On Hilbert-Schmidt Operator Formulation of Noncommutative Quantum Mechanics

Aremua, Isiaka; Baloïtcha, Ezinvi; Hounkonnou, Mahouton Norbert; Sodoga, Komi

doi:10.1007/978-3-319-97175-9_3

Cited by 4 publications

(5 citation statements)

References 28 publications

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“…The Landau problem [29]- [17]- [9] is related to the motion of a charged particle on the flat plane xy in the presence of a constant magnetic field along the z-axis. In metals, the electrons occupy many Landau levels…”

Section: Density Operator Approach In Cs For the Exotic Landau Problemmentioning

confidence: 99%

“…Here, we give two examples of thermal states as known from the literature. For more details, see [45,44,11,1,9].…”

Section: Thermal Statementioning

confidence: 99%

“…The behavior of an electron in an external magnetic field was extensively studied [29]- [9]. This implied a great interest to other similar physical systems describing, for instance, the quantum Hall effect [23].…”

Section: Introductionmentioning

confidence: 99%

“…In [8], the density operator was achieved in the Barut-Girardello CS representation for Landau levels of a gas of spinless charged particles, subject to a perpendicular magnetic field confined in a harmonic potential; the Husimi distribution and Wehrl entropy were investigated. Recently [9], the Hilbert-Schmidt operators and the Tomita-Takesaki modular theories were recast for noncommutative quantum mechanics formulation, and the density matrix formalism related to von Neumann algebras, displayed by the Landau problem [1], was revisited.…”

Section: Introductionmentioning

confidence: 99%

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Density operator approach for Landau problem quantum Hamiltonians

Aremua,

Hounkonnou,

Baloitcha

2017

Preprint

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In this work, the definition of the density operator on quantum states in Hilbert spaces and some of its aspects relevant in thermodynamics and information-theoretical entropy calculations are given. In this framework, a physical model describing an electron in a magnetic field is investigated. The so-called exotic Landau problem in noncommutative plane is also considered. Then, a model related to the fractional quantum Hall effect is revisited. Thanks to the completeness relations verified by the coherent states (CS) in these models, the thermodynamics is discussed by using the diagonal P -representation of the density operator. Specifically, the Q-Husimi distribution and the Wehrl entropy are determined.

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Section: Density Operator Approach In Cs For the Exotic Landau Problemmentioning

confidence: 99%

“…Here, we give two examples of thermal states as known from the literature. For more details, see [45,44,11,1,9].…”

Section: Thermal Statementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Density operator approach for Landau problem quantum Hamiltonians

Aremua,

Hounkonnou,

Baloitcha

2017

Preprint

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“…In quantum mechanics, one of the fundamental problems is to characterize the quantum and classical features of physical system of interest. One generic characteristic of the quantum system is the presence of noncommutative observables in the operator algebra of the system-the set of all operators forming a vector space [1][2][3][4]. The non-commutative structure is the basic ingredient for deriving the Heisenberg uncertainty principle [5,6], and the violation of Bell's inequality [7], as well as classical-quantum discord and related measures [8], all of which are used to testify the quantumness of the underlying system.…”

Section: Introductionmentioning

confidence: 99%

Statistics of projective measurement on a quantum probe as a witness of noncommutativity of algebra of a probed system

Sakuldee¹,

Cywiński²

2021

Preprint

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We consider a quantum probe P undergoing pure dephasing due to its interaction with a quantum system S. The dynamics of P is then described by a well-defined sub-algebra of operators of S, i.e. the "accessible" algebra on S from the point of view of P. We consider sequences of n measurements on P, and investigate the relationship between Kolmogorov consistency of probabilities of obtaining sequences of results with various n, and commutativity of the accessible algebra. For a finitedimensional S we find conditions under which the Kolmogorov consistency of measurement on P, given that the state of S can be arbitrarily prepared, is equivalent to the commutativity of this algebra. These allow us to describe witnesses of non-classicality (understood here as noncommutativity) of part of S that affects the probe. For P being a qubit, the witness is particularly simple: observation of breaking of Kolmogorov consistency of sequential measurements on a qubit coupled to S means that the accessible algebra of S is non-commutative.

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