Density matrix

A density matrix, also known as a density operator, is a mathematical representation used in quantum mechanics to describe the statistical state of a quantum system. It provides a way to capture both pure and mixed states of a quantum system, allowing for a more general formulation than the state vector (wavefunction) approach.

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A finite-dimensional density operator

ρ

represents a quantum state on a Hilbert space. It is Hermitian, positive semidefinite, and has trace one:

ρ = ρ^{†}, ⟨ v ∣ ρ ∣ v ⟩ \geq 0 for every ∣ v ⟩, Tr ρ = 1.

(1)

The density matrix is its matrix in a chosen orthonormal basis. In dimension

N

it has

N^{2} - 1

real parameters: a Hermitian matrix has

N^{2}

, and fixing its trace removes one; positivity restricts the allowed values.

A quantum state ensemble of normalized pure states

∣ ψ_{i} ⟩

with positive probabilities

p_{i}

gives

ρ = \sum_{i} p_{i} ∣ ψ_{i} ⟩ ⟨ ψ_{i} ∣, p_{i} > 0, \sum_{i} p_{i} = 1.

(2)

Zero-weight terms can be omitted. Conversely, the spectral theorem gives such an ensemble for every density operator, using its orthonormal eigenvectors and nonnegative eigenvalues, with zero eigenvalues omitted. A pure state has

ρ = ∣ ψ ⟩ ⟨ ψ ∣

; a mixed state needs at least two nonzero eigenvalues. ^[ref]

For a projective measurement with orthogonal projections

Π_{a}

, the Born rule is

P (a) = Tr (ρ Π_{a}) = \sum_{i} p_{i} ⟨ ψ_{i} ∣ Π_{a} ∣ ψ_{i} ⟩ .

(3)

This is the average of the component pure state probabilities. Positivity makes each outcome probability nonnegative, and

\sum_{a} Π_{a} = I

makes them sum to one. The same density operator therefore predicts every measurement on the system without requiring a unique preparation history. ^[ref]

For a closed system, Schrödinger picture evolution under a unitary operator is

ρ_{S} (t) = U (t) ρ_{H} U (t)^{†}

(4)

when

ρ_{H} = ρ_{S} (0)

denotes the time-independent Heisenberg picture state and

U (0) = I

. In the Schrödinger picture the density operator evolves; in the Heisenberg picture the observables evolve instead, with identical measurement predictions. This unitary operator preserves the eigenvalues and hence the purity of a density operator. ^[ref]

A historical source for the probabilistic formulation of quantum mechanics is von Neumann's 1927 paper Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik. ^[ref]

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