Variance uncertainty relations without covariances for three and four observables

Dodonov, V. V.

doi:10.1103/physreva.97.022105

Cited by 24 publications

(27 citation statements)

References 50 publications

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“…Several interesting questions and problems arise in connection with our study. For example, is it possible to obtain more precise limits for the high-order uncertainty products for several degrees of freedom, generalizing known results for the products of variances [ 2 , 29 , 76 , 77 , 78 , 79 , 80 , 81 , 82 ]? Probably, applications to the problem of intensity-difference squeezing [ 83 , 84 ] can be found.…”

Section: Discussionmentioning

confidence: 99%

Minimal Products of Coordinate and Momentum Uncertainties of High Orders: Significant and Weak High-Order Squeezing

Freitas

Dantas

Dodonov

2020

Entropy

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We consider the problem of minimization of products of mean values of the high powers of operators x and p. From this point of view, we study several two-term superpositions of the Fock states, as well as three popular families of infinite superpositions: squeezed states, even/odd coherent states, and orthogonal even coherent states (or compass states). The new element is the analysis of products of the corresponding (co)variances and the related generalized (Robertson–Schrödinger) intelligent states (RSIS). In particular, we show that both Fock and pure Gaussian homogeneous states are RSIS for the fourth powers (but not for the sixth ones). We show that lower bounds of the high-order uncertainty products can be significantly below the vacuum values. In this connection, the concept of significant and weak high-order squeezing is introduced.

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Section: Discussionmentioning

confidence: 99%

Minimal Products of Coordinate and Momentum Uncertainties of High Orders: Significant and Weak High-Order Squeezing

Freitas

Dantas

Dodonov

2020

Entropy

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“…SUR was initially derived from the Cauchy-Schwarz inequality, and can only be used to describe the uncertainty relation for two incompatible obervables. Since then, lots of work has been done along the way of Schrödinger regime [51][52][53][54][55] , and most of them mainly focused on extending SUR to uncertainty relations for more incompatible observables 56 . We refer to these uncertainty relations as the Schrödinger's spirit, and these uncertainty relations can be uniformly derived as follows.…”

Section: Uncertainty Relation Class-c0mentioning

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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“…Denoting ℳ =̂1 −̂2 − 〈̂̂1〉̂/〈̂̂〉 + 〈̂̂2〉̂/〈̂̂〉, where ̂=̂ or ̂ , and taking advantage of 〈ℳℳ † 〉 ≥ 0 [30] and Eq. (1), one can obtain a lower bound for ∆(̂1 −̂2) 2 (for more details, please see the appendix):…”

Section: Deduction Of the Local Criterions For Quantum Phase Syncmentioning

confidence: 99%

Criterion of quantum phase synchronization in continuous variable systems by local measurement

Zheng

2020

Quantum Inf Process

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Phase synchronization was proved to be unbounded in quantum level, but the witness of phase synchronization is always expensive in terms of the quantum resource and non-local measurements involved. Based on the quantum uncertainty relation, we construct two local criterions for the phase synchronization in this paper. The local criterions indicate that the phase synchronization in the quantum level can be witnessed only by the local measurements, and the deduction has been verified in the optomechanics system in numerical way. Besides, by analyzing the physical essence of the phase synchronization in quantum level, we show that one can prepare a state, which describes two synchronized oscillators with no entanglement between them. Thus, the entanglement resource is not necessary in the occurrence of the ideal phase synchronization, and also the reason for this phenomenon is discussed.PACS number(s): 03.65.Yz; 03.65.Ud I. IntroductionSpontaneous synchronization refers to the phenomenon that the two or more weakly coupled systems with different natural frequencies synchronize their motions only due to their mutual interaction [1][2][3][4]. The occurrence of synchronization is first discovered by Huygens in two coupled pendulum clocks, and then has been observed in so many different settings, such as the collective lightning of fireflies, the beating of heart cells and chemical reaction [2]. In classical mechanics, the spontaneous synchronization has been widely studied [5][6][7][8], and there exist standard methods to verify whether the motion of two systems is synchronized [2].In quantum level, the spontaneous synchronization has been considered from different aspects: clock synchronization [9][10][11][12], synchronization in oscillator networks [13][14][15][16][17][18][19][20][21], and synchronization between two atomic ensembles [22]. Notably, due to the absence of the phase space trajectories, the extension of the notion of phase synchronization from classical mechanics to its quantum counterpart is not straightforward [23]. Lots of work has been done and great progress has been

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Variance uncertainty relations without covariances for three and four observables

Cited by 24 publications

References 50 publications

Minimal Products of Coordinate and Momentum Uncertainties of High Orders: Significant and Weak High-Order Squeezing

Minimal Products of Coordinate and Momentum Uncertainties of High Orders: Significant and Weak High-Order Squeezing

Unified and Exact Framework for Variance-Based Uncertainty Relations

Criterion of quantum phase synchronization in continuous variable systems by local measurement

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