⚛️ quantum physics

On a discrete version of the position-momentum commutation relation

This paper investigates the set of pure quantum states in high-dimensional qudit systems that approximately satisfy a discrete position-momentum commutation relation, identifying specific families such as discrete Gaussian states, coherent states, and Hermite-Gauss states while suggesting potential experimental methods for their realization.

Original authors: Nicolae Cotfas

Published 2026-02-05

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Nicolae Cotfas

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to describe the motion of a car. In the real world (the "continuous" world), the car can be at any exact spot on the road, and its speed can be any exact number. Physics has a famous rule, the Position-Momentum Commutation Relation, which acts like a fundamental law of nature for these smooth, flowing movements. It says that you can't know the car's exact location and exact speed at the same time with perfect precision; there's a built-in fuzziness to the universe.

Now, imagine you are playing a video game. In a game, the world isn't smooth; it's made of a grid of pixels. You can't be "halfway" between two pixels; you are either on one or the next. This is what physicists call a discrete system (like a "qudit," a quantum bit with many possible states instead of just two).

For a long time, scientists thought this fundamental "fuzziness" rule (the commutation relation) simply didn't exist in these pixelated, grid-based quantum worlds. The math didn't add up perfectly because you can't have a smooth curve on a jagged grid.

The Big Discovery
Nicolae Cotfas, a physicist from the University of Bucharest, asks a simple question: What if we look at a grid that is really, really big?

Think of a low-resolution image (like a tiny 10x10 pixel picture). It looks very blocky and jagged. But if you zoom in on a 4K or 8K image (a massive grid), the pixels become so small that the image looks almost perfectly smooth to the naked eye.

Cotfas shows that in these "high-resolution" quantum grids (systems with a large number of dimensions), the fundamental fuzziness rule does appear, but only as an approximation. It's not perfect, but it's close enough that for most practical purposes, the rule holds true for a specific group of special states.

The "Special States" (The Smooth Pixels)
The paper explores which specific quantum states behave like they are in a smooth world, even though they are actually in a pixelated one. Cotfas finds several families of these "smooth" states:

Discrete Gaussian States: Imagine a bell curve (a smooth hill). In the pixelated world, this hill is made of blocks. Cotfas shows that for certain "tall and wide" hills, the blocks are so small and numerous that the hill looks smooth, and the fuzziness rule works.
Coherent States: Think of these as the most "classical" looking quantum states. They are like a wave that moves smoothly across the grid without breaking apart.
Hermite-Gauss States: These are like the different "notes" a guitar string can play. In the smooth world, these notes are perfectly distinct. In the pixelated world, the lower notes (the ones with less energy) still sound very much like the real notes, obeying the fuzziness rule.
Harper and Kravchuk States: These are more complex mathematical shapes, but Cotfas finds that even these "weird" shapes can mimic the smooth world if the grid is big enough.

The "Magic" of the Grid Size
The paper uses a lot of numbers to prove this. It shows that if your quantum system is small (like a 11-pixel grid), the rule fails miserably. But if you increase the size to 31, 61, or 101 pixels, the rule starts working for a huge percentage of the possible states.

It's like trying to draw a circle on graph paper. On a small sheet, it looks like a jagged staircase. On a massive sheet with tiny squares, it looks like a perfect circle. The paper proves that for these "large" quantum systems, the jagged staircase is so fine that the universe's fundamental laws of motion and uncertainty apply to it.

Why This Matters (According to the Paper)
The author suggests that because these special states exist naturally as the "low energy" (calmest) states of these systems, we might be able to create them in a lab. If we can build quantum computers that use these "large grid" systems (qudits) instead of simple switches (qubits), we might be able to use the familiar, smooth rules of physics to design better algorithms and simpler experiments.

In Summary
The paper is a mathematical proof that if you make a quantum system big enough, the jagged, pixelated nature of the universe starts to look smooth. In this "smooth" zone, the famous rules of quantum mechanics (like the uncertainty between position and speed) start to work again, even though the system is technically discrete. The author maps out exactly which "shapes" of quantum states fit into this smooth zone, offering a new way to think about how quantum computers might work in the future.

Technical Summary: On a Discrete Version of the Position-Momentum Commutation Relation

Problem Statement
In continuous-variable quantum mechanics, the canonical commutation relation $[\hat{q}, \hat{p}] = i\hbar$ is fundamental to the mathematical description of quantum systems. However, in finite-dimensional Hilbert spaces (qudits), such a pair of Hermitian operators cannot exist exactly because the trace of the commutator $[\hat{q}, \hat{p}]$ must be zero, whereas the trace of $i\hbar I$ is non-zero. Previous approaches have attempted to resolve this by relaxing the definition to hold only on a proper subspace. This paper investigates a different approach: determining whether a discrete version of the commutation relation can be satisfied with high approximation within a specific subspace of a high-dimensional Hilbert space, and characterizing the quantum states that inhabit this subspace.

Methodology
The author considers a discrete-variable quantum system described by an odd-dimensional Hilbert space $\mathcal{H}$ of dimension $d = 2s+1$ , identified with functions on the set $\{-s, \dots, s\}$ . The study employs the following mathematical framework:

Operator Definitions: The position operator $\hat{q}$ is defined as a multiplication operator in the computational basis, while the momentum operator $\hat{p}$ is defined via the discrete Fourier transform (DFT) as $\hat{p} = F^\dagger \hat{q} F$ .
Displacement and Pauli Operators: Generalized Pauli operators $Z$ and $X$ are defined, leading to discrete displacement operators $D(n,k)$ .
Approximation Metric: The deviation from the canonical relation is quantified by the operator $C = [\hat{q}, \hat{p}] - i \frac{d}{2\pi}$ . The paper analyzes the eigenvalues of $C$ and defines a set $S_\epsilon$ of normalized states $\psi$ such that $\|C\psi\| < \epsilon$ .
State Characterization: The paper numerically evaluates whether various classes of discrete quantum states (Gaussian, coherent, harmonic oscillator eigenstates, etc.) belong to $S_\epsilon$ for increasing dimensions $d$ . It also examines the spectral properties of discrete Hamiltonians and the behavior of the uncertainty relation $\Delta \hat{q} \Delta \hat{p} \ge \frac{d}{4\pi}$ within these states.

Key Contributions and Results
The paper demonstrates that for sufficiently large dimensions $d$ , a significant portion of the Hilbert space satisfies the approximate commutation relation $[\hat{q}, \hat{p}] \approx i \frac{d}{2\pi}$ . Specifically:

Eigenvalue Analysis: Numerical data shows that as $d$ increases, the percentage of eigenvalues of $C$ with modulus $|\lambda| < 0.001$ grows significantly (e.g., from 36% at $d=11$ to 84% at $d=101$ ).
Discrete Gaussian States: A family of discrete-variable Gaussian states $g_\kappa$ , depending on a continuous parameter $\kappa$ , is shown to satisfy the approximate commutation relation and the corresponding uncertainty principle. These states are Fourier invariant ( $F[g_\kappa] = g_{1/\kappa}$ ).
Discrete Fock and Harmonic Oscillator States: Eigenstates of discrete harmonic oscillator Hamiltonians (defined as $\hat{H} = (\hat{p}^2 + \hat{q}^2)/2$ ) and states generated by discrete creation/annihilation operators are found to belong to $S_\epsilon$ for low energy levels. The eigenvalues of these Hamiltonians exhibit an almost equidistant distribution in the lower part of the spectrum, analogous to the continuous case.
Discrete Coherent States: States generated by applying discrete displacement operators to a vacuum state (discrete coherent states) satisfy the approximate relation.
Specialized States: The study extends to other specific families, including:
- Mehta states: Discrete versions of Hermite-Gauss functions, which are eigenstates of the discrete Fourier transform.
- Harper states: Eigenfunctions of a Hamiltonian constructed using finite-difference operators.
- Kravchuk states: States based on Kravchuk polynomials.
Uncertainty Relation: For states in $S_\epsilon$ , the Robertson-Schrödinger uncertainty relation $\Delta \hat{q} \Delta \hat{p} \ge \frac{d}{4\pi}$ is satisfied with high approximation.

Significance and Claims
The paper claims to identify a new property of important discrete-variable quantum states: their ability to approximately satisfy the position-momentum commutation relation when the Hilbert space dimension is sufficiently large.

Theoretical Implication: The results suggest that a mathematical formalism for discrete-variable systems, analogous to the theory of continuous-variable Gaussian states used in Quantum Information, may be constructed using only these particular states.
Physical Interpretation: The author proposes that the operators $\hat{q}$ and $\hat{p}$ could physically correspond to observables with equidistant eigenvalues (e.g., $J_z$ in an $SU(2)$ system) and their Fourier conjugates.
Experimental Relevance: The paper suggests that because these states appear naturally as low-energy eigenstates of certain discrete Hamiltonians, they may be experimentally accessible.
Modesty and Open Questions: The author explicitly states that there is currently no satisfactory mathematical explanation for why these specific states satisfy the relation or why they correspond to low-energy states. The connection between the approximate commutation relation and the equidistant distribution of eigenvalues is noted as a correlation requiring further theoretical investigation. The paper does not propose specific experimental setups but highlights the potential for building useful models in discrete-variable quantum computing by focusing on these particular states.

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