Novel method for one-way local distinguishability of generalized Bell states in arbitrary dimension
Ying‐Hui Yang, Rang-Yang Yan, Xiaoli Wang, Jiangtao Yuan, Hui-Juan Zuo · Journal of Physics A Mathematical and Theoretical · 2021
Abstract In this paper the local distinguishability of generalized Bell states in arbitrary dimension is investigated. We firstly study the decomposition of a basis which consists of d 2 number of generalized Pauli matrices. We discover that this basis is equal to the union of D number of different sets, where D = 2 ϕ ( d ) ∑ t ∈ Z d gcd ( t , d ) = 1 ∑ i = 2 ⌊ d t ⌋ ϕ ( i ) + 1 and ϕ is Euler ϕ -function. Then we define the generator of the matrices in this decomposition, and exhibit an algorithm to calculate generators of a given set of matrices. This algorithm shows that generators of a given set can be calculated simply and efficiently. Secondly, we show that a set L of GBSs can be distinguished by one-way LOCC if the cardinality of G L is less than Dϕ ( d ), where G L is a set of generators of all the elements in difference set of a set L of GBSs. The previous results in [2004 Phys. Rev. Lett. 92 177905; 2019 Phys. Rev. A 99 022307; 2021 Quantum Inf. Process. 20 52] can be covered by our result. Finally, for the uncovered cases in [2021 Quantum Inf. Process. 20 52], we give a new result to partly solve that problem.