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arXiv:quant-ph/0107149 · Physical Review A

Exact uncertainty relations

Michael J. W. Hall

Published Oct 4, 2001• 89 citations
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Abstract

The Heisenberg inequality $\ensuremath{\Delta}X\ensuremath{\Delta}P>~\ensuremath{\Elzxh}/2$ can be replaced by an exact equality, for suitably chosen measures of position and momentum uncertainty, which is valid for all wave functions. The statistics of complementary observables are thus connected by an ``exact'' uncertainty relation. Results may be generalized to angular momentum and phase, photon number and phase, time and frequency, and to states described by density operators. Connections to energy bounds, entanglement, Wigner functions, and optimal estimation of an observable from the measurement of a second observable are also given.

DOI
10.1103/physreva.64.052103

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The Heisenberg inequality \ensuremathΔX\ensuremathΔP> \ensuremath\Elzxh/2\ensuremath{\Delta}X\ensuremath{\Delta}P>~\ensuremath{\Elzxh}/2\ensuremathΔX\ensuremathΔP> \ensuremath\Elzxh/2 can be replaced by an exact equality, for suitably chosen measures of position and momentum uncertainty, which is valid for all wave functions. The statistics of complementary observables are thus connected by an ``exact'' uncertainty relation. Results may be generalized to angular momentum and phase, photon number and phase, time and frequency, and to states described by density operators. Connections to energy bounds, entanglement, Wigner functions, and optimal estimation of an observable from the measurement of a second observable are also given.
The Heisenberg inequality ΔX ΔP \geq \hbar/2 can be replaced by an exact equality, for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. The significance of this "exact" uncertainty relation is discussed, and results are generalised to angular momentum and phase, photon number and phase, time and frequency, and to states described by density operators. Connections to optimal estimation of an observable from the measurement of a second observable, Wigner functions, energy bounds, and entanglement are also given.
The usual Heisenberg uncertainty relation, ΔX ΔP ≥ ℏ/2, may be replaced by an exact equality for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. This exactuncertainty relation, δX ΔPnc ≡ ℏ/2, can be generalised to other pairs of conjugate observables such as photon number and phase, and is sufficiently strong to provide the basis for moving from classical mechanics to quantum mechanics. In particular, the assumption of a nonclassical momentum fluctuation, having a strength which scales inversely with uncertainty in position, leads from the classical equations of motion to the Schrödinger equation.
Quantum-mechanical uncertainty relations for position and momentum are expressed in the form of inequalities involving the R\'enyi entropies. The proof of these inequalities requires the use of the exact expression for the (p,q)(p,q)(p,q)-norm of the Fourier transformation derived by Babenko and Beckner. Analogous uncertainty relations are derived for angle and angular momentum and also for a pair of complementary observables in NNN-level systems. All these uncertainty relations become more attractive when expressed in terms of the symmetrized R\'enyi entropies.
The mathematical content of the uncertainty principle is contained in special states akin to, and including, states of minimum uncertainty. Specific uncertainty relations are produced by averaging observables in these states. The approach is shown to give more flexible and complete expressions of indeterminacy than the familiar generalized Heisenberg inequality: statistical measures other than variances can be employed, the method is not limited to describing uncertainties in terms of products of statistical dispersions, and the resulting inequalities have the most stringent bounds attainable independent of the system state. Applications are then made to find uncertainty relations for position-momentum average displacements from the means, time-energy indeterminacy, and uncertainties in angular momentum components.
The sum of the information gains corresponding to measurements of position and momentum is bounded by $\mathrm{log}2\ensuremath{\Delta}X\ensuremath{\Delta}P/\ensuremath{\hbar}$ for a quantum ensemble with position and momentum uncertainties $\ensuremath{\Delta}X$ and $\ensuremath{\Delta}P$. The bound implies the Heisenberg uncertainty principle and that the gain of position information can be maximized only at the expense of momentum information, and vice versa. This exclusion principle for the information contents of complementary observables is extended to finite Hilbert spaces, and to the quadrature, number, and phase observables of bosonic fields degraded by Gaussian noise.