Framed Hilbert space: hanging the quasi-probability pictures of quantum theory

Abstract

Building on earlier work, we further develop a formalism based on the mathematical theory of frames that defines a set of possible phase-space or quasi-probability representations of finite-dimensional quantum systems. We prove that an alternate approach to defining a set of quasi-probability representations, based on a more natural generalization of a classical representation, is equivalent to our earlier approach based on frames, and therefore is also subject to our no-go theorem for a non-negative representation. Furthermore, we clarify the relationship between the contextuality of quantum theory and the necessity of negativity in quasi-probability representations and discuss their relevance as criteria for non-classicality. We also provide a comprehensive overview of known quasi-probability representations and their expression within the frame formalism.

arXiv comment46 pages, 1 table, contains a review of finite dimensional quasi-probability functions

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• Continuous phase-space representations for finite-dimensional quantum states and their tomography
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• Symmetric Informationally Complete Measurements Identify the Irreducible Difference between Classical and Quantum Systems
• Contextuality Test of the Nonclassicality of Variational Quantum Eigensolvers
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• Necessity of negativity in quantum theory
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• Discrete Wigner Formalism for Qubits and Non-Contextuality of Clifford Gates on Qubit Stabilizer States
• Conditional Hybrid Nonclassicality
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• A structure theorem for generalized-noncontextual ontological models
• Classicality without local discriminability: decoupling entanglement and complementarity
• Classical Simulation of Noncontextual Pauli Hamiltonians
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• Kirkwood-Dirac representations beyond quantum states (and their relation to noncontextuality)
• Time evolution of coupled spin systems in a generalized Wigner representation
• Operational quasiprobabilities for qudits
• Fast computation of spherical phase-space functions of quantum many-body states
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• Wigner distributions for finite state systems without redundant phase point operators
• Minimal informationally complete measurements for probability representation of quantum dynamics
• Quantum Bayesian Inference in Quasiprobability Representations
• Symmetry-adapted decomposition of tensor operators and the visualization of coupled spin systems
• Semiclassical Formulation of Gottesman-Knill and Universal Quantum Computation
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• Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits
• Faster Born probability estimation via gate merging and frame optimisation
• Measurement Contextuality and Planck's Constant
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• Simulations of quantum nonlocality with local negative bits
• Qubit from the classical collision entropy
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• Quantum Bochner's theorem for phase spaces built on projective representations
• Wigner representation of the rotational dynamics of rigid tops
• Nonexistence of sharply covariant mutually unbiased bases in odd prime dimensions
• Algebraic Probability-Theoretic Characterization of Quantum Correlations
• From the Attempt of Certain Classical Reformulations of Quantum Mechanics to Quasi-Probability Representations
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• On Hidden Variables: Value and Expectation No-Go Theorems
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• Frame representations of qudit quantum mechanics
• Reexamination of the Kochen-Specker theorem: Relaxation of the completeness assumption
• A Theory of Inaccessible Information
• Noncontextual Pauli Hamiltonians
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• Ideal stochastic process modeling with post-quantum quasiprobabilistic theories
• Common Denominator for Value and Expectation No-go Theorems: Extended Abstract