The declared one-body reduced operator and the one-body readout of a sector preparation

(a) Adopted readings. Adopt the state and composition postulate State and composition postulate P1: Hilbert spaces, preparations and composite systems (P1) and the observable and probability postulate Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes (P2) in the state reading of The state space of the framework: preparations, pure rays and density operators and the observable reading of Assigned sharp observables and the outcome law of the framework. Work in the finite-mode realization of a declared species sector of Declared species sectors and their preparations: the framework's reading of P6: the one-particle space is a finite-dimensional complex Hilbert space $K$ of dimension $d\ge 1$ with its declared orthonormal modes $(f_1,\dots ,f_d)$, and the $N$-particle sector space is ${\mathfrak{H}}_N^{\sigma }=\mathrm{ran}{P}_{\sigma }^{(N)}\subseteq K^{\otimes N}$ with the declared sector preparations of The declared finite-mode model and its occupation readouts. Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

(b) The one-body reduced operator. Let $\rho$ be a declared sector preparation, that is, a density operator on ${\mathfrak{H}}_N^{\sigma }$ (Density operators and scalar spectral probability clause (a)). The one-body reduced preparation of $\rho$ is the density operator ${\gamma }_{\rho }$ on $K$ of clause (e) of The one-body reduction trace law of the finite-mode permutation sectors (with $H:=K$, $n:=N$ and the same sector projections). Explicitly, for the sector inclusion $j$ put $\tilde{\rho }=j\rho j^{\ast }$ on $K^{\otimes N}$, and for the regrouping unitary $J_N:K^{\otimes N}\to K\widehat{\otimes }{K}^{\otimes (N-1)}$ put ${\gamma }_{\rho }={\mathrm{Tr}}_{K^{\otimes (N-1)}}(J_N\tilde{\rho }{J}_N^{-1})$. Here $K^{\otimes 0}=\mathbb{C}$, so $N=1$ gives ${\gamma }_{\rho }=\rho$. A zero sector has no preparation and makes this clause vacuous. By that lemma ${\gamma }_{\rho }$ is a density operator on $K$ independent of which factor is traced, and for every $O\in \mathcal{B}(K)$ the pairings are related by $\mathrm{tr}(\tilde{\rho }\mathrm{\Sigma }(O))=N\mathrm{tr}\left({\gamma }_{\rho }O\right),$ where $\tilde{\rho }$ is the zero extension of $\rho$ and $\mathrm{\Sigma }(O)$ is the sum over the $N$ labelled slots of the slot operator $O^{(i)}$.

(c) The one-body readout. For a declared bounded self-adjoint $O\in \mathcal{B}(K)$ with a declared unit $u_O$ in the sense of Dimensions, SI unit conventions and quantity types of the framework, the model reads the sector operator $A_O:=\mathrm{\Sigma }(O){\mathrm{\mid }}_{{\mathfrak{H}}_N^{\sigma }}=P_{\sigma }^{(N)}\mathrm{\Sigma }(O)P_{\sigma }^{(N)}{\mathrm{\mid }}_{{\mathfrak{H}}_N^{\sigma }}$ as an assigned sharp observable $(A_O,u_O)$ in the sense of Assigned sharp observables and the outcome law of the framework clause (a): it is bounded and self-adjoint on the nonzero sector space and its outcome law in a declared sector preparation $\rho$ is the law ${\mu }_{\rho }^{A_O}(B)=\mathrm{tr}(\rho E_{A_O}(B))$ of clause (b) of that definition. The one-body readout of $O$ in $\rho$ is the mean of that law when it exists.

(d) Scope. The reduced preparation ${\gamma }_{\rho }$ is a mathematical partial trace of the sector state; it is not asserted to be the state of any individual particle. The reduction need not determine the sector preparation in higher-particle sectors; no universal noninjectivity is asserted. For $N=1$ it is exactly ${\gamma }_{\rho }=\rho$ by clause (b), and a one-dimensional sector has only one density preparation. The definition asserts no separability or factorization of $\rho$, no result for the spatial (infinite-dimensional) realization, whose reduction would require the separable partial trace, and no measurement of individual particle labels, which are mathematical factor labels by P6 clause (i). Whether a declared laboratory preparation realizes a sector preparation $\rho$ and a quantity realizes the pair $(A_O,u_O)$ are tested identifications with uncertainty under P7, not consequences of this definition.

• Normalization. The one-body reduced preparation is normalized as a probability state, $\mathrm{tr}{\gamma }_{\rho }=1$, while the mean of the sum $A_O$ carries the factor $N$; the sum $\mathrm{\Sigma }(O)$ is a many-body quantity, not a single-particle quantity, and its mean is $N$ times the single-particle pairing with ${\gamma }_{\rho }$.
• Choice of retained factor. Lemma clause (e) shows that reducing over any $N-1$ of the labelled factors gives the same operator on $K$; this is the precise sense in which the reduction is label independent inside the model.