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Nicolae Cotfas

Publications and source records attributed to Nicolae Cotfas.

At least 19 recordsLinked to original sources
arXiv2025

On a discrete version of the position-momentum commutation relation

The usual position-momentum commutation relation plays a fundamental role in the mathematical description of continuous-variable quantum systems. In the case of a qudit described by a Hilbert space of a high enough dimension, there exists a discrete version of the position-momentum commutation relation satisfied with approximation of a large part of the pure quantum states. Our purpose is to explore in more details the set of these states. We show that it contains a family of discrete-variable Gaussian states depending on a continuous parameter and certain discrete coherent states. It also contains various discrete-variable versions of the Hermite-Gauss states, defined either as eigenstates of certain discrete versions of the harmonic oscillator Hamiltonian or generated by using a discrete version of the creation or annihilation operator. The presented results suggest a possible way to experimentally obtain some of the investigated states.

quant-ph↗
arXiv2025

Frame-dependent coherence of a quantum state

The tight frames can be regarded as a particular case of POVMs (positive operator-valued measures describing generalized measurements), namely the case when all the operators are rank-one. Each orthonormal basis is a tight frame, and every tight frame, after the embedding into a higher-dimensional space, is the orthogonal projection of an orthonormal basis. There exist several POVM-based definitions of coherence, and they are well-investigated. Our aim is to identify properties specific to the particular case of tight frames, and to look for some applications. All the POVM-based definitions use a Naimark extension. The frame-dependent coherence can be regarded as a particular case of POVM-based coherence, but it can be defined directly, without to use a Naimark extension. Its definition is a direct generalization of the basis-dependent $\ell_1$-norm of coherence, and it offers a more accurate description because we can use a frame containing several orthogonal bases. A frame-invariant definition of coherence for qubits and multi-qubit systems is presented.

quant-ph↗
arXiv2024

Frame representations of qudit quantum mechanics

There exist many attempts to define a Wigner function for qudits, each of them coming with its advantages and limitations. The existing finite versions have simple definitions, but they are artificial in their construction and do not allow an intuitive state analysis. The continuous versions have more complicated definitions, but they are similar to the original Wigner function and allow a visualization of the quantum states. The version based on the concept of tight frame we present is finite, but it has certain properties and applications similar to those of continuous versions. Based on the frame representation, we present several graphical representations of qubit states, and define two new parameters concerning them. We show that, from a mathematical point of view, the qubit is the orthogonal projection of qutrit.

quant-ph↗
arXiv2022

Mixed discrete variable Gaussian states

The quantum systems with finite-dimensional Hilbert space have several applications and are intensively explored theoretically and experimentally. The mathematical description of these systems follows the analogy with the usual infinite-dimensional case. There exist finite versions for most of the elements used in the continuous case, but (to our knowledge) there does not exist a finite version corresponding to the mixed Gaussian states. Our aim is to fill this gap. The definition we propose for the mixed discrete Gaussian states is based on the explicit formulas we have obtained in the case of pure discrete variable Gaussian states.

quant-ph↗
arXiv2022

Quantum states defined by using the finite frame quantization

Finite frame quantization is a discrete version of the coherent state quantization. In the case of a quantum system with finite-dimensional Hilbert space, the finite frame quantization allows us to associate a linear operator to each function defined on the discrete phase space of the system. We investigate the properties of the density operators which can be defined by using this method.

quant-ph↗
arXiv2020

On the Gaussian functions of two discrete variables

A remarkable discrete counterpart of the Gaussian function of one continuous variable can be defined by using a Jacobi theta function, that is, as the sum of a convergent series. We extend this approach to Gaussian functions of two variables, and investigate the Fourier transform and Wigner function of the functions of discrete variable defined in this way.

math.CA↗
arXiv2016

Linear representations of SU(2) described by using Kravchuk polynomials

We show that a new unitary transform with characteristics almost similar to those of the finite Fourier transform can be defined in any finite-dimensional Hilbert space. It is defined by using the Kravchuk polynomials, and we call it Kravchuk transform. Some of its properties are investigated and used in order to obtain a simple alternative description for the irreducible representations of the Lie algebra su(2) and group SU(2). Our approach offers a deeper insight into the structure of the linear representations of SU(2) and new possibilities of computation, very useful in applications in quantum mechanics, quantum information, signal and image processing.

math-ph↗
arXiv2015

Gaussian type functions defined on a finite set and finite oscillators

The mathematical description of the quantum harmonic oscillator is essentially based on the Gaussian function. In the case of a quantum oscillator with finite-dimensional Hilbert space, the position space consists in a finite number of points, and the ground state can be regarded as a finite version of the Gaussian function. Conversely, each finite Gaussian can be used in order to define some finite oscillators. We investigate certain finite Gaussians as concern their Fourier transform, Wigner function and associated oscillators.

math-ph↗
arXiv2014

Exact solution of a generalized version of the Black-Scholes equation

We analyze a generalized version of the Black-Scholes equation depending on a parameter $a\!\in \!(-\infty,0)$. It satisfies the martingale condition and coincides with the Black-Scholes equation in the limit case $a\nearrow 0$. We show that the generalized equation is exactly solvable in terms of Hermite polynomials and numerically compare its solution with the solution of the Black-Scholes equation.

q-fin.CP↗
arXiv2013

Quantum harmonic oscillator in option pricing

The Black-Scholes model anticipates rather well the observed prices for options in the case of a strike price that is not too far from the current price of the underlying asset. Some useful extensions can be obtained by an adequate modification of the coefficients in the Black-Scholes equation. We investigate from a mathematical point of view an extension directly related to the quantum harmonic oscillator. In the considered case, the solution is the sum of a series involving the Hermite-Gauss functions. A finite-dimensional version is obtained by using a finite oscillator and the Harper functions. This simplified model keeps the essential characteristics of the continuous one and uses finite sums instead of series and integrals.

q-fin.CP↗
arXiv2013

Finite oscillator obtained through finite frame quantization

The Hamiltonian of the harmonic oscillator is usually defined as a differential operator, but an integral representation can be obtained by using the coherent state quantization. The finite frame quantization is a finite counterpart of the coherent state quantization and it allows us to define a finite oscillator by starting from the integral representation of the harmonic oscillator. Our purpose is to investigate the oscillator obtained in this way, and to present a possible application to the discrete fractional Fourier transform.

math-ph↗
arXiv2012

Properties of finite Gaussians and the discrete-continuous transition

Weyl's formulation of quantum mechanics opened the possibility of studying the dynamics of quantum systems both in infinite-dimensional and finite-dimensional systems. Based on Weyl's approach, generalized by Schwinger, a self-consistent theoretical framework describing physical systems characterised by a finite-dimensional space of states has been created. The used mathematical formalism is further developed by adding finite-dimensional versions of some notions and results from the continuous case. Discrete versions of the continuous Gaussian functions have been defined by using the Jacobi theta functions. We continue the investigation of the properties of these finite Gaussians by following the analogy with the continuous case. We study the uncertainty relation of finite Gaussian states, the form of the associated Wigner quasi-distribution and the evolution under free-particle and quantum harmonic oscillator Hamiltonians. In all cases, a particular emphasis is put on the recovery of the known continuous-limit results when the dimension $d$ of the system increases.

math-ph↗
arXiv2011

Hypergeometric type operators and their supersymmetric partners

The generalization of the factorization method performed by Mielnik [J. Math. Phys. {\bf 25}, 3387 (1984)] opened new ways to generate exactly solvable potentials in quantum mechanics. We present an application of Mielnik's method to hypergeometric type operators. It is based on some solvable Riccati equations and leads to a unitary description of the quantum systems exactly solvable in terms of orthogonal polynomials or associated special functions.

math-ph↗
arXiv2010

Complex and real Hermite polynomials and related quantizations

It is known that the anti-Wick (or standard coherent state) quantization of the complex plane produces both canonical commutation rule and quantum spectrum of the harmonic oscillator (up to the addition of a constant). In the present work, we show that these two issues are not necessarily coupled: there exists a family of separable Hilbert spaces, including the usual Fock-Bargmann space, and in each element in this family there exists an overcomplete set of unit-norm states resolving the unity. With the exception of the Fock-Bargmann case, they all produce non-canonical commutation relation whereas the quantum spectrum of the harmonic oscillator remains the same up to the addition of a constant. The statistical aspects of these non-equivalent coherent states quantizations are investigated. We also explore the localization aspects in the real line yielded by similar quantizations based on real Hermite polynomials.

quant-ph↗
arXiv2009

Finite tight frames and some applications

A finite-dimensional Hilbert space is usually described in terms of an orthonormal basis, but in certain approaches or applications a description in terms of a finite overcomplete system of vectors, called a finite tight frame, may offer some advantages. The use of a finite tight frame may lead to a simpler description of the symmetry transformations, to a simpler and more symmetric form of invariants or to the possibility to define new mathematical objects with physical meaning, particularly in regard with the notion of a quantization of a finite set. We present some results concerning the use of integer coefficients and frame quantization, several examples and suggest some possible applications.

math-ph↗
arXiv2008

Gazeau-Klauder type coherent states for hypergeometric type operators

The hypergeometric type operators are shape invariant, and a factorization into a product of first order differential operators can be explicitly described in the general case. Some additional shape invariant operators depending on several parameters are defined in a natural way by starting from this general factorization. The mathematical properties of the eigenfunctions and eigenvalues of the operators thus obtained depend on the values of the involved parameters. We study the parameter dependence of orthogonality, square integrability and of the monotony of eigenvalue sequence. The obtained results allow us to define certain systems of Gazeau-Klauder coherent states and to describe some of their properties. Our systematic study recovers a number of well-known results in a natural unified way and also leads to new findings.

math-ph↗
arXiv2008

Symmetry properties of Penrose type tilings

The Penrose tiling is directly related to the atomic structure of certain decagonal quasicrystals and, despite its aperiodicity, is highly symmetric. It is known that the numbers 1, $-τ$, $(-τ)^2$, $(-τ)^3$, ..., where $τ=(1+\sqrt{5})/2$, are scaling factors of the Penrose tiling. We show that the set of scaling factors is much larger, and for most of them the number of the corresponding inflation centers is infinite.

math-ph↗