Von Neumann Entropy and Quantum Algorithmic Randomness
Tejas Bhojraj
TL;DR
This work studies how the entropy of initial segments of infinite quantum states governs algorithmic randomness in the quantum setting. By defining quantum s-tests and leveraging the singular value decomposition, it links the per-qubit entropy rate to randomness notions and introduces uniform integrability as a tensorial analogue of complexity. It proves that for computable states, quantum Schnorr randomness forces an entropy rate of 1, that high initial entropy infinitely often yields strong quantum randomness, and that quantum Schnorr randomness is equivalent to uniform integrability of the eigenvalue distributions; these results are shown to be strict. The findings illuminate the relationship between entropy, randomness tests, and computability in quantum information, with implications for oracle-relativized randomness notions and information-theoretic characterizations.
Abstract
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.