Quantum algorithmic randomness

TL;DR

This paper explores quantum algorithmic randomness, introducing quantum Solovay and Schnorr randomness, establishing their equivalences, and demonstrating foundational properties like a quantum law of large numbers for random states.

Contribution

It defines and relates new quantum randomness notions, proves their equivalence, and extends classical probabilistic laws to the quantum setting.

Findings

Quantum Solovay randomness is equivalent to quantum Martin-Löf randomness.

Random quantum states form a convex set.

A quantum law of large numbers holds for quantum Schnorr random states.

Abstract

Quantum Martin-L\"of randomness (q-MLR) for infinite qubit sequences was introduced by Nies and Scholz. We define a notion of quantum Solovay randomness which is equivalent to q-MLR. The proof of this goes through a purely linear algebraic result about approximating density matrices by subspaces. We then show that random states form a convex set. Martin-L\"of absolute continuity is shown to be a special case of q-MLR. Quantum Schnorr randomness is introduced. A quantum analogue of the law of large numbers is shown to hold for quantum Schnorr random states.