Can a trace function be represented as a Hilbert-Schmidt product with the density matrix?

Question

Given a trace function, say the von Neumann entropy $S(rho) = - mathrm{tr}(rho log rho)$, can it be express as a matrix $M$ where $S(rho) =  mathrm{tr}(M^daggerrho)$ is given by the Hilbert-Schmidt inner product?

NDewolf · Accepted Answer

In the case of the entropy, no. It's a nonlinear function in $rho$, while your matrix multiplication is linear in $rho$.

Can a trace function be represented as a Hilbert-Schmidt product with the density matrix?

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