= Quantum state
{title2=$\rho$}
{wiki}
A <quantum state> describes a preparation through its predicted <quantum measurement> <probabilities>. In the <Hilbert space> formulation, a <pure state> is a <quantum state ray>: a normalized <vector> representative $|\psi\rangle$ satisfies $\langle\psi|\psi\rangle=1$, and representatives differing by a <global phase> describe the same <quantum state>. <Foundations I: States and Ensembles by John Preskill>[Preskill, Foundations I: States and Ensembles, §2.1, printed pp. 1–3, equation (2.1) and the definition of a ray]{ref}
Both <pure states> and <mixed states> are represented by a <density operator> on the chosen complex <Hilbert space>:
$$
\rho=\rho^\dagger,\qquad \langle v|\rho|v\rangle\geq0\text{ for every }|v\rangle,\qquad \operatorname{Tr}\rho=1.
$$
Thus $\rho$ is <Hermitian>, <positive semidefinite> and of unit <trace>; in infinite <dimension> it must also be a <trace-class operator>. For a normalized <pure state>, $\rho=|\psi\rangle\langle\psi|$ is a rank-one <orthogonal projection>. A <mixed state> has no single such <vector> representative. Its <density matrix> is the <matrix> of $\rho$ in a chosen <orthonormal basis>. <The Theory of Quantum Information by John Watrous/Mathematical preliminaries>[Watrous, The Theory of Quantum Information, §1.1.2, printed pp. 17–18, equations (1.94)–(1.97), Hermitian, positive semidefinite and density operators; finite-dimensional setting]{ref} For a discrete <POVM> with effects $E_a\geq0$ and $\sum_aE_a=I$, the <Born rule> reads
$$
P(a)=\operatorname{Tr}(\rho E_a),\qquad
P(a)=\langle\psi|E_a|\psi\rangle\text{ for a pure state}.
$$
These constraints and <Born rule> probabilities describe arbitrary finite-dimensional preparations. <The Theory of Quantum Information by John Watrous/Basic notions of quantum information>[Watrous, The Theory of Quantum Information, §2.1.2, printed pp. 62–64, Definition 2.5 and pure states; §2.3, Definition 2.34, measurement probabilities]{ref}
Randomly choosing normalized <pure states> with <probabilities> $w_j$ and forgetting the label gives
$$
\rho=\sum_j w_j|\psi_j\rangle\langle\psi_j|,\qquad w_j\geq0,\quad\sum_jw_j=1.
$$
This <quantum state ensemble> averages <probabilities>; a <quantum superposition> adds <probability amplitudes>. Several ensembles can give the same <density operator>. <Quantum entanglement> can make a <reduced density matrix> mixed while the joint <quantum state> is pure. <Foundations I: States and Ensembles by John Preskill>[Preskill, Foundations I: States and Ensembles, §2.3.1, printed pp. 14–18, equations (2.51)–(2.61), partial trace and ensemble interpretation]{ref}
The preparation, the choice of <basis>, and the measurement are separate ingredients. A <wavefunction> $\psi(x)=\langle x|\psi\rangle$ is the position representation of a <pure state>; a general <mixed state> requires a <density operator>. For a closed <quantum system>, a <unitary operator> evolves it as $\rho\mapsto U\rho U^\dagger$, preserving its <eigenvalues> and <purity of a density operator>. <Foundations I: States and Ensembles by John Preskill>[Preskill, Foundations I: States and Ensembles, §2.3.4, printed pp. 22–23, equation (2.79)]{ref} The <Bloch sphere> and <Bloch ball> give the <qubit> geometry; <quantum state tomography> connects repeated preparations to an estimated <density matrix>; <projective measurement> treats the conditional state after an outcome.
Back to article page
