Quantum Bit Strings and Prefix-Free Hilbert Spaces

TL;DR

This paper develops a mathematical framework for indeterminate-length quantum bit strings, introduces prefix-free Hilbert spaces without basis assumptions, and explores their implications for quantum lossless compression.

Contribution

It generalizes the concept of prefix-free Hilbert spaces beyond basis-dependent models, providing a foundation for quantum data compression techniques.

Findings

Defined prefixes, fragments, tensor products, and concatenation for quantum bit strings.

Proved a quantum analogue of the Kraft inequality.

Illustrated the framework with examples and discussed compression relevance.

Abstract

We give a mathematical framework for manipulating indeterminate-length quantum bit strings. In particular, we define prefixes, fragments, tensor products and concatenation of such strings of qubits, and study their properties and relationships. The results are then used to define prefix-free Hilbert spaces in a more general way than in previous work, without assuming the existence of a basis of length eigenstates. We prove a quantum analogue of the Kraft inequality, illustrate the results with some examples and discuss the relevance of prefix-free Hilbert spaces for lossless compression.