Universal Properties in Quantum Theory

Abstract

We argue that notions in quantum theory should have universal properties in the sense of category theory. We consider the completely positive trace preserving (CPTP) maps, the basic notion of quantum channel. Physically, quantum channels are derived from pure quantum theory by allowing discarding. We phrase this in category theoretic terms by showing that the category of CPTP maps is the universal monoidal category with a terminal unit that has a functor from the category of isometries. In other words, the CPTP maps are the affine reflection of the isometries.

In Proceedings QPL 2018, arXiv:1901.09476

References in corpus (7)

• Probabilistic theories with purification
• Interacting Hopf Algebras
• Axiomatizing complete positivity
• Paschke Dilations
• Operational Theories of Physics as Categories
• Purity through Factorisation
• QPEL: Quantum Program and Effect Language

Cited by in corpus (10)

• Dagger and Dilation in the Category of Von Neumann algebras
• Bennett and Stinespring, Together at Last
• Universal Properties of Partial Quantum Maps
• Quantum de Finetti Theorems as Categorical Limits, and Limits of State Spaces of C*-algebras
• Compositional Semantics for Probabilistic Programs with Exact Conditioning
• Proceedings 18th International Conference on Quantum Physics and Logic
• Quantum channels as a categorical completion
• Quantum Information Effects
• Compositional Reversible Computation
• A decompositional framework for process theories in spacetime