Laure Gouba - Academia.edu (original) (raw)

Papers by Laure Gouba

Cornell University - arXiv, Aug 1, 2022

The separability problem is one of the basic and emergent problems in present and future quantum ... more The separability problem is one of the basic and emergent problems in present and future quantum information processing. The latter focuses on information and computing based on quantum mechanics and uses quantum bits as its basic information units. In this paper we present an overview of the progress in the separability problem in bipartite systems, more specifically in two quantum bits (qubits) system, from the criterion based on the Bell's inequalities in 1964 to the Li-Qiao criterion and the enhanced entanglement criterion based on the SIC POVMs in 2018.

When dealing with the classical limit of two quantum mechanical oscillators on a noncommutative c... more When dealing with the classical limit of two quantum mechanical oscillators on a noncommutative configuration space, the limits corresponding to the removal of configuration-space noncommutativity and position-momentum noncommutativity do not commute. We address this behaviour from the point of view of the phase-space localisation properties of the Wigner functions of coherent states under the two limits.

A unifying perspective on the Moyal and Voros products and their physical meanings has been recen... more A unifying perspective on the Moyal and Voros products and their physical meanings has been recently presented in the literature, where the Voros formulation admits a consistent physical interpretation. We define a star product , in terms of an antisymmetric fixed matrix Θ, and an arbitrary symmetric matrix Φ, that is a generalization of the Moyal and the Voros products. We discuss the quantum mechanics and the physical meaning of the generalized star product.

In this paper we consider two kinds of noncommutative space-time commutation relations in two-dim... more In this paper we consider two kinds of noncommutative space-time commutation relations in two-dimensional configuration space and feature the absolute value of the minimal length from the generalized uncertainty relations associated to the particular commutation relations. We study the problem of the two-dimensional gravitational quantum well in the new Hermitian variables and confront the experimental results for the first lowest energy state of the neutrons in the Earth's gravitational field to estimate the upper bounds on the noncommutativity parameters. The absolute value of the minimum length is smaller than a few nanometers.

We investigate properties of generalized time-dependent q-deformed coherent states for a noncommu... more We investigate properties of generalized time-dependent q-deformed coherent states for a noncommutative harmonic oscillator. The states are shown to satisfy a generalized version of Heisenberg's uncertainty relations. For the initial value in time the states are demonstrated to be squeezed, i.e. the inequalities are saturated, whereas when time evolves the uncertainty product oscillates away from this value albeit still respecting the relations. For the canonical variables on a noncommutative space we verify explicitly that Ehrenfest's theorem hold at all times. We conjecture that the model exhibits revival times to infinite order. Explicit sample computations for the fractional revival times and superrevival times are presented.

We consider a model of non-commutative Quantum Mechanics given by two harmonic oscillators over a... more We consider a model of non-commutative Quantum Mechanics given by two harmonic oscillators over a non-commutative two dimensional configuration space. We study possible ways of removing the non-commutativity based on the classical limit context known as anti-Wick quantization. We show that removal of non-commutativity from the configuration space and from the canonical operators are not commuting operations.

We demonstrate that dynamical noncommutative space-time will give rise to deformed oscillator alg... more We demonstrate that dynamical noncommutative space-time will give rise to deformed oscillator algebras. In turn, starting from some q-deformations of these algebras in a two dimensional space for which the entire deformed Fock space can be constructed explicitly, we derive the commutation relations for the dynamical variables in noncommutative space-time. We compute minimal areas resulting from these relations, i.e. finitely extended regions for which it is impossible to resolve any substructure in form of measurable knowledge. The size of the regions we find is determined by the noncommutative constant and the deformation parameter q. Any object in this type of space-time structure has to be of membrane type or in certain limits of string type.

We provide a systematic procedure to relate a three dimensional q-deformed oscillator algebra to ... more We provide a systematic procedure to relate a three dimensional q-deformed oscillator algebra to the corresponding algebra satisfied by canonical variables describing noncommutative spaces. The large number of possible free parameters in these calculations is reduced to a manageable amount by imposing various different versions of PT-symmetry on the underlying spaces, which are dictated by the specific physical problem under consideration. The representations for the corresponding operators are in general non-Hermitian with regard to standard inner products and obey algebras whose uncertainty relations lead to minimal length, areas or volumes in phase space. We analyze in particular one three dimensional solution which may be decomposed to a two dimensional noncommutative space plus one commuting space component and also into a one dimensional noncommutative space plus two commuting space components. We study some explicit models on these type of noncommutative spaces.

We present an original approach to quantization based on operator-valued measure that generalizes... more We present an original approach to quantization based on operator-valued measure that generalizes the so-called Berezin-Klauder-Toeplitz quantization, and more generally coherent state quantization approches.

We consider an isotropic two dimensional harmonic oscillator with arbitrarily time-dependent mass... more We consider an isotropic two dimensional harmonic oscillator with arbitrarily time-dependent mass M(t) and frequency Ω(t) in an arbitrarily time-dependent magnetic field B(t). We determine two commuting invariant observables (in the sense of Lewis and Riesenfeld) L,I in terms of some solution of an auxiliary ordinary differential equation and an orthonormal basis of the Hilbert space consisting of joint eigenvectors ϕ_λ of L,I. We then determine time-dependent phases α_λ(t) such that the ψ_λ(t)=e^iα_λϕ_λ are solutions of the time-dependent Schrödinger equation and make up an orthonormal basis of the Hilbert space. These results apply, in particular to a two dimensional Landau problem with time-dependent M,B, which is obtained from the above just by setting Ω(t) ≡ 0. By a mere redefinition of the parameters, these results can be applied also to the analogous models on the canonical non-commutative plane.

We generalize the Milne quantization condition to non-Hermitian systems. In the general case the ... more We generalize the Milne quantization condition to non-Hermitian systems. In the general case the underlying nonlinear Ermakov-Milne-Pinney equation needs to be replaced by a nonlinear integral differential equation. However, when the system is PT-symmetric or/and quasi/pseudo-Hermitian the equations simplify and one may employ the original energy integral to determine its quantization. We illustrate the working of the general framework with the Swanson model and two explicit examples for pairs of supersymmetric Hamiltonians. In one case both partner Hamiltonians are Hermitian and in the other a Hermitian Hamiltonian is paired by a Darboux transformation to a non-Hermitian one.

The nonabelian global chiral symmetries of the two-dimensional N flavour massless Schwinger model... more The nonabelian global chiral symmetries of the two-dimensional N flavour massless Schwinger model are realised through bosonisation and a vertex operator construction.

In this paper the two dimensional Abelian Higgs model is revisited. We show that in the physical ... more In this paper the two dimensional Abelian Higgs model is revisited. We show that in the physical sector, this model describes the coupling of the electric field to the radial part, in field space, of the complex scalar field. 1

We demonstrate that dynamical noncommutative space-time will give rise to deformed oscillator alg... more We demonstrate that dynamical noncommutative space-time will give rise to deformed oscillator algebras. In turn, starting from some q-deformations of these algebras in a two dimensional space for which the entire deformed Fock space can be constructed explicitly, we derive the commutation relations for the dynamical variables in noncommutative space-time. We compute minimal areas resulting from these relations, i.e. finitely extended regions for which it is impossible to resolve any substructure in form of measurable knowledge. The size of the regions we find is determined by the noncommutative constant and the deformation parameter q. Any object in this type of space-time structure has to be of membrane type or in certain limits of string type.

Journal of Physics Communications, 2021

In this paper, we construct the coherent states for a system of an electron moving in a plane und... more In this paper, we construct the coherent states for a system of an electron moving in a plane under uniform external magnetic and electric fields. These coherent states are built in the context of both discrete and continuous spectra and satisfy the Gazeau-Klauder coherent state properties Gazeau and Klauder (1999 J. Phys. A: Math. Gen. 32, 123–132).

In this paper the two dimensional Abelian Higgs model is revisited. We show that in the physical ... more In this paper the two dimensional Abelian Higgs model is revisited. We show that in the physical sector, this model describes the coupling of the electric field to the radial part, in field space, of the complex scalar field. Comment: Contribution to the Proceedings of the Fifth International Workshop on Contemporary Problems in Mathematical Physics, October 27 - November 2, 2007, Cotonou (Republic of Benin), 6 pages

We introduce a new set of noncommutative space-time commutation relations in two space dimensions... more We introduce a new set of noncommutative space-time commutation relations in two space dimensions. The space-space commutation relations are deformations of the standard flat noncommutative space-time relations taken here to have position dependent structure constants. Some of the new variables are non-Hermitian in the most natural choice. We construct their Hermitian counterparts by means of a Dyson map, which also serves to introduce a new metric operator. We propose PT like symmetries, i.e. antilinear involutory maps, respected by these deformations. We compute minimal lengths and momenta arising in this space from generalized versions of Heisenberg’s uncertainty relations and find that any object in this two dimensional space is string like, i.e. having a fundamental length in one direction beyond which a resolution is impossible. Subsequently we formulate and partly solve some simple models in these new variables, the free particle, its PT -symmetric deformations and the harmon...

arXiv: Mathematical Physics, 2019

This paper is a slightly reduced and concise version of a course given at Geonet2019 school on Ne... more This paper is a slightly reduced and concise version of a course given at Geonet2019 school on New Trends in Mathematical Methods for Physics at IMSP in Rep. of Benin, May 2019. The notes are intended to provide the reader with an introduction of a more recent procedure of quantization that is a generalization of the coherent states quantization procedure, highlighting the link between symplectic geometry and classical mechanics. The topic and the goal of the Geonet2019 school motivated the choice of the title of the course as New Trend in Quantization Methods. It is also a coincidence of a current research interest in integral quantization.

Four formulations of noncommutative quantum mechanics are reviewed. These are the canonical, path... more Four formulations of noncommutative quantum mechanics are reviewed. These are the canonical, path-integral, Weyl-Wigner and systematic formulations. The four formulations are charaterized by a deformed Heisenberg algebra but differ in mathematical and conceptual overview.

Cornell University - arXiv, Aug 1, 2022

The separability problem is one of the basic and emergent problems in present and future quantum ... more The separability problem is one of the basic and emergent problems in present and future quantum information processing. The latter focuses on information and computing based on quantum mechanics and uses quantum bits as its basic information units. In this paper we present an overview of the progress in the separability problem in bipartite systems, more specifically in two quantum bits (qubits) system, from the criterion based on the Bell's inequalities in 1964 to the Li-Qiao criterion and the enhanced entanglement criterion based on the SIC POVMs in 2018.

When dealing with the classical limit of two quantum mechanical oscillators on a noncommutative c... more When dealing with the classical limit of two quantum mechanical oscillators on a noncommutative configuration space, the limits corresponding to the removal of configuration-space noncommutativity and position-momentum noncommutativity do not commute. We address this behaviour from the point of view of the phase-space localisation properties of the Wigner functions of coherent states under the two limits.

A unifying perspective on the Moyal and Voros products and their physical meanings has been recen... more A unifying perspective on the Moyal and Voros products and their physical meanings has been recently presented in the literature, where the Voros formulation admits a consistent physical interpretation. We define a star product , in terms of an antisymmetric fixed matrix Θ, and an arbitrary symmetric matrix Φ, that is a generalization of the Moyal and the Voros products. We discuss the quantum mechanics and the physical meaning of the generalized star product.

In this paper we consider two kinds of noncommutative space-time commutation relations in two-dim... more In this paper we consider two kinds of noncommutative space-time commutation relations in two-dimensional configuration space and feature the absolute value of the minimal length from the generalized uncertainty relations associated to the particular commutation relations. We study the problem of the two-dimensional gravitational quantum well in the new Hermitian variables and confront the experimental results for the first lowest energy state of the neutrons in the Earth's gravitational field to estimate the upper bounds on the noncommutativity parameters. The absolute value of the minimum length is smaller than a few nanometers.

We investigate properties of generalized time-dependent q-deformed coherent states for a noncommu... more We investigate properties of generalized time-dependent q-deformed coherent states for a noncommutative harmonic oscillator. The states are shown to satisfy a generalized version of Heisenberg's uncertainty relations. For the initial value in time the states are demonstrated to be squeezed, i.e. the inequalities are saturated, whereas when time evolves the uncertainty product oscillates away from this value albeit still respecting the relations. For the canonical variables on a noncommutative space we verify explicitly that Ehrenfest's theorem hold at all times. We conjecture that the model exhibits revival times to infinite order. Explicit sample computations for the fractional revival times and superrevival times are presented.

We consider a model of non-commutative Quantum Mechanics given by two harmonic oscillators over a... more We consider a model of non-commutative Quantum Mechanics given by two harmonic oscillators over a non-commutative two dimensional configuration space. We study possible ways of removing the non-commutativity based on the classical limit context known as anti-Wick quantization. We show that removal of non-commutativity from the configuration space and from the canonical operators are not commuting operations.

We demonstrate that dynamical noncommutative space-time will give rise to deformed oscillator alg... more We demonstrate that dynamical noncommutative space-time will give rise to deformed oscillator algebras. In turn, starting from some q-deformations of these algebras in a two dimensional space for which the entire deformed Fock space can be constructed explicitly, we derive the commutation relations for the dynamical variables in noncommutative space-time. We compute minimal areas resulting from these relations, i.e. finitely extended regions for which it is impossible to resolve any substructure in form of measurable knowledge. The size of the regions we find is determined by the noncommutative constant and the deformation parameter q. Any object in this type of space-time structure has to be of membrane type or in certain limits of string type.

We provide a systematic procedure to relate a three dimensional q-deformed oscillator algebra to ... more We provide a systematic procedure to relate a three dimensional q-deformed oscillator algebra to the corresponding algebra satisfied by canonical variables describing noncommutative spaces. The large number of possible free parameters in these calculations is reduced to a manageable amount by imposing various different versions of PT-symmetry on the underlying spaces, which are dictated by the specific physical problem under consideration. The representations for the corresponding operators are in general non-Hermitian with regard to standard inner products and obey algebras whose uncertainty relations lead to minimal length, areas or volumes in phase space. We analyze in particular one three dimensional solution which may be decomposed to a two dimensional noncommutative space plus one commuting space component and also into a one dimensional noncommutative space plus two commuting space components. We study some explicit models on these type of noncommutative spaces.

We present an original approach to quantization based on operator-valued measure that generalizes... more We present an original approach to quantization based on operator-valued measure that generalizes the so-called Berezin-Klauder-Toeplitz quantization, and more generally coherent state quantization approches.

We consider an isotropic two dimensional harmonic oscillator with arbitrarily time-dependent mass... more We consider an isotropic two dimensional harmonic oscillator with arbitrarily time-dependent mass M(t) and frequency Ω(t) in an arbitrarily time-dependent magnetic field B(t). We determine two commuting invariant observables (in the sense of Lewis and Riesenfeld) L,I in terms of some solution of an auxiliary ordinary differential equation and an orthonormal basis of the Hilbert space consisting of joint eigenvectors ϕ_λ of L,I. We then determine time-dependent phases α_λ(t) such that the ψ_λ(t)=e^iα_λϕ_λ are solutions of the time-dependent Schrödinger equation and make up an orthonormal basis of the Hilbert space. These results apply, in particular to a two dimensional Landau problem with time-dependent M,B, which is obtained from the above just by setting Ω(t) ≡ 0. By a mere redefinition of the parameters, these results can be applied also to the analogous models on the canonical non-commutative plane.

We generalize the Milne quantization condition to non-Hermitian systems. In the general case the ... more We generalize the Milne quantization condition to non-Hermitian systems. In the general case the underlying nonlinear Ermakov-Milne-Pinney equation needs to be replaced by a nonlinear integral differential equation. However, when the system is PT-symmetric or/and quasi/pseudo-Hermitian the equations simplify and one may employ the original energy integral to determine its quantization. We illustrate the working of the general framework with the Swanson model and two explicit examples for pairs of supersymmetric Hamiltonians. In one case both partner Hamiltonians are Hermitian and in the other a Hermitian Hamiltonian is paired by a Darboux transformation to a non-Hermitian one.

The nonabelian global chiral symmetries of the two-dimensional N flavour massless Schwinger model... more The nonabelian global chiral symmetries of the two-dimensional N flavour massless Schwinger model are realised through bosonisation and a vertex operator construction.

In this paper the two dimensional Abelian Higgs model is revisited. We show that in the physical ... more In this paper the two dimensional Abelian Higgs model is revisited. We show that in the physical sector, this model describes the coupling of the electric field to the radial part, in field space, of the complex scalar field. 1

We demonstrate that dynamical noncommutative space-time will give rise to deformed oscillator alg... more We demonstrate that dynamical noncommutative space-time will give rise to deformed oscillator algebras. In turn, starting from some q-deformations of these algebras in a two dimensional space for which the entire deformed Fock space can be constructed explicitly, we derive the commutation relations for the dynamical variables in noncommutative space-time. We compute minimal areas resulting from these relations, i.e. finitely extended regions for which it is impossible to resolve any substructure in form of measurable knowledge. The size of the regions we find is determined by the noncommutative constant and the deformation parameter q. Any object in this type of space-time structure has to be of membrane type or in certain limits of string type.

Journal of Physics Communications, 2021

In this paper, we construct the coherent states for a system of an electron moving in a plane und... more In this paper, we construct the coherent states for a system of an electron moving in a plane under uniform external magnetic and electric fields. These coherent states are built in the context of both discrete and continuous spectra and satisfy the Gazeau-Klauder coherent state properties Gazeau and Klauder (1999 J. Phys. A: Math. Gen. 32, 123–132).

In this paper the two dimensional Abelian Higgs model is revisited. We show that in the physical ... more In this paper the two dimensional Abelian Higgs model is revisited. We show that in the physical sector, this model describes the coupling of the electric field to the radial part, in field space, of the complex scalar field. Comment: Contribution to the Proceedings of the Fifth International Workshop on Contemporary Problems in Mathematical Physics, October 27 - November 2, 2007, Cotonou (Republic of Benin), 6 pages

We introduce a new set of noncommutative space-time commutation relations in two space dimensions... more We introduce a new set of noncommutative space-time commutation relations in two space dimensions. The space-space commutation relations are deformations of the standard flat noncommutative space-time relations taken here to have position dependent structure constants. Some of the new variables are non-Hermitian in the most natural choice. We construct their Hermitian counterparts by means of a Dyson map, which also serves to introduce a new metric operator. We propose PT like symmetries, i.e. antilinear involutory maps, respected by these deformations. We compute minimal lengths and momenta arising in this space from generalized versions of Heisenberg’s uncertainty relations and find that any object in this two dimensional space is string like, i.e. having a fundamental length in one direction beyond which a resolution is impossible. Subsequently we formulate and partly solve some simple models in these new variables, the free particle, its PT -symmetric deformations and the harmon...

arXiv: Mathematical Physics, 2019

This paper is a slightly reduced and concise version of a course given at Geonet2019 school on Ne... more This paper is a slightly reduced and concise version of a course given at Geonet2019 school on New Trends in Mathematical Methods for Physics at IMSP in Rep. of Benin, May 2019. The notes are intended to provide the reader with an introduction of a more recent procedure of quantization that is a generalization of the coherent states quantization procedure, highlighting the link between symplectic geometry and classical mechanics. The topic and the goal of the Geonet2019 school motivated the choice of the title of the course as New Trend in Quantization Methods. It is also a coincidence of a current research interest in integral quantization.

Four formulations of noncommutative quantum mechanics are reviewed. These are the canonical, path... more Four formulations of noncommutative quantum mechanics are reviewed. These are the canonical, path-integral, Weyl-Wigner and systematic formulations. The four formulations are charaterized by a deformed Heisenberg algebra but differ in mathematical and conceptual overview.