Density Matrix

A mathematical representation of quantum states that can describe both pure and mixed states, essential for open quantum systems and quantum noise.

A density matrix (also called a density operator) is a more general way to describe quantum states than state vectors. It’s essential when dealing with classical uncertainty, entangled subsystems, or noise.

Definition

For a pure state $|\psi ⟩$, the density matrix is:

$$\rho =|\psi ⟩⟨\psi |$$

For a mixed state (classical probability distribution over pure states):

$$\rho =\sum _i{p}_i|{\psi }_i⟩⟨{\psi }_i|$$

where $p_i$ is the probability of being in state $|{\psi }_i⟩$.

Example

Pure state $|+⟩=\frac{1}{\sqrt{2}}(|0⟩+|1⟩)$:

$$\rho =|+⟩⟨+|=\frac{1}{2}\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$$

Mixed state (50% $|0⟩$, 50% $|1⟩$):

$$\rho =\frac{1}{2}|0⟩⟨0|+\frac{1}{2}|1⟩⟨1|=\frac{1}{2}\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$$

Note: These are different states! The pure state has off-diagonal elements (coherence), the mixed state doesn’t.

Properties

A valid density matrix must satisfy:

1. Hermitian: $\rho ={\rho }^{†}$
2. Positive semi-definite: All eigenvalues $\ge 0$
3. Trace one: $\text{Tr}(\rho )=1$

Pure vs Mixed

How to tell if a state is pure or mixed:

Measurement Probabilities

Using the Born rule with density matrices:

$$P(k)=\text{Tr}(|k⟩⟨k|\rho )=⟨k|\rho |k⟩$$

Time Evolution

Density matrices evolve via:

$$\rho (t)=U\rho (0)U^{†}$$

For open systems (with noise), we use quantum channels:

$${\rho }^{\prime }=\mathcal{E}(\rho )$$

Why It Matters

Density matrices are essential for:

• Describing noise and decoherence: Real quantum systems interact with their environment
• Partial traces: Describing part of an entangled system
• Quantum error correction: Tracking how errors affect states
• Quantum information measures: Entropy, fidelity, trace distance

See also: Pure State, Mixed State, Quantum Channel