Density matrix $\rho$ is a mathematical representation of a quantum state that generalizes state vectors to include mixed states (statistical ensembles of pure states). A pure state has density matrix $\rho = |\psi\rangle\langle\psi|$. A mixed state cannot be written as a single state vector and represents incomplete knowledge or entanglement with an external environment.

Representation: A density matrix is Hermitian ($\rho = \rho^\dagger$), has trace 1 ($\text{Tr}(\rho) = 1$), and is positive semidefinite.

• Pure state: can be written $|\psi\rangle$ with $\rho = |\psi\rangle\langle\psi|$; has $\text{Tr}(\rho^2) = 1$
• Mixed state: cannot be written as a single state vector; has $\text{Tr}(\rho^2) < 1$
• Maximally mixed: $\rho = I/2^n$ (no information about state)

For observable with eigenvalues $\{a_i\}$ and eigenstates $\{|a_i\rangle\}$, the probability of outcome $a_i$ is $p_i = \langle a_i|\rho|a_i\rangle$. The expectation value is $\langle A \rangle = \text{Tr}(\rho A)$.

A separable pure state $|\psi\rangle = |\psi_A\rangle \otimes |\psi_B\rangle$ has $\rho = |\psi_A\rangle\langle\psi_A| \otimes |\psi_B\rangle\langle\psi_B|$. An entangled state like the Bell state $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ has density matrix with off-diagonal coherence terms that cannot be factored.

In real quantum systems, interactions with the environment cause decoherence: pure states become mixed. A qubit exposed to noise may transition from $\rho = |0\rangle\langle 0|$ toward the mixed state $\rho = \frac{1}{2}I$.

Unitary operations: $\rho' = U\rho U^\dagger$

Partial trace (tracing out subsystem B): $\rho_A = \text{Tr}_B(\rho_{AB})$ gives the reduced density matrix of subsystem A alone.

POVM (positive-operator-valued measure) for generalized measurements: outcome probabilities are $p_i = \text{Tr}(M_i \rho)$ where $\sum_i M_i = I$.