Task-Dependent Predictive Sufficiency in a Switchable Three-Qubit System

This technical note investigates how much information about a present quantum state must be retained to predict future measurement outcomes when the allowed quantum operations can change. We study a three-qubit system consisting of a locally measured qubit coupled through switchable XY exchange interactions to two environmental qubits. Using established quantum observability methods, we determine the minimum number of real linear expectation coordinates required to predict terminal local measurements and joint probabilities involving retained intermediate measurement outcomes. Seven control menus yield exact predictive dimensions of 7, 11, 22, 22, 41, 54, and 63. Switching between exchange interactions requires 22 coordinates. Allowing unread local Z measurements does not increase this dimension, whereas retaining their outcomes raises it to 41. Permitting up to two retained local Pauli measurement outcomes makes the complete 63-dimensional traceless joint-state description necessary across the specified control schedules. The results are established using commutator closure and an independent exact finite-angle rational calculation. We additionally determine the minimum numbers of fixed product-Pauli measurement settings, obtaining 5, 5, 14, 14, 19, 22, and 27 for the respective control menus. Explicit pairs of physical quantum states demonstrate the limitations of smaller predictive descriptions. Frozen synthetic validation involving 16 reference states and 126 scheduled effects yields a maximum clean-shot probability error of 0.006981 and a simultaneous 95% absolute joint-probability error bound of 0.075776 under trusted readout. The accompanying materials provide mathematical derivations, computational verification scripts, exact algebraic certificates, archived simulation data, and reproducibility instructions. The contribution is a reproducible finite-model application of established quantum observability theory, clarifying how future control choices and retained measurement records determine which present-state information is sufficient for prediction. No experimental hardware results, novel fundamental dynamics, or claims of retrocausality are presented.