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 experimental requirements for observing quantum phenomena –specifically, preparing physical systems in coherent states under strict conditions, such as ultralow temperatures or high fields– effectively constrain the number of independent variables to one, thereby enforcing single-variable behavior. This first-principles, information-theoretic derivation establishes that quantum theory fundamentally describes the physics of single-variable systems and provides a concrete realization of Wheeler’s “it from bit” idea.