Understanding Quantum Instruments Through the Analysis of $C^{\ast }$-Convexity and Their Marginals

Quantum instruments are mathematical devices introduced to describe the conditional state change during a quantum process. They are completely positive map valued measures on measurable spaces. We may also view them as non-commutative analogues of joint probability measures. We analyze the $C^{\ast }$-convexity structure of spaces of quantum instruments. A complete description of the $C^{\ast }$-extreme instruments in finite dimensions has been established. Further, the implications of $C^{\ast }$-extremity between quantum instruments and their marginals has been explored.