The state space of the framework: preparations, pure rays and density operators

(a) State space. Adopt the state postulate State and composition postulate P1: Hilbert spaces, preparations and composite systems for a declared system of the declared massive-matter sector, with its assigned nonzero separable complex Hilbert space $H$. The state space of the system in the model is the set $D(H)=\{\rho :\rho \text{ is a density operator on }H\}$ of The state space: density operators, pure and mixed states, and purity: the self-adjoint positive trace-class operators on $H$ of trace one. A state of the system is an element of $D(H)$; this is the geometry vocabulary for the general preparations of clause (iii) of the state postulate. The model's adopted assignment of a state to each preparation is not produced by this definition; it is the content of that postulate.

(b) Pure and mixed states. A state is pure when it is one of the operators $T_{u,u}$ with $u\in S(H)$, and mixed when it is not pure, as in The state space: density operators, pure and mixed states, and purity. By Rays, unit vectors and rank-one Hilbert projections the map $j:R(H)\to D(H),j([\psi ])=T_{\nu (\psi ),\nu (\psi )},$ is a bijection of the ray space onto the set of pure states; and by clause (ii) of the state postulate the pure preparations are the unit rays of $H$, so the model's pure preparations are exactly its pure states.

(c) State assigned to a state vector. For a nonzero vector $\psi \in H^{\times }$ the state assigned to $\psi$ is the normalized rank-one operator $\widehat{\rho }(\psi )=\frac{T_{\psi ,\psi }}{\mathrm{\parallel }\psi {\mathrm{\parallel }}^2}=T_{\nu (\psi ),\nu (\psi )}\in D(H),$ which by Rays, unit vectors and rank-one Hilbert projections depends only on the ray $[\psi ]$; clause (ii) of the state postulate assigns the same operator to the pure preparation $[\psi ]$. The zero vector is excluded here because $\mathrm{\parallel }\psi {\mathrm{\parallel }}^{-1}$ is defined only for $\psi \mathrm{\ne }0$; no state is assigned to it, in agreement with the state postulate.

(d) Mixtures and purity. A statistical mixture of preparations with weights is the trace-norm convergent sum of clause (iii) of the state postulate; its value is the corresponding convex combination of states. The purity of a state $\rho$ is $\mathrm{tr}({\rho }^2)$, and by Purity, spectral weights and the criterion for a pure state a state is pure exactly when its purity equals $1$.

(e) Status. Clauses (a)-(d) record the readings of the state postulate in the geometry vocabulary of this page and fix the terminology used by the later pages of the category. The cited mathematical suppliers of the ray-to-state correspondence and of the purity criterion assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), which is carried as a hypothesis of these readings. They adopt no assumption beyond the state postulate and that hypothesis, assert no dynamics, no probability law and no measurement model, and make no empirical claim: in particular, whether a given laboratory preparation is pure or mixed, and whether a given apparatus realizes a prescribed mixture, are physical identifications with uncertainty, governed by the preparation and apparatus assumptions of the framework, not consequences of this definition. Superselection restrictions and identical-particle symmetry sectors, where imposed, are separate assumptions by clause (v) of the state postulate and are not part of the state space fixed here.