Preparation spread, apparatus resolution and disturbance are different objects

Preparation spread. For an assigned sharp observable $(A,u_A)$ and a preparation $\rho$, the spread ${\mathrm{\Delta }}_{\rho }A$ of Preparation variance and spread of an assigned sharp observable is the standard deviation of the outcome law that the observable and probability postulate assigns to the pair $(\rho ,A)$: it is a property of the preparation and the assigned observable, computed from the Born law alone, and it exists only under the stated moment hypotheses. The inequalities of The Robertson relation for preparation spreads of assigned sharp observables concern these spreads and nothing else.

Apparatus resolution or error. A finite-resolution apparatus is modeled in this category by a declared instrument with effects and an update rule (Declared ideal measurements and instruments of the framework; clause (iv) of P2 and the ideal conditional measurement postulate P3 in the readings of Ideal conditional measurement postulate P3: the projective Lueders update and discarded outcomes). Whether a declared effect family approximates a target observable, and by how much, is a property of the chosen instrument: the postulates do not attach a universal error parameter to a sharp observable, and the effects alone do not determine the update (Outcome effects do not determine the conditional update, and general instruments need not be repeatable).

Disturbance. The state change produced by a measurement is likewise part of the declared instrument: for the framework's declared ideal sharp measurement the conditional and discarded-outcome updates are specified in Consequences of a declared ideal sharp measurement: repeatability and discarded ensembles, and they are idealizations, not claims about every apparatus. The framework's postulates assign no universal disturbance to "a measurement of position" as such.

No conflation. The Robertson relation bounds the product of two preparation spreads; it does not bound an apparatus error, it does not bound a disturbance, and it is not a statement that measuring one quantity "must" perturb another by a fixed amount. The heuristic accounts of measurement intrusion in the literature are accordingly used on this page at most as motivation: the exact statements are the preparation-spread results, and the identification of a concrete apparatus with a declared instrument is a tested identification with uncertainty under the preparation and apparatus postulate P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions).

No universal error-disturbance relation is recorded. This page and the measurement page contain no theorem of the form "error times disturbance is at least $\mathrm{\hslash }\mathrm{/}2$": such a statement would require declared error and disturbance quantifiers for an instrument and its own proof. What is recorded here is only the separation of the three objects and the conditional results actually proved on the two pages.

• Why the distinction is load-bearing. An experiment that reports a finite resolution, a detection efficiency or a background model is reporting apparatus properties; a preparation spread is a model quantity of the state and the assigned observable. Agreement of a measured histogram with a predicted distribution tests the combined preparation, apparatus and observable identification, not the inequality alone.
• Disturbance is model-dependent. Two declared instruments with the same outcome effects can have different updates and hence different disturbances, as Outcome effects do not determine the conditional update, and general instruments need not be repeatable records; a disturbance claim must therefore name its instrument.