On G\'acs' quantum algorithmic entropy

TL;DR

This paper introduces a modified concept of quantum algorithmic entropy based on Gács' approach, explores the existence of universal operators under certain conditions, and discusses properties and examples to guide future research.

Contribution

It proposes a new infinite-dimensional modification of Gács' quantum algorithmic entropy and demonstrates the existence of a universal operator under specific assumptions.

Findings

Existence of universal operator under additional assumptions

Properties and examples stimulating further research

Universal operator has a nontrivial form if it exists

Abstract

We define an infinite dimensional modification of lower-semicomputability of density operators by G\'acs with an attempt to fix some problem in the paper. Our attempt is partly achieved by showing the existence of universal operator under some additional assumption. It is left as a future task to eliminate this assumption. We also see some properties and examples which stimulate further research. In particular, we show that universal operator has certain nontrivial form if it exists.