Theorem: Exact record law and mean entropy

For a fixed pure input and fixed $U$,

The averaged cumulative record entropy is

It increases strictly with $k$. At the endpoint,

For every fixed $U,\psi$, not merely on average,

Proof. Later operations are SWAPs on other cells. The projectors defining the full outcome history therefore correspond to the commuting tensor-basis projectors on the original scrambled state. Their Born law is (eq:recordlaw); no postselection is used.

Represent a Haar vector by independent standard complex Gaussians divided by their Euclidean norm. The sums of squared moduli over the $b_k$ entries associated with each prefix are independent Gamma variables of shape $b_k$. After normalization, the $r_k$ prefix probabilities have the Dirichlet distribution with all parameters $b_k$. For real $t$ near one,

Differentiation at $t=1$ yields

where ${\psi }_0=(\ln \Gamma {)}^{\prime }$. This proves (eq:recordmean). Its increment is $({\mathsf{H}}_{b_k}-{\mathsf{H}}_{b_{k+1}})/\ln 2>0$.

For a fixed process, Shannon's chain rule gives

and the conditional entropy is nonnegative.