Computing norms of positive maps Suppose $\Phi: S_n \to S_n$ is a positive linear map on symmetric matrices. As a consequence of Russo-Dye theorem we know that
$$\|\Phi\|=\|\Phi(I)\|$$
where $\|\cdot\|$ is the operator norm, ie section 2.5 here.
I'm instead interested in $\|\cdot\|_1$, the trace norm (sum of singular values), how do I compute induced operator norm in such case?
 A: I'll suppose that, contrary to what your reference indicates, you are indeed interested in a map that is positive over the real symmetric matrices.
The "brute force" approach to computing the trace norm would be as follows. Let $E_{ij}$ denote the size $n$ matrix with a $1$ in the $i,j$ entry and zeros elsewhere. Define
$$
B_{ij} = \begin{cases}
E_{ii} & i=j\\
\frac 1{\sqrt{2}}(E_{ij} + E_{ji}) & i \neq j.
\end{cases}
$$
We see that $B_{ij}$ forms an orthonormal basis on $S_n$ relative to the inner product defined by $\langle A,B \rangle = \operatorname{tr}(AB)$. Let $\mathcal B$ denote the basis $\mathcal B = \{B_{ij} : 1 \leq i \leq j \leq n\}$, where the tuples $i,j$ are taken in lexicographical order. Define $f : \{(i,j): 1 \leq i \leq j \leq n\} \to \{1,\dots,n(n+1)/2\}$ to be the associated counting function, so that
$$
f(1,1) = 1, \quad f(1,2) = 2,\dots, f(2,1) = n+1, \dots, f(n,n) = n(n+1)/2.
$$
Let $M$ denote the size $n(n+1)/2$ matrix of $\Phi$ relative to $\mathcal B$. The entries of $M$ satisfy
$$
M_{\phi(i,j),\phi(p,q)} = \langle \Phi(B_{ij}),\Phi(B_{pq})\rangle.
$$
The trace norm of $\Phi$ is equal to the trace norm of $M$.
