Optimizing over interactions

Our goal will be to optimize over all behaviors for Bob, maximizing the probability with which he causes Alice to output the measurement outcome a. (Later we will consider a variant of this scenario where Bob also produces a measurement outcome, but for now our focus will be on the case where only Alice produces a measurement outcome.) In the remainder of this lecture we will discuss how such an optimization problem can be expressed as a semidefinite program. There are, in fact, at least two ways to represent such an optimization as a semidefinite program, and we will discuss a second way to do this later in the course.