Local discrimination of qudit lattice states

In this paper, we investigate the distinguishability of qudit lattice states under local operations and classical communication (LOCC) in C d ⊗ C d with d = ∏ j = 1 r p j . Firstly, we give a decomposition of the basis B d of lattice unitary matrices. This basis B d can be decomposed into ∏ i = 1 r ( p i + 1 ) maximal commuting sets which are all cyclic groups. Meanwhile, we define a generator of each lattice unitary matrix and present that the generator can be easily calculated. Secondly, based on the decomposition of lattice unitary matrices, we show that a set L of qudit lattice states can be distinguished by one-way LOCC if | G L | < ∏ i = 1 r ( p i + 1 ) , where G L is the set of the generators of all the elements in the difference set of L . Our method can be used to distinguish lattice states which cannot be distinguished by previous results in Li et al. (Sci China-Phys Mech Astron 63:280312, 2020). Applying our method, we also find an interesting phenomenon, that is, in C d ⊗ C d there exists a set of qudit lattice states which has such property: although a set consisting of i -th subsystem of qudit lattice states cannot be locally distinguished in C p i ⊗ C p i , the set of qudit lattice states can be distinguished by one-way LOCC. Finally, we give a necessary and sufficient condition for d qudit lattice states to be indistinguishable by LOCC.