB that all the POVM elements are proportional to the identity operator. In fact, we give a weaker condition to prove the local indistinguishability of an incomplete PB, which can be seen obviously by the proof of Theorem 1.
It is noted that the product basis (1) is completable since the 4p − 4 states of (1) can become a completed orthogonal product basis in m ⊗ n (m ≥ 3 and n ≥ 3) by adding the following mn − 4p + 4 states: From Theorem 1, we know that the parameter p can be an arbitrary integer from 3 to min(m, n). That is, we actually construct a series of orthogonal product bases that are locally indistinguishable in m ⊗ n (m ≥ 3 and n ≥ 3). In particular, we have the following corollary by Theorem 1 when p = 3. corollary 1. In an m ⊗ n quantum system, the eight orthogonal product states As a special case, we can get eight states (2) that cannot be perfectly distinguished by LOCC in a d ⊗ d quantum system when m = n = d ≥ 3. This fact shows that the conjecture 22 , i.e., any set of no more than 2(d − 1) product states is locally distinguishable in a d ⊗ d (d ≥ 3) quantum system, is not true. By Refs 22-24, we know that the product basis composed by the eight states (2) is so far the smallest completable and locally indistinguishable orthogonal product basis in m ⊗ n (m ≥ 3 and n ≥ 3).
Local indistinguishability of small orthogonal product basis. Now we give a small orthogonal product basis with 2p − 1 members that cannot be perfectly distinguished by LOCC in an m ⊗ n quantum system, where m ≥ 3, n ≥ 3 and 3 ≤ p ≤ min(m, n). Then we give a simple proof for its local indistinguishability. theorem 2. In an m ⊗ n quantum system, the 2p − 1 orthogonal product states Proof. Similar to the proof of Theorem 1, one of the two parties (Alice and Bob) has to start with a nondisturbing measurement to distinguish these states, i.e., the postmeasurement states should be mutually orthogonal. Without loss of generality, suppose that Alice goes first with a set of general for j = 1, 2, ···, p − 1 by the same way as the proof of Theorem 1 since the postmeasurement states should be mutually orthogonal. For the states ψ | 〉 according to the completeness of the measurement operators. This means that any one of the 2p − 1 states can lead to the outcome that is corresponding to M t with the same probability a t 00 , i.e., the measurement {M t ⊗ I B | t = 1, 2, ···, l} is trivial to the 2p − 1 states. That is, Alice cannot get any useful information about which the measured state will be by the measurement {M t }. In fact, if Bob goes first with a nondisturbing measurement, they cannot distinguish the 2p − 1 states, either. Therefore, the 2p − 1 states cannot be perfectly distinguished by LOCC. This completes the proof.
□ The parameter p can be an arbitrary integer from 3 to min(m, n) in Theorem 2. That means that we construct a series of orthogonal product bases that are locally indistinguishable in m ⊗ n (m ≥ 3 and n ≥ 3). We have the following corollary directly by Theorem 2 when p = 3.  are locally indistinguishable, where m ≥ 3 and n ≥ 3. When m = n = 3, the five states of (4) form a UPB. In Ref. 21, Bennett et al. exhibits two results. One is that a UPB is not completable even in a locally extended Hilbert space. The other is that if a set of orthogonal product states is exactly measurable by LOCC, then the set can be completed in some extended space. Thus it is obvious that the five states of (4) in m ⊗ n are locally indistinguishable by the two results, which is coincident with Corollary 2. Since any four orthogonal product states are shown to be locally distinguishable 20 , it is easy to see that five is the smallest number of uncompletable and locally indistinguishable product states. Now we consider whether or not the 2p − 1 states of Theorem 2 are uncompletable in the m ⊗ n quantum system, where m ≥ 3, n ≥ 3 and p is an arbitrary integer from 3 to min(m, n). By the analysis of the last paragraph, we know that the 2p − 1 states of Theorem 2 are uncompletable in the m ⊗ n (m ≥ 3 and n ≥ 3) quantum system when p = 3. Then we prove that the 2p − 1 states of Theorem 2 are uncompletable in the m ⊗ n (m ≥ 4 and n ≥ 4) quantum system when p = 4. It is noted that some proof techniques are borrowed from Ref. 20. By Theorem 2, we get the following seven orthogonal product states that cannot be locally indistinguishable in the 4 ⊗ 4 quantum system.
Let S = {|ψ 1 〉 , |ψ 2 〉 , ···, |ψ 7 〉 }. The density matrix ρ S has rank 16 − 7 = 9. We can enumerate the product states that are orthogonal to the members of S, which are not all mutually orthogonal: These six vectors are not enough to span the full Hilbert space ⊥ H S . This means that the range of ρ S contains only six product states, whereas ρ S has rank 9. Therefore ρ S must be entangled. By Lemma 1, we can get S is a SUCPB because ρ S is entangled. That is, the 2p − 1 states of Theorem 2 are uncompletable in the m ⊗ n (m ≥ 4 and n ≥ 4) quantum system when p = 4. On the other hand, we can prove that the 2p − 1 states of Theorem 2 are uncompletable in the m ⊗ n (m ≥ 5 and n ≥ 5) quantum system by the same method, where 5 ≤ p ≤ min(m, n). By Theorem 2, we get the following 2p − 1 states in the p ⊗ p (p ≥ 5) quantum system. space ′ ⊥ H S . This means that the range of ρ ′ S contains only 2p − 2 product states, whereas ρ ′ S has rank (p − 1) 2 . Therefore ρ ′ S must be entangled. By Lemma 1, we can get S′ is a SUCPB because ρ ′ S is entangled. That is, the 2p − 1 states of Theorem 2 are uncompletable in the m ⊗ n (m ≥ 5 and n ≥ 5) quantum system when 5 ≤ p ≤ min(m, n). Therefore, the 2p − 1 orthogonal product states of (3) are uncompletable in the m ⊗ n quantum system, where m ≥ 3, n ≥ 3 and p is an arbitrary integer from 3 to min(m, n).

Discussion
In this paper, we construct a completable orthogonal product basis with 4p − 4 members that cannot be perfectly distinguished by LOCC in an m ⊗ n quantum system, where m ≥ 3, n ≥ 3 and p is an arbitrary integer from 3 to min(m, n), and give a simple but quite effective proof. As a special case, we get eight orthogonal product states that can be completable but cannot be locally distinguished in m ⊗ n (m ≥ 3 and n ≥ 3). On the other hand, we construct a samll locally indistinguishable orthogonal product basis with 2p − 1 members in m ⊗ n, which are uncompletable, where m ≥ 3, n ≥ 3 and p is an arbitrary integer from 3 to min(m, n). Our work is useful to understand the structures both of completable and uncompletable product bases that cannot be distinguished by LOCC in arbitrary bipartite quantum system.