Abstract: In quantum computing a distinction is made between separable and entangled states on the tensor product BxB of a matrix algebra B over the complex numbers. Among the entangled states are the maximally entangled pure states: these are the pure states f on BxB that satisfy f(axI) = tr(a) = f(Ixa), for all matrices a in the algebra B, where tr is the normalized tracial state on B. Any state (pure or not) that satisfies the equation above is called a marginal tracial state. We will look at some results on the classification of the extremal marginal tracial states on BxB. This is joint work with Professor S. Sakai of Sendai, Japan.