Strong converse for the capacity of quantum Gaussian channels

We prove that a strong converse theorem holds for the classical capacity of all phase-insensitive bosonic Gaussian channels, when imposing a maximum photon number constraint on the inputs of the channel. This class is a natural extension of classical continuous Gaussian channels, and the well studied pure-loss, thermal, additive noise, and amplifier channels are all in this class of channels. The statement of the strong converse theorem is that the probability of correctly decoding a classical message rapidly converges to zero in the limit of many channel uses if the communication rate exceeds the classical capacity. We prove this theorem by relating the success probability of any code with its rate of data transmission, the effective dimension of the channel output space, and the purity of the channel as quantified by the minimum output entropy. Our result bolsters the understanding of the classical capacity of these channels by establishing it as a sharp dividing line between possible and impossible communication rates over them.