Strong Converse Exponent for Entanglement-Assisted Communication

We determine the exact strong converse exponent for entanglement-assisted classical communication of a quantum channel. Our main contribution is the derivation of an upper bound for the strong converse exponent which is characterized by the sandwiched R\'enyi divergence. It turns out that this upper bound coincides with the lower bound of Gupta and Wilde (Commun Math Phys 334:867--887, 2015). Thus, the strong converse exponent follows from the combination of these two bounds. Our result has two implications. Firstly, it implies that the exponential bound for the strong converse property of quantum-feedback-assisted classical communication, derived by Cooney, Mosonyi and Wilde (Commun Math Phys 344:797--829, 2016), is optimal. This answers their open question in the affirmative. Hence, we have determined the exact strong converse exponent for this problem as well. Secondly, due to an observation of Leung and Matthews, it can be easily extended to deal with the transmission of quantum information under the assistance of entanglement or quantum feedback, yielding similar results. The above findings provide, for the first time, a complete operational interpretation to the channel's sandwiched R\'enyi information of order $\alpha > 1$.