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I am currently reading Nielsen & Chuang, and one of the questions asks to calculate a density matrix with the following mixed state,
$$ \frac{1}{9}\begin{bmatrix} 5 & 1 & −i \\ 1 & 2 & -2i \\ i & 2i & 2 \end{bmatrix}$$
and retrieving the probability of measuring
$$ |0\rangle = \begin{pmatrix} 1 \\ 0\\ 0\\ \end{pmatrix}$$
How would I do this?
Let $\rho$ denote the density matrix specified in the question. The probability of obtaining the result corresponding to $|0\rangle$ when measuring $\rho$ in the computational basis is
$$ p(0|\rho) = \mathrm{tr}(P_0\rho) = \mathrm{tr}(|0\rangle\langle 0|\rho) = \langle 0|\rho|0 \rangle = \frac{5}{9} $$
where $P_0=|0\rangle\langle 0|$ is the projector onto the subspace spanned by $|0\rangle$. See equation $(2.159)$ on page 102 in Nielsen & Chuang where $M_m = P_0$ and note that $P_0=P_0^\dagger$ and $P_0^2 = P_0$.
