On the Cauchy-Schwarz inequality and its reverse in semi-inner product C*-modules

Ilišević, Dijana; Varošanec, Sanja

doi:10.15352/bjma/1240321557

Cited by 32 publications

(15 citation statements)

References 7 publications

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“…Here, for the first time, we introduce the reverse bound, i.e., the upper bound to the product and the sum of variances of two incompatible observables. To prove the reverse uncertainty relation for the product of variances of two observables, we use the reverse Cauchy-Schwarz inequality for positive real numbers [36][37][38][39]. This states that for two sets of positive real numbers c 1 , ..., c n and d 1 , ...d n , if 0 < c ≤ c i ≤ C < ∞, 0 < d ≤ d i ≤ D < ∞ for some constants c, d, C and D for all i = 1, ...n, then i,j…”

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confidence: 99%

Tighter uncertainty and reverse uncertainty relations

2017

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We prove a few novel state-dependent uncertainty relations for product as well the sum of variances of two incompatible observables. These uncertainty relations are shown to be tighter than the Roberson-Schrödinger uncertainty relation and other ones existing in the current literature. Also, we derive state dependent upper bound to the sum and the product of variances using the reverse Cauchy-Schwarz inequality and the Dunkl-Williams inequality. Our results suggest that not only we cannot prepare quantum states for which two incompatible observables can have sharp values, but also we have both, lower and upper limits on the variances of quantum mechanical observables at a fundamental level.Introduction.-Quantum mechanics has many distinguishing features from classical mechanics in the microscopic world. One of these distinguished features is the existence of incompatible observables. As a result of this incompatibility, we have the uncertainty principle and the uncertainty relations. Owing to the seminal works by Heisenberg [1], Robertson [2] and Schrödinger [3], lower bounds were shown to exist for the product of variances of two non-commuting observables. Recently, Maccone and Pati have shown stronger uncertainty relations for all incompatible observables [4]. The stronger uncertainty relations have also been experimentally tested [5]. In addition, the entropic uncertainty and the reverse uncertainty relations also capture the essence of quantum uncertainty [6][7][8][9][10][11] and the incompatibility between two observables, but in a state-independently way.

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confidence: 99%

Tighter uncertainty and reverse uncertainty relations

2017

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“…First we are going to develop the upper bound in uncertainty for the linear model followed by the non -linear model. For this we are going to use reverse Cauchy-Schwarz inequality [63][64][65][66]. It is defined as:…”

Section: Reverse Uncertainty Relations For Nc Spacementioning

confidence: 99%

Probing Uncertainty Relations in Non-Commutative Space

2019

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In this paper, we compute uncertainty relations for non-commutative space and obtain a better lower bound than the standard one obtained from Heisenberg's uncertainty relation. We also derive the reverse uncertainty relation for product and sum of uncertainties of two incompatible variables for one linear and another nonlinear model of the harmonic oscillator. The non-linear model in non-commutating space yields two different expressions for Schrödinger and Heisenberg uncertainty relation. This distinction does not arise in commutative space, and even in the linear model of non-commutative space. * goutam.paul@isical.ac.in

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“…Moreover, Niculescu [15] and Joi¸ta [10] have investigated the reverse of the Cauchy-Schwarz inequalities in the framework of C * -algebras and Hilbert C * -modules, see also [14] and references therein. We also refer to another interesting paper by Ilišević and Varošanec [9] of this type. Some operator versions of the Cauchy-Schwarz inequality with simple conditions for the case of equality are presented in [6].…”

Section: Introductionmentioning

confidence: 99%

A treatment of the Cauchy–Schwarz inequality in C⁎-modules

Arambašić

Bakić

Moslehian

2011

Journal of Mathematical Analysis and Applications

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We study the Cauchy-Schwarz and some related inequalities in a semi-inner product module over a C * -algebra A . The key idea is to consider a semi-inner product A -module as a semi-inner product A -module with respect to another semi-inner product. In this way, we improve some inequalities such as the Ostrowski inequality and an inequality related to the Gram matrix. The induced semi-inner products are also related to the notion of covariance and variance. Furthermore, we obtain a sequence of nested inequalities that emerges from the Cauchy-Schwarz inequality. As a consequence, we derive some interesting operator-theoretical corollaries. In particular, we show that the sequence arising from our construction, when applied to a positive invertible element of a C * -algebra, converges to its inverse.

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On the Cauchy-Schwarz inequality and its reverse in semi-inner product C*-modules

Cited by 32 publications

References 7 publications

Tighter uncertainty and reverse uncertainty relations

Tighter uncertainty and reverse uncertainty relations

Probing Uncertainty Relations in Non-Commutative Space

A treatment of the Cauchy–Schwarz inequality in C⁎-modules

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