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Quantum mechanics as the quadratic Taylor approximation of classical mechanics: The finite-dimensional case

Andrei Yu. Khrennikov

Theoretical and Mathematical PhysicsAug 1, 20077 citations

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

Christopher Ferrie, Joseph Emerson

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 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.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 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:0711.2658Journal of Physics A Mathematical and TheoreticalJul 25, 2008111 citations

Some remarks on classical representations of quantum mechanics

Werner Stulpe

Foundations of PhysicsJul 1, 19945 citations

On phase-space representations of quantum mechanics

Joachim J. Włodarz

Physics Letters AJul 1, 20017 citations

Contrasting classical and quantum theory in the context of quasi-probability

Christopher Ferrie

Several finite dimensional quasi-probability representations of \nquantum states have been proposed to study various problems in \nquantum information theory and quantum foundations. These \nrepresentations are often defined only on restricted dimensions and \ntheir physical significance in contexts such as drawing \nquantum-classical comparisons is limited by the non-uniqueness of \nthe particular representation. In this thesis it is shown how the mathematical \ntheory of frames provides a unified formalism which accommodates all \nknown quasi-probability representations of finite dimensional \nquantum systems. \n \nIt is also shown that any quasi-probability \nrepresentation is equivalent to a frame representation and it is \nproven that any such representation of quantum mechanics must exhibit \neither negativity or a deformed probability calculus. \n \nAlong the way, the connection between negativity and two other famous notions of non-classicality, namely contextuality and nonlocality, is clarified.Several finite dimensional quasi-probability representations of \nquantum states have been proposed to study various problems in \nquantum information theory and quantum foundations. These \nrepresentations are often defined only on restricted dimensions and \ntheir physical significance in contexts such as drawing \nquantum-classical comparisons is limited by the non-uniqueness of \nthe particular representation. In this thesis it is shown how the mathematical \ntheory of frames provides a unified formalism which accommodates all \nknown quasi-probability representations of finite dimensional \nquantum systems. \n \nIt is also shown that any quasi-probability \nrepresentation is equivalent to a frame representation and it is \nproven that any such representation of quantum mechanics must exhibit \neither negativity or a deformed probability calculus. \n \nAlong the way, the connection between negativity and two other famous notions of non-classicality, namely contextuality and nonlocality, is clarified.

Aug 22, 20080 citations

Conceptual inconsistencies in finite-dimensional quantum and classical mechanics

Denys I. Bondar, Renán Cabrera, HERSCHEL A. RABITZ

Utilizing operational dynamic modeling [D. I. Bondar et al., Phys. Rev. Lett. 109, 190403 (2012)], we demonstrate that any finite-dimensional representation of quantum and classical dynamics violates the Ehrenfest theorems. Other peculiarities are also revealed, including the nonexistence of the free particle and ambiguity in defining potential forces. Non-Hermitian mechanics is shown to have the same problems. This work compromises a popular belief that finite-dimensional mechanics is a straightforward discretization of the corresponding infinite-dimensional formulation.Utilizing operational dynamic modeling [D. I. Bondar et al., Phys. Rev. Lett. 109, 190403 (2012)], we demonstrate that any finite-dimensional representation of quantum and classical dynamics violates the Ehrenfest theorems. Other peculiarities are also revealed, including the nonexistence of the free particle and ambiguity in defining potential forces. Non-Hermitian mechanics is shown to have the same problems. This work compromises a popular belief that finite-dimensional mechanics is a straightforward discretization of the corresponding infinite-dimensional formulation.

arXiv:1112.3679Physical Review AJul 16, 20138 citations

On classical representations of finite-dimensional quantum mechanics

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