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SHANNON ENTROPY AS A MEASURE OF UNCERTAINTY IN POSITIONS AND MOMENTA

Łukasz Rudnicki

Published Aug 16, 2016• 0 citations
OpenAlex

Abstract

This paper is prepared as a contribution to the proceedings after the 12th ICSSUR/Feynfest Conference held in Foz do Iguaçu (Brazil) from 2 to 6 May 2011. In the first part I briefly report the topic of entropic uncertainty relations for position and momentum variables. Then I investigate the discrete Shannon entropies related to the case of finite number of detectors set to measure probability distributions in position and momentum spaces. I derive an uncertainty relation for the sum of the Shannon entropies which generalizes previous approaches [Phys. Lett. 103 A, 253 (1984)] based on an infinite number of detectors (bins).

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This paper is prepared as a contribution to the proceedings after the 12th ICSSUR/Feynfest Conference held in Foz do Iguaçu (Brazil) from 2 to 6 May 2011. In the first part I briefly report the topic of entropic uncertainty relations for position and momentum variables. Then I investigate the discrete Shannon entropies related to the case of finite number of detectors set to measure probability distributions in position and momentum spaces. I derive an uncertainty relation for the sum of the Shannon entropies which generalizes previous approaches [Phys. Lett. 103 A, 253 (1984)] based on an infinite number of detectors (bins).
This paper is prepared as a contribution to the proceedings after the 12th ICSSUR/Feynfest Conference held in Foz do Iguacu (Brazil) from 2 to 6 May 2011. In the first part I briefy report the topic of entropic uncertainty relations for position and momentum variables. Then I investigate the discrete Shannon entropies related to the case of fnite number of detectors set to measure probability distributions in position and momentum spaces. I derive an uncertainty relation for the sum of the Shannon entropies which generalizes previous approaches [Phys. Lett. 103 A, 253 (1984)] based on an infnite number of detectors (bins).
CONFINED HARMONIC OSCILLATOR.The aim of this study was a didactic presentation of the Shannon entropy in the quantum theory context, followed by application to the case of a one-dimensional harmonic oscillator in its ground state, both in the free case and confined case.The study of these systems allows us to highlight notions such as location or delocation of a particle, a possible interpretation that the Shannon entropy can adopt.The Shannon entropy in position (S x ) and momentum (S p ) spaces were calculated for both systems, beyond the entropic sum (S t = S x + S p ).With this procedure it was possible to identify trends in the behavior of the Shannon entropy (S x and S p ) and test the compliance of the entropic uncertainty relation (S t = S x + S p ≥ n(1 + ln(p))).
A prominent formulation of the uncertainty principle identifies the fundamental quantum feature that no particle may be prepared with certain outcomes for both position and momentum measurements. Often the statistical uncertainties are thereby measured in terms of entropies providing a clear operational interpretation in information theory and cryptography. Recently, entropic uncertainty relations have been used to show that the uncertainty can be reduced in the presence of entanglement and to prove security of quantum cryptographic tasks. However, much of this recent progress has been focused on observables with only a finite number of outcomes not including Heisenberg’s original setting of position and momentum observables. Here, we show entropic uncertainty relations for general observables with discrete but infinite or continuous spectrum that take into account the power of an entangled observer. As an illustration, we evaluate the uncertainty relations for position and momentum measurements, which is operationally significant in that it implies security of a quantum key distribution scheme based on homodyne detection of squeezed Gaussian states.
Uncertainty principle, which was first introduced by Werner Heisenberg in 1927, forms a fundamental component of quantum mechanics. A graceful aspect of quantum mechanics is that the uncertainty relations between incompatible observables allow for succinct quan- titative formulations of this revolutionary idea: it is impossible to simultaneously measure two complementary variables of a particle in precision. In particular, information theory offers two basic ways to express the Heisenberg’s principle: variance-based uncertainty relations and entropic uncertainty relations. We first investigate the uncertainty relations based on the sum of variances and derive a family of weighted uncertainty relations to provide an optimal lower bound for all situations. Our work indicates that it seems unreasonable to assume a priori that incompatible observables have equal contribution to the variance-based sum form uncertainty relations. We also study the role of mutually exclusive physical states in the recent work and generalize the variance-based uncertainty relations to mutually exclusive uncertainty relations. Next, we develop a new kind of entanglement detection criteria within the framework of marjorization theory and its matrix representation. By virtue of majorization uncertainty bounds, we are able to construct the entanglement criteria which have advantage over the scalar detect- ing algorithms as they are often stronger and tighter. Furthermore, we explore various expression of entropic uncertainty relations, including sum of Shannon entropies, majorization uncer- tainty relations and uncertainty relations in presence of quantum memory. For entropic uncertainty relations without quantum side information, we provide several tighter bounds for multi-measurements, with some of them also valid for Rényi and Tsallis entropies besides the Shannon entropy. We employ majorization theory and actions of the symmetric group to obtain an admixture bound for entropic uncertainty relations with multi-measurements. Comparisons among existing bounds for multi-measurements are also given. However,classical entropic uncertainty relations assume there has only classical side information. For modern uncertainty relations, those who allowed for non-trivial amount of quantum side information, their bounds have been strengthened by our recent result for both two and multi- measurements. Finally, we propose an approach which can extend all uncertainty relations on Shannon entropies to allow for quantum side information and discuss the applications of our entropic framework. Combined with our uniform entanglement frames, it is possible to detect entanglement via entropic uncertainty relations even if there is no quantum side in- formation. With the rising of quantum information theory, uncertainty relations have been established as important tools for a wide range of applications, such as quantum cryptography, quantum key distribution, entanglement detection, quantum metrology, quantum speed limit and so on. It is thus necessary to focus on the study of uncertainty relations.
Abstract Besides the well-known Shannon entropy, there is a set of Shannon-like entropies which have applications in statistical and quantum physics. These entropies are functions of certain parameters and converge toward Shannon entropy when these parameters approach the value 1. We describe briefly the most important Shannon-like entropies and present their graphical representations. Their graphs look almost identical, though by superimposing them it appears that they are distinct and characteristic of each Shannon-like entropy. We try to formulate the alternative entropic uncertainty relations by means of the Shannon-like entropies and show that all of them equally well express the uncertainty principle of quantum physics.
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.