Theoretical formulation of finite-dimensional discrete phase spaces: I. Algebraic structures and uncertainty principles

Marchiolli, Marcelo A.; Ruzzi, M.

doi:10.1016/j.aop.2012.02.015

Cited by 23 publications

(38 citation statements)

References 92 publications

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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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“…The Wigner-Weyl-Moyal (WWM) formalism is a particularly powerful representation of quantum mechanics based on quasi-probability functions. Starting from Wootters' original derivation of discrete Wigner functions [1], there has been much work on analyzing states and operators in finite Hilbert spaces by considering their quasi-probability representation on continuous and discrete support [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The WWM formalism gives a discrete quasi-probability representation in terms of a classical phase space and explicitly introduces classical mechanics and quantum corrections in terms of higher order corrections with respect to ÿ.…”

Section: Introductionmentioning

confidence: 99%

Measurement contextuality and Planck’s constant

Kocia

Love

2018

New J. Phys.

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Contextuality is a necessary resource for universal quantum computation and non-contextual quantum mechanics can be simulated efficiently by classical computers in many cases. Orders of Planck's constant, ÿ, can also be used to characterize the classical-quantum divide by expanding quantities of interest in powers of ÿ-all orders higher than ÿ 0 can be interpreted as quantum corrections to the order ÿ 0 term. We show that contextual measurements in finite-dimensional systems have formulations within the Wigner-Weyl-Moyal (WWM) formalism that require higher than order ÿ 0 terms to be included in order to violate the classical bounds on their expectation values. As a result, we show that contextuality as a resource is closely related to orders of ÿ as a resource within the WWM formalism. This offers an explanation for why qubits can only exhibit state-independent contextuality under Pauli observables as in the Peres-Mermin square while odd-dimensional qudits can also exhibit state-dependent contextuality. In particular, qubit states exhibit contextuality when measured by qubit Pauli observables regardless of the state being measured and so the Weyl symbol of these observables lack an order ÿ 0 contribution altogether. On the other hand, odddimensional qudit states exhibit contextuality when measured by qudit observables depending on the state measured and so odd-dimensional qudit observables generally possess non-zero order ÿ 0 terms, and higher, in their WWM formulation: odd-dimensional qudit states that exhibit measurement contextuality have an order ÿ 1 contribution in their expectation values with the observable that allows for the violation of classical bounds while states that have insufficiently large order ÿ 1 contributions do not exhibit measurement contextuality.

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“…II). There are also plenty of other ways to visualize states [23][24][25][26][27][28][29][30]. We refer the reader to Refs.…”

Section: Toroidal Doubly-discrete Quantum Phase Space (Dv)mentioning

confidence: 99%

General phase spaces: from discrete variables to rotor and continuum limits

Albert

Pascazio

Devoret

2017

J. Phys. A: Math. Theor.

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We provide a basic introduction to discrete-variable, rotor, and continuous-variable quantum phase spaces, explaining how the latter two can be understood as limiting cases of the first. We extend the limit-taking procedures used to travel between phase spaces to a general class of Hamiltonians (including many local stabilizer codes) and provide six examples: the Harper equation, the Baxter parafermionic spin chain, the Rabi model, the Kitaev toric code, the Haah cubic code (which we generalize to qudits), and the Kitaev honeycomb model. We obtain continuous-variable generalizations of all models, some of which are novel. The Baxter model is mapped to a chain of coupled oscillators and the Rabi model to the optomechanical radiation pressure Hamiltonian. The procedures also yield rotor versions of all models, five of which are novel many-body extensions of the almost Mathieu equation. The toric and cubic codes are mapped to lattice models of rotors, with the toric code case related to U (1) lattice gauge theory.

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Theoretical formulation of finite-dimensional discrete phase spaces: I. Algebraic structures and uncertainty principles

Cited by 23 publications

References 92 publications

Strong unitary uncertainty relations

Strong unitary uncertainty relations

Measurement contextuality and Planck’s constant

General phase spaces: from discrete variables to rotor and continuum limits

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