Density Matrix

A density matrix is a mathematical representation of a state in quantum mechanics that determines the probabilities of measurement outcomes. Unlike a wave function or state vector alone, it can describe both pure states and mixed states, including statistical mixtures and subsystems entangled with an environment. Conventionally denoted $\rho$, it provides a common framework for quantum uncertainty and classical randomness. The term density operator emphasizes the basis-independent object; density matrix refers to its representation in a chosen basis. (quantum.cloud.ibm.com)

Mathematical definition

For a finite-dimensional system with Hilbert space $\mathcal H$, a density matrix is a square matrix satisfying

Here $\dagger$ denotes the conjugate transpose, positivity holds for every vector $|v\rangle$, and the trace is the sum of diagonal entries. Thus $\rho$ is Hermitian, positive semidefinite, and normalized. Its entries may be complex numbers, but its eigenvalues are real, nonnegative, and sum to one. In infinite-dimensional Hilbert spaces, the corresponding definition requires a positive trace-class operator with unit trace. (quantum.cloud.ibm.com)

By the spectral theorem,

where the eigenvectors form an orthonormal basis. The eigenvalues constitute a probability distribution. Density matrices form a convex set: any probabilistically weighted average of valid density matrices is another valid density matrix. (learning.quantum.ibm.com)

Pure states, mixtures, and coherence

A normalized vector $|\psi\rangle$ defines a pure state through

This is a rank-one projection, satisfying $\rho^2=\rho$. Its rank is one, and multiplying $|\psi\rangle$ by an overall phase does not change $\rho$. A statistical preparation choosing $|\psi_k\rangle$ with probability $p_k$ instead gives

The vectors need not be orthogonal. Distinct preparation ensembles can produce exactly the same density matrix and therefore cannot be distinguished by measurements on that system alone. A mixed state has rank greater than one. (preskill.caltech.edu)

A mixture differs from a quantum superposition. For a qubit, compare the pure state $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$ with an equal mixture of $|0\rangle$ and $|1\rangle$:

Both yield equal computational-basis probabilities, but measuring in the $|+\rangle,|-\rangle$ basis distinguishes them. Off-diagonal entries encode coherence relative to the chosen basis and affect interference. Diagonality is basis-dependent: every density matrix is diagonal in an eigenbasis, so diagonality alone does not make a system classical. (quantum.cloud.ibm.com)

Measurements and observable quantities

The expectation value of an observable $A$ is

For a projective measurement with outcome projector $P_a$, the Born rule becomes $p(a)=\operatorname{Tr}(\rho P_a)$. More generally, a positive-operator-valued measurement uses positive operators $E_a$ summing to the identity, with $p(a)=\operatorname{Tr}(\rho E_a)$. These formulas apply equally to pure and mixed states. (preskill.caltech.edu)

The outcome probability does not, by itself, specify the post-measurement state. For an ideal projective measurement conditioned on outcome $a$,

provided the denominator is nonzero. Experimental reconstruction of a density matrix from measurements on many identically prepared systems is called quantum state tomography. (fab.cba.mit.edu)

Composite systems and reduced states

The joint Hilbert space of systems $A$ and $B$ is their tensor product, $\mathcal H_A\otimes\mathcal H_B$. Their joint state is described by $\rho_{AB}$. When only $A$ is considered, its reduced state is obtained by the partial trace:

This operation discards access to $B$ while preserving all measurement predictions concerning $A$. (learning.quantum.ibm.com)

Entanglement allows a globally pure state to have mixed reduced states. For the Bell state $(|00\rangle+|11\rangle)/\sqrt2$, each individual qubit has density matrix $I/2$, although the pair is pure. This local mixedness does not require an observer actually to measure the other qubit. Consequently, a mixed density matrix need not represent ignorance of a separately existing pure state of the subsystem. (quantum.cloud.ibm.com)

Time evolution and thermal states

For an isolated system, unitary evolution gives $\rho(t)=U(t)\rho(0)U^\dagger(t)$. The Schrödinger equation implies the Liouville–von Neumann equation,

where $H$ is the Hamiltonian operator. The equation applies to arbitrary mixtures because it is linear in $\rho$. (ocw.mit.edu)

Environmental interactions require a reduced-system description and can change purity. In quantum decoherence, coherence relative to a relevant basis is suppressed; this need not imply that all off-diagonal entries vanish in every basis. Density matrices therefore provide a natural language for quantum noise. (quantum.cloud.ibm.com)

In statistical mechanics, the canonical equilibrium state at temperature $T$ is

where $k_B$ is the Boltzmann constant. This expression assumes that the partition function $Z$ is finite. In the energy eigenbasis, its populations follow Boltzmann weights. (fab.cba.mit.edu)

Purity and entropy

Purity is $\operatorname{Tr}(\rho^2)$. In dimension $d$, it lies between $1/d$ and $1$; the upper limit characterizes pure states, while the lower limit characterizes the maximally mixed state $I/d$. (preskill.caltech.edu)

The von Neumann entropy, central to quantum information theory, is

with $0\log0=0$. It vanishes for pure states and reaches $\log d$ for the maximally mixed state. Using base-two logarithms measures it in bits. For a bipartite pure state, the entropy of either reduced density matrix quantifies entanglement; for general mixed states, entropy is not by itself an entanglement measure. (preskill.caltech.edu)