Any quasi-probability representation of a no-signaling system – including quantum systems – can be simulated via a purely classical scheme by allowing signed events and a cancellation procedure. This raises a fundamental question: What properties of the non-classical system does such a classical simulation fail to replicate? We answer by using large deviation theory to show that the probability of a large fluctuation under the classical simulation can be strictly greater than under the actual non-classical system. The key finding driving our result is that negativity in probability relaxes the data processing inequality of information theory. We propose this potential large deviation stability of quantum (and no-signaling) systems as a novel form of quantum advantage.

We modify the Rényi (1961) axioms for entropy to apply to negative (“signed”) measures as arise, for example, in phase-space representations of quantum mechanics. We obtain two new measures of (lack of) information about a system – which we propose as signed analogs to classical Shannon entropy and classical Rényi entropy, respectively. We show that signed Rényi entropy witnesses non-classicality of a system. Specifically, a measure has at least one negative component if and only if signed Rényi α-entropy is negative for some α > 1. The corresponding non-classicality test does not work with signed Shannon entropy. We next show that signed Rényi 2k-entropy, when k is a positive integer, is Schurconcave. (An example shows that signed Shannon entropy is not Schur-concave.) We then establish an abstract quantum H-theorem for signed measures. We prove that signed Rényi 2k-entropy is nondecreasing under classical (“decohering”) evolution of a signed measure, where the latter could be a Wigner function or other phase-space representation of a quantum system. (An example shows that signed Shannon entropy may be non-monotonic.) We also provide a characterization of the Second Law for signed Rényi 2-entropy in terms of what we call eventual classicalization of evolution of a system. We conclude with an argument that signed Rényi 2-entropy of the Wigner function is constant under Moyal bracket evolution.

Is the world quantum? An active research line in quantum foundations is devoted to exploring what constraints can rule out the postquantum theories that are consistent with experimentally observed results. We explore this question in the context of epistemics, and ask whether agreement between observers can serve as a physical principle that must hold for any theory of the world. Aumann’s seminal Agreement Theorem states that two observers (of classical systems) cannot agree to disagree. We propose an extension of this theorem to no-signaling settings. In particular, we establish an Agreement Theorem for observers of quantum systems, while we construct examples of (postquantum) no-signaling boxes where observers can agree to disagree. The PR box is an extremal instance of this phenomenon. These results make it plausible that agreement between observers might be a physical principle, while they also establish links between the fields of epistemics and quantum information that seem worthy of further exploration.

We show how quantum entanglement may be able to improve the joint performance of a system of telescopes, cameras, or other sensors which are widely separated in space. The improvement is relative to any observation strategy that uses only classical coordinating devices. Potential application domains include space-based observatories and multi-frequency interferometry.

We study team decision problems where communication is not possible, but coordination among team members can be realized via signals in a shared eironment. We consider a variety of decision problems that differ in what team members know about one another’s actions and knowledge. For each type of decision problem, we iestigate how different assumptions on the available signals affect team performance. Specifically, we consider the cases of perfectly correlated, i.i.d., and exchangeable classical signals, as well as the case of quantum signals. We find that, whereas in perfect-recall trees (Kuhn [15, 1950], [16, 1953]) no type of signal improves performance, in imperfect-recall trees quantum signals may bring an improvement. Isbell [13, 1957] proved that in non-Kuhn trees, classical i.i.d. signals may improve performance. We show that further improvement may be possible by use of classical exchangeable or quantum signals. We include an example of the effect of quantum signals in the context of high-frequency trading.