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IEEE Transactions on Information Theory

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

Itamar Katz, Alex Samorodnitsky, Yuval Kochman

Published Nov 16, 2023• 1 citation
Publisher

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.

DOI
10.1109/tit.2023.3333414

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

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Technical Program

Dolinar's receiver is an architecture for distinguishing between two possible coherent states, using a photon detector and a local signal which may depend on past detector measurements.The optimal local signal satisfies very favorable properties: it is independent of the time horizon, and the resulting error probability is independent of the measurements.It was also shown that the same signal is optimal in the sense of maximizing the mutual information between the identity of the state and the measurements.In this work we show that the same signal is optimal for the optimization of the expected value of a wide class of objective functions.Our proof is based entirely on convex optimization and functional analysis, without resorting to any "quantum" arguments.

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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.
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 111 and above (resp. 1/21/21/2 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.
We consider the problem of discrimination between two pure quantum states. 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. The well-known results for the error probability, log-loss and “erasure-distortion” criteria are obtained as special cases. We also treat a special setting of great practical interest, the discrimination between two coherent-light waveforms. Dolinar showed that a simple detector, with an optimal local signal, obtains the quantum-optimal error probability. By applying the convexity approach, we obtain in a unified manner the optimal local signal for the same criteria as in the general setting.
Dolinar's receiver is an architecture for distinguishing between two possible coherent states, using a photon detector and a local signal which may depend on past detector measurements. The optimal local signal satisfies very favorable properties: it is independent of the time horizon, and the resulting error probability is independent of the measurements. It was also shown that the same signal is optimal in the sense of maximizing the mutual information between the identity of the state and the measurements. In this work we show that the same signal is optimal for the optimization of the expected value of a wide class of objective functions. Our proof is based entirely on convex optimization and functional analysis, without resorting to any "quantum" arguments.
Dolinar's receiver is an architecture for distinguishing between two possible coherent states, using a photon detector and a local signal which may depend on past detector measurements.The optimal local signal satisfies very favorable properties: it is independent of the time horizon, and the resulting error probability is independent of the measurements.It was also shown that the same signal is optimal in the sense of maximizing the mutual information between the identity of the state and the measurements.In this work we show that the same signal is optimal for the optimization of the expected value of a wide class of objective functions.Our proof is based entirely on convex optimization and functional analysis, without resorting to any "quantum" arguments.
Quantum state discrimination is a fundamental task in the field of quantum communication and quantum information theory. Unless the states to be discriminated are mutually orthogonal, there will be some error in any attempt to determine which state was sent. Several strategies to optimally discriminate between quantum states exist, each maximising some figure of merit. In this thesis we mainly investigate the minimum-error strategy, in which the probability of correctly guessing the signal state is maximised. We introduce a method for constructing the optimal Positive-Operator Valued Measure (POVM) for this figure of merit, which is applicable for arbitrary states and arbitrary prior probabilities. We then use this method to solve minimum-error state discrimination for the so-called trine states with arbitrary prior probabilities - the first such general solution for a set of quantum states since the two-state case was solved when the problem of state discrimination was first introduced. We also investigate the difference between local and global measurements for a bipartite ensemble of states, and find that in certain circumstances the local measurement is superior. We conclude by finding a bipartite analogue to the Helstrom conditions, which indicate when a POVM satisfies the minimum-error criteria.