The state space: density operators, pure and mixed states, and purity

Density operators. Let $H\mathrm{\ne }\{0\}$ be a complex Hilbert space. A density operator on $H$ is a self-adjoint positive trace-class operator $\rho$ with $\mathrm{tr}\rho =1$, in the sense of Density operators and scalar spectral probability clause (a) (Trace class operator, Trace of a trace class operator, Self-adjoint, positive, unitary and normal operators). The state space of $H$ is the set of all density operators on $H$ and is denoted $D(H):=\{\rho :\rho \text{ is a density operator on }H\}\mathrm{.}$ An element of $D(H)$ is called a state of $H$ in the mathematical sense fixed here.

Pure states. A state $\rho \in D(H)$ is pure when there is a unit vector $u\in S(H)$ with $\rho =T_{u,u},T_{u,u}x=⟨x,u⟩u(x\in H),$ where $T_{u,u}$ is the rank-one operator of Rank-one operator calculus and $S(H)$ is the unit sphere of State vectors, unit vectors, rays, normalization and phases of a Hilbert space. By Rays, unit vectors and rank-one Hilbert projections the pure states are exactly the Hilbert orthogonal projections of $H$ with one-dimensional range, and the map $j:R(H)⟶D(H),j([\psi ]):=T_{\nu (\psi ),\nu (\psi )},$ on the ray space $R(H)$, with $\nu$ the normalization map, is well defined and injective with image the set of pure states. A state is mixed when it is not pure. For a nonzero vector $\psi$ the normalized rank-one operator of $\psi$ is $\widehat{\rho }(\psi ):=\frac{T_{\psi ,\psi }}{\mathrm{\parallel }\psi {\mathrm{\parallel }}^2}=\frac{T_{\psi ,\psi }}{\mathrm{tr}(T_{\psi ,\psi })},$ which equals $T_{\nu (\psi ),\nu (\psi )}$; thus $\widehat{\rho }(\psi )$ is the pure state of the ray $[\psi ]$, and the vector $\psi$ enters only through its ray.

Purity. For $\rho \in D(H)$ the operator ${\rho }^2$ is self-adjoint, positive and trace class: it is self-adjoint by the adjoint algebra of The Hilbert-space adjoint of a bounded operator, positive because $⟨{\rho }^{2}x,x⟩=⟨\rho x,\rho x⟩\ge 0$, and trace class because $\rho$ is a bounded trace-class operator and the trace-class operators form a two-sided ideal (Trace class is a two sided Banach operator ideal clauses 1 and 2). The purity of $\rho$ is the real number $\mathcal{P}(\rho ):=\mathrm{tr}({\rho }^2)\in \mathbb{R}\mathrm{.}$ The numerical range of the purity and the equivalence of purity with purity of the state are established by the purity criterion of this page; this definition records the object only.

Scope. This definition fixes the mathematical state space of a Hilbert space together with the pure states, the mixed states, the normalized rank-one operator of a nonzero vector and the purity functional. The cited rank-one calculus and the density-operator expansion assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), and the definition is read under that assumption. It asserts no dynamics, no measurement, no probability law and no physical interpretation, and it introduces no new assumptions beyond the cited mathematics and that choice hypothesis. In particular, whether a given physical preparation is described by a pure or a mixed state is a question about the model's assignments, not about this definition.