State and composition postulate P1: Hilbert spaces, preparations and composite systems

Adopt as physical assumptions of the framework, for systems in the declared massive-matter sector, the following statements about states and their composition. Assume throughout the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), which is the choice hypothesis of the mathematical suppliers cited below.

(i) State space. To each physical system of the sector the model assigns a complex Hilbert space $H$ that is nonzero and separable: a complex Hilbert space in the sense of Hilbert space which admits a sequence whose range is dense in $H$, so that the hypothesis of A Hilbert space with a dense sequence has a finite or countable orthonormal basis holds and $H$ has a finite or countable orthonormal basis. The assignment of a Hilbert space to a physical system is a model assumption; the linear structure used to combine descriptions mathematically is not an empirical claim. The physical system to which $H$ is assigned, and the sector in which the assignment is made, are declared with it.

(ii) Pure preparations. A pure preparation of the system is a unit ray $[\psi ]=\{z\psi :z\in \mathbb{C}\setminus \{0\}\}$ of a vector $\psi \in H$ with $\mathrm{\parallel }\psi \mathrm{\parallel }=1$; that is, two nonzero vectors describe the same pure preparation exactly when they differ by a nonzero complex factor. The zero vector is not a state and no preparation is assigned to it: the normalization $\mathrm{\parallel }\psi \mathrm{\parallel }=1$ is part of the assignment. The ray allows arbitrary nonzero rescaling; between unit representatives the factor has modulus one and is a global phase, which has no observable consequence (P2). The rank-one operator $T_{\psi ,\psi }$ of Rank-one operator calculus is the density operator associated with the ray of $\psi$.

(iii) General preparations and mixing. A general preparation of the system is a density operator: a self-adjoint positive trace-class operator $\rho$ on $H$ with $\mathrm{tr}\rho =1$ (Trace class operator, Trace of a trace class operator, Density operators and scalar spectral probability clause (a)). Pure preparations are the special case $\rho =T_{\psi ,\psi }$ with $\mathrm{\parallel }\psi \mathrm{\parallel }=1$. A statistical mixture of preparations ${\rho }_n$ with weights $w_n\ge 0$, ${\sum }_n{w}_n=1$, is the trace-norm convergent sum $\rho ={\sum }_n{w}_n{\rho }_n$ as follows directly in the trace-class Banach space (Trace class is a two sided Banach operator ideal): each density has trace norm one, so ${\sum }_n\mathrm{\parallel }{w}_n{\rho }_n{\mathrm{\parallel }}_1=1$ gives trace-norm convergence. Every partial sum is positive; operator-norm convergence preserves its nonnegative quadratic forms, and continuity of trace gives $\mathrm{tr}\rho ={\sum }_n{w}_n=1$ (Trace is absolutely convergent and basis independent). Thus the limit is again a density operator, without requiring the components to be orthogonal pure states. The weights are a classical preparation uncertainty and are not amplitudes. Distinct weighted ensembles of pure states may define the same $\rho$.

(iv) Composition. Two independent distinguishable subsystems with state spaces $H_A$ and $H_B$ are composed by the completed Hilbert tensor product $H_A\widehat{\otimes }{H}_B$ of Hilbert tensor construction: inner product, completion and product bases. Product preparations ${\psi }_A\otimes {\psi }_B$ are simple tensors and are the special case in which each subsystem has its own specified preparation. Their finite linear span is dense in $H_A\widehat{\otimes }{H}_B$ by the construction; a pure composite preparation may be any unit ray in this completion, including a norm limit of finite linear combinations of product vectors, and need not factor. General composite preparations are density operators on this space and include the statistical mixtures of clause (iii). A quantity of subsystem $A$ alone is represented on the composite by an operator of the form $A\otimes I_B$ and a quantity of $B$ alone by $I_A\otimes B$, in the sense of Bounded tensor maps: extension, norm estimate, adjoint and commutation (bounded case; unbounded local operators are treated where they occur). The existence of the completed tensor product is mathematics; the assignment of that product to a composite physical system is the physical assumption here.

(v) Restrictions that are separate assumptions. Restrictions of the state space by superselection rules, identical-particle symmetry sectors, and the replacement of the tensor product by more general algebraic structures are not consequences of the clauses above: where such a restriction is used it is adopted separately, and the identical-species sectors are the subject of the postulate P6 of this page.

(vi) Status. Clauses (i)-(v) are adopted assumptions of the model within its declared sector. They are not derived from experiment, and they do not assert that a physical system is exactly described by such a Hilbert space; no numerical measurement is recorded. The compatibility of the framework with observation is a question about fully specified models and reported data, in the sense of P7.

• Sector. The assignment is made for systems of nonrelativistic massive matter with a fixed finite particle number, each system treated once and for all with a declared state space; quantum field theory, gravity and systems whose particle number is not fixed are outside this postulate.
• No derivation. The state space, the ray equivalence and the tensor composition are adopted. In particular the tensor composition is not a theorem about physical composition, and the nonuniqueness of mixtures recorded in clause (iii) is a mathematical property of the adopted objects, not an empirical claim.
• Choice. The cited mathematical suppliers assume Countable Choice; all consequences of this postulate that use them inherit that assumption, and any choice-free special case must be proved separately where it is claimed.
• Composition is not a measurement postulate. Clause (iv) fixes how descriptions of independent subsystems are combined; it asserts nothing about interactions, correlations or measurements of the composite.