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A note on entropic uncertainty relations of position and momentum

Thomas Schürmann

arXiv:1006.3987Journal of Russian Laser ResearchJan 1, 20123 citations

On entropic uncertainty relations in the presence of a minimal length

Alexey E. Rastegin

arXiv:1607.08512Annals of PhysicsMay 15, 201715 citations

Entropic Uncertainty Relations for Mutually Unbiased Operator Frames

Jesni Shamsul Shaari, Stefano Mancini

We develop an operator-frame formulation of entropic uncertainty relations in the Hilbert-Schmidt space of operators. For general continuous indexed operator frames, we derive an entropic uncertainty relation for the associated coefficient distributions by combining endpoint norm estimates with Riesz-Thorin interpolation. We then identify a distinguished class of mutually unbiased operator frames, defined through constant-modulus trace overlaps. Under suitable structural conditions, the corresponding coefficient amplitudes are related by a bilinear Fourier transform, leading to a stronger Hirschman-Beckner-type entropic uncertainty relation. As canonical realizations, we consider Weyl displacement operators and Wigner kernels, as well as Cartesian dyadic frames generated by position and momentum eigenstates. These examples recover familiar continuous-variable Fourier dualities while extending entropic uncertainty relations beyond measurement outcomes to operator representations themselves.We develop an operator-frame formulation of entropic uncertainty relations in the Hilbert-Schmidt space of operators. For general continuous indexed operator frames, we derive an entropic uncertainty relation for the associated coefficient distributions by combining endpoint norm estimates with Riesz-Thorin interpolation. We then identify a distinguished class of mutually unbiased operator frames, defined through constant-modulus trace overlaps. Under suitable structural conditions, the corresponding coefficient amplitudes are related by a bilinear Fourier transform, leading to a stronger Hirschman-Beckner-type entropic uncertainty relation. As canonical realizations, we consider Weyl displacement operators and Wigner kernels, as well as Cartesian dyadic frames generated by position and momentum eigenstates. These examples recover familiar continuous-variable Fourier dualities while extending entropic uncertainty relations beyond measurement outcomes to operator representations themselves.

quant-pharXiv:2606.23185Journal of Physics A Mathematical and TheoreticalJun 23, 20260 citations

Entropic formulation of the uncertainty principle for the number and annihilation operators

Alexey E. Rastegin

An entropic approach to formulating uncertainty relations for the number–annihilation pair is considered. We construct some normal operator that traces the annihilation operator as well as commuting quadratures with a complete system of common eigenfunctions. Expanding the measured wave function with respect to them, one obtains a relevant probability distribution. Another distribution is naturally generated by measuring the number operator. Due to the Riesz–Thorin theorem, there exists a nontrivial inequality between corresponding functionals of the above distributions. We find the bound in this inequality and further derive uncertainty relations in terms of both the Rényi and Tsallis entropies. Entropic uncertainty relations for a continuous distribution as well as relations for a discretized one are presented.An entropic approach to formulating uncertainty relations for the number–annihilation pair is considered. We construct some normal operator that traces the annihilation operator as well as commuting quadratures with a complete system of common eigenfunctions. Expanding the measured wave function with respect to them, one obtains a relevant probability distribution. Another distribution is naturally generated by measuring the number operator. Due to the Riesz–Thorin theorem, there exists a nontrivial inequality between corresponding functionals of the above distributions. We find the bound in this inequality and further derive uncertainty relations in terms of both the Rényi and Tsallis entropies. Entropic uncertainty relations for a continuous distribution as well as relations for a discretized one are presented.

arXiv:1007.1519Physica ScriptaOct 21, 201124 citations

On a generalized entropic uncertainty relation in the case of the qubit

Steeve Zozor, G. M. Bosyk, M. Portesi

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 squareWe 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

Jan 1, 201314 citations

Entropic formulation of uncertainty relations

M. D. Srinivas

PramanaOct 1, 19856 citations

Notes on Entropic Uncertainty Relations Beyond the Scope of Riesz’s Theorem

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