Von Neumann Entropy and Quantum Algorithmic Randomness

TL;DR

This paper explores the relationship between von Neumann entropy and quantum algorithmic randomness, establishing conditions under which quantum states exhibit various forms of randomness and their entropy behaviors.

Contribution

It introduces quantum s-tests, characterizes quantum Schnorr randomness via uniform integrability, and links entropy growth to strong quantum randomness, advancing understanding of quantum algorithmic randomness.

Findings

Quantum Schnorr random states have entropy growth rate approaching 1.

Quantum s-tests provide bounds on entropy liminf for certain states.

States with high entropy infinitely often are strong quantum random.

Abstract

A state $\rho=(\rho_n)_{n=1}^{\infty}$ is a sequence such that $\rho_n$ is a density matrix on $n$ qubits. It formalizes the notion of an infinite sequence of qubits. The von Neumann entropy $H(d)$ of a density matrix $d$ is the Shannon entropy of its eigenvalue distribution. We show: (1) If $\rho$ is a computable quantum Schnorr random state then $\lim_n [H(\rho_n )/n] = 1$. (2) We define quantum s-tests for $s\in [0,1]$, show that $\liminf_n [H(\rho_n)/n]\geq \{ s: \rho$ is covered by a quantum s-test $\}$ for computable $\rho$ and construct states where this inequality is an equality. (3) If $\exists c \exists^\infty n H(\rho_n)> n-c$ then $\rho$ is strong quantum random. Strong quantum randomness is a randomness notion which implies quantum Schnorr randomness relativized to any oracle. (4) A computable state $(\rho_n)_{n=1}^{\infty}$ is quantum Schnorr random iff the family of distributions of the $\rho_n$'s is uniformly integrable. We show that the implications in (1) and (3) are strict.