Wigner function for a particle in an infinite lattice

Abstract

We study the Wigner function for a quantum system with a discrete, infinite dimensional Hilbert space, such as a spinless particle moving on a one dimensional infinite lattice. We discuss the peculiarities of this scenario and of the associated phase space construction, propose a meaningful definition of the Wigner function in this case, and characterize the set of pure states for which it is non-negative. We propose a measure of non-classicality for states in this system which is consistent with the continuum limit. The prescriptions introduced here are illustrated by applying them to localized and Gaussian states, and to their superpositions.

19 pages (single column), 7 figures

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• Normalizer Circuits and Quantum Computation
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• Normalizer circuits and a Gottesman-Knill theorem for infinite-dimensional systems
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• Quasicondensation and off-diagonal long-range order of hard-core bosons during a free expansion
• Limitations of the Markovian approximation in the harmonic oscillator
• Bath-induced correlations in an infinite-dimensional Hilbert space
• Absence of diagonal force constants in cubic Coulomb crystals
• Entanglement Hamiltonian during a domain wall melting in the free Fermi chain
• Inhomogeneous quenches and GHD in the $\nu =1$ QSSEP model
• Hubble tension problem encompassed by phase-space quantum cosmology
• Twisted Fourier analysis and pseudo-probability distributions