Maximum nonlocality and minimum uncertainty using magic states

Abstract

We prove that magic states from the Clifford hierarchy give optimal solutions for tasks involving nonlocality and entropic uncertainty with respect to Pauli measurements. For both the nonlocality and uncertainty tasks, stabilizer states are the worst possible pure states so our solutions have an operational interpretation as being highly non-stabilizer. The optimal strategy for a qudit version of the Clauser-Horne-Shimony-Holt (CHSH) game in prime dimensions is achieved by measuring maximally entangled states that are isomorphic to single-qudit magic states. These magic states have an appealingly simple form and our proof shows that they are "balanced" with respect to all but one of the mutually unbiased stabilizer bases. Of all equatorial qudit states, magic states minimize the average entropic uncertainties for collision entropy and also, for small prime dimensions, min-entropy -- a fact that may have implications for cryptography.