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arXiv:1103.2316 · Journal of Mathematical Physics

Entropic uncertainty relations and the stabilizer formalism

Sönke Niekamp, Matthias Kleinmann, Otfried Gühne

Published Jan 1, 2012• 12 citations
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Abstract

Entropic uncertainty relations express the quantum mechanical uncertainty principle by quantifying uncertainty in terms of entropy. Central questions include the derivation of lower bounds on the total uncertainty for given observables, the characterization of observables that allow strong uncertainty relations, and the construction of such relations for the case of several observables. We demonstrate how the stabilizer formalism can be applied to these questions. We show that the Maassen–Uffink entropic uncertainty relation is tight for the measurement in any pair of stabilizer bases. We compare the relative strengths of variance-based and various entropic uncertainty relations for dichotomic anticommuting observables.

DOI
10.1063/1.3678200

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Entropic uncertainty relations express the quantum mechanical uncertainty principle by quantifying uncertainty in terms of entropy. Central questions include the derivation of lower bounds on the total uncertainty for given observables, the characterization of observables that allow strong uncertainty relations, and the construction of such relations for the case of several observables. We demonstrate how the stabilizer formalism can be applied to these questions. We show that the Maassen–Uffink entropic uncertainty relation is tight for the measurement in any pair of stabilizer bases. We compare the relative strengths of variance-based and various entropic uncertainty relations for dichotomic anticommuting observables.
Entropic uncertainty relations place nontrivial lower bounds to the sum of Shannon information entropies for noncommuting observables. Here we obtain a lower bound on the entropy sum for general pairs of observables in finite-dimensional Hilbert space, which improves on the best bound known to date [H. Maassen and J. B. M. Uffink, Phys. Rev. Lett. 60, 1103 (1988)] for a wide class of observables. This result follows from another formulation of the uncertainty principle, the Landau-Pollak inequality, whose relationship to the Maassen-Uffink entropic uncertainty relation is discussed.
How much of the uncertainty in predicting measurement outcomes for noncommuting quantum observables is genuinely quantum mechanical? We provide a natural decomposition of the total entropic uncertainty of two noncommuting observables into a classical component and an intrinsically quantum mechanical component. We show that the total quantum component in a state is never lower or upper bounded by any state-independent quantities, but instead admits ``purity-based'' lower bounds that generalize entropic formulations such as the Maassen-Uffink relation. These relations reveal a nontrivial interplay between quantum and classical randomness in any finite-dimensional state.
The entropic uncertainty relation proven by Maassen and Uffink for arbitrary pairs of two observables is known to be nonoptimal. Here, we call an uncertainty relation optimal, if the lower bound can be attained for any value of either of the corresponding uncertainties. In this work, we establish optimal uncertainty relations by characterizing the optimal lower bound in scenarios similar to the Maassen–Uffink type. We disprove a conjecture by Englert et al. and generalize various previous results. However, we are still far from a complete understanding and, based on numerical investigation and analytical results in small dimension, we present a number of conjectures.
Entropic uncertainty relations H(A)+H(B)⩾γH(A)+H(B)\geqslant γH(A)+H(B)⩾γ give a nonzero lower bound γγγ to the sum of the Shannon entropies HHH of the outcome probabilities of incompatible observables AAA and BBB. They are better than the variance-based uncertainty relations because they only depend on the Born statistics of the outcomes and not on the outcomes themselves, and because bounds γγγ typically are state independent. Here we provide a state-independent lower bound γsγ_sγs​ that is better than the textbook Maassen-Uffink bound and, in the limit of the parameter s→2s\to 2s→2, becomes asymptotically tight for all A,BA,BA,B. The bound can be extended to Renyi entropies. After the submission of this paper, we found out that the main results presented here were already on page 6 of https://quantum-journal.org/papers/q-2018-03-30-59/pdf/
Abstract Quantum states with minimum or maximum uncertainty are of special significance due to their extreme properties. Celebrated examples are coherent states induced from certain Lie groups and intelligent states for various uncertainty relations. In this work, by virtue of the Maassen-Uffink entropic uncertainty relation, we introduce an entropic quantifier of uncertainty and use it to characterize several important families of states in the stabilizer formalism of quantum computation. More specifically, we show that the stabilizer states and T-type magic states stand at the two extremes of the entropic quantifier of uncertainty: The former are precisely the minimum entropic uncertainty states, while the latter are precisely the maximum entropic uncertainty states. Moreover, interpolating between the above two extremes, the H-type magic states are the saddle points of the entropic quantifier of uncertainty. These entropic characterizations reveal some intrinsic features of stabilizer states, H- and T-type magic states, and cast novel light on the resource-theoretic viewpoint of regarding the stabilizer states as free states and the T-type magic states as the most precious source states in the stabilizer quantum theory.