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

• URL: http://arxiv.org/abs/2311.02366v1
• Date: Sat, 4 Nov 2023 10:29:35 GMT
• Title: Optimal Discrimination Between Two Pure States and Dolinar-Type Coherent-State Detection
• Authors: Itamar Katz, Alex Samorodnitsky and Yuval Kochman
• Abstract summary: 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. We present a unified approach which finds the optimal measurement under any distortion measure.
• Score: 8.901073744693315
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 Renyi 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.

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