General existence of locally distinguishable maximally entangled states only with two-way classical communication

It is known that there exist two locally operational settings, local operations with one-way and two-way classical communication. And recently, some sets of maximally entangled states have been built in specific dimensional quantum systems, which can be locally distinguished only with two-way classical communication. In this paper, we show the existence of such sets is general, through constructing such sets in all the remaining quantum systems. Specifically, such sets including p or n maximally entangled states will be built in the quantum system of (np − 1) ⊗ (np − 1) with n ≥ 3 and p being a prime number, which completes the picture that such sets do exist in every possible dimensional quantum system.