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.
