= Diamond norm of the transposition map
For the <transposition map> $\Theta:M_d\to M_d$, $\Theta(X)=X^T$, the <induced trace norm> is one, while the <diamond norm> is
$$
\|\Theta\|_{1\to1}=1,\qquad\|\Theta\|_\diamond=d.
$$
The first equality follows because <transposition> preserves every <singular value> and hence the <trace norm>.
For the lower bound, apply $\Theta\otimes\mathrm{id}_d$ to the normalized <maximally entangled state> projector $\rho=|\Phi_d\rangle\langle\Phi_d|$. Its <partial transpose> is $F/d$, with $F$ the <swap operator>. All $d^2$ <singular values> of $F$ are one, so $\|F/d\|_1=d$ while $\|\rho\|_1=1$.
For the upper bound, <partial transpose> preserves the <Frobenius norm> $\|\cdot\|_2$. On matrices of size $d^2$,
$$
\|(\Theta\otimes\mathrm{id}_d)(X)\|_1\leq d\|(\Theta\otimes\mathrm{id}_d)(X)\|_2=d\|X\|_2\leq d\|X\|_1.
$$
A $d$-dimensional ancilla suffices in the definition of the <diamond norm>, proving equality. <The Theory of Quantum Information by John Watrous>[Chapter 3, Example 3.42, equations 3.249–3.255, and Definition 3.43]{ref}
For $d>1$, <transposition> is not <completely positive>: $F/d$ has a negative <eigenvalue> on the antisymmetric subspace. Thus this is a mathematical example of <norm> amplification by an <entangled> ancilla; transposition itself is not a physically implementable <quantum channel>.
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