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.

13 pages, published version

References in corpus (4)

• Negativity and contextuality are equivalent notions of nonclassicality
• Classicality in discrete Wigner functions
• Discrete Wigner functions and quantum computational speedup
• Quantum Margulis expanders

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• Spekkens's Symmetric No-Go Theorem
• Minimizing the negativity of quantum circuits in overcomplete quasiprobability representations
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• Conspiracy in ontological models with measurement contextuality
• First considerations on the generalized uncertainty principle for finite-dimensional discrete phase spaces
• Noncontextual Pauli Hamiltonians
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