Achieving the Holevo capacity of a pure state classical-quantum channel via unambiguous state discrimination

Abstract

We prove that the ultimate channel capacity, the Holevo bound, for sending classical data on a quantum channel (the so-called classical-quantum, or cq channel) can be achieved for a pure-state cq channel by decoding codewords via a collective quantum measurement based on unambiguous state discrimination (USD). In cq communication theory, the channel decoder acts directly on the modulated codeword waveform in the quantum (viz., electromagnetic or optical) domain, and it is known that collective measurements on long codeword blocks are needed to attain the Holevo capacity, which is strictly larger than the Shannon capacity of the classical channel induced by any specific measurement choice on each channel use. The USD measurement based channel decoder we propose, can distinguish finite blocklength codeword quantum states unambiguously (i.e., an incorrect codeword is never chosen) provided one allows for a finite probability of obtaining an inconclusive (erasure) outcome. We show that the probability of the inconclusive outcome goes to zero for asymptotically long codewords whenever the code rate is below the Holevo bound. The USD channel decoder is an addition to a small list of other collective measurements known to achieve the Holevo capacity (such as, the square root measurement, the minimum probability of error measurement, the sequential decoding measurement, and the quantum successive cancellation decoder for the cq polar code). A structured optical receiver design is not known yet for any of these decoders. What makes the USD decoder special is that there is no classical analogue to truly unambiguous discrimination (say, of samples drawn from a set of probability distributions). Secondly, the erasures-only decoding of USD is likely to result in a better channel reliability function. Finally, the USD measurement seems more likely to lead naturally to a structured optical receiver design and implementation.