Local discrimination of qudit lattice states

Abstract

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.