Theorem: Identified complete-isometry comparison

Let $V,W:{\mathcal{H}}_C\to {\mathcal{H}}_A\otimes {\mathcal{H}}_B$ be isometries on the same input and identified complete output spaces. Then

The allowance also bounds either partial-trace channel.

If $V^{†}W=z{I}_C$, then

If $z\in [0,1]$ is real, then

Proof. For every amplified trace-class operator $X$,

Its trace norm is at most $2{\Vert V-W\Vert }_{\infty }{\Vert X\Vert }_1$. Divide by two and use partial-trace contractivity.

For any pure code–reference input $\psi$, the output vectors have overlap

Their trace distance is $\sqrt{1-|z{|}^2}$. Convexity gives the same upper bound for mixed inputs, and every pure input attains it. The finite-input channel-discrimination characterization therefore proves (eq:scalaroverlap).

Finally,

when $z$ is real.