Local discrimination of qudit lattice states via commutativity

Abstract

Qudit lattice states, as the generalization of qubit lattice states, are the maximally entangled states determined by qudit lattice unitaries in a ${p}^{r}\ensuremath{\bigotimes}{p}^{r}$ quantum system with $p$ being a prime and $r$ being an integer. Based on the partitions of qudit lattice unitaries into commuting sets, we present a sufficient condition for local discrimination of qudit lattice states, in which the commutativity plays an efficient role. It turns out that any set of $l$ qudit lattice states with $2\ensuremath{\le}l\ensuremath{\le}{p}^{r}$, including $k\ensuremath{\le}l$ mutually commuting qudit lattice unitaries and satisfying $l(l\ensuremath{-}1)\ensuremath{-}(k+1)(k\ensuremath{-}2)\ensuremath{\le}2{p}^{r}$, can be locally distinguished, not only extending Fan's result [H. Fan, Phys. Rev. Lett. 92, 177905 (2004)] to the prime power quantum system but also involving the local discrimination of a larger number of maximally entangled states.