Preparation variance and spread of an assigned sharp observable

(a) Setting. Adopt the state and composition postulate P1 and the observable and probability postulate P2 in the readings of The state space of the framework: preparations, pure rays and density operators and Assigned sharp observables and the outcome law of the framework (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes), and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Let $(A,u_A)$ be an assigned sharp observable with declared unit $u_A$ and outcome law ${\mu }_{\rho }^A$ in a preparation $\rho$, in the sense of Assigned sharp observables and the outcome law of the framework clause (b).

(b) Preparation variance. When the mean $m(\rho ):={\int }_{\mathbb{R}}\lambda d{\mu }_{\rho }^A(\lambda )$ and the second moment ${\int }_{\mathbb{R}}{\lambda }^{2}d{\mu }_{\rho }^A(\lambda )$ of the outcome law are finite, the preparation variance of $(A,u_A)$ in $\rho$ is ${\mathrm{Var}}_{\rho }(A):={\int }_{\mathbb{R}}(\lambda -m(\rho ){)}^{2}d{\mu }_{\rho }^A(\lambda ),$ read in the square $u_A^2$ of the declared unit, and the preparation spread is ${\mathrm{\Delta }}_{\rho }A:={\mathrm{Var}}_{\rho }(A{)}^{1\mathrm{/}2}\ge 0$, read in the unit $u_A$. When either quantity is not finite the variance and the spread of the pair in $\rho$ are not defined; the definition does not extend them by $+\mathrm{\infty }$.

(c) Values and hypotheses. The variance and its hypotheses are exactly those of the moment theorem: for a pure preparation of a unit vector $\psi \in D(A)$ the spread is ${\mathrm{\Delta }}_{\psi }A=\mathrm{\parallel }(A-⟨A\psi ,\psi ⟩I)\psi \mathrm{\parallel }$, equal to $(⟨A^2\psi ,\psi ⟩-⟨A\psi ,\psi {⟩}^2{)}^{1\mathrm{/}2}$ when $\psi \in D(A^2)$; for a preparation $\rho$ the identities and existence criteria are those of Mean, second moment and variance of an assigned observable in a preparation and Mean, second moment and variance of the scalar spectral law. In particular, for an orthonormal density expansion $\rho ={\sum }_n{w}_n{T}_{u_n,u_n}$, the mean is finite exactly when ${\sum }_{n:w_n>0}{w}_n{\int }_{\mathbb{R}}\mathrm{\mid }\lambda \mathrm{\mid }d{E}_{A,u_n}(\lambda )<\mathrm{\infty },$ by Mean, second moment and variance of the scalar spectral law clause (d); zero-weight indices are omitted before evaluating their component integrals. For a pure unit vector the corresponding criterion is membership in $D(\mathrm{\mid }A{\mathrm{\mid }}^{1\mathrm{/}2})$ by clause (b). Componentwise domain membership alone does not imply convergence of the weighted sum, and finite moments are not automatic.

(d) Products of spreads. For two assigned sharp observables $(A,u_A)$ and $(B,u_B)$ in a common preparation $\rho$ in which both spreads are defined, the product ${\mathrm{\Delta }}_{\rho }A{\mathrm{\Delta }}_{\rho }B$ is a nonnegative real number read in the product unit $u_A{u}_B$; for a pure preparation of a unit vector in $D(A)\cap D(B)$ it is $\mathrm{\parallel }(A-a)\psi \mathrm{\parallel }\mathrm{\parallel }(B-b)\psi \mathrm{\parallel }$ with the real means $a=⟨A\psi ,\psi ⟩$ and $b=⟨B\psi ,\psi ⟩$.

(e) Reading and scope. The spread ${\mathrm{\Delta }}_{\rho }A$ is a property of the pair (preparation, assigned observable): it is the standard deviation of the outcome law that P2 assigns. It is not an apparatus resolution, not a measurement error and not a disturbance of the preparation; those are further objects that require a declared instrument or apparatus model and are not introduced here. Clause (b) makes no claim that a preparation realizes a specific laboratory state, that a laboratory procedure measures $A$ with resolution $u_A$, or that the spread is observed in any experiment; the identification of an apparatus with the pair $(A,u_A)$ is a tested identification under the preparation and apparatus postulate P7. Dimensions, SI unit conventions and quantity types of the framework fixes the units in which the displays are read.

• Vanishing spread and eigenstates. For a pure preparation of a unit vector $\psi \in D(A)$, clause (c) shows ${\mathrm{\Delta }}_{\psi }A=0$ exactly when $A\psi =⟨A\psi ,\psi ⟩\psi$, that is exactly when $\psi$ is an eigenvector of $A$; this is the framework's reading of the textbook statement that the uncertainty of an observable vanishes in an eigenstate. The identification of a laboratory state with such a preparation is again a tested identification, not a consequence of the definition.
• The product is not a joint distribution. Clause (d) forms a product of two spreads of the same preparation; it does not introduce a joint outcome law of $A$ and $B$, and the product can be defined even when no joint sharp observable of the pair exists. Joint sharp observables are a separate notion with its own criterion.