Quantum Theory from Probability Conservation

The standard formalism of quantum theory is derived by analyzing the behavior of single-variable physical systems. These systems, which have a minimal information capacity of only one piece of information, exhibit indeterministic behavior under independent measurements but can be described probabilistically for dependent measurements. By enforcing the principle of probability conservation in the transformations of outcome probabilities across various measurement scenarios, we derive the core components of standard quantum theory, including the Born rule, the Hilbert space structure, and the Schrödinger equation. Furthermore, we demonstrate that the requirements for conducting quantum experiments –specifically, preparing physical systems in coherent states– effectively reduce the number of independent variables to one, thereby transforming these systems into single-variable ones in practice. This completes our first-principles, information-theoretic derivation of quantum theory as the physics of single-variable physical systems.