Quantum Bochner's theorem for phase spaces built on projective representations
arXiv:1410.5755 · doi:10.1088/1751-8113/48/11/115305
Abstract
Bochner's theorem gives the necessary and sufficient conditions on a function such that its Fourier transform corresponds to a true probability density function. In the Wigner phase space picture, quantum Bochner's theorem gives the necessary and sufficient conditions on a function such that it is a quantum characteristic function of a valid (and possibly mixed) quantum state and such that its Fourier transform is a true probability density. We extend this theorem to discrete phase space representations which possess enough symmetry. More precisely, we show that discrete phase space representations that are built on projective unitary representations of abelian groups, with a slight restriction on admissible 2-cocycles, enable a quantum Bochner's theorem.
References in corpus (15)
- Symmetric Informationally Complete Quantum Measurements
- Positive Wigner functions render classical simulation of quantum computation efficient
- Discrete phase space based on finite fields
- Negative Quasi-Probability as a Resource for Quantum Computation
- Hudson's Theorem for finite-dimensional quantum systems
- Negativity and contextuality are equivalent notions of nonclassicality
- Decoherence, Chaos, and the Correspondence Principle
- Quasi-probability representations of quantum theory with applications to quantum information science
- Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations
- Framed Hilbert space: hanging the quasi-probability pictures of quantum theory
- Necessity of negativity in quantum theory
- Nonnegative subtheories and quasiprobability representations of qubits
- Inadequacy of a classical interpretation of quantum projective measurements via Wigner functions
- Non-classicality of positive relativistic Wigner function
- From the Attempt of Certain Classical Reformulations of Quantum Mechanics to Quasi-Probability Representations