Consider an observable $O = \sum_i \lambda_i P_i$ decomposed into Paulistrings $P_i$ and a unitary $U$ each acting on $n$ qubits. The Pauli-norm of $O$ is defined as the 1-norm of the Pauli vector, i.e. $$\|O\|_P = \sum_i |\lambda_i|.$$ I can write the transformed operator again as a sum of Paulistrings using the transformed Pauli-basis $$U P_i U^\dagger = \sum_{j} u_{ij} P_j \qquad \implies \qquad UOU^\dagger = \sum_{i,j} \lambda_i u_{ij} P_j$$ with $u_{ij} \in \mathbb{R}$ describing an adjoint representation of $U$. Since the Pauli-norm is not unitary-invariant, there is a question about its minimality under unitary transformations. I am trying to prove $$\|D\|_P \leq \|UOU^\dagger\|_P \qquad \forall U\in \text{U}(2^n), $$ with $D = WOW^\dagger$ being a diagonal matrix that contains the eigenvalues of $O$. Is there an easy way to see this?
EDIT: I changed the lower case $p$ to an upper case $P$ to not confuse the Pauli-norm $\|\cdot\|_P$ with the $p$-norm $\|\cdot\|_p$.
