The measured relative entropies of quantum states and channels find operational significance in quantum information theory as achievable error rates in hypothesis testing tasks. They are of interest in the near term, as they correspond to hybrid quantum--classical strategies with technological requirements far less challenging to implement than required by the most general strategies allowed by quantum mechanics. In this paper, we prove that these measured relative entropies can be calculated efficiently by means of semi-definite programming, by making use of variational formulas for the measured relative entropies of states and semi-definite representations of the weighted geometric mean and the operator connection of the logarithm. Not only do the semi-definite programs output the optimal values of the measured relative entropies of states and channels, but they also provide numerical characterizations of optimal strategies for achieving them, which is of significant practical interest for designing hypothesis testing protocols.

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