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 circle times C-d with d = Pi(r)(j=1) p(j). Firstly, we give a decomposition of the basis B-d of lattice unitary matrices. This basis B-d can be decomposed into Pi(r)(i=1) (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 vertical bar G(L)vertical bar < Pi(r)(i=1) (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 circle times 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-pi circle times C-pi, 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.