Local discrimination of qudit lattice states via commutativity

Qudit lattice states, as the generalization of qubit lattice states, are the maximally entangled states determined by qudit lattice unitaries in a pʳpʳ 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≤l≤pʳ, including k≤l mutually commuting qudit lattice unitaries and satisfying l(l-1)-(k+1)(k-2)≤2pʳ, 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.