Abstract
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.
Highlights
The uncertainty principle (UP) is a fundamental concept in physics that states the impossibility to predict with absolute certainty and simultaneously the outcomes of measurements for pairs of noncommuting quantum observables.In its primary quantitative formulation, the principle is described by the existence of a nontrivial lower bound for the product of the variances of the operators [1, 2, 3]
For pure states of qubit systems, we obtain the most general entropic formulation of the uncertainty principle in terms of the sum of Renyi entropies associated with any given pair of quantum observables, namely an inequality of the form
Our derivation focusses on obtaining the minimum of the entropies sum and we do not use Riesz–Thorin theorem in contrast to many results in the literature
Summary
The uncertainty principle (UP) is a fundamental concept in physics that states the impossibility to predict with absolute certainty and simultaneously the outcomes of measurements for pairs of noncommuting quantum observables. In its primary quantitative formulation, the principle is described by the existence of a nontrivial lower bound for the product of the variances of the operators [1, 2, 3]. Extensions for nonconjugated indices exist, based on the decreasing property of Renyi entropy versus its index, leading to suboptimal bounds [20, 28]. These bounds have been refined in the case of 2-level systems (or qubits) when the entropic indices coincide and have the value [29] or 2 [30, 31]. The proofs of our results are given in the appendices
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