We develop a semidenite programming method for the optimization of quantum networks, including both causal networks and networks with indenite causal structure. Our method applies to a broad class of performance measures, dened operationally in terms of interactive tests set up by a verier. We show that the optimal performance is equal to a max relative entropy, which quanties the informativeness of the test. Building on this result, we extend the notion of conditional min-entropy from quantum states to quantum causal networks. 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 a proof-of-principle application to the maximization of the winning probability in a non-causal quantum game.
['Giulio Chiribella', 'Daniel Ebler']
Optimal quantum networks and one-shot entropies