Secure Certification of Mixed Quantum States with Application to Two-Party Randomness Generation

We investigate sampling procedures that certify that an arbitrary quantum state on $n$ subsystems is close to an ideal mixed state $\varphi^{\otimes n}$ for a given reference state $\varphi$, up to errors on a few positions. This task makes no sense classically: it would correspond to certifying that a given bitstring was generated according to some desired probability distribution. However, in the quantum case, this is possible if one has access to a prover who can supply a purification of the mixed state. In this work, we introduce the concept of mixed-state certification, and we show that a natural sampling protocol offers secure certification in the presence of a possibly dishonest prover: if the verifier accepts then he can be almost certain that the state in question has been correctly prepared, up to a small number of errors. We then apply this result to two-party quantum coin-tossing. Given that strong coin tossing is impossible, it is natural to ask "how close can we get". This question has been well studied and is nowadays well understood from the perspective of the bias of individual coin tosses. We approach and answer this question from a different---and somewhat orthogonal---perspective, where we do not look at individual coin tosses but at the global entropy instead. We show how two distrusting parties can produce a common high-entropy source, where the entropy is an arbitrarily small fraction below the maximum (except with negligible probability).