{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.

Diamond norm

For the <transposition map> $\Theta:M_d\to M_d$, the <induced trace norm> is $\lVert\Theta\rVert_1=1$, while stabilization gives
$$
\lVert\Theta\rVert_\diamond=d.
$$
Thus an entangled <quantum ancilla> can amplify the distinguishability witnessed by transposition by a factor of $d$.

Diamond norm of the transposition map

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

{title2=$\lVert T\rVert_1$}

The induced trace norm of a <linear map> $T$ between matrix spaces is
$$
\lVert T\rVert_1=\sup_{X\ne0}\frac{\lVert T(X)\rVert_1}{\lVert X\rVert_1}.
$$
It measures the largest trace-norm amplification available without an auxiliary tensor factor.

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