Quantifying randomness with measurement incompatibility

Abstract

We present a trade-off between the amount of observed measurement incompatibility and the capabilities of a classical Eavesdropper in a prepare-and-measure scenario. The result is based on a qualitative connection between measurement incompatibility and randomness generation together with the utilization of incompatibility witnesses as randomness certificates. This allows one to use a geometric measure of incompatibility, the generalised robustness, to bound Eve's strategies through a semi-definite program, while providing an explicit protocol for generating randomness from any set of incompatible measurements. By translating the result to quantum steering, we find a tight connection between steerability and randomness generation in a setting using any finite number of measurement inputs. We further show how our techniques can be generalised to scenarios where Eve has a quantum memory by using a dimensional generalisation of joint measurability.