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Sci Rep. 2016; 6: 28864.
Published online 2016 July 5. doi:  10.1038/srep28864
PMCID: PMC4932555

LOCC indistinguishable orthogonal product quantum states

Abstract

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 An external file that holds a picture, illustration, etc.
Object name is srep28864-m1.jpg2k+i [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m2.jpg2l+j (i, j [set membership] {0, 1} and i  j ) and An external file that holds a picture, illustration, etc.
Object name is srep28864-m3.jpg3k+i [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m4.jpg3l+j (i, j [set membership] {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 An external file that holds a picture, illustration, etc.
Object name is srep28864-m5.jpg3k+i [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m6.jpg3l+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 An external file that holds a picture, illustration, etc.
Object name is srep28864-m7.jpg2k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m8.jpg2l and An external file that holds a picture, illustration, etc.
Object name is srep28864-m9.jpg2k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m10.jpg2l but also An external file that holds a picture, illustration, etc.
Object name is srep28864-m11.jpg2k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m12.jpg2l+1 and An external file that holds a picture, illustration, etc.
Object name is srep28864-m13.jpg2k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m14.jpg2l+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 An external file that holds a picture, illustration, etc.
Object name is srep28864-m15.jpg2k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m16.jpg2l 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 performed1. When many global operations cannot be performed, LOCC becomes very useful. The phenomenon of quantum nonlocality without entanglement2 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 distinguished3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 or not. In refs 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, they focus on the local distinguishability of quantum states such as multipartite orthogonal product states can be exactly distinguished10 or how to distinguish two quantum pure states11,12. Moreover, locally indistinguishability13,14,15,16,17,18,19,20,21,22,23 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 C3 [multiply sign in circle] C3 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.’s16. Yu and Oh22 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 An external file that holds a picture, illustration, etc.
Object name is srep28864-m17.jpg2k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m18.jpg2l, An external file that holds a picture, illustration, etc.
Object name is srep28864-m19.jpg2k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m20.jpg2l+1 and An external file that holds a picture, illustration, etc.
Object name is srep28864-m21.jpg2k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m22.jpg2l+1. The smallest dimension of An external file that holds a picture, illustration, etc.
Object name is srep28864-m23.jpg2k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m24.jpg2l can be constructed is An external file that holds a picture, illustration, etc.
Object name is srep28864-m25.jpg6 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m26.jpg6 in Wang et al.’s paper20. However, the smallest dimension of An external file that holds a picture, illustration, etc.
Object name is srep28864-m27.jpg2k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m28.jpg2l can be constructed is An external file that holds a picture, illustration, etc.
Object name is srep28864-m29.jpg4 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m30.jpg4 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 An external file that holds a picture, illustration, etc.
Object name is srep28864-m31.jpg2k+i [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m32.jpg2l+j with i, j [set membership] {0, 1} (i  j) and An external file that holds a picture, illustration, etc.
Object name is srep28864-m33.jpg3k+i [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m34.jpg3l+j with i, j [set membership] {0, 1, 2} and the two families of orthogonal product quantum states cannot be exactly distinguished by LOCC but can be distinguished by separable operations. Our constructions give the smaller dimension of quantum system in quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m35.jpg2k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m36.jpg2l than Wang et al.’s20. Wang et al.’s construction can be extended, but our construction in quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m37.jpg2k+i [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m38.jpg2l+j with i, j [set membership] {0, 1} (i  j) is a complete unextendible product bases (i.e. UPB). Therefore, our construction is trivial. The indistinguishability of a complete UPB can be directly judged by performing projective measurements and classical communication, but not Wang et al.’s20. In quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m39.jpg3k+i [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m40.jpg3l+j (i, j = 0, 1, 2), it contains not only An external file that holds a picture, illustration, etc.
Object name is srep28864-m41.jpg2k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m42.jpg2l and An external file that holds a picture, illustration, etc.
Object name is srep28864-m43.jpg2k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m44.jpg2l but also An external file that holds a picture, illustration, etc.
Object name is srep28864-m45.jpg2k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m46.jpg2l+1 and An external file that holds a picture, illustration, etc.
Object name is srep28864-m47.jpg2k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m48.jpg2l+1, so our construction in quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m49.jpg3k+i [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m50.jpg3l+j with i, j [set membership] {0, 1, 2} is more generalized than Zhang et al.19 and Wang et al.20. We also use a simple method to judge the local indistinguishable by calculating the non-commutativity to quantify the quantumness of a quantum ensemble24, but not Zhang et al. and Wang et al. We also generalize the Theorem 2 in Ma et al.24 to Corollary 1 in Methods in this paper. Our work is a necessary complement to understand the phenomenon of quantum nonlocality without entanglement.

Results

LOCC indistinguishable orthogonal product quantum states in quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m51.jpg 2k+i  [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m52.jpg 2l+j with k  1, l  1 and i, j [set membership] {0, 1} (i  j)

Case 1. Firstly, we construct LOCC indistinguishable orthogonal product quantum states in quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m53.jpg2k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m54.jpg2l (k, l  2) (see Fig. 1(a)) and give an example in the smallest dimension (see Fig. 2(a)).

Figure 1
The tiling structure of orthogonal product quantum states in quantum system of (a) An external file that holds a picture, illustration, etc.
Object name is srep28864-m278.jpg2k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m279.jpg2l with k, l  2 and (b) An external file that holds a picture, illustration, etc.
Object name is srep28864-m280.jpg2k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m281.jpg2l+1 with k, l  1.
Figure 2
The tiling structure of orthogonal product quantum states in quantum system of (a) An external file that holds a picture, illustration, etc.
Object name is srep28864-m282.jpg4 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m283.jpg4, (b) An external file that holds a picture, illustration, etc.
Object name is srep28864-m284.jpg3 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m285.jpg3, (c) An external file that holds a picture, illustration, etc.
Object name is srep28864-m286.jpg5 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m287.jpg6, and (d) An external file that holds a picture, illustration, etc.
Object name is srep28864-m288.jpg5 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m289.jpg5.
An external file that holds a picture, illustration, etc.
Object name is srep28864-m55.jpg
An external file that holds a picture, illustration, etc.
Object name is srep28864-m56.jpg
An external file that holds a picture, illustration, etc.
Object name is srep28864-m57.jpg
An external file that holds a picture, illustration, etc.
Object name is srep28864-m58.jpg
An external file that holds a picture, illustration, etc.
Object name is srep28864-m59.jpg

Here An external file that holds a picture, illustration, etc.
Object name is srep28864-m60.jpg. For example, An external file that holds a picture, illustration, etc.
Object name is srep28864-m61.jpg.

Proposition 1. In quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m62.jpg2k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m63.jpg2l, there are 4 kl orthogonal product quantum states |ψiright angle bracket (in Eq. (1)) can not be exactly distinguished by LOCC whatever Alice measures firstly or Bob.

Proof. We only discuss the case of Alice measures firstly and the same as Bob. We consider the subspace An external file that holds a picture, illustration, etc.
Object name is srep28864-m64.jpg2 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m65.jpgm to determine POVM elements An external file that holds a picture, illustration, etc.
Object name is srep28864-m66.jpg. A set of general An external file that holds a picture, illustration, etc.
Object name is srep28864-m67.jpg2k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m68.jpg2k POVM elements An external file that holds a picture, illustration, etc.
Object name is srep28864-m69.jpg under the basis An external file that holds a picture, illustration, etc.
Object name is srep28864-m70.jpg can be expressed as follows

An external file that holds a picture, illustration, etc.
Object name is srep28864-m71.jpg

where aij  0 and An external file that holds a picture, illustration, etc.
Object name is srep28864-m72.jpg Firstly, this selected sets {|0right angle bracket, |1right angle bracket}A, {|1right angle bracket, |2right angle bracket}A, …, and {|2k  2right angle bracket, |2k  1right angle bracket}A of states are of dimension An external file that holds a picture, illustration, etc.
Object name is srep28864-m73.jpg2 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m74.jpg2k, Alice cannot find appropriate basis to express them in the form of Eq. (23) in Methods according to the necessary and sufficient condition of Lemma 1.

For example, we consider the subspace {|0right angle bracket, |1right angle bracket}A, there are quantum states

An external file that holds a picture, illustration, etc.
Object name is srep28864-m75.jpg

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 a00 = a11 and a10 = a01 = 0 in the subspace {|0right angle bracket, |1right angle bracket}A. Suppose, the form An external file that holds a picture, illustration, etc.
Object name is srep28864-m76.jpg is set up in Eq. (23), where An external file that holds a picture, illustration, etc.
Object name is srep28864-m77.jpgAn external file that holds a picture, illustration, etc.
Object name is srep28864-m78.jpg, An external file that holds a picture, illustration, etc.
Object name is srep28864-m79.jpg. The two sets An external file that holds a picture, illustration, etc.
Object name is srep28864-m80.jpg and An external file that holds a picture, illustration, etc.
Object name is srep28864-m81.jpg satisfy An external file that holds a picture, illustration, etc.
Object name is srep28864-m82.jpg if ij. However, there also exist quantum states |0 ± 1right angle bracketA in the subspace {|0right angle bracket, |1right angle bracket}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

An external file that holds a picture, illustration, etc.
Object name is srep28864-m83.jpg

where α > β  0. For the quantum state |0right angle bracketA, it collapses into α|0right angle bracketA after measurement. For the quantum state |1right angle bracketΑ, it collapses into β|1right angle bracketA after measurement. For the quantum states An external file that holds a picture, illustration, etc.
Object name is srep28864-m84.jpg, they collapse into An external file that holds a picture, illustration, etc.
Object name is srep28864-m85.jpg. Hence, if and only if α = β, An external file that holds a picture, illustration, etc.
Object name is srep28864-m86.jpg 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

An external file that holds a picture, illustration, etc.
Object name is srep28864-m87.jpg

is entirely specified by those elements a00, a11, a01, a10 draw from the {|0right angle bracket, |1right angle bracket}A subspace. It means that Alice cannot perform a nontrivial measurement upon the {|0right angle bracket, |1right angle bracket}A subspace. Thus, the corresponding submatrix must be proportional to the identity. Then, we obtain a00 = a11 = a, a01 = a10 = 0. For the states

An external file that holds a picture, illustration, etc.
Object name is srep28864-m88.jpg

and the subspace {|1right angle bracket, |2right angle bracket}A, we make the same argument. Then we get the result a11 = a22 = a, a12 = a21 = 0. For the states

An external file that holds a picture, illustration, etc.
Object name is srep28864-m89.jpg

and subspace {|2right angle bracket, |3right angle bracket}A, we get the result a22 = a33 = a, a23 = a32 = 0. In the same way, for the subspace {|3right angle bracket, |4right angle bracket}A, …, the subspace An external file that holds a picture, illustration, etc.
Object name is srep28864-m90.jpg, we get the result

An external file that holds a picture, illustration, etc.
Object name is srep28864-m91.jpg

Because POVM elements An external file that holds a picture, illustration, etc.
Object name is srep28864-m92.jpg is Hermitian, the equation An external file that holds a picture, illustration, etc.
Object name is srep28864-m93.jpg is correct. Then we obtain

An external file that holds a picture, illustration, etc.
Object name is srep28864-m94.jpg

Now An external file that holds a picture, illustration, etc.
Object name is srep28864-m95.jpg can be rewritten as

An external file that holds a picture, illustration, etc.
Object name is srep28864-m96.jpg

where a is a real number.

We now consider the states An external file that holds a picture, illustration, etc.
Object name is srep28864-m97.jpg with f = 3, i3 = 0, 2 and the subspace {|0right angle bracket, |2right angle bracket}A. After Alice measures, the result is either the states orthogonal or distinguishing them outright. If the states are orthogonal, we demand that An external file that holds a picture, illustration, etc.
Object name is srep28864-m98.jpg. So, we get An external file that holds a picture, illustration, etc.
Object name is srep28864-m99.jpg. For the states An external file that holds a picture, illustration, etc.
Object name is srep28864-m100.jpg with ib = 3, jb = 2l−2 and An external file that holds a picture, illustration, etc.
Object name is srep28864-m101.jpg, we get the same argument and we get An external file that holds a picture, illustration, etc.
Object name is srep28864-m102.jpg. For the subspace {|0right angle bracket, |4right angle bracket}A, {|0right angle bracket, |5right angle bracket}A, … and the subspace {|2k  3right angle bracket, |2k  1right angle bracket}A, we get the results

An external file that holds a picture, illustration, etc.
Object name is srep28864-m103.jpg

Now the An external file that holds a picture, illustration, etc.
Object name is srep28864-m104.jpg is proportional to the identity. However, if Alice distinguishes the state An external file that holds a picture, illustration, etc.
Object name is srep28864-m105.jpg with f = 3, if = 0, 2, we get the result An external file that holds a picture, illustration, etc.
Object name is srep28864-m106.jpg. We can also have the result An external file that holds a picture, illustration, etc.
Object name is srep28864-m107.jpg, therefore a = 0. It produces contradictory with the theorem of Walgate16. So, An external file that holds a picture, illustration, etc.
Object name is srep28864-m108.jpg 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 An external file that holds a picture, illustration, etc.
Object name is srep28864-m109.jpg4 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m110.jpg4 (see Fig. 2(a)).

An external file that holds a picture, illustration, etc.
Object name is srep28864-m111.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep28864-m112.jpg with An external file that holds a picture, illustration, etc.
Object name is srep28864-m113.jpg.

Case 2. Secondly, we construct LOCC indistinguishable orthogonal product quantum states in quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m114.jpg2k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m115.jpg2l+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)).

An external file that holds a picture, illustration, etc.
Object name is srep28864-m116.jpg
An external file that holds a picture, illustration, etc.
Object name is srep28864-m117.jpg
An external file that holds a picture, illustration, etc.
Object name is srep28864-m118.jpg
An external file that holds a picture, illustration, etc.
Object name is srep28864-m119.jpg
An external file that holds a picture, illustration, etc.
Object name is srep28864-m120.jpg

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.

Proposition 2. In quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m121.jpg2k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m122.jpg2l+1, there are (2k + 1)(2l + 1) orthogonal product quantum states |ϕiright angle bracket (in Eq. (13)) can not be exactly distinguished by LOCC whatever Alice measures firstly or Bob.

Example 2. Now we will give 9 orthogonal product quantum states in quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m123.jpg3 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m124.jpg3 (see Fig. 2(b)).

An external file that holds a picture, illustration, etc.
Object name is srep28864-m125.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep28864-m126.jpg with 0  i < j  2.

Case 3. Thirdly, we consider the indistinguishable orthogonal product states in quantum system An external file that holds a picture, illustration, etc.
Object name is srep28864-m127.jpg2k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m128.jpg2l with k  2, l  3 (see Fig. 3) and give an example in the smallest dimension (see Fig. 2(c)).

Figure 3
The tiling structure of orthogonal product quantum states in quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m290.jpg2k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m291.jpg2l with k  2, l  3.
An external file that holds a picture, illustration, etc.
Object name is srep28864-m129.jpg
An external file that holds a picture, illustration, etc.
Object name is srep28864-m130.jpg
An external file that holds a picture, illustration, etc.
Object name is srep28864-m131.jpg
An external file that holds a picture, illustration, etc.
Object name is srep28864-m132.jpg
An external file that holds a picture, illustration, etc.
Object name is srep28864-m133.jpg

Proposition 3. In quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m134.jpg2k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m135.jpg2l, there are 2l(2k + 1) orthogonal product quantum states An external file that holds a picture, illustration, etc.
Object name is srep28864-m136.jpg (in Eq. (15)) can not be exactly distinguished by LOCC whatever Alice measures firstly or Bob.

Example 3. Now we will give 30 orthogonal product quantum states in quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m137.jpg5 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m138.jpg6 (see Fig. 3(c)).

An external file that holds a picture, illustration, etc.
Object name is srep28864-m139.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep28864-m140.jpg with 0  i  4 and 0  j  5.

LOCC indistinguishable orthogonal product quantum states in quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m141.jpg 3k+i  [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m142.jpg 3k+j with i, j [set membership] {0, 1, 2}

We give LOCC indistinguishable orthogonal product quantum states in quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m143.jpgm [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m144.jpgn.

An external file that holds a picture, illustration, etc.
Object name is srep28864-m145.jpg

In quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m146.jpg3k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m147.jpg3l, An external file that holds a picture, illustration, etc.
Object name is srep28864-m148.jpg3k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m149.jpg3l+2, An external file that holds a picture, illustration, etc.
Object name is srep28864-m150.jpg3k+2 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m151.jpg3l with k, l  2, An external file that holds a picture, illustration, etc.
Object name is srep28864-m152.jpg3k+2 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m153.jpg3l+2 with k, l  1, An external file that holds a picture, illustration, etc.
Object name is srep28864-m154.jpg3k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m155.jpg3l and An external file that holds a picture, illustration, etc.
Object name is srep28864-m156.jpg3k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m157.jpg3l+2 with k, l  2.

An external file that holds a picture, illustration, etc.
Object name is srep28864-m158.jpg

In quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m159.jpgm [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m160.jpgn including An external file that holds a picture, illustration, etc.
Object name is srep28864-m161.jpg3k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m162.jpg3l+1, An external file that holds a picture, illustration, etc.
Object name is srep28864-m163.jpg3k+2 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m164.jpg3l+1, An external file that holds a picture, illustration, etc.
Object name is srep28864-m165.jpg3k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m166.jpg3l+1 with k, l  2.

An external file that holds a picture, illustration, etc.
Object name is srep28864-m167.jpg

In quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m168.jpgm [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m169.jpgn including An external file that holds a picture, illustration, etc.
Object name is srep28864-m170.jpg3k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m171.jpg3l, An external file that holds a picture, illustration, etc.
Object name is srep28864-m172.jpg3k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m173.jpg3l+2, An external file that holds a picture, illustration, etc.
Object name is srep28864-m174.jpg3k+2 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m175.jpg3l, An external file that holds a picture, illustration, etc.
Object name is srep28864-m176.jpg3k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m177.jpg3l+1 and An external file that holds a picture, illustration, etc.
Object name is srep28864-m178.jpg3k+2 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m179.jpg3l+1 with An external file that holds a picture, illustration, etc.
Object name is srep28864-m180.jpg, An external file that holds a picture, illustration, etc.
Object name is srep28864-m181.jpg3k+2 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m182.jpg3l+2 with An external file that holds a picture, illustration, etc.
Object name is srep28864-m183.jpg.

An external file that holds a picture, illustration, etc.
Object name is srep28864-m184.jpg

In quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m185.jpgm [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m186.jpgn including An external file that holds a picture, illustration, etc.
Object name is srep28864-m187.jpg3k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m188.jpg3l, An external file that holds a picture, illustration, etc.
Object name is srep28864-m189.jpg3k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m190.jpg3l+2, An external file that holds a picture, illustration, etc.
Object name is srep28864-m191.jpg3k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m192.jpg3l+1 with k, l  2.

An external file that holds a picture, illustration, etc.
Object name is srep28864-m193.jpg

The equation k = 2ρ (or k = 2μ + 1) expresses that k is even (or odd).

Proposition 4. In quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m194.jpgm [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m195.jpgn, there are 3(n + m)  9 orthogonal product quantum states |θiright angle bracket (in Eqs (17, 18, 19, 20, 21)) can not be exactly distinguished by LOCC whatever Alice measures firstly or Bob, where m = 3k + i, n = 3l + j with i, j [set membership] {0, 1, 2}.

For the proof of the proposition 2, 3, 4, we make the same arguments to prove the indistinguishability only by LOCC. We only need to modify some relevant places.

Example 4. Now we will give the 21 orthogonal product quantum states in quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m196.jpg5 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m197.jpg5 (see Fig. 2(d)).

An external file that holds a picture, illustration, etc.
Object name is srep28864-m198.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep28864-m199.jpg with 0  i  4 and 0  j  4.

Discussion

The orthogonal product quantum states constructed by us are indistinguishable by performing local operation and classical communication, but not separable operations25. Now, we discuss whether the separable operations can distinguish these product quantum states or not.

LOCC indistinguishable orthogonal product quantum states in quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m200.jpg 2k+i  [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m201.jpg 2l+j with i, j [set membership] {0, 1} (i  j)

Obviously, these states in Eqs (1, 12, 13, 14, 15, 16) can be distinguished by separable operations. The orthogonal quantum states are not extended. Suppose, the mn quantum states are An external file that holds a picture, illustration, etc.
Object name is srep28864-m202.jpg respectively. Now, we give the measurement operations An external file that holds a picture, illustration, etc.
Object name is srep28864-m203.jpg. Because the set An external file that holds a picture, illustration, etc.
Object name is srep28864-m204.jpg is an orthogonal product normal base of An external file that holds a picture, illustration, etc.
Object name is srep28864-m205.jpgm [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m206.jpgn, the equation An external file that holds a picture, illustration, etc.
Object name is srep28864-m207.jpg satisfies the completeness and An external file that holds a picture, illustration, etc.
Object name is srep28864-m208.jpg is a separable measurement. Due to An external file that holds a picture, illustration, etc.
Object name is srep28864-m209.jpg, where 1  i  mn, 1  j  mn, if the measurement outcome is |[var phi]iright angle bracket, the quantum state is |[var phi]iright angle bracket. Therefore, the mn quantum states in Eqs (1, 12, 13, 14, 15, 16) can be distinguished by the separable operations.

Similar to Zhang et al.’s paper19, the multipartite quantum systems can be constructed when m = n = d. Such as in the quantum system An external file that holds a picture, illustration, etc.
Object name is srep28864-m210.jpgd [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m211.jpgd [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m212.jpgd, we give the orthogonal indistinguishable product states An external file that holds a picture, illustration, etc.
Object name is srep28864-m213.jpg, where An external file that holds a picture, illustration, etc.
Object name is srep28864-m214.jpg and An external file that holds a picture, illustration, etc.
Object name is srep28864-m215.jpg in Eqs (1, 12) of An external file that holds a picture, illustration, etc.
Object name is srep28864-m216.jpg2k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m217.jpg2l and An external file that holds a picture, illustration, etc.
Object name is srep28864-m218.jpg2k+1 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m219.jpg2l+1. However, Wang et al.’s construction20 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 An external file that holds a picture, illustration, etc.
Object name is srep28864-m220.jpg 3k+i  [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m221.jpg 3l+j with i, j [set membership] {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 116. Alice and Bob share a An external file that holds a picture, illustration, etc.
Object name is srep28864-m222.jpg2 [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m223.jpgn 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 {|[var phi]iright angle bracket} is exactly locally distinguishable if and only if there is a basis {|0right angle bracket, |1right angle bracket}A such that in that basis

An external file that holds a picture, illustration, etc.
Object name is srep28864-m224.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep28864-m225.jpg if ij. The Lemma 1 is used to prove the indistinguishability of An external file that holds a picture, illustration, etc.
Object name is srep28864-m226.jpg2k+i [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m227.jpg2l+j quantum system with i, j [set membership] {0, 1} (i  j) and An external file that holds a picture, illustration, etc.
Object name is srep28864-m228.jpg3k+i [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m229.jpg3l+j quantum system with i, j [set membership] {0, 1, 2} in Results.

Definition24. Let An external file that holds a picture, illustration, etc.
Object name is srep28864-m230.jpg be a set of operators. The total non-commutativity for this set is defined

An external file that holds a picture, illustration, etc.
Object name is srep28864-m231.jpg

where [A, B] = AB  BA, An external file that holds a picture, illustration, etc.
Object name is srep28864-m232.jpg is the trace norm of the operator A, An external file that holds a picture, illustration, etc.
Object name is srep28864-m233.jpg.

In the Methods of Ma et al.’s24, they give the concrete calculation formula, i.e. suppose An external file that holds a picture, illustration, etc.
Object name is srep28864-m234.jpg and An external file that holds a picture, illustration, etc.
Object name is srep28864-m235.jpg. Denote An external file that holds a picture, illustration, etc.
Object name is srep28864-m236.jpg with x [set membership] [0, 1], An external file that holds a picture, illustration, etc.
Object name is srep28864-m237.jpg. HenceAn external file that holds a picture, illustration, etc.
Object name is srep28864-m238.jpg. When An external file that holds a picture, illustration, etc.
Object name is srep28864-m239.jpg or 1, An external file that holds a picture, illustration, etc.
Object name is srep28864-m240.jpg, when An external file that holds a picture, illustration, etc.
Object name is srep28864-m241.jpg, An external file that holds a picture, illustration, etc.
Object name is srep28864-m242.jpg and when An external file that holds a picture, illustration, etc.
Object name is srep28864-m243.jpg, An external file that holds a picture, illustration, etc.
Object name is srep28864-m244.jpg. Nextly, we give Lemma 2 as a standard of judging the indistinguishability of complete orthogonal product states.

Lemma 224. For a complete set of An external file that holds a picture, illustration, etc.
Object name is srep28864-m245.jpg POPS, An external file that holds a picture, illustration, etc.
Object name is srep28864-m246.jpg with An external file that holds a picture, illustration, etc.
Object name is srep28864-m247.jpg, the ε cannot be completely locally distinguished if and only if there exist subsets An external file that holds a picture, illustration, etc.
Object name is srep28864-m248.jpg, such that An external file that holds a picture, illustration, etc.
Object name is srep28864-m249.jpg and An external file that holds a picture, illustration, etc.
Object name is srep28864-m250.jpg are all single sets, i.e. there exist An external file that holds a picture, illustration, etc.
Object name is srep28864-m251.jpg linear independent An external file that holds a picture, illustration, etc.
Object name is srep28864-m252.jpg in An external file that holds a picture, illustration, etc.
Object name is srep28864-m253.jpg and An external file that holds a picture, illustration, etc.
Object name is srep28864-m254.jpg linear independent An external file that holds a picture, illustration, etc.
Object name is srep28864-m255.jpg in An external file that holds a picture, illustration, etc.
Object name is srep28864-m256.jpg satisfying

An external file that holds a picture, illustration, etc.
Object name is srep28864-m257.jpg

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 in24 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 εA and εB.

An external file that holds a picture, illustration, etc.
Object name is srep28864-m258.jpg
An external file that holds a picture, illustration, etc.
Object name is srep28864-m259.jpg
An external file that holds a picture, illustration, etc.
Object name is srep28864-m260.jpg
An external file that holds a picture, illustration, etc.
Object name is srep28864-m261.jpg

Some duplicate items are removed in εA and εB. Nextly, we concretely calculate the non-commutativity N to quantify the quantumness of a quantum ensemble. There are 2k = (spanεA) linear independent states in εA.

An external file that holds a picture, illustration, etc.
Object name is srep28864-m262.jpg

For the last two non-commutativity 1.74k + 0.26 and 1.74k + 0.61, we obtain that the difference (1.74k + 0.26) −(1.74k  0.61) = 0.87 > 0. Hence, we obtain the inequality as follows

An external file that holds a picture, illustration, etc.
Object name is srep28864-m263.jpg

So εA is a single set according to Lemma 2.

There are 2l = dim(spanεB) linear independent states in εB.

An external file that holds a picture, illustration, etc.
Object name is srep28864-m264.jpg

For the last two non-commutativity 1.74l + 0.26 and 1.74l + 0.61, we obtain that the difference (1.74l + 0.26)−(1.74l  0.61) = 0.87 > 0. Hence, we obtain the inequality as follows

An external file that holds a picture, illustration, etc.
Object name is srep28864-m265.jpg

So εB is also a single set according to Lemma 2. According to the necessary and sufficient condition of Lemma 2, we make a conclusion that the set of complete orthogonal product quantum states in Eq. (1) is indistinguishable. Similarly, for the orthogonal product states in Eqs (13, 15), we obtain the same conclusion. The quantum orthogonal product states in Eqs (17, 18, 19, 20, 21) are incomplete but can be extended into a complete set, we can also judge the indistinguishability by Corollary 1. Now we will introduce the Corollary 1.

Corollary 1. For a incomplete set of orthogonal product states in quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m266.jpgm [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m267.jpgn, it firstly should be extended into a complete set An external file that holds a picture, illustration, etc.
Object name is srep28864-m268.jpg with An external file that holds a picture, illustration, etc.
Object name is srep28864-m269.jpg if and only if it is completable. The indistinguishability of its complete set can be judged by Lemma 2.

The Corollary 1 is used to judge the indistinguishability of a set of incomplete orthogonal product states which is completable. The second family construction in quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m270.jpg3k+i [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m271.jpg3l+j with An external file that holds a picture, illustration, etc.
Object name is srep28864-m272.jpg is incomplete but can be completable, so we can use the Corollary 1 to judge the indistinguishability. For example, for the quantum system of An external file that holds a picture, illustration, etc.
Object name is srep28864-m273.jpg3k [multiply sign in circle] An external file that holds a picture, illustration, etc.
Object name is srep28864-m274.jpg3l when k, l are all even, quantum states |0right angle bracketA|0right angle bracketB, |1right angle bracketA|n1right angle bracketB with n1 = 3, 4, 5, 6, …, 3l  3, |mright angle bracketA|n2right angle bracketB with An external file that holds a picture, illustration, etc.
Object name is srep28864-m275.jpg, An external file that holds a picture, illustration, etc.
Object name is srep28864-m276.jpg with An external file that holds a picture, illustration, etc.
Object name is srep28864-m277.jpg and |3k  1right angle bracketA|2right angle bracketB are added into the original incomplete set. The original incomplete set becomes a complete set. And its indistinguishability can be judged by using Corollary 1.

Additional Information

How to cite this article: Zhang, X. et al. LOCC indistinguishable orthogonal product quantum states. Sci. Rep. 6, 28864; doi: 10.1038/srep28864 (2016).

Acknowledgments

The research is funded by National Natural Science Foundation of China, under Grant Nos 61370228, 61472165 and 61502200, and Science and Technology Planning Project of Guangdong Province, China, under Grant Nos 2013B010401018 and 2015B010128008, and Natural Science Foundation of Guangdong province, China, under Grant Nos 2014A030310245 and 2016A030313090.

Footnotes

Author Contributions X.Z. and X.T. proposed and wrote the main manuscript text. J.W. and Y.L. reviewed the manuscript and provided funding support.

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