Variations on the Choi–Jamiołkowski isomorphism

Abstract

The Choi-Jamiolkowski isomorphism is an essential component in every quantum information theorist’s toolkit: it allows to identify linear maps between two quantum systems with linear operators on the composite system. Here, we analyse this widely used gadget from a new perspective. Namely, we explicitly distinguish between its kinematical and dynamical properties, that is, we study the isomorphism on two different levels: Jordan algebras and the different C∗-algebras they arise from, which are distinguished by their order of composition.
A number of important and novel insights stem from our analysis. We find that Choi’s theorem, which asserts that Choi’s version of the isomorphism [M.-D. Choi, Lin. Alg. Appl., 10, 285 (1975)] further maps the positive cone of completely positive linear maps (such as quantum channels) to the cone of positive linear operators (such as quantum states) on the composite system, crucially
depends on the dynamical (compositional) structure in C∗-algebras. This in turn lies at the heart of the mismatch between the basis-dependence of Choi’s version of the isomorphism, and the basis-independent version by Jamiolkowski [A. Jamiolkowski, Rep. Math. Phys., 3, 275 (1972)]. Here, we overcome this subtle but pervasive issue in a number of ways: first, we prove a version
of Choi’s theorem for Jamiolkowski’s isomorphism, second, we define a basis-independent variant of Choi’s isomorphism and, third, by making explicit the dynamical distinction between Jordan and C∗-algebras, we combine the different variants of the isomorphism into a unified description, that subsumes their individual features. We also embed and interpret our results in the graphical calculus of categorical quantum mechanics.