Density matrix (written as $\rho$) is a matrix representation of a quantum state.

It's a more general way to represent a quantum state compared to the state vector $\lvert\psi\rangle$. While a state vector can only represent pure states, a density matrix can represent mixed states as well, making it the correct tool for open quantum systems, noisy circuits, and statistical ensembles of quantum states.

Every pure state $\lvert\psi\rangle$ has a corresponding density matrix constructed by the outer product of the ket with its bra.

$$\rho = \lvert\psi\rangle\langle\psi\rvert$$

For example, the density matrices of the two computational basis states $\lvert 0\rangle$ and $\lvert 1\rangle$ are the projectors onto those states.

$$\rho_0 = \lvert 0\rangle\langle 0\rvert = \begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}, \qquad \rho_1 = \lvert 1\rangle\langle 1\rvert = \begin{pmatrix}0 & 0 \\ 0 & 1\end{pmatrix}$$

A density matrix $\rho$ always satisfies three properties: it is Hermitian ($\rho^\dagger = \rho$), positive semidefinite ($\rho \geq 0$), and has unit trace ($\text{tr}(\rho) = 1$). These three conditions are necessary and sufficient for $\rho$ to represent a valid quantum state. The purity $\text{tr}(\rho^2)$ ranges from $1/d$ (maximally mixed state in dimension $d$) to $1$ (pure state).

A mixed state arises when a quantum system is in state $\lvert\psi_k\rangle$ with classical probability $p_k$. Its density matrix is the convex combination of the individual pure-state density matrices.

$$\rho = \sum_k p_k\lvert\psi_k\rangle\langle\psi_k\rvert, \qquad \sum_k p_k = 1, \quad p_k \geq 0$$

Under Hamiltonian evolution, the density matrix evolves according to the von Neumann equation $d\rho/dt = -i[H,\rho]/\hbar$. For open systems with environmental noise, this generalizes to the Lindblad master equation, which adds dissipative terms describing decoherence and relaxation.