Quantum information recast via multiresolution in $L_2(0,1]$

TL;DR

This paper introduces a multiresolution mathematical framework for quantum information using classical constructions, enabling analysis of infinite qubit arrays and revealing calculus concepts within quantum structures.

Contribution

It develops a novel approach combining Borel isomorphism and Haar basis to analyze infinite qubit arrays within $L_2(0,1]$, connecting quantum operations to geometric operators.

Findings

Representation of quantum operations via geometric operators

Identification of $L_2(0,1]$ with infinite qubit array Hilbert space

Antiderivative as a natural operator in quantum array analysis

Abstract

We present a multiresolution approach to the theory of quantum information. It arose from an effort to develop a systematic mathematical approach to the analysis of an infinite array of qubits, i.e., a structure that may be interpreted as a quantum metamaterial. Foundational to our approach are two mathematical constructions with classical roots: the Borel isomorphism and the Haar basis. Here, these constructions are intertwined to establish an identification between $L_2(0,1]$ and the Hilbert space of an infinite array of qubits and to enable analysis of operators that act on arrays of qubits (either finite or infinite). The fusion of these two concepts empowers us to represent quantum operations and observables through geometric operators. As an unexpected upshot, we observe that the fundamental concept of calculus is inherent in an infinite array of qubits; indeed, the antiderivative arises as a natural and indispensable operator in this context.