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arXiv:quant-ph/9708037

Moments of the Wigner Distribution and a Generalized Uncertainty Principle

Richard M Simon, Narasimhaiengar Mukunda

Published Aug 22, 1997• 4 citations
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Abstract

The nonnegativity of the density operator of a state is faithfully coded in its Wigner distribution, and this places constraints on the moments of the Wigner distribution. These constraints are presented in a canonically invariant form which is both concise and explicit. Since the conventional uncertainty principle is such a constraint on the first and second moments, our result constitutes a generalization of the same to all orders. Possible application in quantum state reconstruction using optical homodyne tomography is noted.

DOI
10.48550/arxiv.quant-ph/9708037

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The nonnegativity of the density operator of a state is faithfully coded in its Wigner distribution, and this places constraints on the moments of the Wigner distribution. These constraints are presented in a canonically invariant form which is both concise and explicit. Since the conventional uncertainty principle is such a constraint on the first and second moments, our result constitutes a generalization of the same to all orders. Possible application in quantum state reconstruction using optical homodyne tomography is noted.
The non-negativity of the density operator of a state is faithfully coded in its Wigner distribution, and this coding places on the moments of the Wigner distribution constraints arising from the non-negativity of the density operator. Working in a monomial basis for the algebra of operators on the Hilbert space of a bosonic mode, we formulate these constraints in a canonically covariant form which is both concise and explicit. Since the conventional uncertainty relation is such a constraint on the first and second moments, our result constitutes a generalization of the same to all orders. The structure constants of , in the monomial basis, are shown to be essentially the SU (2) Clebsch–Gordan coefficients. Our results have applications in quantum state reconstruction using optical homodyne tomography and, when generalized to the n -mode case, which will be done in the second part of this work, will have applications also for continuous variable quantum information systems involving non-Gaussian states.
The first part is a systematical representation of the Wigner quasiprobability in representations by canonical variables and pairs of complex conjugate variables with particular attention to the reconstruction of the density operator from the Wigner quasiprobability to show their equivalence. All formulae are derived from basic postulates of quantum mechanics. Problems of operator orderings are touched. The second part considers the integration over the amplitude in the Wigner quasiprobability and its specialization for coherent states and for even and odd coherent states and provides phase distributions which are presented in graphics that are new in their kind. It is demonstrated that the Heitler number phase uncertainty relation is incorrect. The problem of the transition from quantum to classical distribution functions by ℏ→0 is discussed with the conjecture that the inertial point of the distributions plays a basic role.
With the correspondence between operator ordering rules and probability distribution functions, moments can be calculated in many ways. We show here the independence of minimum uncertainty relations on the choice of Hermitian ordering rule. Generalized Wigner functions were expressed in terms of ordering real-valued function ϴ(x) for x ϵ R satisfying ϴ(0)=1. Uncertainties of position and momentum are obtained for a Gaussian state. Results confirm that the minimum uncertainty relation holds even when the Hermitian ordering rule is modified.