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.

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