General existence of locally distinguishable maximally entangled states only with two-way classical communication

It is known that there exist two locally operational settings, local operations with one-way and two-way classical communication. And recently, some sets of maximally entangled states have been built in specific dimensional quantum systems, which can be locally distinguished only with two-way classical communication. In this paper, we show the existence of such sets is general, through constructing such sets in all the remaining quantum systems. Specifically, such sets including p or n maximally entangled states will be built in the quantum system of (np − 1) ⊗ (np − 1) with n ≥ 3 and p being a prime number, which completes the picture that such sets do exist in every possible dimensional quantum system.

However, all the existing results do not cover every possible dimensional quantum system, that is, it is still unknown whether the existence of 2-LOCC sets is general or restricted to specific dimensions. In this paper, we answer this question in positive. We start with presenting a decomposition of all the dimension numbers D Z where p is a prime number. Combined with the previous result that there exists a 2-LOCC set in even dimensional quantum system, we only need to build 2-LOCC sets in the remaining dimensions {np − 1, n ≥ 3} because there is no 2-LOCC sets in 3 ⊗ 3 system. The detailed form, np − 1, n ≥ 3, motivates us to consider using the second 2-LOCC distinguishing protocol in ref. 15 to complete the picture. Before constructing more 2-LOCC sets, we should make a simplification of the referred 2-LOCC distinguishing protocol. Specifically, Bob's local measurement elements become less, which makes the following construction more efficient. Next, we build a + 1 mutually orthogonal MESs in d ⊗ d quantum system with d = a + (a + 1)t base on the simplified 2-LOCC distinguishing protocol. Finally, because of np − 1 = p(n − 1) + (p − 1) = n(p − 1) + (n − 1), n ≥ 3, we can construct p or n MESs as 2-LOCC sets easily in the remaining quantum systems, which ensures the general existence of 2-LOCC sets in every possible dimensional quantum system.

Notations. Consider a bipartite quantum system
} d be the computational basis, and the standard bipartite maximally entangled state in this system is expressed as We can also define the bit flip and phase flip operators to be , where |Φ 〉 is the standard MES. There has already been a necessary and sufficient condition 15 , which will be employed in the following part repeatedly. Decomposition of all the dimension numbers. So far, it has already been proved in refs 15 and 16 that there exist 2-LOCC sets in the quantum system of 2m ⊗ 2m, (3r + 2) ⊗ (3r + 2), 4m ⊗ 4m, 2Rm ⊗ 2Rm and 4Rm ⊗ 4Rm, where R = 2 r with r and m being a positive integer. But obviously, the union of {2m}, {3r + 2}, {4m}, {2Rm} and {4Rm} does not cover all the dimension numbers, or to say all the positive integers greater than one, which motivates us to generalize the result. That is, we want to look for some 2-LOCC set(s) in every-dimension quantum system. Before that, we should be clear about the mathematical structure of all the dimension numbers. Therefore, we will start with a decomposition of all the positive integers, denoted by  + , as the following lemma, which may not be the best one but can satisfy our requirements.

Lemma 2. The set of all dimension numbers, denoted by
Proof. It is obvious that every number in the union Next, we will explain each positive number in  + can be expressed as an element in the union.
As the fundamental theorem of arithmetic stated, every integer greater than one either is prime itself or is the product of prime numbers, and that this product is unique, up to the order of the factors. Thus we have  20 . Thus the dimensions of quantum system, denoted by , we consider to build 2-LOCC sets should be not smaller than 4, that is, And when the dimension number belongs to {p − 1}, which must be even, 2-LOCC sets have been presented in ref. 15. Therefore, our objective becomes to construct 2-LOCC sets in the quantum system of dimension numbers d ∈ {np − 1, n ≥ 3}, where p is a prime number. We rewritten the form of "np − 1, n ≥ 3" to which makes us associate the structure of the second kind of 2-LOCC sets in ref. 15 in the (3r + 2) ⊗ (3r + 2) quantum system. However, to make full use of that 2-LOCC distinguishing protocol efficiently, we will give it a simpler presentation. That is, a simplified 2-LOCC protocol, we prefer omitting the word "distinguishing" for simplicity, will be discussed as in the following subsection.
Simplified 2-LOCC protocol. In ref. 15, the author has already proposed two triple MESs sets, which can be distinguished by 2-LOCC but not by 1-LOCC. The 1-LOCC indistinguishability of both sets has been assured by the above sufficient and necessary condition, while the 2-LOCC distinguishability has been obtained by two different protocols. We have successfully employed the first 2-LOCC protocol to construct more 2-LOCC sets in ref. 16. As for the second 2-LOCC protocol to distinguish the three MESs below, actually, it can be simplified further.
We will present the detailed 2-LOCC distinguishing protocol, which is simpler in Bob's choice of local measurement. First, Alice does the same measurement as is shown in ref. 15 If Alice's outcome is k = 0, without loss of generality, then the original 3 local unitaries U i will become Next, we will build a simpler W, in which each column acts as Bob's measurement element, which satisfies the condition that † † W U A U W ( ) j T k i T has diagonal element equal to zero. After Bob transforming this measurement result to Alice, she can finish the local discrimination finally. The whole distinguishing process can be described in the following Fig. 1.
The above 2-LOCC protocol has been much simplified compared to the one given in the ref. 15, because we need only one basis to form the rank-1 measurement operators. That is, we do not need that many measurement elements (u, v) to distinguish the three MESs using local operations and two-way classical communication.
Constructing more 2-LOCC sets. Taking use of the above simplified 2-LOCC protocol, we will directly build more 2-LOCC sets of orthogonal MESs in this subsection, which can help us to generalize the existence of 2-LOCC in every possible dimensional quantum system in the next subsection.
Consider the following a + 1 mutually orthogonal MESs in d ⊗ d quantum system with d = a + (a + 1)r and r being an interger, where Proof. We will prove the case of r = 1 without loss of generality, because the similar proof method can work for other cases of r > 1. To show the 1-LOCC indistinguishability of the a + 1 maximally entangled states, we assume there exists a POVM measurement {M k } with every operators M k all rank-1 to discriminate the above set of states perfectly, which is the necessary and sufficient condition 15

Tr U M U Tr A S S Tr B P P
Because the specific forms of S i ,  ∈ i a and P i ,  ∈ + i a 1 , we can write Eq. (4) partly in detail.
due to the first two eqnarrays in Eq. (3), and further  . Therefore, It is a contradiction to the local indistinguishability of a +1 maximally entangled states in a ⊗ a system. So the above set is one-way LOCC indistinguishable.
Then we will prove the 2-LOCC distinguishability through presenting a detailed protocol as follows, which is based on the protocol given in ref. 15  If her outcome is k, then the original a + 1 local unitaries U i will become Step 2. To discriminate the present a + 1 states with certainty, Bob should find out rank-1 measurement operators, one column of W, such that . As is referred, the eigenvectors |e l 〉 of S a can make 〈 | | 〉 = e S S e 0 l i j T l , for So according to Eq. (5), if we take v k = λ l , where λ l is the eigenvalue of S a corresponding to |e l 〉 , then Therefore, the sum of all these rank-1 operators is Step 3. According to the necessary and sufficient condition in ref. 15 General existence of 2-LOCC sets. Based on the analysis of all dimension numbers previously, we only need to construct 2-LOCC sets in the quantum system of dimensions d ∈ {np − 1, n ≥ 3} actually. The above "a + 1" orthogonal MESs in the quantum system of [(a + 1)r + a] ⊗ [(a + 1)r + a] can help us to explain the existence of 2-LOCC sets in d ⊗ d quantum system with d ∈ {np − 1, n ≥ 3}, which will prove the fact that 2-LOCC sets are ubiquitous regardless of the dimension of quantum system.
As has already pointed out, we suppose either a = p − 1, r = n − 1 or a = n − 1, r = p − 1, and both cases work successfully. That means, there are at least two 2-LOCC sets can be built in the quantum system of (np − 1) ⊗ (np − 1). If the dimension number plus one, i.e., np, have two or more decompositions, then we have more choices to find out 2-LOCC sets. The relationship between the dimension number and the cardinality of a 2-LOCC set in the corresponding quantum system can be shown completely in Table 1. That is, if the dimension is p − 1 (p ≥ 5), then there exists a 2-LOCC sets including 3 MESs 15 . While if the dimension can be decomposed to np − 1 (n ≥ 3, p ≥ 2), then there exist at least two 2-LOCC sets of p or n MESs. Until now, we can claim the fact that 2-LOCC sets are general existed regardless of the dimension of quantum system.  10 9 I