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On a discrete version of the position-momentum commutation relation

Nicolae Cotfas

TL;DR

This work addresses how to realize a discrete analogue of the canonical position–momentum commutation relation in finite-dimensional quantum systems by focusing on a high-dimensional subspace where the commutator behaves as $[\hat{\mathfrak{q}},\hat{\mathfrak{p}}]\approx i\frac{d}{2\pi}$. The authors establish the subspace $\mathcal{S}_\varepsilon$ of states for which this approximation holds and derive an associated uncertainty bound $\Delta\hat{\mathfrak{q}}\Delta\hat{\mathfrak{p}}\ge\frac{d}{4\pi}$. They catalog and analyze several discrete-state families—discrete Gaussian states, discrete coherent states, and discrete analogs of Hermite–Gauss states (Mehta, Harper, Kravchuk)—and develop a discrete coherent-state quantization that yields a discrete harmonic-oscillator Hamiltonian with eigenstates in $\mathcal{S}_\varepsilon$. The results lay groundwork for new discrete-variable formalisms and potential quantum-information applications that leverage approximately canonical structures in high-dimensional qudits.

Abstract

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 approximately satisfied by 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. As a direct consequence, a discrete version of the incertitude relation is satisfied by the considered quantum states.

On a discrete version of the position-momentum commutation relation

TL;DR

This work addresses how to realize a discrete analogue of the canonical position–momentum commutation relation in finite-dimensional quantum systems by focusing on a high-dimensional subspace where the commutator behaves as . The authors establish the subspace of states for which this approximation holds and derive an associated uncertainty bound . They catalog and analyze several discrete-state families—discrete Gaussian states, discrete coherent states, and discrete analogs of Hermite–Gauss states (Mehta, Harper, Kravchuk)—and develop a discrete coherent-state quantization that yields a discrete harmonic-oscillator Hamiltonian with eigenstates in . The results lay groundwork for new discrete-variable formalisms and potential quantum-information applications that leverage approximately canonical structures in high-dimensional qudits.

Abstract

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 approximately satisfied by 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. As a direct consequence, a discrete version of the incertitude relation is satisfied by the considered quantum states.

Paper Structure

This paper contains 16 sections, 93 equations, 3 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Quantum systems with finite-dimensional Hilbert space
  3. Discrete Fourier transform
  4. Discrete version of position and momentum operators
  5. Discrete version of the position-momentum commutation relation
  6. A discrete version of the uncertainty relation
  7. Discrete-variable Gaussian functions
  8. Discrete creation operator
  9. Discrete version of the harmonic oscillator
  10. Discrete coherent states
  11. Discrete coherent state quantization
  12. Discrete annihilation operator
  13. Mehta states
  14. Harper states
  15. Kravchuk states
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Discrete Gaussian functions: $\mathbf{g}_3$ and $\mathbf{g}_{1/3}$ in the case $d\!=\!31$.
  • Figure 2: Relation $[\hat{\mathfrak{q}},\hat{\mathfrak{p}}]\approx{\rm i}\hbox{\small $\frac{d}{2\pi }$}$ in Gaussian case : $\parallel ([\hat{\mathfrak{q}},\hat{\mathfrak{p}}]-{\rm i}\frac{d}{2\pi })\mathbf{g}_k||$ for $d\!=\!11$ and $d\!=\!31$.
  • Figure 3: Relation $\Delta\hat{\mathfrak{q}}\ \Delta\hat{\mathfrak{p}}\geq \hbox{\small $\frac{d}{4\pi }$}$ in Gaussian case : $|\ |\langle \mathbf{g}_k |[\hat{\mathfrak{q}},\hat{\mathfrak{p}}]|\mathbf{g}_k\rangle |-\frac{d}{2\pi }\ |$ for $d\!=\!11$ and $d\!=\!31$.