Uncertainty relations for general unitary operators

Bagchi, Shrobona; Pati, Arun Kumar

doi:10.1103/physreva.94.042104

Cited by 39 publications

(40 citation statements)

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“…(1) was improved by Schrödinger [5]. Recently, variancebased uncertainty relations have been intensely studied in [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. Because of their relevance in quantum information theory, the entropies [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45] have been employed to quantify the uncertainty relations between incompatible observables.…”

Section: Introductionmentioning

confidence: 99%

Strong unitary uncertainty relations

Jing

Li‐Jost

2019

Phys. Rev. A

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In this paper we provide a new set of uncertainty principles for unitary operators using a sequence of inequalities with the help of the geometric-arithmetic mean inequality. As these inequalities are "fine-grained" compared with the well-known Cauchy-Schwarz inequality, our framework naturally improves the results based on the latter. As such, the unitary uncertainty relations based on our method outperform the best known bound introduced in [Phys. Rev. Lett. 120, 230402 (2018)] to some extent. Explicit examples of unitary uncertainty relations are provided to back our claims. arXiv:1908.05053v1 [quant-ph]

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Section: Introductionmentioning

confidence: 99%

Strong unitary uncertainty relations

Jing

Li‐Jost

2019

Phys. Rev. A

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“…There exist two kinds of operators in quantum mechanics: Hermitian and non-Hermitian operators, but it should be paid particular attention that the previous uncertainty relations are contradictory with the non-Hermitian operators, i.e., lots of uncertainty relations will be violated when applied to non-Hermitian operators [68][69][70] 2 for all qubit systems, where the non-Hermitian operator σ + (σ − ) is the raising (lowering) operator of the single qubit system. That is to say, different from the Hermitian operators, the uncertainties of the non-Hermitian operators are not lower-bounded by the quantities related with the commutator and anti-commutator.…”

Section: The Applicability Of the Unified And Exact Framework To Non-mentioning

confidence: 99%

Unified and Exact Framework for Variance-Based Uncertainty Relations

Zheng

Fan

et al. 2020

Sci Rep

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We provide a unified and exact framework for the variance-based uncertainty relations. This unified framework not only recovers some well-known previous uncertainty relations, but also fixes the deficiencies of them. Utilizing the unified framework, we can construct the new uncertainty relations in both product and sum form for two and more incompatible observables with any tightness we require. Moreover, one can even construct uncertainty equalities to exactly express the uncertainty relation by the unified framework, and the framework is therefore exact in describing the uncertainty relation. Some applications have been provided to illustrate the importance of this unified and exact framework. Also, we show that the contradiction between uncertainty relation and non-Hermitian operator, i.e., most of uncertainty relations will be violated when applied to non-Hermitian operators, can be fixed by this unified and exact framework. Quantum uncertainty relations 1-3 , expressing the impossibility of the joint sharp preparation of the incompatible observables 4,5 , are the most fundamental differences between quantum and classical mechanics 6-9. The uncertainty relation has been widely used in the quantum information science 10,11 , such as quantum non-cloning theorem 12,13 , quantum cryptography 14-17 , entanglement detection 18-22 , quantum spins squeezing 23-26 , quantum metrology 27-29 , quantum synchronization 30,31 and mixedness detection 32,33. In general, the improvement in uncertainty relations will greatly promote the development of quantum information science 18,28,34-36. The variance-based uncertainty relations for two incompatible observables A and B can be divided into two forms: the product form ΔA 2 ΔB 2 ≥ LB p 2,3,5,37,38 and the sum form ΔA 2 + ΔB 2 ≥ LB s 39-42 , where LB p and LB s represent the lower bounds of the two forms uncertainty relations, and ΔQ 2 is the variance of Q (To make sure that the quantity measuring the uncertainty will be a real number, the variance is taken as 〈(Q − 〈Q〉) † (Q − 〈Q〉) for non-Hermitian operators. Here the 〈Q〉 represents the expected value of Q). The product form uncertainty relation cannot fully capture the concept of the incompatible observables, because it can be trivial; i.e., the lower bound LB p can be null even for incompatible observables 39,40,43,44. This deficiency is referred to as the triviality problem of the product form uncertainty relation. In order to fix the triviality problem, Maccone and Pati deduced a sum form uncertainty relation with a non-zero lower bound for incompatible observables 44 , firstly showing that the triviality problem can be addressed by the sum form uncertainty relation. Thus, the sum form uncertainty relations were considered to be stronger than the product form uncertainty relations, and since then, lots of effort has been made to investigate the uncertainty relation in the sum form 18,39,45-48. However, most of the sum form uncertainty relations depend on the orthogonal state to the state of the system, and thus are diffic...

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“…In 2014, Mcconne and Pati [27] proposed a sum form of variancebased uncertainty relation, which captures the essence of noncommutativity and also raises the question of compatibility (see [28] for further discussion). In [29] the sum form was strengthened considerably. In [30] the authors showed that there are no nontrivial unconditional joint-measurement bounds for state-dependent errors in the conceptual framework, while Heisenberg-type measurement uncertainty relations for state-independent errors have been proven (cf.…”

Section: Introductionmentioning

confidence: 99%

Improved unitary uncertainty relations

Hu,

Jing

2022

Preprint

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We derive strong variance-based uncertainty relations for arbitrary two and more unitary operators by re-examining the mathematical foundation of the uncertainty relation. This is achieved by strengthening the celebrated Cauchy-Schwarz inequality using a method of brackets and convex functions. The unitary uncertainty relations outperform several strong unitary uncertainty relations, notably better than some recent best lower bounds such as [

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Uncertainty relations for general unitary operators

Cited by 39 publications

References 50 publications

Strong unitary uncertainty relations

Strong unitary uncertainty relations

Unified and Exact Framework for Variance-Based Uncertainty Relations

Improved unitary uncertainty relations

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