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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 $S(\rho) = \mathrm{tr}(M^\dagger\rho)$ is given by the Hilbert-Schmidt inner product?

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    $\begingroup$ What is a "trace function"?! $\endgroup$ Oct 29, 2020 at 20:25

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

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