Limitation of maximally entangled probes for single-shot distinguishability of unitaries

There have been many instances where the maximally entangled state as a probe acts better than the product and the non-maximally entangled states in the task of distinguishing quantum channels. We provide a proof that for single-shot discrimination of two unitary channels, entangled and product states are operationally equivalent. But we identify pairs of unitaries that are perfectly distinguishable using a non-maximally entangled state, but not with a maximally entangled one. This contrast becomes more pronounced when the number of unitaries exceeds two. In every dimension $\geqslant 3$, we show that there exists a class of unitaries that are indistinguishable under maximally entangled probes, yet perfectly distinguishable using product or non-maximally entangled inputs. Another interesting set of unitaries in every dimension $\geqslant 3$ has been presented where only non-maximally entangled state acts as the successful probe, while product states and maximally entangled states cannot.