Quantum Kolmogorov complexity and quantum correlations in deterministic-control quantum Turing machines

TL;DR

This paper explores the extension of Kolmogorov complexity to quantum states within a deterministic-control quantum Turing machine framework, analyzing quantum information, copying limits, and correlations.

Contribution

It introduces a model for quantum Kolmogorov complexity with mixed states and develops a correlation-aware mutual information measure.

Findings

Quantum states' complexity compared with classical representations

Limits of quantum state copying analyzed through complexity

Correlation-aware mutual information defined and studied

Abstract

This work presents a study of Kolmogorov complexity for general quantum states from the perspective of deterministic-control quantum Turing Machines (dcq-TM). We extend the dcq-TM model to incorporate mixed state inputs and outputs, and define dcq-computable states as those that can be approximated by a dcq-TM. Moreover, we introduce (conditional) Kolmogorov complexity of quantum states and use it to study three particular aspects of the algorithmic information contained in a quantum state: a comparison of the information in a quantum state with that of its classical representation as an array of real numbers, an exploration of the limits of quantum state copying in the context of algorithmic complexity, and study of the complexity of correlations in quantum systems, resulting in a correlation-aware definition for algorithmic mutual information that satisfies symmetry of information property.