Introduction: Beyond Pure States
Sometimes we don't know the exact quantum state—we only have probabilistic knowledge. The density matrix \(\hat{\rho}\) generalizes quantum mechanics to handle such mixed states, combining quantum and classical uncertainty.
Definition
For a pure state: \(\hat{\rho} = |\psi\rangle\langle\psi|\)
For a statistical mixture of states \(|\psi_i\rangle\) with classical probabilities \(p_i\):
\[\hat{\rho} = \sum_i p_i |\psi_i\rangle\langle\psi_i|\]Properties
- \(\hat{\rho}^\dagger = \hat{\rho}\) (Hermitian)
- \(\text{Tr}(\hat{\rho}) = 1\) (normalization)
- \(\hat{\rho} \geq 0\) (positive semidefinite)
- Pure state: \(\text{Tr}(\hat{\rho}^2) = 1\)
- Mixed state: \(\text{Tr}(\hat{\rho}^2) < 1\)
Worked Examples
Example 1: Pure State Density Matrix
For \(|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\):
\[\hat{\rho} = \frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}\]Check: \(\text{Tr}(\hat{\rho}^2) = 1\) ✓ (pure)
Example 2: Maximally Mixed State
50% chance of \(|0\rangle\), 50% chance of \(|1\rangle\):
\[\hat{\rho} = \frac{1}{2}|0\rangle\langle 0| + \frac{1}{2}|1\rangle\langle 1| = \frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\]Check: \(\text{Tr}(\hat{\rho}^2) = 1/2 < 1\) (mixed)
Example 3: Expectation Values
\[\langle\hat{A}\rangle = \text{Tr}(\hat{\rho}\hat{A})\]This works for both pure and mixed states.
The Quantum Connection
The density matrix is essential for quantum statistical mechanics, open quantum systems, and entanglement. When you trace out part of an entangled system, the remaining part is described by a mixed-state density matrix. This is how decoherence works—entanglement with the environment makes pure states appear mixed.