For any quantum ensemble and any positive score function, the optimal expected gain is no larger than the square root of the gain of the generalized pretty good measurement, implying a two-fold mean-square-error bound in Bayesian estimation.
On average output entropy of a quantum channel
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abstract
We describe analytical properties of the average output entropy of a quantum channel as a function of a pair (channel, input ensemble). In particular, tight semicontinuity bounds for this function with the rank/energy constraints are obtained by using the modified semicontinuity bounds for the quantum conditional entropy of quantum-classical states and a special approximation technique. Several applications are considered. New semicontinuity and continuity bounds for the output Holevo information of a channel as a function of a pair (channel, input ensemble) are obtained. The semicontinuity bound for the entanglement of formation with the rank constraint obtained in [1] is improved. In the preliminary part, some results concerning ensembles of quantum states are presented. In particular, a new useful metric on the set of generalized ensembles is proposed and explored. The concept of passive energy of an ensemble introduced here plays an important role in the article.
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Near-optimal performance of square-root measurement for general score functions and quantum ensembles
For any quantum ensemble and any positive score function, the optimal expected gain is no larger than the square root of the gain of the generalized pretty good measurement, implying a two-fold mean-square-error bound in Bayesian estimation.