Quantum Mechanics · Chapter 17

Density matrices

A state known only with probabilities is not a superposition, and needs an object of its own. Pure and mixed states, the density matrix and its rules, purity and the Bloch ball, how a density matrix evolves, von Neumann entropy, and the composite systems whose parts are described by a partial trace and by the Schmidt decomposition.

ρ^= k \sum p_{k} ∣ ψ_{k} ⟩ ⟨ ψ_{k} ∣

tr ρ^= 1

tr (ρ^^{2}) \leq 1

⟨ A^⟩ = tr (ρ^A^)

Lessons

Build the foundation

Read lesson →

06Composite systemsA system of two parts lives in the tensor product of their spaces: basis states $\ket{a}\ket{b}$, one from each part, so dimensions multiply.Read lesson →

07Partial trace and reduced statesWhen the whole is in $\hat{\rho}$ and only part $A$ is looked at, everything $A$’s measurements can show is in the reduced density matrix $\hat{\rho}_{A}=\operatorname{tr}_{B}\hat{\rho}$, found by tracing out $B$. For a product state it is $A$’s own state. For an entangled pure state it is mixed: each spin of the singlet, looked at alone, is completely mixed, $\tfrac12\cdot1$, up or down at random along every axis. So a mixed state need not come from ignorance of a preparation: it can come from entanglement with something else. This is one way into entanglement: a part whose reduced state is not pure.Read lesson →

08The Schmidt decompositionEvery pure state of two systems can be written as a single sum $\sum_{i}\sqrt{\lambda_{i}}\ket{a_{i}}\ket{b_{i}}$, with orthonormal states on each side and positive weights adding to one — its Schmidt decomposition. It shows at a glance how entangled the state is: one term means a product, more terms mean entanglement. The two reduced states have the same nonzero eigenvalues, the weights $\lambda_{i}$, so the two parts are exactly as mixed as each other, however different their dimensions. The number of terms, the Schmidt rank, is at most the smaller of the two dimensions.Read lesson →

09Chapter practiceBring the chapter together: ensembles, the density matrix and its rules, purity and the Bloch ball, the von Neumann equation, entropy, composite systems, the partial trace and the Schmidt decomposition. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Density matrices

Build the foundation