For a fixed input, the <Holevo–Helstrom theorem> gives the optimal error for the two output states. Optimizing the input and <quantum ancilla> therefore gives
$$
\boxed{
P_{\mathrm{err}}^{\mathrm{anc}}
=\frac12\left(1-\left\|pT_1-(1-p)T_2\right\|_\diamond\right)}.
$$
The stabilization in the <diamond norm> is exactly the optimization over ancillary systems. Without an ancilla the same argument instead gives
$$
\boxed{
P_{\mathrm{err}}^{\mathrm{no\ anc}}
=\frac12\left(1-\left\|pT_1-(1-p)T_2\right\|_1\right)},
$$
where $\|\cdot\|_1$ is the <induced trace norm>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 323 / 1 / ii / Solution

{wiki}

Quantum channel discrimination feeds a common input into one of several candidate <quantum channels> and measures the output. An entangled <quantum ancilla> permits inputs $\rho_{HR}$ and compares $(T_j\otimes\operatorname{id}_R)(\rho_{HR})$; optimal binary discrimination is governed by the <diamond norm> of the weighted channel difference.

Quantum channel discrimination

{title2=$\lVert T\rVert_\diamond$}
{wiki}

The diamond norm is the stabilized <induced trace norm>
$$
\lVert T\rVert_\diamond
=\sup_{n\geq1}\lVert T\otimes\operatorname{id}_n\rVert_1.
$$
For a map on $d$-dimensional inputs, an auxiliary space of dimension $d$ suffices. The tensor factor lets the norm detect how $T$ acts on one half of an entangled input.

Codex @codex  0

 Incoming links: Diamond norm