Quantum Information is Physical
What do ultra-secure communications, a longstanding open problem in operator algebra and boson sampling have in common? These are all problems that are solved using quantum techniques. In this colloquium aimed at a general mathematical audience, we discuss how quantum information resembles more a physical object than a digital one, and how this translates to a method for unforgeable money and ultra-secure communications. We then link this to quantum interactive proofs, the study of which have recently led to the unravelling of a 50-year old mathematical puzzle called the Connes embedding problem. Finally, we present boson sampling as the first ever demonstration of a computational advantage of quantum computers over conventional ones.