Optimal Discrimination Between Two Pure States and Dolinar-Type Coherent-State Detection

Itamar Katz, Alex Samorodnitsky, Yuval Kochman

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)2701-2712
Number of pages12
JournalIEEE Transactions on Information Theory
Issue number4
StatePublished - 1 Apr 2024
Externally publishedYes

Bibliographical note

Publisher Copyright:
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  • Quantum communication
  • information entropy
  • optical signal detection
  • quantum state


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