Density matrix
= Density matrix
{wiki}
= Density operator
{synonym}
A finite-dimensional <density operator> $\rho$ represents a <quantum state> on a <Hilbert space>. It is <Hermitian>, <positive semidefinite>, and has <trace> one:
$$
\rho=\rho^\dagger,\qquad \langle v|\rho|v\rangle\geq0\text{ for every }|v\rangle,\qquad \operatorname{Tr}\rho=1.
$$
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> $|\psi_i\rangle$ with positive <probabilities> $p_i$ gives
$$
\rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|,\qquad p_i>0,\qquad \sum_i p_i=1.
$$
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 $\rho=|\psi\rangle\langle\psi|$; a <mixed state> needs at least two nonzero <eigenvalues>. https://cs.uwaterloo.ca/~watrous/TQI/TQI.2.pdf[Watrous, The Theory of Quantum Information, chapter 2, section 2.1.2, pp. 62–64]{ref}
For a <projective measurement> with <orthogonal projections> $\Pi_a$, the <Born rule> is
$$
P(a)=\operatorname{Tr}(\rho\Pi_a)=\sum_i p_i\langle\psi_i|\Pi_a|\psi_i\rangle.
$$
This is the average of the component <pure state> <probabilities>. Positivity makes each outcome <probability> nonnegative, and $\sum_a\Pi_a=I$ makes them sum to one. The same <density operator> therefore predicts every measurement on the system without requiring a unique preparation history. https://www.preskill.caltech.edu/ph229/notes/chap2.pdf[Preskill, Physics 229 notes, chapter 2, section 2.3.1, equations (2.51)–(2.59)]{ref}
For a closed system, <Schrödinger picture> evolution under a <unitary operator> is
$$
\rho_S(t)=U(t)\rho_HU(t)^\dagger
$$
when $\rho_H=\rho_S(0)$ 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>. https://www.preskill.caltech.edu/ph229/notes/chap2.pdf[Preskill, Physics 229 notes, chapter 2, section 2.3.4, equation (2.79)]{ref}
A historical source for the probabilistic formulation of <quantum mechanics> is von Neumann's 1927 paper Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik. https://geodesic.mathdoc.fr/item/NGWGMP_1927__1927_59230/[von Neumann, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1927, pp. 245–272]{ref}