www.nature.com/scientificreports
OPEN
LOCC indistinguishable orthogonal product quantum states Xiaoqian Zhang1, Xiaoqing Tan1, Jian Weng2 & Yongjun Li3
received: 14 January 2016 accepted: 06 June 2016 Published: 05 July 2016
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 C 2k+i ⊗ C 2l+j (i, j ∈ {0, 1} and i ≥ j ) and C 3k+i ⊗ C 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 C 3k+i ⊗ C 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 C 2k ⊗ C 2l and C 2k+1 ⊗ C 2l but also C 2k ⊗ C 2l+1 and C 2k+1 ⊗ C 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 C 2k ⊗ C 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 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 distinguished3–23 or not. In refs 3–12, 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 indistinguishability13–23 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 ⊗ 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 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 paper20. 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 1
Department of Mathematics, Jinan University, Guangzhou, P.R. China. 2Department of Computer Science, Jinan University, Guangzhou, P.R. China. 3School of Computer Science and Engineering, South China University of Technology, Guangzhou, P.R. China. Correspondence and requests for materials should be addressed to X.T. (email: [email protected]) Scientific Reports | 6:28864 | DOI: 10.1038/srep28864
1
www.nature.com/scientificreports/
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. 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 C 2k ⊗ C 2l than Wang et al.’s20. Wang et al.’s construction can be extended, but our construction in quantum system of C 2k+i ⊗ C 2l+j with i, j ∈ {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 C 3k+i ⊗ C 3l+j (i, j = 0, 1, 2), it contains not only C 2k ⊗ C 2l and C 2k+1 ⊗ C 2l but also C 2k ⊗ C 2l+1 and C 2k+1 ⊗ C 2l+1, so our construction in quantum system of C 3k+i ⊗ C 3l+j with i, j ∈ {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 C2k+i ⊗ C2l+j with k ≥ 1, l ≥ 1 and i, j ∈ {0, 1} (i ≥ j). Case 1. Firstly, we construct LOCC indistinguishable orthogonal product quantum states in quantum system of C 2k ⊗ C 2l (k, l ≥ 2) (see Fig. 1(a)) and give an example in the smallest dimension (see Fig. 2(a)).
Scientific Reports | 6:28864 | DOI: 10.1038/srep28864
2
www.nature.com/scientificreports/
Figure 2. The tiling structure of orthogonal product quantum states in quantum system of (a) C 4 ⊗ C 4, (b) C 3 ⊗ C 3, (c) C 5 ⊗ C 6, and (d) C 5 ⊗ C 5.
|ψib j 〉 = |ib 〉A |jb + (jb + 1)〉B , where b = 1, 2, 3 and b
i1 = 0, 2, 4, , min {2k , 2l} − 4, j1 = i1 + 1,i1 + 3, 2l − 3;
i2 = 1, 3, 5, , min {2k , 2l} − 3, j2 = i2 + 1, i2 + 3, , 2l − 2; i3 = 2k − 1,
j3 = 0;
|ψic j 〉 = |ic 〉A |jc − (jc + 1) 〉B , where c = 1, 2, 3 and c
i1 = 0, 2, 4, , min {2k , 2l} − 4, j1 = i1 + 1, i1 + 3, 2l − 3;
i2 = 1, 3, 5, , min {2k , 2l} − 3, j2 = i2 + 1, i2 + 3, , 2l − 2; i3 = 2k − 1,
j3 = 0;
|ψ j id 〉 = |id + (id + 1)〉A |jd 〉B , where d = 1, 2, 3 and d
j1 = 1, 3, 5, , min{2k − 3, 2l − 1}, i1 = j1 , j1 + 2, , 2k − 3;
j2 = 2, 4, 6, , min{2k − 2,2l − 2}, i2 = j2 , j2 + 2, , 2k − 2;
j3 = 0,
i3 = 0,2,4, , 2k − 4;
|ψ j ie 〉 = |ie − (ie + 1)〉A |je 〉B , where e = 1, 2, 3 and e
j1 = 1, 3, 5, , min{2k − 3, 2l − 1}, i1 = j1 , j1 + 2, , 2k − 3;
j2 = 2, 4, 6, , min{2k − 2, 2l − 2}, i2 = j2 , j2 + 2, , 2k − 2; j3 = 0,
Scientific Reports | 6:28864 | DOI: 10.1038/srep28864
i3 = 0, 2, 4, , 2k − 4;
3
www.nature.com/scientificreports/
|ψ j
f if
〉 = |i f 〉A |jf 〉B , where f = 1, 2, 3 and
j1 = 0, 2k − 1, 2k, , 2l − 1,
i1 = 2k − 2;
j3 = 2l − 1,
i3 =0, 2, 4, , min {2l − 2, 2k − 4} .
j2 = 3, 5, , 2k − 3, 2k − 1, 2k, 2k + 1, 2l − 1, i2 = 2k − 1;
Here |α ± β〉 =
1 2
(|α〉 ± |β〉 ). For example, |ie − (ie + 1) 〉A =
1 2
(1)
(|ie 〉 − |ie + 1〉 )A.
Proposition 1. In quantum system of C 2k ⊗ C 2l, there are 4 kl orthogonal product quantum states |ψi〉 (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 C 2 ⊗ C m to determine POVM elements Mm† Mm . A set of general C 2k ⊗ C 2k POVM elements Mm† Mm under the basis {|0〉 , |1〉 , , |2k − 1〉 }A can be expressed as follows a01 a00 a a11 10 Mm† Mm = a 2k−2,0 a2k−2,1 a2k−1,0 a2k−1,1
a0,2k−1 a1,2k−1
, a2k−2,2k−1 a2k−1,2k−1
(2)
where aij ≥ 0 and i, j ∈ {0, 1, 2, , 2k − 1} . Firstly, this selected sets {|0〉, |1〉}A, {|1〉, |2〉}A, …, and {|2k − 2〉, |2k − 1〉}A of states are of dimension C 2 ⊗ C 2k, 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 {|0〉, |1〉}A, there are quantum states |ψib j 〉 = |ib 〉A |jb + (jb + 1)〉B , where b = 1, 2 and b
i1 = 0, i2 = 1,
j1 = 1, 3, , 2l − 3;
j2 = 2, 4, , 2l − 2;
|ψ j id 〉 = |0 ± 1〉A |0〉B ,
(3)
d
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 {|0〉, |1〉}A. Suppose, the form |0〉A |η0i 〉B + |1〉A |η1i 〉B is set up in Eq. (23), where {|η0i 〉 } = {|1 + 2〉, |3 + 4〉, |5 + 6〉, , |(2l − 3)+(2l − 2)〉 }, {|η1i 〉 } = {|2 + 3〉 , |4 + 5〉 , |6 + 7〉 , , |(2l − 2) + (2l − 1)〉 } . The two sets {|η0i 〉 } and {|η1i 〉 } satisfy 〈η0i η0j 〉 = 〈η1i η1j 〉 = 0 if i ≠ j. 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 α 0 , Mm† Mm = 0 β
(4)
where α > β ≥ 0. For the quantum state |0〉A, it collapses into α|0〉A after measurement. For the quantum state |1〉Α, it collapses into β|1〉A after measurement. For the quantum states |0 ± 1〉A = 1/ 2 (|0〉 ± |1〉 ), they collapse into α|0〉 + β|1〉 . Hence, if and only if α = β, α|0〉 + β|1〉 = 2 ⋅ 1 (|0〉 + |1〉 ) holds. It produces contradiction between α2 + β 2
α2 + β 2
2
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 |ψi j 〉 b b i1 |ψi j 〉 c c i1 | ψ jd id 〉 |ψ j i 〉 e e |ψ j i 〉 f f
= |ib 〉A |jb + (jb + 1)〉B , = 0,
j1 = i1 + 1, i1 + 3, 2l − 3;
i2 = 1,
j1 = i1 + 1, i1 + 3, 2l − 3;
i2 = 1,
= |ic 〉A |jc − (jc + 1)〉B , = 0,
where b = 1, 2 and where c = 1, 2 and
j2 = i2 + 1, i2 + 3, , 2l − 2, j2 = i2 + 1, i2 + 3, , 2l − 2,
= |id + (id + 1) 〉A |jd 〉B ,
where d = 3 and i3 = 0, j3 = 0,
= |ie − (ie + 1) 〉A |je 〉B ,
where e = 3 and i3 = 0, j3 = 0,
= |i f 〉A |jf 〉B ,
where f = 3 and i3 = 0, j3 = 2l − 1
(5)
is entirely specified by those elements a00, a11, a01, a10 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 a00 = a11 = a, a01 = a10 = 0. For the states
Scientific Reports | 6:28864 | DOI: 10.1038/srep28864
4
www.nature.com/scientificreports/
|ψi j 〉 b b i1 |ψi j 〉 cc i1 |ψ j i 〉 d d |ψ j i 〉 f f
= |ib 〉A |jb + (jb + 1) 〉B , = 2,
j1 = i1 + 1, i1 + 3, 2l − 3;
i2 = 1,
j1 = i1 + 1, i1 + 3, 2l − 3;
i2 = 1,
= |ic 〉A |jc − (jc + 1) 〉B , = 2,
where b = 1, 2 and
= |id + (id + 1) 〉A |jd 〉B , = |i f 〉A |jf 〉B ,
where c = 1, 2 and
j2 = i2 + 1, i2 + 3, , 2l − 2, j2 = i2 + 1, i2 + 3, , 2l − 2,
where d = 1 and i1 = j1 = 1,
where f = 3 and i3 = 2, j3 = 2l − 1
(6)
and the subspace {|1〉, |2〉}A, we make the same argument. Then we get the result a11 = a22 = a, a12 = a21 = 0. For the states |ψi j 〉 bb i1 |ψ 〉 ic jc i1 |ψ j i 〉 dd
= |ib 〉A |jb + (jb + 1)〉B , = 2,
i2 = 3,
j1 = i1 + 1, i1 + 3, 2l − 3;
i2 = 3,
= |ic 〉A |jc − (jc + 1)〉B , = 2,
where b = 1, 2 and
j1 = i1 + 1, i1 + 3, 2l − 3;
= |id + (id + 1)〉A |jd 〉B ,
where c = 1, 2 and
j2 = i2 + 1, i2 + 3, , 2l − 2, j2 = i2 + 1, i2 + 3, , 2l − 2,
where d = 1 and i2 = j2 = 2
(7)
and subspace {|2〉, |3〉}A, we get the result a22 = a33 = a, a23 = a32 = 0. In the same way, for the subspace {|3〉, |4〉}A, …, the subspace {|2k − 2〉, |2k − 1〉 }A, we get the result a44 = = a2k−2,2k−2 = a2k−1,2k−1 = a,
Because POVM elements Mm† Mm ⁎
a = a,
a34 = a43 = = a2k−2,2k−1 = a2k−1,2k−2 = 0 .
is Hermitian, the equation (Mm† Mm )†
a20 =
⁎ a02 ,
a30 =
⁎ a03 ,
=
Mm† Mm
, a2k−1,2k−3 =
(8)
is correct. Then we obtain
a2⁎k−3,2k−1.
(9)
Now Mm† Mm can be rewritten as a 0 0 a Mm† Mm = ⁎ ⁎ a0,2k−2 a1,2k−2 ⁎ ⁎ a 0,2k−1 a1,2k−1
a02 a0,2k−1 0 a1,2k−1 , a 0 0 a
(10)
where a is a real number. We now consider the states |ψ j i 〉 = |i f 〉A |2l − 1〉B with f = 3, i3 = 0, 2 and the subspace {|0〉, |2〉}A. After f f Alice measures, the result is either the states orthogonal or distinguishing them outright. If the states are orthogonal, ⁎ we demand that 0|Mm+Mm |2〉 2l − 1|2l − 1〉 = a02 = 0 . So, we get a02 = a02 = 0 . For the states |ψib j 〉 = |ib 〉|jb + (jb + 1)〉 with ib = 3, jb = 2l−2 and |ψ j i f 〉 = |0〉A |2l − 1〉B, we get the same argument and we get b f ⁎ a03 = a03 = 0. For the subspace {|0〉, |4〉}A, {|0〉, |5〉}A, … and the subspace {|2k − 3〉, |2k − 1〉}A, we get the results ⁎ ⁎ ⁎ a04 = a04 = a05 = a05 = = a13 = a13 = = a2k−3,2k−1 = a2⁎k−3,2k−1 = 0 .
(11)
Now the Mm† Mm is proportional to the identity. However, if Alice distinguishes the state |ψi j 〉 = |i f 〉A |2l − 1〉B f f with f = 3, if = 0, 2, we get the result 〈ψi j |Mm† Mm |ψi j 〉 = 0. We can also have the result 〈ψi j |Mm† Mm |ψi j 〉 = a, f f f f f f f f therefore a = 0. It produces contradictory with the theorem of Walgate16. So, Mm† Mm 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)). |ψ1,2 〉 = |0〉A |1 ± 2〉B , |ψ3,4 〉 = |1〉A |2 ± 3〉B , |ψ5,6 〉 = |3〉A |0 ± 1〉B , |ψ7,8 〉 = |0 ± 1〉A |0〉B , |ψ9,10 〉 = |1 ± 2〉A |1〉B , |ψ11,12 〉 = |2 ± 3〉A |2〉B , |ψ13 〉 = |0〉A |3〉B ,
where |i ± j〉 =
1 2
|ψ14 〉 = |2〉A |0〉B ,
|ψ15 〉 = |2〉A |3〉B ,
|ψ16 〉 = |3〉A |3〉B ,
(12)
(|i〉 ± |j〉 ) with 0 ≤ i < j ≤ 3.
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)). |φ i g j 〉 = |ig 〉A |jg + (jg + 1)〉B , where g = 1, 2 and g
i1 = 0, 1, 2, , k − 1,
j1 = i1, i1 + 2, , 2l − 2 − i1;
i2 = 2k , 2k − 1, , k + 1, j2 = 2l − 3 − i2 , 2l − 1 − i2, , 3 + i2, Scientific Reports | 6:28864 | DOI: 10.1038/srep28864
5
www.nature.com/scientificreports/
|φ ih j 〉 = |ih 〉A |jh − (jh + 1)〉B , where h = 1, 2 and h
i1 = 0, 1, 2, , k − 1,
j1 = i1, i1 + 2, , 2l − 2 − i1;
i2 = 2k , 2k − 1, , k + 1, j2 = 2l − 3 − i2 , 2l − 1 − i2, , 3 + i2, |φ il j 〉 = |il + (il + 1)〉A |jl 〉B , where l = 1, 2 and l
j1 = 0, 1, , k − 1,
i1 = j1 + 1, j1 + 3, , 2k − 1 − j1 ;
j2 = 2l , 2l − 1, , 2l − k + 1, i2 = 2l − j2 , 2l + 2 − j2 , , 2k − 2 − (2l − j2 ), |φ i p j 〉 = |ip − (ip + 1)〉A |jp 〉B , where p = 1, 2 and p
j1 = 0, 1, , k − 1,
i1 = j1 + 1, j1 + 3, , 2k − 1 − j1 ;
j2 = 2l , 2l − 1, , 2l − k + 1, i2 = 2l − j2 , 2l + 2 − j2 , , 2k − 2 − (2l − j2 ), |φj 〉 = |k〉|j〉 , where j = k , k + 1, , 2l − k .
(13)
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 C 2k+1 ⊗ C 2l+1, there are (2k + 1)(2l + 1) orthogonal product quantum states |φi〉 (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 C 3 ⊗ C 3 (see Fig. 2(b)).
where |i ± j〉 =
|φ1,2 〉 = |0〉A |0 ± 1〉B ,
|φ3,4 〉 = |2〉A |1 ± 2〉B ,
|φ 7,8 〉 = |0 ± 1〉A |2〉B ,
|φ9 〉 = |1〉A |1〉B ,
1 2
|φ5,6 〉 = |1 ± 2〉A |0〉B , (14)
(|i〉 ± |j〉 ) with 0 ≤ i j
‖ [A i , A j ] ‖ ,
(24)
where [A, B] = AB − BA, A is the trace norm of the operator A, A = Tr AA† . In the Methods of Ma et al.’s24, they give the concrete calculation formula, i.e. suppose A = |φ〉 φ| and B = |ψ〉 ψ|. Denote φ ψ〉 = xe iθ with x ∈ [0, 1], θ ∈ [0, 2π ). Hence [A , B] = 2x 1 − x 2 . When φ ψ〉 = o or 1, [|φ〉, |ψ〉 ] = 0, when φ ψ〉 = 1/ 2 , [|φ〉, |ψ〉 ] = 1 and when φ ψ〉 = 1/2, [|φ〉, |ψ〉 ] = 0.87. Nextly, we give Lemma 2 as a standard of judging the indistinguishability of complete orthogonal product states. Lemma 224. For a complete set of C m ⊗ C n POPS, ε = {|ψi 〉 = |ai 〉 ⊗ |bi 〉 } with 〈ψi |ψj 〉 = 0,i ≠ j, the ε cannot be completely locally distinguished if and only if there exist subsets {|ψi′ 〉 = |ai′ 〉 ⊗ |bi′ 〉 } ⊆ ε, such that |ai′ 〉 and |bi′ 〉 are ′ ′ ′ all single sets, i.e. there exist m′ = dim (span {|ai′ 〉 }) linear independent {|aik 〉 }m k= 1 in {|ai 〉 } and n′ = dim (span {|bi 〉 }) n′ linear independent {|b j 〉 }l =1 in {|bi′ 〉 } satisfying l
0 < N (|ai1 〉 , |ai 2 〉 ) < N (|ai1 〉 , |ai 2 〉 , |ai 3 〉 ) < < N (|ai1 〉 , |ai 2 〉 , , |ai m′ 〉 ),
0 < N (|b j 〉 , |b j 〉 ) < N (|b j 〉 , |b j 〉 , |b j 〉 ) < < N (| b j 〉 , | b j 〉 , , | b j 〉 ) . 1
2
1
2
3
1
2
n′
(25)
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.
Scientific Reports | 6:28864 | DOI: 10.1038/srep28864
9
www.nature.com/scientificreports/
ε A = {|a1〉 = |0〉 , |a2 〉 = |1〉 , |a3 〉 = |2〉 , |a4 〉 = |3〉 , |a5 〉 = |4〉 , , |a2k−3 〉 = |2k − 4〉 , |a2k−2 〉 = |2k − 3〉 , |a2k−1〉 = |2k − 2〉 , |a2k 〉 = |2k − 1〉 , |a2k+1〉 = 1/ 2 (|0〉 + |1〉 ), |a2k+2 〉 = 1/ 2 (|0〉 − |1〉 ), |a2k+3 〉 = 1/ 2 (|1〉 + |2〉 ), |a2k+4 〉 = 1/ 2 (|1〉 − |2〉 ), |a2k+5 〉 = 1/ 2 (|2〉 + |3〉 ), |a2k+6 〉 = 1/ 2 (|2〉 − |3〉 ), |a2k+7 〉 = 1/ 2 (|3〉 + |4〉 ), |a2k+8 〉=1/ 2 (|3〉 − |4〉 ), |a2k+9 〉 = 1/ 2 (|4〉 + |5〉 ), |a2k+10 〉=1/ 2 (|4〉 − |5〉 ), |a2k+11〉 = 1/ 2 (|5〉 + |6〉 ), |a2k+12 〉 = 1/ 2 (|5〉 − |6〉 ), , |a6k−5 〉 = 1/ 2 (|2k − 3〉 + |2k − 2〉 ), |a6k−4 〉 = 1/ 2 (|2k − 3〉 − |2k − 2〉 ), |a6k−3 〉 = 1/ 2 (|2k − 2〉 + |2k − 1〉 ), |a6k−2 〉 = 1/ 2 (|2k − 2〉 − |2k − 1〉 )}. εB = {|b1〉 = 1/ 2 (|1〉 + |2〉 ), |b2 〉 = 1/ 2 (|1〉 − |2〉 ), |b3 〉 = 1/ 2 (|3〉 + |4〉 ), |b4 〉=1/ 2 (|3〉 − |4〉 ), |b5 〉 = 1/ 2 (|5〉 + |6〉 ), |b6 〉 = 1/ 2 (|5〉 − |6〉 ), |b 7 〉 = 1/ 2 (|7〉 + |8〉 ), |b8 〉 = 1/ 2 (|7〉 − |8〉 ), |b9 〉 = 1/ 2 (|9〉 + |10〉 ), |b10 〉 = 1/ 2 (|9〉 − |10〉 ), , |b2l−3 〉 = 1/ 2 (|2l − 3〉 + |2l − 2〉 ), |b2l−2 〉 = 1/ 2 (|2l − 3〉 − |2l − 2〉 ), b2l−1 = 1/ 2 (|2〉 + |3〉 ), b2l = 1/ 2 (|2〉 − |3〉 ), |b2l +1〉 = 1/ 2 (|4〉 + |5〉 ), |b2l +2 〉 = 1/ 2 (|4〉 − |5〉 ), |b2l +3 〉 = 1/ 2 (|6〉 + |7〉 ), |b2l +4 〉 = 1/ 2 (|6〉 − |7〉 ), , |b4l−5 〉 = 1/ 2 (|2l − 2〉 + |2l − 1〉 ), |b4l−4 〉 = 1/ 2 (|2l − 2〉 − |2l − 1〉 ), |b4l−3 〉 = 1/ 2 (|0〉 + |1〉 ), |b4l−2 〉 = 1/ 2 (|0〉 − |1〉 ), |b4l−1〉 = |0〉 , |b4l 〉 = |1〉 , |b4l +1〉 = |2〉 , |b4l +2 〉 = |3〉 , |b4l +3 〉 = |4〉 , |b4l +4 〉 = |5〉 , |b4l +5 〉 = |6〉 , , |b6l−3 〉 = |2l − 2〉 , |b6l−2 〉 = |2l − 1〉 }
(26)
Some duplicate items are removed in ε and ε . 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. A
B
N (|a2 〉 , |a2k+1〉 ) = 1, N (|a2 〉 , |a2k+1〉, |a2k+3 〉 ) = 2.87, N (|a2 〉, |a2k+1〉, |a2k+3 〉, |a2k+5 〉 ) = 3.74, N (|a2 〉, |a2k+1〉, |a2k+3 〉, |a2k+5 〉, |a2k+7 〉 ) = 4.61, , , N (|a2 〉, |a2k+1〉, |a2k+3 〉, , |a6k−5 〉 ) = 2 + (2k − 3) × 0.87 = 1.74k − 0.61, N (|a2 〉, |a2k+1〉, |a2k+3 〉, , |a6k−5 〉, |a6k−3 〉 ) = 2 + (2k − 2) × 0.87 = 1.74k + 0.26 .
(27)
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
Scientific Reports | 6:28864 | DOI: 10.1038/srep28864
10
www.nature.com/scientificreports/
N (| a 2 〉 , | a 2 k + 1 〉 ) = 1 < N (|a2 〉 , |a2k+1〉 , |a2k+3 〉 ) = 2.87 < N (|a2 〉 , |a2k+1〉 , |a2k+3 〉 , |a2k+5 〉 ) = 3.74 < N (|a2 〉 , |a2k+1〉 , |a2k+3 〉 , |a2k+5 〉, |a2k+7 〉 ) = 4.61
OPEN
LOCC indistinguishable orthogonal product quantum states Xiaoqian Zhang1, Xiaoqing Tan1, Jian Weng2 & Yongjun Li3
received: 14 January 2016 accepted: 06 June 2016 Published: 05 July 2016
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 C 2k+i ⊗ C 2l+j (i, j ∈ {0, 1} and i ≥ j ) and C 3k+i ⊗ C 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 C 3k+i ⊗ C 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 C 2k ⊗ C 2l and C 2k+1 ⊗ C 2l but also C 2k ⊗ C 2l+1 and C 2k+1 ⊗ C 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 C 2k ⊗ C 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 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 distinguished3–23 or not. In refs 3–12, 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 indistinguishability13–23 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 ⊗ 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 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 paper20. 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 1
Department of Mathematics, Jinan University, Guangzhou, P.R. China. 2Department of Computer Science, Jinan University, Guangzhou, P.R. China. 3School of Computer Science and Engineering, South China University of Technology, Guangzhou, P.R. China. Correspondence and requests for materials should be addressed to X.T. (email: [email protected]) Scientific Reports | 6:28864 | DOI: 10.1038/srep28864
1
www.nature.com/scientificreports/
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. 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 C 2k ⊗ C 2l than Wang et al.’s20. Wang et al.’s construction can be extended, but our construction in quantum system of C 2k+i ⊗ C 2l+j with i, j ∈ {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 C 3k+i ⊗ C 3l+j (i, j = 0, 1, 2), it contains not only C 2k ⊗ C 2l and C 2k+1 ⊗ C 2l but also C 2k ⊗ C 2l+1 and C 2k+1 ⊗ C 2l+1, so our construction in quantum system of C 3k+i ⊗ C 3l+j with i, j ∈ {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 C2k+i ⊗ C2l+j with k ≥ 1, l ≥ 1 and i, j ∈ {0, 1} (i ≥ j). Case 1. Firstly, we construct LOCC indistinguishable orthogonal product quantum states in quantum system of C 2k ⊗ C 2l (k, l ≥ 2) (see Fig. 1(a)) and give an example in the smallest dimension (see Fig. 2(a)).
Scientific Reports | 6:28864 | DOI: 10.1038/srep28864
2
www.nature.com/scientificreports/
Figure 2. The tiling structure of orthogonal product quantum states in quantum system of (a) C 4 ⊗ C 4, (b) C 3 ⊗ C 3, (c) C 5 ⊗ C 6, and (d) C 5 ⊗ C 5.
|ψib j 〉 = |ib 〉A |jb + (jb + 1)〉B , where b = 1, 2, 3 and b
i1 = 0, 2, 4, , min {2k , 2l} − 4, j1 = i1 + 1,i1 + 3, 2l − 3;
i2 = 1, 3, 5, , min {2k , 2l} − 3, j2 = i2 + 1, i2 + 3, , 2l − 2; i3 = 2k − 1,
j3 = 0;
|ψic j 〉 = |ic 〉A |jc − (jc + 1) 〉B , where c = 1, 2, 3 and c
i1 = 0, 2, 4, , min {2k , 2l} − 4, j1 = i1 + 1, i1 + 3, 2l − 3;
i2 = 1, 3, 5, , min {2k , 2l} − 3, j2 = i2 + 1, i2 + 3, , 2l − 2; i3 = 2k − 1,
j3 = 0;
|ψ j id 〉 = |id + (id + 1)〉A |jd 〉B , where d = 1, 2, 3 and d
j1 = 1, 3, 5, , min{2k − 3, 2l − 1}, i1 = j1 , j1 + 2, , 2k − 3;
j2 = 2, 4, 6, , min{2k − 2,2l − 2}, i2 = j2 , j2 + 2, , 2k − 2;
j3 = 0,
i3 = 0,2,4, , 2k − 4;
|ψ j ie 〉 = |ie − (ie + 1)〉A |je 〉B , where e = 1, 2, 3 and e
j1 = 1, 3, 5, , min{2k − 3, 2l − 1}, i1 = j1 , j1 + 2, , 2k − 3;
j2 = 2, 4, 6, , min{2k − 2, 2l − 2}, i2 = j2 , j2 + 2, , 2k − 2; j3 = 0,
Scientific Reports | 6:28864 | DOI: 10.1038/srep28864
i3 = 0, 2, 4, , 2k − 4;
3
www.nature.com/scientificreports/
|ψ j
f if
〉 = |i f 〉A |jf 〉B , where f = 1, 2, 3 and
j1 = 0, 2k − 1, 2k, , 2l − 1,
i1 = 2k − 2;
j3 = 2l − 1,
i3 =0, 2, 4, , min {2l − 2, 2k − 4} .
j2 = 3, 5, , 2k − 3, 2k − 1, 2k, 2k + 1, 2l − 1, i2 = 2k − 1;
Here |α ± β〉 =
1 2
(|α〉 ± |β〉 ). For example, |ie − (ie + 1) 〉A =
1 2
(1)
(|ie 〉 − |ie + 1〉 )A.
Proposition 1. In quantum system of C 2k ⊗ C 2l, there are 4 kl orthogonal product quantum states |ψi〉 (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 C 2 ⊗ C m to determine POVM elements Mm† Mm . A set of general C 2k ⊗ C 2k POVM elements Mm† Mm under the basis {|0〉 , |1〉 , , |2k − 1〉 }A can be expressed as follows a01 a00 a a11 10 Mm† Mm = a 2k−2,0 a2k−2,1 a2k−1,0 a2k−1,1
a0,2k−1 a1,2k−1
, a2k−2,2k−1 a2k−1,2k−1
(2)
where aij ≥ 0 and i, j ∈ {0, 1, 2, , 2k − 1} . Firstly, this selected sets {|0〉, |1〉}A, {|1〉, |2〉}A, …, and {|2k − 2〉, |2k − 1〉}A of states are of dimension C 2 ⊗ C 2k, 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 {|0〉, |1〉}A, there are quantum states |ψib j 〉 = |ib 〉A |jb + (jb + 1)〉B , where b = 1, 2 and b
i1 = 0, i2 = 1,
j1 = 1, 3, , 2l − 3;
j2 = 2, 4, , 2l − 2;
|ψ j id 〉 = |0 ± 1〉A |0〉B ,
(3)
d
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 {|0〉, |1〉}A. Suppose, the form |0〉A |η0i 〉B + |1〉A |η1i 〉B is set up in Eq. (23), where {|η0i 〉 } = {|1 + 2〉, |3 + 4〉, |5 + 6〉, , |(2l − 3)+(2l − 2)〉 }, {|η1i 〉 } = {|2 + 3〉 , |4 + 5〉 , |6 + 7〉 , , |(2l − 2) + (2l − 1)〉 } . The two sets {|η0i 〉 } and {|η1i 〉 } satisfy 〈η0i η0j 〉 = 〈η1i η1j 〉 = 0 if i ≠ j. 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 α 0 , Mm† Mm = 0 β
(4)
where α > β ≥ 0. For the quantum state |0〉A, it collapses into α|0〉A after measurement. For the quantum state |1〉Α, it collapses into β|1〉A after measurement. For the quantum states |0 ± 1〉A = 1/ 2 (|0〉 ± |1〉 ), they collapse into α|0〉 + β|1〉 . Hence, if and only if α = β, α|0〉 + β|1〉 = 2 ⋅ 1 (|0〉 + |1〉 ) holds. It produces contradiction between α2 + β 2
α2 + β 2
2
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 |ψi j 〉 b b i1 |ψi j 〉 c c i1 | ψ jd id 〉 |ψ j i 〉 e e |ψ j i 〉 f f
= |ib 〉A |jb + (jb + 1)〉B , = 0,
j1 = i1 + 1, i1 + 3, 2l − 3;
i2 = 1,
j1 = i1 + 1, i1 + 3, 2l − 3;
i2 = 1,
= |ic 〉A |jc − (jc + 1)〉B , = 0,
where b = 1, 2 and where c = 1, 2 and
j2 = i2 + 1, i2 + 3, , 2l − 2, j2 = i2 + 1, i2 + 3, , 2l − 2,
= |id + (id + 1) 〉A |jd 〉B ,
where d = 3 and i3 = 0, j3 = 0,
= |ie − (ie + 1) 〉A |je 〉B ,
where e = 3 and i3 = 0, j3 = 0,
= |i f 〉A |jf 〉B ,
where f = 3 and i3 = 0, j3 = 2l − 1
(5)
is entirely specified by those elements a00, a11, a01, a10 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 a00 = a11 = a, a01 = a10 = 0. For the states
Scientific Reports | 6:28864 | DOI: 10.1038/srep28864
4
www.nature.com/scientificreports/
|ψi j 〉 b b i1 |ψi j 〉 cc i1 |ψ j i 〉 d d |ψ j i 〉 f f
= |ib 〉A |jb + (jb + 1) 〉B , = 2,
j1 = i1 + 1, i1 + 3, 2l − 3;
i2 = 1,
j1 = i1 + 1, i1 + 3, 2l − 3;
i2 = 1,
= |ic 〉A |jc − (jc + 1) 〉B , = 2,
where b = 1, 2 and
= |id + (id + 1) 〉A |jd 〉B , = |i f 〉A |jf 〉B ,
where c = 1, 2 and
j2 = i2 + 1, i2 + 3, , 2l − 2, j2 = i2 + 1, i2 + 3, , 2l − 2,
where d = 1 and i1 = j1 = 1,
where f = 3 and i3 = 2, j3 = 2l − 1
(6)
and the subspace {|1〉, |2〉}A, we make the same argument. Then we get the result a11 = a22 = a, a12 = a21 = 0. For the states |ψi j 〉 bb i1 |ψ 〉 ic jc i1 |ψ j i 〉 dd
= |ib 〉A |jb + (jb + 1)〉B , = 2,
i2 = 3,
j1 = i1 + 1, i1 + 3, 2l − 3;
i2 = 3,
= |ic 〉A |jc − (jc + 1)〉B , = 2,
where b = 1, 2 and
j1 = i1 + 1, i1 + 3, 2l − 3;
= |id + (id + 1)〉A |jd 〉B ,
where c = 1, 2 and
j2 = i2 + 1, i2 + 3, , 2l − 2, j2 = i2 + 1, i2 + 3, , 2l − 2,
where d = 1 and i2 = j2 = 2
(7)
and subspace {|2〉, |3〉}A, we get the result a22 = a33 = a, a23 = a32 = 0. In the same way, for the subspace {|3〉, |4〉}A, …, the subspace {|2k − 2〉, |2k − 1〉 }A, we get the result a44 = = a2k−2,2k−2 = a2k−1,2k−1 = a,
Because POVM elements Mm† Mm ⁎
a = a,
a34 = a43 = = a2k−2,2k−1 = a2k−1,2k−2 = 0 .
is Hermitian, the equation (Mm† Mm )†
a20 =
⁎ a02 ,
a30 =
⁎ a03 ,
=
Mm† Mm
, a2k−1,2k−3 =
(8)
is correct. Then we obtain
a2⁎k−3,2k−1.
(9)
Now Mm† Mm can be rewritten as a 0 0 a Mm† Mm = ⁎ ⁎ a0,2k−2 a1,2k−2 ⁎ ⁎ a 0,2k−1 a1,2k−1
a02 a0,2k−1 0 a1,2k−1 , a 0 0 a
(10)
where a is a real number. We now consider the states |ψ j i 〉 = |i f 〉A |2l − 1〉B with f = 3, i3 = 0, 2 and the subspace {|0〉, |2〉}A. After f f Alice measures, the result is either the states orthogonal or distinguishing them outright. If the states are orthogonal, ⁎ we demand that 0|Mm+Mm |2〉 2l − 1|2l − 1〉 = a02 = 0 . So, we get a02 = a02 = 0 . For the states |ψib j 〉 = |ib 〉|jb + (jb + 1)〉 with ib = 3, jb = 2l−2 and |ψ j i f 〉 = |0〉A |2l − 1〉B, we get the same argument and we get b f ⁎ a03 = a03 = 0. For the subspace {|0〉, |4〉}A, {|0〉, |5〉}A, … and the subspace {|2k − 3〉, |2k − 1〉}A, we get the results ⁎ ⁎ ⁎ a04 = a04 = a05 = a05 = = a13 = a13 = = a2k−3,2k−1 = a2⁎k−3,2k−1 = 0 .
(11)
Now the Mm† Mm is proportional to the identity. However, if Alice distinguishes the state |ψi j 〉 = |i f 〉A |2l − 1〉B f f with f = 3, if = 0, 2, we get the result 〈ψi j |Mm† Mm |ψi j 〉 = 0. We can also have the result 〈ψi j |Mm† Mm |ψi j 〉 = a, f f f f f f f f therefore a = 0. It produces contradictory with the theorem of Walgate16. So, Mm† Mm 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)). |ψ1,2 〉 = |0〉A |1 ± 2〉B , |ψ3,4 〉 = |1〉A |2 ± 3〉B , |ψ5,6 〉 = |3〉A |0 ± 1〉B , |ψ7,8 〉 = |0 ± 1〉A |0〉B , |ψ9,10 〉 = |1 ± 2〉A |1〉B , |ψ11,12 〉 = |2 ± 3〉A |2〉B , |ψ13 〉 = |0〉A |3〉B ,
where |i ± j〉 =
1 2
|ψ14 〉 = |2〉A |0〉B ,
|ψ15 〉 = |2〉A |3〉B ,
|ψ16 〉 = |3〉A |3〉B ,
(12)
(|i〉 ± |j〉 ) with 0 ≤ i < j ≤ 3.
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)). |φ i g j 〉 = |ig 〉A |jg + (jg + 1)〉B , where g = 1, 2 and g
i1 = 0, 1, 2, , k − 1,
j1 = i1, i1 + 2, , 2l − 2 − i1;
i2 = 2k , 2k − 1, , k + 1, j2 = 2l − 3 − i2 , 2l − 1 − i2, , 3 + i2, Scientific Reports | 6:28864 | DOI: 10.1038/srep28864
5
www.nature.com/scientificreports/
|φ ih j 〉 = |ih 〉A |jh − (jh + 1)〉B , where h = 1, 2 and h
i1 = 0, 1, 2, , k − 1,
j1 = i1, i1 + 2, , 2l − 2 − i1;
i2 = 2k , 2k − 1, , k + 1, j2 = 2l − 3 − i2 , 2l − 1 − i2, , 3 + i2, |φ il j 〉 = |il + (il + 1)〉A |jl 〉B , where l = 1, 2 and l
j1 = 0, 1, , k − 1,
i1 = j1 + 1, j1 + 3, , 2k − 1 − j1 ;
j2 = 2l , 2l − 1, , 2l − k + 1, i2 = 2l − j2 , 2l + 2 − j2 , , 2k − 2 − (2l − j2 ), |φ i p j 〉 = |ip − (ip + 1)〉A |jp 〉B , where p = 1, 2 and p
j1 = 0, 1, , k − 1,
i1 = j1 + 1, j1 + 3, , 2k − 1 − j1 ;
j2 = 2l , 2l − 1, , 2l − k + 1, i2 = 2l − j2 , 2l + 2 − j2 , , 2k − 2 − (2l − j2 ), |φj 〉 = |k〉|j〉 , where j = k , k + 1, , 2l − k .
(13)
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 C 2k+1 ⊗ C 2l+1, there are (2k + 1)(2l + 1) orthogonal product quantum states |φi〉 (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 C 3 ⊗ C 3 (see Fig. 2(b)).
where |i ± j〉 =
|φ1,2 〉 = |0〉A |0 ± 1〉B ,
|φ3,4 〉 = |2〉A |1 ± 2〉B ,
|φ 7,8 〉 = |0 ± 1〉A |2〉B ,
|φ9 〉 = |1〉A |1〉B ,
1 2
|φ5,6 〉 = |1 ± 2〉A |0〉B , (14)
(|i〉 ± |j〉 ) with 0 ≤ i j
‖ [A i , A j ] ‖ ,
(24)
where [A, B] = AB − BA, A is the trace norm of the operator A, A = Tr AA† . In the Methods of Ma et al.’s24, they give the concrete calculation formula, i.e. suppose A = |φ〉 φ| and B = |ψ〉 ψ|. Denote φ ψ〉 = xe iθ with x ∈ [0, 1], θ ∈ [0, 2π ). Hence [A , B] = 2x 1 − x 2 . When φ ψ〉 = o or 1, [|φ〉, |ψ〉 ] = 0, when φ ψ〉 = 1/ 2 , [|φ〉, |ψ〉 ] = 1 and when φ ψ〉 = 1/2, [|φ〉, |ψ〉 ] = 0.87. Nextly, we give Lemma 2 as a standard of judging the indistinguishability of complete orthogonal product states. Lemma 224. For a complete set of C m ⊗ C n POPS, ε = {|ψi 〉 = |ai 〉 ⊗ |bi 〉 } with 〈ψi |ψj 〉 = 0,i ≠ j, the ε cannot be completely locally distinguished if and only if there exist subsets {|ψi′ 〉 = |ai′ 〉 ⊗ |bi′ 〉 } ⊆ ε, such that |ai′ 〉 and |bi′ 〉 are ′ ′ ′ all single sets, i.e. there exist m′ = dim (span {|ai′ 〉 }) linear independent {|aik 〉 }m k= 1 in {|ai 〉 } and n′ = dim (span {|bi 〉 }) n′ linear independent {|b j 〉 }l =1 in {|bi′ 〉 } satisfying l
0 < N (|ai1 〉 , |ai 2 〉 ) < N (|ai1 〉 , |ai 2 〉 , |ai 3 〉 ) < < N (|ai1 〉 , |ai 2 〉 , , |ai m′ 〉 ),
0 < N (|b j 〉 , |b j 〉 ) < N (|b j 〉 , |b j 〉 , |b j 〉 ) < < N (| b j 〉 , | b j 〉 , , | b j 〉 ) . 1
2
1
2
3
1
2
n′
(25)
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.
Scientific Reports | 6:28864 | DOI: 10.1038/srep28864
9
www.nature.com/scientificreports/
ε A = {|a1〉 = |0〉 , |a2 〉 = |1〉 , |a3 〉 = |2〉 , |a4 〉 = |3〉 , |a5 〉 = |4〉 , , |a2k−3 〉 = |2k − 4〉 , |a2k−2 〉 = |2k − 3〉 , |a2k−1〉 = |2k − 2〉 , |a2k 〉 = |2k − 1〉 , |a2k+1〉 = 1/ 2 (|0〉 + |1〉 ), |a2k+2 〉 = 1/ 2 (|0〉 − |1〉 ), |a2k+3 〉 = 1/ 2 (|1〉 + |2〉 ), |a2k+4 〉 = 1/ 2 (|1〉 − |2〉 ), |a2k+5 〉 = 1/ 2 (|2〉 + |3〉 ), |a2k+6 〉 = 1/ 2 (|2〉 − |3〉 ), |a2k+7 〉 = 1/ 2 (|3〉 + |4〉 ), |a2k+8 〉=1/ 2 (|3〉 − |4〉 ), |a2k+9 〉 = 1/ 2 (|4〉 + |5〉 ), |a2k+10 〉=1/ 2 (|4〉 − |5〉 ), |a2k+11〉 = 1/ 2 (|5〉 + |6〉 ), |a2k+12 〉 = 1/ 2 (|5〉 − |6〉 ), , |a6k−5 〉 = 1/ 2 (|2k − 3〉 + |2k − 2〉 ), |a6k−4 〉 = 1/ 2 (|2k − 3〉 − |2k − 2〉 ), |a6k−3 〉 = 1/ 2 (|2k − 2〉 + |2k − 1〉 ), |a6k−2 〉 = 1/ 2 (|2k − 2〉 − |2k − 1〉 )}. εB = {|b1〉 = 1/ 2 (|1〉 + |2〉 ), |b2 〉 = 1/ 2 (|1〉 − |2〉 ), |b3 〉 = 1/ 2 (|3〉 + |4〉 ), |b4 〉=1/ 2 (|3〉 − |4〉 ), |b5 〉 = 1/ 2 (|5〉 + |6〉 ), |b6 〉 = 1/ 2 (|5〉 − |6〉 ), |b 7 〉 = 1/ 2 (|7〉 + |8〉 ), |b8 〉 = 1/ 2 (|7〉 − |8〉 ), |b9 〉 = 1/ 2 (|9〉 + |10〉 ), |b10 〉 = 1/ 2 (|9〉 − |10〉 ), , |b2l−3 〉 = 1/ 2 (|2l − 3〉 + |2l − 2〉 ), |b2l−2 〉 = 1/ 2 (|2l − 3〉 − |2l − 2〉 ), b2l−1 = 1/ 2 (|2〉 + |3〉 ), b2l = 1/ 2 (|2〉 − |3〉 ), |b2l +1〉 = 1/ 2 (|4〉 + |5〉 ), |b2l +2 〉 = 1/ 2 (|4〉 − |5〉 ), |b2l +3 〉 = 1/ 2 (|6〉 + |7〉 ), |b2l +4 〉 = 1/ 2 (|6〉 − |7〉 ), , |b4l−5 〉 = 1/ 2 (|2l − 2〉 + |2l − 1〉 ), |b4l−4 〉 = 1/ 2 (|2l − 2〉 − |2l − 1〉 ), |b4l−3 〉 = 1/ 2 (|0〉 + |1〉 ), |b4l−2 〉 = 1/ 2 (|0〉 − |1〉 ), |b4l−1〉 = |0〉 , |b4l 〉 = |1〉 , |b4l +1〉 = |2〉 , |b4l +2 〉 = |3〉 , |b4l +3 〉 = |4〉 , |b4l +4 〉 = |5〉 , |b4l +5 〉 = |6〉 , , |b6l−3 〉 = |2l − 2〉 , |b6l−2 〉 = |2l − 1〉 }
(26)
Some duplicate items are removed in ε and ε . 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. A
B
N (|a2 〉 , |a2k+1〉 ) = 1, N (|a2 〉 , |a2k+1〉, |a2k+3 〉 ) = 2.87, N (|a2 〉, |a2k+1〉, |a2k+3 〉, |a2k+5 〉 ) = 3.74, N (|a2 〉, |a2k+1〉, |a2k+3 〉, |a2k+5 〉, |a2k+7 〉 ) = 4.61, , , N (|a2 〉, |a2k+1〉, |a2k+3 〉, , |a6k−5 〉 ) = 2 + (2k − 3) × 0.87 = 1.74k − 0.61, N (|a2 〉, |a2k+1〉, |a2k+3 〉, , |a6k−5 〉, |a6k−3 〉 ) = 2 + (2k − 2) × 0.87 = 1.74k + 0.26 .
(27)
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
Scientific Reports | 6:28864 | DOI: 10.1038/srep28864
10
www.nature.com/scientificreports/
N (| a 2 〉 , | a 2 k + 1 〉 ) = 1 < N (|a2 〉 , |a2k+1〉 , |a2k+3 〉 ) = 2.87 < N (|a2 〉 , |a2k+1〉 , |a2k+3 〉 , |a2k+5 〉 ) = 3.74 < N (|a2 〉 , |a2k+1〉 , |a2k+3 〉 , |a2k+5 〉, |a2k+7 〉 ) = 4.61
