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Journal of Physics A Mathematical and Theoretical

Entropic uncertainty relations and quasi-Hermitian operators

Alexey E. Rastegin

Published Oct 23, 2012• 8 citations
Publisher

Abstract

We discuss a possible treatment of quasi-Hermitian operators from the viewpoint of the uncertainty principle. Here, probabilities are actually determined by the pair containing the square root of a given metric operator and adopted resolution of the identity. For two pairs of such a kind, we derive some inequality between norm-like functionals of generated probability distributions. Based on Riesz’s theorem, this inequality assumes that one enjoys some condition with norms for the squared roots of metric operators and measured density matrix. The derived inequality between norm-like functionals naturally leads to entropic uncertainty relations in terms of the unified entropies. Entropic bounds of both the state-dependent and state-independent forms are presented. The latter form means some implicit dependence, since the measured density matrix is involved in the above condition. The presented entropic bounds are an extension of the previous bounds to the quasi-Hermitian case. The results are discussed within an example of 2×2 quasi-Hermitian matrices. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

DOI
10.1088/1751-8113/45/44/444026

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We discuss a possible treatment of quasi-Hermitian operators from the viewpoint of the uncertainty principle. Here, probabilities are actually determined by the pair containing the square root of a given metric operator and adopted resolution of the identity. For two pairs of such a kind, we derive some inequality between norm-like functionals of generated probability distributions. Based on Riesz’s theorem, this inequality assumes that one enjoys some condition with norms for the squared roots of metric operators and measured density matrix. The derived inequality between norm-like functionals naturally leads to entropic uncertainty relations in terms of the unified entropies. Entropic bounds of both the state-dependent and state-independent forms are presented. The latter form means some implicit dependence, since the measured density matrix is involved in the above condition. The presented entropic bounds are an extension of the previous bounds to the quasi-Hermitian case. The results are discussed within an example of 2×2 quasi-Hermitian matrices. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.
Number-phase uncertainty relations are formulated in terms of unified entropies which form a family of two-parametric extensions of the Shannon entropy. For two generalized measurements, unified-entropy uncertainty relations are given in both the state-dependent and state-independent forms. A few examples are discussed as well. Using the Pegg--Barnett formalism and the Riesz theorem, we obtain a nontrivial inequality between norm-like functionals of generated probability distributions in finite dimensions. The principal point is that we take the infinite-dimensional limit right for this inequality. Hence number-phase uncertainty relations with finite phase resolutions are expressed in terms of the unified entropies. Especially important case of multiphoton coherent states is separately considered. We also give some entropic bounds in which the corresponding integrals of probability density functions are involved.
Number-phase uncertainty relations are formulated in terms of unified entropies which form a family of two-parametric extensions of the Shannon entropy. For two generalized measurements, unified-entropy uncertainty relations are given in both the state-dependent and state-independent forms. A few examples are discussed as well. Using the Pegg--Barnett formalism and the Riesz theorem, we obtain a nontrivial inequality between norm-like functionals of generated probability distributions in finite dimensions. The principal point is that we take the infinite-dimensional limit right for this inequality. Hence number-phase uncertainty relations with finite phase resolutions are expressed in terms of the unified entropies. Especially important case of multiphoton coherent states is separately considered. We also give some entropic bounds in which the corresponding integrals of probability density functions are involved.
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.
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
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 an 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 [0 , 1/2] 2 in the α–β plane, and a semi-analytical expression on the line β = α. It is seen that previous results are included as particular cases. Moreover we present a semi-analytical, suboptimal bound for any couple of indices. In all cases, we provide the minimizing states.
A way to pose the entropic uncertainty principle for trace-preserving super-operators is presented. It is based on the notion of extremal unraveling of a super-operator. For a given input state, different effects of each unraveling result in some probability distribution at the output. As is shown, all Tsallis' entropies of positive order as well as some of Rényi's entropies of this distribution are minimized by the same unraveling of a super-operator. Entropic relations between a state ensemble and the generated density matrix are revisited in terms of both the adopted measures. Using Riesz's theorem, we obtain two uncertainty relations for any pair of generalized resolutions of the identity in terms of the Rényi and Tsallis entropies. The inequality with Rényi's entropies is an improvement of the previous one, whereas the inequality with Tsallis' entropies is a new relation of a general form. The latter formulation is explicitly shown for a pair of complementary observables in a d -level system and for the angle and angular momentum. The derived general relations are immediately applied to extremal unravelings of two super-operators.