One-way local distinguishability of generalized Bell states in arbitrary dimension

Necessary and sufficient conditions for local discrimination of generalized Bell states: finding out all locally indistinguishable sets of generalized Bell states

“…Firstly, some practical and effective sufficient conditions for GBS sets in an arbitrary dimensional system are given. These sufficient conditions are easy to use and widely applicable, they cover the previous conclusions including Fan , s and Wang et al , s results [6,22]. And then we show that these sufficient conditions are also necessary for local discrimination of all 4-GBS sets in C 4 ⊗ C 4 , and we give a table containing all the locally indistinguishable (standard) 4-GBS sets, which can be used to judge the local distinguishability of a 4-GBS set quickly.…”

“…In C 5 ⊗ C 5 , one of the sufficient conditions is proved to be necessary, thus a necessary and sufficient condition for one-way local discrimination of GBS sets is provided. The necessary and sufficient condition is concise and computable, and it gives an affirmative answer to the case d = 5 of the problem proposed by Wang et al [22,25].…”

“…Corollary 1 implies that the following case d = 5 of Wang et al' problem [22,Problem] is true. When d is a composite number, it obvious that Problem 1 is not true (see Examples 1-3).…”

“…When d is a composite number, it obvious that Problem 1 is not true (see Examples 1-3). When d is a prime, Wang et al [22] disproved the case d = 101 of Problem 1 or Conjecture 1 by finding a counterexample in C 101 ⊗ C 101 . The counterexample suggests that Problem 1 (or Conjecture 1) may not be true.…”

“…The researchers have constructed many examples of locally indistinguishable GBS sets [16][17][18][19][20][21], for example, Bandyopadhyay et al [16] showed oneway LOCC indistinguishable 4-GBS sets in C 4 ⊗ C 4 and C 5 ⊗ C 5 . On the other hand, some sufficient conditions for local discrimination of GBSs are obtained by using admissible solution set and nonadmissible solution set [22,24]. Although many advances have been made, they are mainly concrete examples of local indistinguishability and sufficient conditions for local distinguishability.…”