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.

翻译：我们研究对两个纯量子态的区分问题。众所周知，在错误概率和对数损失准则下的最优测量为投影测量，而在“擦除失真”准则下则为三输出正算子取值测度（POVM）。这些结果此前是分别推导得出的。我们提出一种统一方法，可在任意满足与巴塔查里亚距离相关凸性关系的失真度量下找到最优测量。具体而言，当该度量相对凸（凹）时，最优测量即为上述投影测量（三输出POVM）。上述三个经典结果均可作为这一简洁推导的特例得到。对于适用我们结果的其它度量，我们证明阶数$1$及以上（$1/2$及以下）的Rényi熵具有相对凸性（凹性）。一个极具实际重要性的特殊场景是两束相干光波形的区分。在Dolinar的开创性工作中，由光子计数器和反馈控制本地振荡器构成的简单检测器即可获得量子最优错误概率。后续研究表明，同一检测器（使用相同本地信号）在对数损失意义下也是最优的。通过应用类似的凸性方法，我们统一获得了多种准则下的最优信号。