Postselected Quantum Hypothesis Testing

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
$\max \! \big \{ D_{\max }(\rho \|\sigma ),\, D_{\max }(\sigma \| \rho ) \big \}$
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.