Theorem: Optimal score, input ceiling, and data processing

For unrestricted binary measurements after receiving $J$,

For any CPTP channel applied to the trusted ensemble (eq:squareensemble),

If $Z^{\prime }$ is obtained from $Z$ by a message-independent CPTP map, then

Proof. For a fixed query $j$, the two requested-bit hypotheses have equal prior probability and states ${\tau }_{0\mid j}$, ${\tau }_{1\mid j}$. Let $H_j={\tau }_{0\mid j}-{\tau }_{1\mid j}$. For an effect $0\le M\le I$, the binary success probability is

Since $\mathrm{Tr}{H}_j=0$, the supremum is $\frac{1}{2}+\frac{1}{4}{\Vert H_j\Vert }_1$, attained by the positive spectral projector of $H_j$, with an arbitrary decision on its kernel. Averaging over $j$ proves (eq:helstromscore). This is the binary Holevo–Helstrom argument Watrous2018.

At the input,

Each trace norm is $\sqrt{2}$. Contractivity under a CPTP channel proves the upper bound and (eq:scoreprocessing). Random guessing proves the lower bound. An untouched qubit with query-dependent $X$ or $Z$ measurement attains the upper bound.