Postselected quantum hypothesis testing

Understand

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.