Necessary condition for local distinguishability of maximally entangled states: Beyond orthogonality preservation

The (im)possibility of local distinguishability of orthogonal multipartite quantum states still remains an intriguing question. Beyond ${\mathbb{C}}^3\otimes {\mathbb{C}}^3$, the problem remains unsolved even for maximally entangled states (MESs). So far, the only known condition for the local distinguishability of states is the well-known orthogonality preservation (OP). Using an upper bound on the locally accessible information for bipartite states, we derive a very simple necessary condition for any set of pairwise orthogonal MESs in ${\mathbb{C}}^d\otimes {\mathbb{C}}^d$ to be perfectly locally distinguishable. It is seen that particularly when the number of pairwise orthogonal MES states in ${\mathbb{C}}^d\otimes {\mathbb{C}}^d$ is equal to $d$, then this necessary condition, along with the OP condition, imposes more constraints (for said states to be perfectly locally distinguishable) than the OP condition does. When testing this condition for the local distinguishability of all sets of four generalized Bell states in ${\mathbb{C}}^4\otimes {\mathbb{C}}^4$, we find that it is not only necessary but also sufficient to determine their local distinguishability. This demonstrates that the aforementioned upper bound may play a significant role in the general scenario of local distinguishability of bipartite states.