Quaternionic Quantum Dynamics on Complex Hilbert Spaces

Abstract

We consider a quaternionic quantum formalism for the description of quantum states and quantum dynamics. We prove that generalized quantum measurements on physical systems in quaternionic quantum theory can be simulated by usual quantum measurements with positive operator valued measures on complex Hilbert spaces. Furthermore, we prove that quaternionic quantum channels can be simulated by completely positive trace preserving maps on complex matrices. These novel results map all quaternionic quantum processes to algorithms in usual quantum information theory.

Virial theorem and generalized momentum in quaternionic quantum mechanics

“…In fact, these works argue that real Hilbert spaces must be ruled out of quantum mechanics, while other works point out the fundamental role of complex numbers in quantum mechanics [40,41]. Furthermore, the formulation of anti-hermitian HQM in a quaternic Hilbert space is also criticized [42,43]. We understand that our results are in agreement with these previous results.…”

An Introduction to Operational Quantum Dynamics

“…In fact, they could also be used to describe a more general linear theory, such as one based on quaternionic vector spaces [37], as well as other generalised statistical theories [7,54]. It is worth mentioning tensor network calculus as a helpful tool for graphically representing quantum maps (and other linear algebraic objects).…”

Locality for quantum systems on graphs depends on the number field

“…In the study of correlations, quantum mechanics is "sandwiched" between classical mechanics and general probabilistic theories (see [7] for a recent treatment of this point in the graph theoretic framework). When we consider number fields, we seem to face a similar situation: while the choice of number field does not affect the computational power of the theory [15,23], there are cogent arguments about the inadequacy of real numbers (e.g., parameter counting for bipartite mixed states, continuity of time, the quantum de Finetti theorem, the need of superselection rules, etc. [1]).…”

“…In this respect, the consequences of our theorem are significant when Z-locality is not reflected by the use of ad hoc constructions. More generally, Z-locality is not an obstacle from a point of view embracing specific algorithmic applications, because there are methods to translate between different number fields [15,23].…”