Quantum Shannon–McMillan–Breiman conjecture for quantum Martin-Löf random states

Let M_ denote the infinite tensor product of $2\times 2$ complex matrix algebras. For a state $\mu$ on M_, write

$$h(\mu)=\lim_{n\to\infty}\frac{1}{n}H\left(\mu\!\upharpoonright_{M_{2^n}}\right),$$

where $H(S)=-\operatorname{Tr}(S\log S)$. A state $\rho$ is quantum Martin-L\f random with respect to a computable state $\mu$ if it passes every quantum Martin-L\f test relative to $\mu$.

Quantum Shannon\fMcMillan\fBreiman conjecture. If $\mu$ is an ergodic computable state on $M_\e1$ and $\rho$ is quantum Martin-L\f random with respect to $\mu$, then

$$h(\mu)=-\lim_{n\to\infty}\frac{1}{n}\operatorname{Tr}\left(\rho\!\upharpoonright_{M_{2^n}}\log\left(\mu\!\upharpoonright_{M_{2^n}}\right)\right).$$

This is proposed as a quantum analogue of the Shannon\fMcMillan theorem, replacing an almost-everywhere assertion by one about quantum Martin-L\f random states. The source describes it as work in progress, and no resolution is supplied.