2. Density Matrix Formalism

For a pure state $|\psi\rangle$, the density operator is:

For a statistical mixture of states $|\psi_i\rangle$ with probabilities $p_i$:

Where $\sum_i p_i = 1$ and $p_i \geq 0$

Key insight: Density matrix describes our incomplete knowledge about the quantum state

A valid density matrix satisfies three conditions:

• Hermiticity: $\hat{\rho}^\dagger = \hat{\rho}$ (ensures real eigenvalues)
• Positive semidefinite: $\langle\phi|\hat{\rho}|\phi\rangle \geq 0$ for all $|\phi\rangle$
• Unit trace: $\text{Tr}(\hat{\rho}) = 1$ (normalization)

Additional property distinguishing pure vs. mixed states:

For pure states: $\hat{\rho}^2 = \hat{\rho}$ (idempotent)

The expectation value of an observable $\hat{A}$ is:

For pure state $\hat{\rho} = |\psi\rangle\langle\psi|$:

Recovers the standard quantum mechanical formula

Probability of measuring eigenvalue $a_n$:

Where $\hat{P}_n = |a_n\rangle\langle a_n|$ is the projector onto eigenstate $|a_n\rangle$

The density matrix evolves according to the von Neumann equation:

This is the quantum Liouville equation, analogous to classical Liouville theorem

Formal solution for time-independent Hamiltonian:

Where $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$ is the unitary time evolution operator

Conservation: $\text{Tr}(\hat{\rho}^2)$ is conserved under unitary evolution (pure states remain pure)

For a composite system AB with density matrix $\hat{\rho}_{AB}$, the reduced density matrix of subsystem A is:

Where $\{|b_i\rangle\}$ is a basis for subsystem B

Key property: Even if $\hat{\rho}_{AB}$ is pure, $\hat{\rho}_A$ can be mixed (signature of entanglement)

Example: Bell state $|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$

Subsystem A is maximally mixed despite AB being in a pure state

The purity of a state is quantified by:

Properties:

• $\frac{1}{d} \leq \mathcal{P} \leq 1$ for $d$-dimensional Hilbert space
• $\mathcal{P} = 1$: pure state
• $\mathcal{P} = 1/d$: maximally mixed state $\hat{\rho} = \mathbb{I}/d$

Alternative measure - linear entropy:

Measures the degree of mixedness: $S_L = 0$ for pure, $S_L = (d-1)/d$ for maximally mixed

The quantum analog of Shannon entropy:

Where $\lambda_i$ are the eigenvalues of $\hat{\rho}$

Properties:

• $S \geq 0$ with equality iff pure state
• $S \leq \ln d$ with equality iff maximally mixed
• Invariant under unitary transformations: $S(\hat{U}\hat{\rho}\hat{U}^\dagger) = S(\hat{\rho})$
• Concave: $S(\sum_i p_i\hat{\rho}_i) \geq \sum_i p_i S(\hat{\rho}_i)$

For composite systems, satisfies subadditivity:

For a qubit, any density matrix can be written as:

Where $\vec{r} = (r_x, r_y, r_z)$ is the Bloch vector and $\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$ are Pauli matrices

Geometric interpretation:

• Pure states: $|\vec{r}| = 1$ (on surface of Bloch sphere)
• Mixed states: $|\vec{r}| < 1$ (inside Bloch sphere)
• Maximally mixed: $\vec{r} = 0$ (center of sphere)

Purity in terms of Bloch vector:

General quantum operations are described by completely positive trace-preserving (CPTP) maps:

Where Kraus operators $\hat{K}_k$ satisfy $\sum_k \hat{K}_k^\dagger \hat{K}_k = \mathbb{I}$

Examples:

• Unitary evolution: Single Kraus operator $\hat{K} = \hat{U}$
• Measurement: $\hat{K}_k = \hat{P}_k$ (projectors)
• Decoherence: Multiple Kraus operators describing environment interaction

1. Thermal states

Canonical ensemble at temperature $T$:

Where $\beta = 1/(k_B T)$

2. Open quantum systems

System interacting with environment evolves non-unitarily:

Lindblad master equation, where $\mathcal{L}$ describes dissipation and decoherence

3. Quantum information processing

• Quantum channels and communication protocols
• Entanglement quantification via entropy
• Quantum error correction (tracking mixed states through noisy channels)
• Quantum tomography (reconstructing $\hat{\rho}$ from measurements)