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 isThe 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 $\rho =|{\Phi }_d⟩⟨{\Phi }_d|$. Its partial transpose is $F/d$, with $F$ the swap operator. All $d^2$ singular values of $F$ are one, so $\Vert F/d{\Vert }_1=d$ while $\Vert \rho {\Vert }_1=1$.

For the upper bound, partial transpose preserves the Frobenius norm $\Vert \cdot {\Vert }_2$. On matrices of size $d^2$,A $d$-dimensional ancilla suffices in the definition of the diamond norm, proving equality. ^[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.