Local discrimination of generalized Bell states via commutativity

Abstract

We study the distinguishability of generalized Bell states under local operations and classical communication. We introduce the concept of a maximally commutative set (MCS), a subset of generalized Pauli matrices whose elements are mutually commutative, and there is no other generalized Pauli matrix that commutes with all the elements of this set. We find that MCS can be considered a detector for the local distinguishability of a set S of generalized Bell states. In fact, we get an efficient criterion. That is, if the difference set ΔS of S is disjoint with or completely contained in some MCS, then the set S is locally distinguishable. Furthermore, we give a useful characterization of MCS for arbitrary dimensions, which provides great convenience for detecting the local discrimination of generalized Bell states. Our method can be generalized to more general settings which contain the lattice qudit basis. The results of Fan [Phys. Rev. Lett. 92, 177905 (2004)PRLTAO0031-900710.1103/PhysRevLett.92.177905], Tian et al. [Phys. Rev. A 92, 042320 (2015)PLRAAN1050-294710.1103/PhysRevA.92.042320], and a recent work Yuan et al. [arXiv:2109.07390] can be deduced as special cases of our result.