Discretized Gaussian Transitions in Finite Hilbert Spaces

• Discretized Gaussian Transitions are finite-dimensional analogues of continuous Gaussian functions that retain key uncertainty properties and phase-space characteristics.
• They employ Jacobi theta functions and generalized Heisenberg-Weyl algebra to construct discrete uncertainty relations and Wigner quasi-distributions.
• Applications span quantum simulation, digital signal processing, and error correction in finite quantum systems by mirroring continuous dynamical behavior.

Discretized Gaussian Transitions refer to the mathematical, physical, and computational frameworks by which the properties and dynamics of Gaussian functions—central to continuous quantum mechanics, probability theory, and signal processing—are transferred to finite, discrete settings. Central to this topic is the paper of how continuous Gaussian states, their uncertainty relations, Wigner quasi-distributions, and time-evolution operators transform when confined to a finite-dimensional Hilbert space, and in particular, how discrete analogues such as Jacobi theta functions systematically replicate the continuous properties as the system size increases. This construct underlies modern developments in quantum information theory, discrete phase-space analysis, and the simulation of quantum systems with a finite basis.

1. Mathematical Frameworks for Discretizing Gaussian Functions

The core technical challenge addressed in this field is constructing finite-dimensional analogues of the continuous Gaussian function, preserving as much as possible the key properties of the original. According to (Cotfas et al., 2012), the most direct approach leverages Weyl’s group-theoretic formulation of quantum mechanics, which permits the explicit development of discrete phase-space operators by generalizing the Heisenberg-Weyl algebra to a finite-dimensional setting, as initiated by Schwinger.

A critical insight is that the Jacobi theta function provides a discrete, periodic analogue of the continuous Gaussian. The discrete Gaussian $\Theta(m; d)$ is defined over integers modulo the system dimension $d$, with

$$\Theta(m; d) = \frac{1}{\sqrt{d}} \sum_{k=0}^{d-1} \exp\left(-\frac{\pi}{d} k^2 \right) \exp\left(\frac{2\pi i k m}{d}\right),$$

mirroring the structure of the continuous Fourier transform of the Gaussian. This construction ensures the analytic, algebraic, and uncertainty characteristics of the underlying operator algebra are retained in finite dimension.

2. Uncertainty Relations and their Discrete Analogues

In continuous quantum mechanics, the Gaussian saturates the Heisenberg uncertainty relation. For discrete systems, the uncertainty relation adapts to the periodicity and finite basis, with the variance computed over integer indices. The paper establishes that finite Gaussian states, constructed by means of Jacobi theta functions, realize a minimal uncertainty product analogous to the continuous case, with corrections that vanish as $d \rightarrow \infty$. The discrete uncertainty product reflects the finite support and periodic boundary conditions, and the minimal value approaches that of the continuous Gaussian as the system size increases.

3. Discrete Wigner Quasi-Distributions

Wigner quasi-probability distributions play a central role in continuous quantum mechanics for phase-space description. Their discrete analogues in finite Hilbert space, as shown in (Cotfas et al., 2012) and related works ([Opatrný et al. 1995], [Ruzzi 2006]), are constructed via finite sums over orthonormal phase-point operators, yielding

$$W(m, n; d) = \frac{1}{d} \sum_{k=0}^{d-1} \Psi(m+k; d) \Psi^*(m-k; d) \exp\left( -\frac{2\pi i k n}{d} \right),$$

where $\Psi(m; d)$ is the finite Gaussian state. The structure is such that marginals over one index recover the probability densities associated with projections onto discrete coordinate or momentum bases, preserving the quantum-classical correspondence in the finite setting.

4. Dynamics: Discretized Free-Particle and Harmonic Oscillator Evolution

The time evolution of finite Gaussian states under free-particle and harmonic oscillator Hamiltonians is deeply analogous to the continuous case, with the peculiarity that the finite-dimensional operators (such as finite translation and rotation generators) must be used in place of continuous differential operators. Discrete time evolution operators are constructed from exponentials of the finite analogues of the momentum and position operators (see [Zak 1967], [Štoviček & Tolar 1984]). Explicit analytic formulas for the propagation of finite Gaussian wave packets are derived, showing that the familiar spreading and oscillation behaviors of the continuous Gaussian are recovered asymptotically as $d \rightarrow \infty$.

5. Recovery of Continuous Limit and the Discrete-Continuous Transition

A central theme in (Cotfas et al., 2012) is the rigorous analysis of the transition from discrete to continuous as the system dimension $d$ tends to infinity. All key constructs—the discrete Gaussian wave packet, uncertainty products, Wigner function, time evolution—approach their continuous Gaussian analogues as $d$ increases. This property is formalized using asymptotic expansions and limiting arguments, with Jacobi theta functions tending to the standard Gaussian and its Fourier transform in the limit. Discrete operators converge to their continuous counterparts under appropriate scaling.

The following table summarizes these approaches:

Quantity	Discrete Analogue	Limit as $d\to\infty$
Gaussian function	Jacobi theta function over finite cyclic group	Continuous Gaussian
Uncertainty product	Computed via finite basis, minimal for theta-Gaussian	Heisenberg limit
Wigner function	Finite sum over phase-point operators	Continuous Wigner function
Free-particle/harmonic evolution	Discretized via finite operator exponentials	Schrödinger evolution

6. Connections to Phase-Space Transforms and Quantum Information

Discrete analogues of Gaussian functions underpin major developments in quantum information for finite systems, including the construction of discrete phase-space representations ([Opatrný et al. 1996], [Poletti 1988]), the definition of discrete Q and Wigner functions, and the analysis of quantum error correction codes which rely on finite group symmetries.

Work on analytic representations of finite quantum systems ([Zhang & Vourdas 2004]), discrete Fourier transform properties ([Ruzzi 2006]), and the theory of special functions ([Vilenkin & Klimyk 1992]) is deeply entwined with the mathematical apparatus needed for discretizing Gaussian transitions. Moreover, understanding the behavior of these constructs is essential in the design and simulation of finite quantum devices, where only a finite-dimensional Hilbert space is accessible.

7. Broader Implications and Application Domains

The framework for discretized Gaussian transitions is central in a variety of fields:

• Quantum simulation and computation: Modeling dynamics in finite Hilbert spaces for quantum algorithms and error correction.
• Discrete signal processing: Analysis of finite Fourier transforms, Gabor-type filtering, and time-frequency representations for digital signals.
• Mathematical physics: Rigorous understanding of the continuous–discrete correspondence, including the behavior of special functions, spectral properties, and group representations.

These developments tie naturally to foundational work ([Weil 1954], [Whittaker & Watson 2000], [Vourdas 2004]) on the algebraic and analytic structure of finite and infinite-dimensional systems, showing that the methods for discretizing Gaussian functions are not merely technical expedients but provide deep connections between disparate areas of mathematics and physics.

8. Bibliographical Context and Foundational References

The topic is supported by a robust body of foundational literature, including the referenced works:

• Discrete phase-space functions and Q/Wigner representations ([Opatrný et al. 1995], [Opatrný et al. 1996], [Poletti 1988])
• Jacobi theta functions and their role in discrete Fourier analysis ([Ruzzi 2006])
• Group-theoretic and operator algebra foundations ([Weil 1954], [Zak 1967], [Štoviček & Tolar 1984])
• Discrete analogues of special functions and transforms ([Vilenkin & Klimyk 1992], [Whittaker & Watson 2000])
• Analytic representations for finite quantum systems ([Zhang & Vourdas 2004])
• Reviews and broader conceptual frameworks ([Vourdas 2004])

This bibliography establishes the theoretical underpinning and demonstrates the widespread relevance and applicability of discretized Gaussian transitions in modern mathematical physics and engineering.