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Relaxed uncertainty relations and information processing

We consider a range of "theories" that violate the uncertainty relation for anti-commutingobservables derived in [JMP, 49, 062105 (2008)]. We first show that Tsirelson's boundfor the CHSH inequality can be derived from this uncertainty relation, and that relaxingthis relation allows for non-local correlations that are stronger than what can be obtainedin quantum mechanics. We continue to construct a hierarchy of related non-signalingtheories, and show that on one hand they admit superstrong random access encodingsand exponential savings for a particular communication problem, while on the other handit becomes much harder in these theories to learn a state. We show that the existence ofthese effects stems from the absence of certain constraints on the expectation values ofcommuting measurements from our non-signaling theories that are present in quantumtheory.