Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations

Abstract

Several finite dimensional quasi-probability representations of quantum states have been proposed to study various problems in quantum information theory and quantum foundations. These representations are often defined only on restricted dimensions and their physical significance in contexts such as drawing quantum-classical comparisons is limited by the non-uniqueness of the particular representation. Here we show how the mathematical theory of frames provides a unified formalism which accommodates all known quasi-probability representations of finite dimensional quantum systems. Moreover, we show that any quasi-probability representation satisfying two reasonable properties is equivalent to a frame representation and then prove that any such representation of quantum mechanics must exhibit either negativity or a deformed probability calculus.

arXiv comment13 pages, published version

References in corpus (12)

• Contextuality for preparations, transformations, and unsharp measurements
• General Entanglement Breaking Channels
• Quantum Theory From Five Reasonable Axioms
• Discrete phase space based on finite fields
• Negativity and contextuality are equivalent notions of nonclassicality
• Quantum computers in phase space
• Classicality in discrete Wigner functions
• Discrete Wigner functions and quantum computational speedup
• Discrete Moyal-type Representations for a Spin
• Discrete Wigner functions and the phase space representation of quantum teleportation
• Phase-space approach to the study of decoherence in quantum walks
• Quantum Margulis expanders

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