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 nonmaximally entangled states in the task of distinguishing quantum channels. We provide a proof that for the single-shot discrimination of two unitary channels, entangled and product states are operationally equivalent. However, we identify pairs of unitaries that are perfectly distinguishable using a nonmaximally entangled state, but not with a maximally entangled one. This contrast becomes more pronounced when the number of unitaries exceeds two. In every $\text{dimension}\phantom{\rule{0.16em}{0ex}}\ensuremath{\geqslant}3$, we show that there exists a class of unitaries that is indistinguishable under maximally entangled probes, yet perfectly distinguishable using product or nonmaximally entangled inputs. Another interesting set of unitaries in every $\text{dimension}\phantom{\rule{0.16em}{0ex}}\ensuremath{\geqslant}3$ has been presented where only a nonmaximally entangled state acts as a successful probe, while product states and maximally entangled states cannot.