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What is special about quantum entropy?

Howard Barnum, Perimeter Institute for Theoretical Physics

(Session 3 : Friday from 2:30-3:00)

Abstract. What is special about quantum entropy? Howard Barnum, Perimeter Institute for Theoretical Physics Pawlowski et. al. recently showed that stronger-than-quantum correlations (ones that violate the Tsirel'son bound on the strength of CHSH/Bell correlations) violate a principle they call Information Causality. The principle states that, using some shared correlations plus classical communication as a resource, the total mutual information Alice can make available to Bob about a set S of classical bits cannot exceed the number of bits of classical communication she uses, even though this total mutual information is the sum of alternative, possibly mutually exclusive strategies Bob can use to get each of the bits of S ("random-access coding", in computer science jargon). They also showed that quantum theory satisfies this principle. We examine the question of what properties of a theory may lead to its correlations satisfying information causality. We define measurement and preparation entropies for states of a general class of theories: the minimum, over finegrained measurements on the state, of the Shannon entropy of the probabilities of the outcomes of the measurement, and the minimum, over pure-state ensembles for the state, of the Shannon entropies of the ensemble probabilities. We find sufficient conditions for information causality in terms of these entropies: If the measurement entropy satisfies a data-processing inequality, and if the conditional measurement entropy is positive when the conditioning is on a classical system, then information causality holds. Besides the principle of strong subadditivity (which is closely related to data processing) this focuses attention on another property of quantum entropy, the positivity of entropy conditional on a classical system. We show that this property follows from another very natural one exhibited by quantum theory, but that does not hold in general: the equality of measurement and preparation entropy. We briefly consider the implications of this principle for the structure and information-processing possibilities of theories. Joint work with Jonathan Barrett, Lisa Orloff Clark, Matthew Leifer, Robert Spekkens, Nicholas Stepanik, Alex Wilce, and Robin Wilke.