General quantum theory—unification of classical and modal quantum theories

Abstract

Inspired by classical (‘actual’) quantum theory over 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 København interpretation—over general division rings with involution, in which the inner product ‘is’ a (σ, 1)-Hermitian form φ. 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 and in some MQTs, hold for any general quantum theory. Our approach is totally different (and more general) than Hardy’s axiomatic approach [], and generalized probability theories. 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 U(V, φ), and which carries the geometry of a so-called polar space. This object cannot be seen in classical quantum theory over , but it is present, for instance, in all known MQTs. 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 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 . 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. In this sense, these singular objects are always virtually around in abundance.