2015/01/23 by Stefano Gogioso, Gogioso, Stefano · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Category Theory (math.CT) #FOS: Mathematics #FOS: Physical sciences #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #math.CT #quant-ph
paper · pdf · doi:10.48550/arxiv.1501.06489
Added a section connecting Stone's theorem on 1-parameter unitary groups to the dynamics/Hamiltonians duality presented in the work. Colours adapted to be discernible by protanopes, deuteranopes and tritanopes
openalex publication_date 2015/01/23 · arxiv created 2015/02/25 · arxiv updated 2015/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Applying ideas from monadic dynamics to the well-established framework of categorical quantum mechanics, we provide a novel toolbox for the simulation of finite-dimensional quantum dynamics. We use strongly complementary structures to give a graphical characterisation of quantum clocks, their action on systems and the relevant energy observables, and we proceed to formalise the connection between unitary dynamics and projection-valued spectra. We identify the Weyl canonical commutation relations in the axioms of strong complementarity, and conclude the existence of a dual pair of time/energy observables for finite-dimensional quantum clocks, with the relevant uncertainty principle given by mutual unbias of the corresponding orthonormal bases. We show that Schrödinger's equation can be abstractly formulated as characterising the Fourier transforms of certain Eilenberg-Moore morphisms from a quantum clock to a quantum dynamical system, and we use this to obtain a generalised version of the Feynman's clock construction. We tackle the issue of synchronism of clocks and systems, prove conservation of total energy and give conditions for the existence of an internal time observable for a quantum dynamical system. Finally, we identify our treatment as part of a more general theory of simulated symmetries of quantum systems (of which our clock actions are a special case) and their conservation laws (of which energy is a special case).