Ross Duncan (Brussels, Belgium), Interacting Observables in Categorical Quantum Mechanics


  • Feb. 20, 2012, 14:00 - 15:00


One of the most shocking features of quantum mechanics is the possibility that quantum observables may be incompatible: when one property is well-defined, another cannot have a definite value assigned to it. This is most clearly seen in the case of complementary observables, such as the spin along the X and Z axes. Perfect knowledge of the X spin implies complete ignorance of the Z spin. Historically complementarity has been viewed as a negative property, a failure of some classical behaviour; however, from a different point of view a positive attitude is possible.

Non-degenerate quantum measurements are equivalent to coalgebras whose action is to copy and delete the eigenstates associated with the observable; in this way we may view each observable as embedding a classical data type in the quantum world. When two complementary observables are considered together, they jointly form a structure closely related to a Hopf algebra. These interacting algebras give rise to much of the equational structure exploited in quantum computing. Further, all of the axiomatisation can be performed in an abstract categorical setting, and a beautiful, highly legible graphical notation results.

I will introduce the basic ideas of categorical quantum mechanics, and the the various algebra structures and demonstrate some applications in concerning quantum circuits and measurement-based quantum computing.