Quantum information is a dynamic and exciting field at the interface of mathematics, physics, computer science, and engineering. In this talk, I will survey some ways that the multidisciplinary roots of quantum information have yielded fresh insights and problems of genuine interest to mathematics, both pure and applied. First I will describe how quantum information has led to the development of powerful large-deviation inequalities for matrix-valued random variables. I will show how these methods can be combined with techniques from convex geometry to develop a noncommutative analog of compressed sensing, and describe some applications of this idea. Then I will show how problems arising naturally in quantum information even overlap with number theory. The long-standing problem of finding explicit generators for abelian extensions of number fields is wide open even for the case of real quadratic fields. I will discuss how a geometric problem arising in quantum information leads to a precise number-theoretic conjecture about finding canonical unit generators in infinite towers of ray class fields above every real quadratic field. The emphasis will be on the mathematics (not the physics!) and no prior knowledge of quantum mechanics will be assumed.