Density matrix

A density matrix represents the probabilities of outcomes for every quantum measurement. For a pure state with state vector |ψ⟩, it is ρ = |ψ⟩⟨ψ|; for an ensemble of states |ψᵢ⟩ prepared with probabilities pᵢ, it is ρ = Σᵢ pᵢ|ψᵢ⟩⟨ψᵢ|. Different ensembles can produce the same matrix, so ρ describes the operational state rather than a unique story about how the state was prepared.

A physical density matrix is Hermitian, positive semidefinite, and has unit trace. The probability of an outcome associated with measurement operator E is given by the Born rule, p = Tr(ρE). A pure state satisfies Tr(ρ²) = 1, whereas a mixed state has Tr(ρ²) < 1; this quantity is called purity. Density matrices therefore encode uncertainty without treating it simply as ignorance about an ordinary classical variable.

Partial trace is the central operation for obtaining the state of a subsystem. If a composite system has state ρAB, the state accessible in subsystem A is ρA = TrB(ρAB), formed by summing over subsystem B's basis states.¹ Even when ρAB is pure, ρA can be mixed because A may be entangled with B.

Closed-system evolution transforms a density matrix by unitary conjugation, ρ′ = UρU†. More generally, physical quantum channels are represented by completely positive, trace-preserving maps, often written with Kraus operators as ρ′ = Σₖ KₖρKₖ†. This framework covers measurement, decoherence, relaxation, and noise while preserving positivity and total probability. A measurement generally changes the state as well as producing an outcome.

Density matrices are the standard language for quantum information because real devices rarely remain isolated or perfectly known. They describe qubits in quantum computing, noisy communication channels, thermal states, and the reduced states used to quantify entanglement. The von Neumann entropy, S(ρ) = −Tr(ρ log ρ), measures the mixedness of a state and contributes to definitions of information, entanglement, and thermodynamic entropy.

In experiments, quantum-state tomography estimates ρ from many measurements performed on identically prepared systems. The reconstruction must enforce physical constraints such as positivity and unit trace; otherwise statistical noise can yield an unphysical matrix.² Density operators also appear in quantum optics, nuclear magnetic resonance, atomic physics, and open-quantum-system models, where ensemble averages are often more practical than individual state vectors.

Density matrices distinguish proper mixtures from entanglement-induced mixtures only when the larger preparation context is available. The same reduced matrix may arise from many different global states, so local measurements alone cannot reveal every correlation with an environment or an unobserved partner.¹

The matrix need not be diagonal in the basis used to describe it: off-diagonal elements represent quantum coherences, while diagonalization identifies its eigenstates and probabilities. Under environmental interaction, those coherences can decay through decoherence, making some preferred states unusually stable without requiring an instantaneous classical collapse.³ A density matrix can also be infinite-dimensional, as in a harmonic oscillator, where technical conditions on trace and operators become essential. In relativistic quantum field theory, local density matrices are subtler because sharply localized subsystems and ultraviolet entanglement require regularization.

Pure state

A quantum state representable by one state vector and by a density matrix satisfying Tr(ρ²) = 1.

Mixed state

A state described by a nontrivial statistical mixture, with Tr(ρ²) less than 1.

Partial trace

The operation that removes unobserved degrees of freedom and produces a subsystem's reduced density matrix.

Von Neumann entropy

The quantum entropy S(ρ) = −Tr(ρ log ρ), determined by the eigenvalues of the density matrix.