Theoretical formulation of finite-dimensional discrete phase spaces: II. On the uncertainty principle for Schwinger unitary operators

Marchiolli, Marcelo A.; Mendonça, Paulo E. M. F.

doi:10.1016/j.aop.2013.05.009

Cited by 13 publications

(10 citation statements)

References 66 publications

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“…A few other results can be found in Ref. [17][18][19][20], where the authors have discussed the uncertainty lower bounds for the unitary operators related by the discrete Fourier transform. However, there are no uncertainty relation for general unitary operators.…”

Section: Introductionmentioning

confidence: 94%

Uncertainty relations for general unitary operators

Bagchi

Pati

2016

Phys. Rev. A

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We derive several uncertainty relations for two arbitrary unitary operators acting on physical states of a Hilbert space. We show that our bounds are tighter in various cases than the ones existing in the current literature. Using the uncertainty relation for the unitary operators we obtain the tight state-independent lower bound for the uncertainty of two Pauli observables and anticommuting observables in higher dimensions. With regard to the minimum uncertainty states, we derive the minimum uncertainty state equation by analytic method, and relate this to the ground-state problem of the Harper Hamiltonian. Furthermore, the higher dimensional limit of the uncertainty relations and minimum uncertainty states are explored. From an operational point of view, we show that the uncertainty in the unitary operator is directly related to the visibility of quantum interference in an interferometer where one arm of the interferometer is affected by a unitary operator. This shows a principle of preparation uncertainty, i.e., for any quantum system, the amount of visibility for two general non-commuting unitary operators is non-trivially upper bounded.

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Section: Introductionmentioning

confidence: 94%

Uncertainty relations for general unitary operators

Bagchi

Pati

2016

Phys. Rev. A

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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].…”

Section: Introductionmentioning

confidence: 99%

“…Other applications of Masser-Spiandel's uncertainty relations include modular variables [7] and signal processing [8,9]. Several further uncertainty relations for unitary operators related by DFT have been investigated in [11][12][13][14]. Later Bagchi and Pati [19] derived sum-form variancebased uncertainty relations for two general unitary operators, which have been tested experimentally with photonic qutrits [24].…”

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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“…where Û and V correspond to the Schwinger unitary operators [48] defined in an N -dimensional state vector space, whose mathematical properties were extensively explored in Refs. [28,49].…”

Section: Preliminaries On the Discrete Wigner Functions For Su(n)mentioning

confidence: 99%

Representations of two-qubit and ququart states via discrete Wigner functions

Marchiolli,

Galetti

2019

Preprint

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By means of a well-grounded mapping scheme linking Schwinger unitary operators and generators of the special unitary group SU(N), it is possible to establish a self-consistent theoretical framework for finite-dimensional discrete phase spaces which has the discrete SU(N) Wigner function as a legitimate by-product. In this paper, we apply these results with the aim of putting forth a detailed study on the discrete SU(2) ⊗ SU(2) and SU(4) Wigner functions, in straight connection with experiments involving, among other things, the tomographic reconstruction of density matrices related to the two-qubit and ququart states. Next, we establish a formal correspondence between both the descriptions that allows us to visualize the quantum correlation effects of these states in finite-dimensional discrete phase spaces. Moreover, we perform a theoretical investigation on the twoqubit X-states, which combines discrete Wigner functions and their respective marginal distributions in order to obtain a new function responsible for describing qualitatively the quantum correlation effects. To conclude, we also discuss possible extensions to the discrete Husimi and Glauber-Sudarshan distribution functions, as well as future applications on spin chains.

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Theoretical formulation of finite-dimensional discrete phase spaces: II. On the uncertainty principle for Schwinger unitary operators

Cited by 13 publications

References 66 publications

Uncertainty relations for general unitary operators

Uncertainty relations for general unitary operators

Strong unitary uncertainty relations

Representations of two-qubit and ququart states via discrete Wigner functions

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