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Why is the maximum entropy of $n$ qubits (in $\log_2$ units) equal to $n$? How does one calculate $\operatorname{Tr}(\rho \operatorname{log} \rho)$ (since $\log \rho$ must be expanded). What is even the density matrix of this state?

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Write $\rho$ in its eigenbasis. The eigenvalues are probabilities $p_i$, $\sum p_i=1$. Then, $$ -\mathrm{tr}(\rho\log\rho)=-\sum p_i\log p_i\ . $$ You should now be able to easily see which $p_i$ maximizes this quantity (which is just the classical Shannon entropy).

(If you want to formalize this, you need to use the concept of Schur-convexity/concavity and majorization: $-\sum p_i\log p_i$ is Schur-concave and thus maximized by the distribution which is majorized by all other ones.)

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