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 \({\mathbb {C}}^{d}\otimes {\mathbb {C}}^{d}\) with \(d=\prod _{j=1}^{r} p_{j}\). Firstly, we give a decomposition of the basis \({\mathcal {B}}_{d}\) of lattice unitary matrices. This basis \({\mathcal {B}}_{d}\) can be decomposed into \(\prod _{i=1}^{r}(p_{i}+1)\) 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 \({\mathcal {L}}\) of qudit lattice states can be distinguished by one-way LOCC if \(|{\mathcal {G}}_{{\mathcal {L}}}|<\prod _{i=1}^{r}(p_{i}+1)\), where \({\mathcal {G}}_{{\mathcal {L}}}\) is the set of the generators of all the elements in the difference set of \({\mathcal {L}}\). 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 \({\mathbb {C}}^{d}\otimes {\mathbb {C}}^{d}\) 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 \(\mathbb {C}^{p_{i}}\otimes {\mathbb {C}}^{p_{i}}\), 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.