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arXiv:1308.5051 · Communications in Theoretical Physics

Uncertainty and Certainty Relations for Pauli Observables in Terms of Rényi Entropies of Order α ∈ (0; 1]

Alexey E. Rastegin

Published Mar 1, 2014• 10 citations
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Abstract

We obtain uncertainty and certainty relations of state-independent form for the three Pauli observables with use of the Rényi entropies of order α ∈ (0; 1]. It is shown that these entropic bounds are tight in the sense that they are always reached with certain pure states. A new result is the condition for equality in Rényi-entropy uncertainty relations for the Pauli observables. Upper entropic bounds in the pure-state case are also novel. Combining the presented bounds leads to a band, in which the rescaled average Rényi α-entropy ranges for a pure measured state. A width of this band is compared with the Tsallis formulation derived previously.

DOI
10.1088/0253-6102/61/3/04

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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
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Abstract Uncertainty relations are fundamental in quantum mechanics, distinguishing it from classical physics by setting limits on the precision of incompatible measurements. For qubit systems, considering the three incompatible Pauli observables, we propose two entropic quantifiers of uncertainty exploiting the Tsallis entropies and Rényi entropies. We analyze the minimum and maximum uncertainty states and uncover their close connections with some important families of states in the stabilizer formalism. Explicitly, our findings show that for the two entropic quantifiers of uncertainties with suitable parameters, the stabilizer states correspond to the minimum uncertainty states, while the T-type magic states correspond to the maximum uncertainty states. Additionally, we identify that the H-type magic states serve as the saddle points in the Tsallis entropy framework. These results expand the recent proposed characterizations of stabilizer and magic states via the Shannon entropic uncertainty relation [Li et al 2024 Phys. Scr. 99, 035117].