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Physical Review A

Characterizing quantum coherence

John E. Sipe, Nima Arkani-hamed A

Published Sep 1, 1992• 8 citations

Abstract

For a system with canonical variables x and p, [x,p]=i\ensuremath{\Elzxh}, we associate with each density operator \ensuremath{\rho} an x coherence \ensuremath{\delta}x and a p coherence \ensuremath{\delta}p which specify, respectively, the characteristic distance and momentum range over which the underlying quantum-mechanical nature of the ensemble cannot be neglected. These coherences are less than or equal to the associated uncertainties \ensuremath{\delta}x\ensuremath{\le}\ensuremath{\Delta}x, \ensuremath{\delta}p\ensuremath{\le}\ensuremath{\Delta}p; so the ratios ${\mathit{c}}_{\mathit{x}}$\ensuremath{\equiv}\ensuremath{\delta}x/\ensuremath{\Delta}x, ${\mathit{c}}_{\mathit{p}}$\ensuremath{\equiv}\ensuremath{\delta}p/\ensuremath{\Delta}p give measures of the quantum character of the ensemble with respect to x and p. Examples are presented.

DOI: 10.1103/physreva.46.2317

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For a system with canonical variables x and p, [x,p]=i\ensuremath{\Elzxh}, we associate with each density operator \ensuremath{\rho} an x coherence \ensuremath{\delta}x and a p coherence \ensuremath{\delta}p which specify, respectively, the characteristic distance and momentum range over which the underlying quantum-mechanical nature of the ensemble cannot be neglected. These coherences are less than or equal to the associated uncertainties \ensuremath{\delta}x\ensuremath{\le}\ensuremath{\Delta}x, \ensuremath{\delta}p\ensuremath{\le}\ensuremath{\Delta}p; so the ratios

{\mathit{c}}_{\mathit{x}}

\ensuremath{\equiv}\ensuremath{\delta}x/\ensuremath{\Delta}x,

{\mathit{c}}_{\mathit{p}}

\ensuremath{\equiv}\ensuremath{\delta}p/\ensuremath{\Delta}p give measures of the quantum character of the ensemble with respect to x and p. Examples are presented.

Quantifying quantum coherence based on the generalized

\alpha

α –z-relative R

\acute{e}

e ´ nyi entropy

The arguments presented by Gibbins (1981) in his Note are based on a sharp distinction between the product Δx·Δp, which refers to the ranges of position and momentum of an individual system, and the uncertainty principle ΔX·ΔP ≥ ħ/2, which expresses a statistical relation for an ensemble of systems. A critical role in Gibbins’ reasoning is played by the theorem T which states that the restriction of the dynamical variable of position x of an individual system to a finite range Δx excludes the possibility of restricting the canonically conjugate dynamical variable of momentum p to a finite range Δp.

(A) The momentum density conjugate to a bosonic quantum field splits naturally into the sum of a classical component and a nonclassical component. It is shown that the field and the nonclassical component of the momentum density satisfy an_exact_ uncertainty relation, i.e., an equality, which underlies the Heisenberg-type uncertainty relation for fields. (B) The above motivates a new approach to deriving and interpreting bosonic quantum fields, based on an exact uncertainty principle. In particular, the postulate that an ensemble of classical fields is subject to nonclassical momentum fluctuations, of a strength determined by the field uncertainty, leads from the classical to the quantum field equations. Examples include scalar, electromagnetic and gravitational fields. For the latter case the exact uncertainty principle specifies a unique (non-Laplacian) operator ordering for the Wheeler-deWitt equation.