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 discriminate the hypotheses. The error probabilities are then conditioned on successful attempt, with inconclusive trials disregarded. completely characterise this task in both single-shot and asymptotic regimes, providing exact formulas for optimal probabilities. In particular, we prove that exponent discriminating any two states ρ σ given by Hilbert projective metric Dmax(ρ∥σ)+Dmax(σ∥ρ) asymmetric testing, Thompson max {Dmax(ρ∥σ), Dmax(σ∥ρ)} symmetric testing. This endows these quantities fundamental operational interpretations state discrimination. Our findings extend composite where show respect convex set density matrices regularisation metric. apply our results also channels, showing no advantage gained employing adaptive or even more general discrimination schemes over parallel ones, settings. make use properties specific mechanics valid probabilistic theories.