Stronger uncertainty relations of mixed states
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Stronger uncertainty relations of mixed states Yajing Fan1
· Huaixin Cao2 · Liang Chen3 · Huixian Meng4
Received: 1 February 2020 / Accepted: 11 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The Heisenberg–Robertson uncertainty relation bounds the product of the variances in the two possible measurement outcomes in terms of the expectation of the commutator of the observables. Notably, it does not capture the concept of incompatible observables because it can be trivial, i.e., the lower bound can be null even for two noncompatible observables. Here, we give two stronger uncertainty relations, relating to the sum of variances with respect to density matrix, whose lower bounds are guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system; moreover, two stronger uncertainty relations in terms of the product of the variances of two observables are established. Also, several stronger uncertainty relations for three observables are established, relating to the sum and product of variances with respect to density matrix, respectively. Keywords Uncertainty relation · Variance · Observable
1 Introduction Uncertainty relations are fundamental in quantum mechanics, underlying many conceptual differences between classical and quantum theories. The Heisenberg– Robertson uncertainty relations [1] are expressed in terms of the product Vρ (A)Vρ (B) of the variances of the measurement results of the observables A and B, and the product can be null even when one of the two variances is different from zero. Here, we provide a different uncertainty relation, based on the sum Vρ (A)+ Vρ (B), that is guaranteed to
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Yajing Fan [email protected]
1
School of Mathematics and Information Science, Ningxia Key Laboratory of Intelligent Information and Big Data Processing, North Minzu University, Yinchuan 750021, China
2
School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China
3
Department of Mathematics, Changji College, Changji 831100, China
4
School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China 0123456789().: V,-vol
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be nontrivial whenever the observables are incompatible on the state. Previous uncertainty relations that provide a bound to the sum of the variances comprise a lower bound in terms of the variance of the sum of observables [2], a lower bound based on the entropic uncertainty relations [3], a sum uncertainty relation for angular momentum observables [4], sum uncertainty relations for arbitrary N observables [5], a series of uncertainty inequalities in the qubit system and a state-independent bound for the sum of variances [6], a unified and exact framework for the variance-based uncertainty relations [7], a lower bound based on the Wigner–Yanase skew information or Wigner–Yanase–Dyson skew information uncertainty relations [8–12]. Uncertainty relations are useful in many areas related or even unrelated to qua
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