Optimal quantum networks and one-shot entropies

Abstract

We develop a semidefinite programming 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 interative tests set up by a verifier. We show that the optimal performance is equal to a max relative entropy, which quantifies 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. IntroductionAdvances in quantum communication [1][2][3] and in the integration of quantum hardware [4-8] are pushing towards the realization of networked quantum information systems, such as quantum communication networks [9-13] and distributed quantum computing [14][15][16]. Networks of interacting quantum devices are attracting interest also at the theoretical level, providing a framework for quantum games [17] and protocols [18][19][20], insights on the foundations of quantum mechanics [18, 21-23], a starting point for a general theory of Bayesian inference [24-31] and for the development of models of higher-order quantum computation [32][33][34].The network scenario motivates a new set of optimization problems, where the goal is not to optimize individual devices, but rather to optimize how different devices interact with one another. In many situations, the devices operate in a well-defined causal order-this is the case, for example, in the circuit model of quantum computing, where computations are implemented by sequences of gates [35,36]. Recently, researchers have started to investigate more general situations, where the causal order can be in a quantum superposition [20,33,34,[37][38][39] or can be indefinite in other more exotic ways, in principle compatible with quantum mechanics [34,[40][41][42][43][44][45]. In these new situations, optimizing quantum networks is important, for at least three reasons: First, in order to establish an advantage, one has to first find the optimal performances achievable in a definite causal order. Second, finding the maximum advantage requires an optimization over all non-causal networks. This is an essential step for assessing the power of the new, non-causal models of information processing. Third, identifying the ultimate performances achieved in the absence of pre-defined causal structure is expected to shed light on the interplay between quantum mechanics and spacetime.In this paper we develop a semidefinite programming (SDP) approach to the optimization of quantum networks. We start by analyzing scenarios with definite causal order, choosing an operational measure of performance, quantified by how much the network scores in a giv...

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