Abstract
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.
Highlights
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
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 in the remaining quantum systems, which ensures the general existence of 2-LOCC sets in every possible dimensional quantum system
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
Summary
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 = 2r with r and m being a positive integer. When the dimension number belongs to {p − 1}, which must be even, 2-LOCC sets have been presented in ref. To show the 1-LOCC indistinguishability of the a + 1 maximally entangled states, we assume there exists a POVM measurement {Mk} with every operators Mk all rank-1 to discriminate the above set of states perfectly, which is the necessary and sufficient condition[15] of 1-LOCC distinguishability. To discriminate the present a + 1 states with certainty, Bob should find out rank-1 measurement operators, one column of W, such that ( ) (u†, v†)(VjT )†ViT u v
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