Von Neumann Entropy and Quantum Algorithmic Randomness

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