Digital representation of continuous observables in Quantum Mechanics

TL;DR

This paper explores how continuous quantum observables can be discretized into digital representations using Hermitean operators, analyzing their properties, commutation relations, and effects of numeral systems on quantum simulations.

Contribution

It introduces a novel method for representing continuous quantum observables as discrete digit operators and examines their mathematical properties and implications for quantum simulation.

Findings

Operators of digits are Hermitean with discrete spectra

The choice of numeral system influences lattice structures and representations

Renormalizations of diverging sums naturally arise during discretization

Abstract

To simulate the quantum systems at classical or quantum computers, it is necessary to reduce continuous observables (e.g. coordinate and momentum or energy and time) to discrete ones. In this work we consider the continuous observables represented in the positional systems as a series of powers of the radix mulitplied over the summands (``digits``), which turn out to be Hermitean operators with discrete spectrum. We investigate the obtained quantum mechanical operators of digits, the commutation relations between them and the effects of choice of numeral system on the lattices and representations. Furthermore, during the construction of the digital representation renormalizations of diverging sums naturally occur.