How to check if a matrix is a valid density matrix?

Question

What conditions must a matrix hold to be considered a valid density matrix?

Answer 1

If a matrix has unit trace and if it is positive semi-definite (and Hermitian) then it is a valid density matrix. More specifically check if the matrix is Hermitian; find the eigenvalues of the matrix , check if they are non-negative and add up to $1$.

Answer 2

Suppose someone has prepared your quantum system in one of an orthogonal set of states $\{|\psi_j\rangle\}$. You don't know which of these states they've prepared it in, but you do know that they prepared state $|\psi_j\rangle$ with probability $p_j$. Your system is then described by the density matrix,

$\rho = \sum_j \, p_j \, |\psi_j\rangle \langle\psi_j|$.

There are some properties that will apply to any density matrix of this form.

• Clearly it is diagonalizable, since it is explicitly written in terms of its eigenvalues $p_j$ and eigenstates $|\psi_j\rangle$.

• Since the $|\psi_j\rangle \langle\psi_j|$ are Hermitian, and since probabilities are real numbers, the density matrix is Hermitian.

• Since probabilities are all either zero or positive, the density matrix is positive semidefinite.

• Since all probabilities must sum to 1, and the trace is a sum of eigenvalues, the density matrix must have a trace of 1.

These are exactly the properties required of all density matrices. Hopefully this derivation of them gives a bit of understanding of why they are required.