Algorithmic complexity of quantum states

TL;DR

This paper introduces a new way to measure the complexity of pure quantum states based on the classical description of procedures needed for their preparation, aligning with the intuitive notion of difficulty in state preparation.

Contribution

It defines the Kolmogorov complexity for quantum states via classical descriptions of preparation procedures, providing a framework for quantifying quantum state complexity.

Findings

Provides an upper bound on the complexity of various quantum states

Aligns quantum complexity with classical description complexity

Offers a new perspective on quantum state preparation difficulty

Abstract

In this paper we give a definition for the Kolmogorov complexity of a pure quantum state. In classical information theory the algorithmic complexity of a string is a measure of the information needed by a universal machine to reproduce the string itself. We define the complexity of a quantum state by means of the classical description complexity of an (abstract) experimental procedure that allows us to prepare the state with a given fidelity. We argue that our definition satisfies the intuitive idea of complexity as a measure of ``how difficult'' it is to prepare a state. We apply this definition to give an upper bound on the algorithmic complexity of a number of states.