John von Neumann

A Hungarian-born mathematician of astonishing speed, who contributed to set theory, game theory, economics, computing and nuclear weapons design.

His 1932 book Mathematische Grundlagen der Quantenmechanik presented quantum theory as geometry in an abstract space of states (Hilbert space), unifying Heisenberg's and Schrödinger's versions. He introduced the density matrix, which handles mixed and uncertain states, and the entropy that bears his name.

He also analysed measurement carefully. A quantum state evolves smoothly under the Schrödinger equation until someone measures it, when it jumps. Von Neumann showed that the cut between the quantum system and the observer can be placed almost anywhere without changing predictions, which is what makes the 'measurement problem' so awkward.

Von Neumann's framework is still the standard textbook form of the theory, and it is the mathematical home of quantum information. His 1932 argument that hidden variables are impossible was widely believed for decades, until Grete Hermann (1935) and John Bell (1966) pointed out that it ruled out far less than people thought.