Empirical Coordination of Quantum Correlations

TL;DR

This paper introduces empirical coordination for quantum correlations, analyzing optimal resource rates for simulating quantum states in networks, with implications for quantum nonlocal games and network communication.

Contribution

It establishes the optimal coordination rates for quantum networks, including classical and quantum links, and introduces a unique quantum optimization feature not present classically.

Findings

Optimal rates for classical cascade networks are derived.

Shared randomness does not reduce the resource rates.

Quantum state decomposition significantly affects performance.

Abstract

We introduce the notion of empirical coordination for quantum correlations. Quantum mechanics enables the calculation of probabilities for experimental outcomes, emphasizing statistical averages rather than detailed descriptions of individual events. This makes empirical coordination a natural and operationally meaningful framework for quantum systems - particularly in the context of nonlocal games, which rely on repeated measurements to assess performance. We begin by analyzing networks with classical links, focusing on the cascade network. For this setting, we establish the optimal coordination rates, which indicate the minimal resources required to simulate a quantum state on average. Providing the users with shared randomness, before communication begins, does not affect the optimal rates for empirical coordination. Our analysis starts with a basic two-node scenario and extends to cascade networks, including the special case of a network with an isolated node. The results can be further generalized to other networks as our analysis includes a generic achievability scheme. The optimal rate formula involves optimization over a collection of state extensions. This is a unique feature of the quantum setting, as the classical parallel does not include optimization. As demonstrated through examples, the performance depends heavily on the choice of decomposition. We then extend the framework to networks with quantum links, focusing on a broadcast setting where the receivers have side information. Finally, we discuss how our results provide new insights into the implementation and simulation of quantum nonlocal games in the empirical regime.