30,544 research outputs found
On Composite Quantum Hypothesis Testing
We extend quantum Stein's lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states ρ^(⊗n) against convex combinations of quantum states σ^(⊗n) is given by a regularized quantum relative entropy distance formula. We prove that in general such a regularization is needed but also discuss various settings where our formula as well as extensions thereof become single-letter. This includes a novel operational interpretation of the relative entropy of coherence in terms of hypothesis testing. For our proof, we start from the composite Stein's lemma for classical probability distributions and lift the result to the non-commutative setting by only using elementary properties of quantum entropy. Finally, our findings also imply an improved Markov type lower bound on the quantum conditional mutual information in terms of the regularized quantum relative entropy - featuring an explicit and universal recovery map
On Composite Quantum Hypothesis Testing
We extend quantum Stein's lemma in asymmetric quantum hypothesis testing to composite null and alternative hypotheses. As our main result, we show that the asymptotic error exponent for testing convex combinations of quantum states ρ^(⊗n) against convex combinations of quantum states σ^(⊗n) is given by a regularized quantum relative entropy distance formula. We prove that in general such a regularization is needed but also discuss various settings where our formula as well as extensions thereof become single-letter. This includes a novel operational interpretation of the relative entropy of coherence in terms of hypothesis testing. For our proof, we start from the composite Stein's lemma for classical probability distributions and lift the result to the non-commutative setting by only using elementary properties of quantum entropy. Finally, our findings also imply an improved Markov type lower bound on the quantum conditional mutual information in terms of the regularized quantum relative entropy - featuring an explicit and universal recovery map
Correlation detection and an operational interpretation of the R�nyi mutual information
A variety of new measures of quantum R�nyi mutual information and quantum R�nyi conditional entropy have recently been proposed, and some of their mathematical properties explored. Here, we show that the R�nyi mutual information attains operational meaning in the context of composite hypothesis testing, when the null hypothesis is a fixed bipartite state and the alternative hypothesis consists of all product states that share one marginal with the null hypothesis. This hypothesis testing problem occurs naturally in channel coding, where it corresponds to testing whether a state is the output of a given quantum channel or of a "useless" channel whose output is decoupled from the environment. Similarly, we establish an operational interpretation of R�nyi conditional entropy by choosing an alternative hypothesis that consists of product states that are maximally mixed on one system. Specialized to classical probability distributions, our results also establish an operational interpretation of R�nyi mutual information and R�nyi conditional entropy
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 and is given by the Hilbert projective
metric in asymmetric
hypothesis testing, and by the Thompson metric 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.Comment: 30 page
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