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arXiv:quant-ph/0608116 · Physical Review A

Formulation of the uncertainty relations in terms of the Rényi entropies

Iwo Białynicki‐Birula

Published Nov 1, 2006• 273 citations
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Abstract

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 $(p,q)$-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 $N$-level systems. All these uncertainty relations become more attractive when expressed in terms of the symmetrized R\'enyi entropies.

DOI
10.1103/physreva.74.052101

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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 position-momentum uncertainty-like inequality based on moments of arbitrary order for ddd-dimensional quantum systems, which is a generalization of the celebrated Heisenberg formulation of the uncertainty principle, is improved here by use of the R\'enyi-entropy-based uncertainty relation. The accuracy of the resulting lower bound is physico-computationally analyzed for the two main prototypes in ddd-dimensional physics: the hydrogenic and oscillator-like systems.
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.
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We revisit generalized entropic formulations of the uncertainty principle for an arbitrary pair of quantum observables in two-dimensional Hilbert space. Rényi entropy is used as uncertainty measure associated with the distribution probabilities corresponding to the outcomes of the observables. We derive a general expression for the tight lower bound of the sum of Rényi entropies for any couple of (positive) entropic indices (α, β). Thus, we have overcome the Hölder conjugacy constraint imposed on the entropic indices by Riesz–Thorin theorem. In addition, we present an analytical expression for the tight bound inside the square