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https://en.wikipedia.org/wiki/Von_Neumann_entropy#Properties

How to show that von Neumann entropy for $p_k$ with basis $|\psi_i\rangle$ is the same for $p_n$ with basis $|\phi_i\rangle$?

That is, to show that von Neumann entropy is invariant under coordinate transformation?

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    $\begingroup$ Invariance of the trace. $\endgroup$ – Cosmas Zachos Oct 12 '18 at 2:55
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The key is diagonalisation $A=Q(\alpha)Q^{-1}$, the matrix $Q$ as orthonormal basis has the property that $QQ^{-1}=I$ which allows us to to this $\log(A)=Q\log(\alpha)Q^{-1}$ Thus using $Tr(AB)=Tr(BA)$ obtain the answer.

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