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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.
The Agreement Theorem Aumann (1976 Ann. Stat. 4, 1236–1239. (doi:10.1214/aos/1176343654)) states that if two Bayesian agents start with a common prior, then they cannot have common knowledge that they hold different posterior probabilities of some underlying event of interest. In short, the two agents cannot ‘agree to disagree’. This result applies in the classical domain where classical probability theory applies. But in non-classical domains, such as the quantum world, classical probability theory does not apply. Inspired principally by their use in quantum mechanics, we employ signed probabilities to investigate the epistemics of the non-classical world. We find that here, too, it cannot be common knowledge that two agents assign different probabilities to an event of interest. However, in a non-classical domain, unlike the classical case, it can be common certainty that two agents assign different probabilities to an event of interest. Finally, in a non-classical domain, it cannot be common certainty that two agents assign different probabilities, if communication of their common certainty is possible—even if communication does not take place.
This article is part of the theme issue ‘Quantum contextuality, causality and freedom of choice’.
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.
The states of the qubit, the basic unit of quantum information, are 2×2 positive semi-definite Hermitian matrices with trace 1. We contribute to the program to axiomatize quantum mechanics by characterizing these states in terms of an entropic uncertainty principle formulated on an eight-point phase space. We do this by employing Rényi entropy (a generalization of Shannon entropy) suitably defined for the signed phase-space probability distributions that arise in representing quantum states.
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.
Is the world quantum? An active research line in quantum foundations is devoted to exploring what constraints can rule out the post-quantum 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. The seminal Agreement Theorem by Aumann (Annals of Statistics, 1976) states that two (classical) agents cannot agree to disagree. We examine the extension of this theorem to no-signalling settings. In particular, we establish an Agreement Theorem for quantum agents. We also construct examples of (post-quantum) no-signalling boxes where agents can agree to disagree. The PR box is an extremal instance of this phenomenon. These results make it plausible that agreement 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.