The Structure of Quantum Questions

Abstract: In classical physics, a single measurement can in principle reveal the state of a system. However, quantum theory permits numerous non-equivalent measurements on a physical system, each providing only limited information about the state. This set of various measurements on a quantum system indicates a rich internal structure. We illuminate this structure for both individual and composite systems by conceptualizing measurements as questions with a finite number of outcomes. We create a mathematical question structure to explore the underlying properties, employing the concept of information as a key tool representing our knowledge gained from asking these questions. We subsequently propose informational assumptions based on properties observed from measurements on qubits, generalizing these to higher dimensional systems. Our informational assumptions shape the correlations between subsystems, which are symbolized as classical logical gates. Interestingly, systems with prime number dimensions exhibit unique property: the logical gate can be expressed simply as a linear equation under modular arithmetic. We also identify structures in quantum theory that correspond to those in the structure of quantum questions. For instance, the questions determining the system correspond to generalized Pauli matrices, and the logical gate connecting questions in subsystems is directly related to the tensor product combining operators. Based on these correspondences, we present two equivalent scenarios regarding the evolution of systems and the change of information within both quantum questions and quantum mechanics.