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arXiv:2001.00799 · New Journal of Physics

Entropic time–energy uncertainty relations: an algebraic approach

Christian Bertoni, Yuxiang Yang, Joseph M. Renes

Published Jun 22, 2020• 3 citations
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Abstract

Abstract We address entropic uncertainty relations between time and energy or, more precisely, between measurements of an observable G and the displacement r of the G-generated evolution e−irG . We derive lower bounds on the entropic uncertainty in two frequently considered scenarios, which can be illustrated as two different guessing games in which the role of the guessers are fixed or not. In particular, our bound for the first game improves the previous result by Coles et al [Phys. Rev. Lett. 122 100401 (2019)]. To derive our bounds, we extend a recently proposed novel algebraic method by Gao et al [arXiv:1710.10038 [quant-ph]] which was used to derive both strong subadditivity and entropic uncertainty relations for measurements.

DOI
10.1088/1367-2630/ab9ee5

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Abstract We address entropic uncertainty relations between time and energy or, more precisely, between measurements of an observable G and the displacement r of the G-generated evolution e−irG . We derive lower bounds on the entropic uncertainty in two frequently considered scenarios, which can be illustrated as two different guessing games in which the role of the guessers are fixed or not. In particular, our bound for the first game improves the previous result by Coles et al [Phys. Rev. Lett. 122 100401 (2019)]. To derive our bounds, we extend a recently proposed novel algebraic method by Gao et al [arXiv:1710.10038 [quant-ph]] which was used to derive both strong subadditivity and entropic uncertainty relations for measurements.
We address entropic uncertainty relations between time and energy or, more precisely, between measurements of an observableGand the displacementrof theG-generated evolution e(-irG). We derive lower bounds on the entropic uncertainty in two frequently considered scenarios, which can be illustrated as two different guessing games in which the role of the guessers are fixed or not. In particular, our bound for the first game improves the previous result by Coleset al[Phys. Rev. Lett.122100401 (2019)]. To derive our bounds, we extend a recently proposed novel algebraic method by Gaoet al[arXiv:1710.10038 [quant-ph]] which was used to derive both strong subadditivity and entropic uncertainty relations for measurements.
Abstract Heisenberg's uncertainty principle in application to energy and time is a powerful heuristics. This statement plays an important role in foundations of quantum theory and statistical physics. If some state exists for a finite interval of time, then it cannot have a completely definite value of energy. It is well known that the case of energy and time principally differs from more familiar examples of two non‐commuting observables. Since quantum theory was originated, many approaches to energy–time uncertainties have been proposed. Entropic way to formulate the uncertainty principle is currently the subject of active researches. Using the Pegg concept of complementarity of the Hamiltonian, uncertainty relations of the “energy–time” type are obtained in terms of Rényi and Tsallis entropies. Although this concept is somehow restricted in scope, derived relations can be applied to systems typically used in quantum information processing. Both the state‐dependent and state‐independent formulations are of interest. Some of the derived state‐independent bounds are similar to the results obtained within a more general approach on the basis of sandwiched relative entropies. The developed method allows us to address the case of detection inefficiencies.
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.
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 are powerful tools, especially in quantum cryptography. They typically bound the amount of uncertainty a third-party adversary may hold on a measurement outcome as a result of the measurement overlap. However, when the two measurement bases are biased towards one another, standard entropic uncertainty relations do not always provide optimal lower bounds on the entropy. Here, we derive a new entropic uncertainty relation, for certain quantum states, which can provide a significantly higher bound even if the two measurement bases are no longer mutually unbiased. We evaluate our bound on two different quantum cryptographic protocols, including BB84 with faulty/biased measurement devices, and show that our new bound can produce substantially higher key-rates under several scenarios when compared with prior work using standard entropic uncertainty relations.