Optimal quantum networks and one-shot entropies

We present a general-purpose method for the optimization of quantum networks, including both causal networks and networks with indefinite causal structure. Our method applies to a broad class of performance measures, defined operationally in terms of tests set up by a verifier. We show that the optimal performance can be quantified by the max relative entropy between a suitable operator, associated to the test, and the set of operators associated to non-informative tests. Building on the connection with the max relative entropy, we extend the notion of conditional minentropy from quantum states to quantum causal networks, providing a measure of the total correlations generated by ordered sequences of quantum channels with memory. The optimization method is illustrated in a number of applications, including the inversion, charge conjugation, and controlization of an unknown unitary dynamics. In the non-causal setting, we show an application to the maximization of the winning probability in a non-causal quantum game. ar X iv :1 60 6. 02 39 4v 1 [ qu an tph ] 8 J un 2 01 6 Optimal quantum networks and one-shot entropies 2