## Abstract

We consider the problem of discrimination between two pure quantum states. It is well known that the optimal measurement under both the error-probability and log-loss criteria is a projection, while under an 'erasure-distortion' criterion it is a three-outcome positive operator-valued measure (POVM). These results were derived separately. We present a unified approach which finds the optimal measurement under any distortion measure that satisfies a convexity relation with respect to the Bhattacharyya distance. Namely, whenever the measure is relatively convex (resp. concave), the measurement is the projection (resp. three-outcome POVM) above. The three above-mentioned results are obtained as special cases of this simple derivation. As for further measures for which our result applies, we prove that Rényi entropies of order 1 and above (resp. 1/2 and below) are relatively convex (resp. concave). A special setting of great practical interest, is the discrimination between two coherent-light waveforms. In a remarkable work by Dolinar it was shown that a simple detector consisting of a photon counter and a feedback-controlled local oscillator obtains the quantum-optimal error probability. Later it was shown that the same detector (with the same local signal) is also optimal in the log-loss sense. By applying a similar convexity approach, we obtain in a unified manner the optimal signal for a variety of criteria.

Original language | English |
---|---|

Pages (from-to) | 2701-2712 |

Number of pages | 12 |

Journal | IEEE Transactions on Information Theory |

Volume | 70 |

Issue number | 4 |

DOIs | |

State | Published - 1 Apr 2024 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 1963-2012 IEEE.

## Keywords

- Quantum communication
- information entropy
- optical signal detection
- quantum state