BilQC | Walker Stern | Symmetric Monoidal Categories (III)

In a symmetric monoidal category, the set of maps from the monoidal unit to itself inherits additional
structure. These scalars in the category are inspired by, for instance, the isomorphism between linear maps
from $\mathbb{C}$ to itself and the field $\mathbb{C}$. However, the additive structure and the notion of traces which are key to
the computation of quantum probabilities require some development. In this talk, we describe biproducts
and distributivity in (symmetric monoidal) category, and the concomitant structures induced on scalars in
a symmetric monoidal category (Primary reference: [1, §5], secondary references: [2,§2.2.3]). We introduce
categorical definitions of traces and partial traces, the latter familiar in quantum information theory as
a procedure for obtaining information about a subsystem (Primary reference: [2, §4.6, §5.1], secondary
reference: [1, §2.1, §2.2], [3]).
References:
1. Abramsky, Samson, and Bob Coecke. Categorical quantum mechanics. Handbook of quantum logic
and quantum structures 2 (2009): 261-325.
2. Heunen, Chris, and Jamie Vicary. Categories for Quantum Theory: an introduction. Oxford University
Press, 2019.
3. Andre Joyal, Ross Street, and Dominic Verity. Traced Monoidal Categories. (doi:10.1017/S0305004100074338).