We propose a framework to variationally obtain detectable capacity bounds for quantum channels. The proposal of this framework is motivated by the difficulty of estimating the von Neumann entropy of an unknown quantum state. Instead of estimating the von Neumann entropy at the channel output, we propose to estimate the state purity---which can be measured by a single measurement setting---followed by bounding the von Neumann entropy from above and below. This procedure leads to new upper and lower bounds on various communication rates of quantum channels for some fixed input states. Then, by utilizing the variational method to find optimal input states we obtain lower bounds on (i) the quantum capacity of arbitrary channels, (ii) the entanglement-assisted classical capacity of arbitrary channels, and (iii) the classical capacity of covariant channels. Corresponding to these lower bounds, we also obtain upper bounds on (i)
N-shot coherent information, (ii) the entanglement-assisted classical capacity, and (iii) the
N-shot Holevo capacity of arbitrary quantum channels. All these bounds can be estimated by a single measurement setting without needing a full process tomography or any further a priori knowledge, e.g., preferred basis of the channel.
The quantum-channel capacity gives the ultimate limit for the rate at which quantum data can be reliably transmitted through a noisy quantum channel. Degradable quantum channels are among the few channels whose quantum capacities are known. Given the quantum capacity of a degradable channel, it remains challenging to find a practical coding scheme which approaches capacity. Here we discuss code designs for the detected-jump channel, a degradable channel with practical relevance describing the physics of spontaneous decay of atoms with detected photon emission. We show that this channel can be used to simulate a binary classical channel with both erasures and bit flips. The capacity of the simulated classical channel gives a lower bound on the quantum capacity of the detected-jump channel. When the jump probability is small, it almost equals the quantum capacity. Hence using a classical capacity-approaching code for the simulated classical channel yields a quantum code which approaches the quantum capacity of the detected-jump channel.