# Prove Von Neumann entropy is invariant under coordinate transformation

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?

• Invariance of the trace. – Cosmas Zachos Oct 12 '18 at 2:55

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.