General Quantum Theory

TL;DR

This paper introduces a unifying framework called General Quantum Theory over division rings, encompassing classical quantum mechanics and modal quantum theories, and explores its mathematical structure, invariants, and potential for new quantum coding schemes.

Contribution

It generalizes quantum theory to division rings with involution, unifies known approaches, and introduces the quantum kernel concept for new coding schemes and geometric insights.

Findings

Many quantum phenomena hold in General Quantum Theories.

The quantum kernel relates to polar space geometry.

Born's rule applies in all such theories.

Abstract

Inspired by classical ("actual") Quantum Theory over $\mathbb{C}$ and Modal Quantum Theory (MQT), which is a model of Quantum Theory over certain finite fields, we introduce General Quantum Theory as a Quantum Theory -- in the K{\o}benhavn interpretation -- over general division rings with involution, in which the inner product "is" a $(\sigma,1)$-Hermitian form $\varphi$. This unites all known such approaches in one and the same theory, and we show that many of the known results such as no-cloning, no-deleting, quantum teleportation and super-dense quantum coding, which are known in classical Quantum Theory over $\mathbb{C}$ and in some MQTs, hold for any General Quantum Theory. On the other hand, in many General Quantum Theories, a geometrical object which we call "quantum kernel" arises, which is invariant under the unitary group $\mathbf{U}(V,\varphi)$, and which carries the geometry of a so-called polar space. We use this object to construct new quantum (teleportation) coding schemes, which mix quantum theory with the geometry of the quantum kernel (and the action of the unitary group). We also show that in characteristic $0$, every General Quantum Theory over an algebraically closed field behaves like classical Quantum Theory over $\mathbb{C}$ at many levels, and that all such theories share one model, which we pin down as the "minimal model," which is countable and defined over $\overline{\mathbb{Q}}$. Moreover, to make the analogy with classical Quantum Theory even more striking, we show that Born's rule holds in any such theory. So all such theories are not modal at all. Finally, we obtain an extension theory for General Quantum Theories in characteristic $0$ which allows one to extend any such theory over algebraically closed fields (such as classical complex Quantum Theory) to larger theories in which a quantum kernel is present.