Martin-L\"of random quantum states

TL;DR

This paper generalizes Martin-L"of randomness from classical infinite sequences to quantum states, aiming to characterize quantum ML-randomness through incompressibility of initial segments.

Contribution

It introduces a quantum analogue of Martin-L"of randomness for infinite quantum states, extending classical concepts to the quantum domain.

Findings

Proposes a framework for quantum ML-randomness

Establishes an analogy with Levin-Schnorr theorem in quantum setting

Lays groundwork for quantum incompressibility characterization

Abstract

We extend the key notion of Martin-L\"of randomness for infinite bit sequences to the quantum setting, where the sequences become states of an infinite dimensional system. We work towards showing an analogy with the Levin-Schnorr theorem to characterise quantum ML-randomness of states by incompressibility (in the sense of quantum Turing machines) of all initial segments.