Density matrix

Remarks

On a complex Hilbert space, clause 1 implies that $\rho$ is self-adjoint. When $\mathcal{H} = \C^n$, a density matrix is a Hermitian positive semidefinite $n \times n$ matrix with trace $1$. In general $\rho = \sum_k p_k \lvert e_k \rangle\langle e_k \rvert$ for an orthonormal family $(e_k)$ of eigenvectors and eigenvalues $p_k \ge 0$ with $\sum_k p_k = 1$.

Example: $\rho = \begin{pmatrix} 2/3 & 1/3 \\ 1/3 & 1/3 \end{pmatrix}$ is Hermitian with trace $1$. Its diagonal entries are positive and its determinant is $2/9 - 1/9 = 1/9 > 0$, so both eigenvalues are positive and $\rho$ is a density matrix.

Non-example: $\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ is Hermitian with trace $1$, but its determinant is $-1 < 0$. So one eigenvalue is negative, and clause 1 fails.