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Operational axioms for diagonalizing states

Chiribella, Giulio and Scandolo, Carlo Maria (2015) Operational axioms for diagonalizing states. EPTCS, 195. pp. 96-115.

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In quantum theory every state can be diagonalized, i.e. decomposed as a convex combination of perfectly distinguishable pure states. This elementary structure plays an ubiquitous role in quantum mechanics, quantum information theory, and quantum statistical mechanics, where it provides the foundation for the notions of majorization and entropy. A natural question then arises: can we reconstruct these notions from purely operational axioms? We address this question in the framework of general probabilistic theories, presenting a set of axioms that guarantee that every state can be diagonalized. The first axiom is Causality, which ensures that the marginal of a bipartite state is well defined. Then, Purity Preservation states that the set of pure transformations is closed under composition. The third axiom is Purification, which allows to assign a pure state to the composition of a system with its environment. Finally, we introduce the axiom of Pure Sharpness, stating that for every system there exists at least one pure effect occurring with unit probability on some state. For theories satisfying our four axioms, we show a constructive algorithm for diagonalizing every given state. The diagonalization result allows us to formulate a majorization criterion that captures the convertibility of states in the operational resource theory of purity, where random reversible transformations are regarded as free operations.

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Item Type: Published Article or Volume
Scandolo, Carlo
Keywords: Foundations of quantum theory, mathematical structure of quantum theory, foundations of quantum thermodynamics
Subjects: Specific Sciences > Physics > Quantum Mechanics
Specific Sciences > Physics > Statistical Mechanics/Thermodynamics
Depositing User: Mr. Carlo Maria Scandolo
Date Deposited: 01 Feb 2016 03:17
Last Modified: 01 Feb 2016 03:17
Item ID: 11891
Journal or Publication Title: EPTCS
DOI or Unique Handle: 10.4204/EPTCS.195.8
Subjects: Specific Sciences > Physics > Quantum Mechanics
Specific Sciences > Physics > Statistical Mechanics/Thermodynamics
Date: 4 November 2015
Page Range: pp. 96-115
Volume: 195

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