Uncertainty relations in the product form

Huang, Xiaofen; Zhang, Tinggui; Jing, Naihuan

doi:10.1088/1674-1056/27/7/070302

Cited by 2 publications

(8 citation statements)

References 29 publications

(43 reference statements)

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“…The lower bound of the Schrödinger inequality (5) holds when both are satisfied simultaneously. Indeed, by using ( 14) and (17), and combining with ( 27) and (28) we find…”

Section: Decompsing the Schr öDinger Inequalitymentioning

confidence: 93%

“…At this point, we must point to the fact that the property (17) has only been shown to be valid under the introductory conditions (i)-(iii) we have been assuming so far. For the purpose of this paper, it is necessary to lift the assumptions (ii) and (iii), which amounts to include the case in which B is degenerate or b i |ψ = 0 for some i, leaving us with the finite dimensional condition (i).…”

Section: Tighter Inequalitiesmentioning

confidence: 99%

“…Now we shall discuss when our extra term E(A, B) is non-vanishing, which amounts to proving the inequality (30) as well. For this, we first note that the key property (17) appears to suggest that the discord between the operators A and A w (B) vanishes identically…”

Section: Tighter Inequalitiesmentioning

confidence: 99%

“…This property follows immediately from the completeness condition i |b i b i | = I, where I is the identity operator in L(H). The equality (17) may at first sound strange, because the r.h.s. is independent of B while the l.h.s.…”

Section: Preliminariesmentioning

confidence: 99%

“…This suggests that our inequality incorporates a novel aspect of uncertainty which is represented neither by the noncommutativity nor the covariance, but by the said discord of observ-ables. Various forms of improvement of the Schrödinger inequality have been suggested recently [12][13][14][15][16][17], some of which require optimization processes to establish their lower bounds. Our new inequality is distinct from all of them in that the improvement is implemented by an explicit term, i.e., the aforementioned discord between the two observables, which was shown in our previous works to be closely related to the compatibility of two observables.…”

Section: Introductionmentioning

confidence: 99%

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Uncertainty Relations of Variances in View of the Weak Value

Lee¹,

Takeuchi²,

Watanabe³

et al. 2020

Preprint

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The Schrödinger inequality is known to underlie the Kennard-Robertson inequality, which is the standard expression of quantum uncertainty for the product of variances of two observables A and B, in the sense that the latter is derived from the former. In this paper we point out that, albeit more subtle, there is yet another inequality which underlies the Schrödinger inequality in the same sense.The key component of this observation is the use of the weak-value operator Aw(B) introduced in our previous works (named after Aharonov's weak value), which was shown to act as the proxy operator for A when B is measured. The lower bound of our novel inequality supplements that of the Schrödinger inequality by a term representing the discord between Aw(B) and A. In addition, the decomposition of the Schrödinger inequality, which was also obtained in our previous works by making use the weak-value operator, is examined more closely to analyze its structure and the minimal uncertainty states. Our results are exemplified with some elementary spin 1 and 3/2 models as well as the familiar case of A and B being the position and momentum of a particle.

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“…The lower bound of the Schrödinger inequality (5) holds when both are satisfied simultaneously. Indeed, by using ( 14) and (17), and combining with ( 27) and (28) we find…”

Section: Decompsing the Schr öDinger Inequalitymentioning

confidence: 93%

Section: Tighter Inequalitiesmentioning

confidence: 99%

Section: Tighter Inequalitiesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Uncertainty Relations of Variances in View of the Weak Value

Lee¹,

Takeuchi²,

Watanabe³

et al. 2020

Preprint

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ZjHXK5 and ZjHXK6 negatively regulate the sugar metabolism of Ziziphus jujuba Mill.

Tong,

Liao,

et al. 2024

Front. Plant Sci.

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Hexokinase (HXK) plays a crucial role in plants, catalyzing the phosphorylation of hexose substances, which is one of the key steps in sugar metabolism and energy production. While HXK genes have been well-studied in model plants, the evolutionary and functional characteristics of HXK gene family in jujube is unknow. In this study, the HXK gene family members were identified by bioinformatics methods, the key members regulating glucose metabolism were identified by transcriptome data, and finally the function of the key genes was verified by instantaneous and stable genetic transformation. Our results showed that seven HXK genes were identified in the jujube genome, all of which were predict located in the chloroplast and contain Hexokinase-1 (PF00349) and Hexokinase-2 (PF03727) conserved domains. Most of HXK proteins were transmembrane protein with stable, lipid-soluble, hydrophilic. The secondary structure of ZjHXK proteins main α-helix, and contains two distinct tertiary structure. All ZjHXK genes contain nine exons and eight introns. Predictions of cis-regulatory elements indicate that the promoter region of ZjHXK contains a large number of MeJA responsive elements. Finally, combined with the analysis of the relationship between the expression and glucose metabolism, found that ZjHXK5 and ZjHXK6 may the key genes regulating sugar metabolism. Transient overexpression of ZjHXK5 and ZjHXK6 on jujube, or allogeneic overexpression of ZjHXK5 and ZjHXK6 on tomato would significantly reduce the content of total sugar and various sugar components. Transient silencing of ZjHXK5 and ZjHXK6 genes results in a significant increase in sucrose and total sugar content. Interestingly, the expression of ZjHXK5 and ZjHXK6 were also affected by methyl jasmonate.

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Uncertainty relations in the product form

Abstract: We study the uncertainty relation in the product form of variances and obtain some new uncertainty relations with weight, which are shown to be tighter than those derived from the Cauchy-Schwarz inequality.

Cited by 2 publications

References 29 publications

Uncertainty Relations of Variances in View of the Weak Value

Uncertainty Relations of Variances in View of the Weak Value

ZjHXK5 and ZjHXK6 negatively regulate the sugar metabolism of Ziziphus jujuba Mill.

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