We study a variant of quantum hypothesis testing wherein an additional 'inconclusive' measurement outcome is added, allowing one to abstain from attempting to discriminate the hypotheses. The error probabilities are then conditioned on a successful attempt, with inconclusive trials disregarded. We completely characterise this task in both the single-shot and asymptotic regimes, providing exact formulas for the optimal error probabilities. In particular, we prove that the asymptotic error exponent of discriminating any two quantum states $\rho$ and $\sigma$ is given by the Hilbert projective metric $D_{\max}(\rho\|\sigma) + D_{\max}(\sigma \| \rho)$ in asymmetric hypothesis testing, and by the Thompson metric $\max \{ D_{\max}(\rho\|\sigma), D_{\max}(\sigma \| \rho) \}$ in symmetric hypothesis testing. This endows these two quantities with fundamental operational interpretations in quantum state discrimination. Our findings extend to composite hypothesis testing, where we show that the asymmetric error exponent with respect to any convex set of density matrices is given by a regularisation of the Hilbert projective metric. We apply our results also to quantum channels, showing that no advantage is gained by employing adaptive or even more general discrimination schemes over parallel ones, in both the asymmetric and symmetric settings. Our state discrimination results make use of no properties specific to quantum mechanics and are also valid in general probabilistic theories.

翻译：研究了一种量子假设检验的变体，其中添加了一个额外的"不确定"测量结果，允许人们放弃对假设的区分尝试。错误概率随后以成功尝试为条件，忽略不确定的试验。我们在单次和渐近两种情况下完全刻画了这一任务，给出了最优错误概率的精确公式。特别地，我们证明区分任意两个量子态$\rho$和$\sigma$的渐近错误指数在非对称假设检验中由希尔伯特投影度量$D_{\max}(\rho\|\sigma) + D_{\max}(\sigma \| \rho)$给出，在对称假设检验中由汤普森度量$\max \{ D_{\max}(\rho\|\sigma), D_{\max}(\sigma \| \rho) \}$给出。这赋予了这两个量在量子态区分中的基本操作解释。我们的发现扩展到复合假设检验，其中我们证明对于任意凸密度矩阵集的非对称错误指数由希尔伯特投影度量的正则化给出。我们还将结果应用于量子信道，表明在非对称和对称设置中，采用自适应或更一般的区分方案相对于并行方案并未获得优势。我们的态区分结果不依赖于量子力学的特定性质，也适用于一般概率论。