Quantum randomness beyond projective measurements

TL;DR

This paper investigates the amount of intrinsic quantum randomness generated by extremal measurements, explicitly characterizing it for qubits and establishing maximal randomness in higher dimensions with SIC measurements.

Contribution

It provides an explicit characterization of the randomness generated by extremal rank-one measurements, including the first complete solution in dimension two, and proves maximal randomness achievable with SIC measurements.

Findings

Four-outcome qubit measurements are tomographic, enabling source-device-dependent randomness.

The tetrahedral SIC measurement has the least intrinsic randomness among extremal measurements.

Maximal $2 \, \log d$ bits of randomness can be generated in any dimension with a SIC measurement.

Abstract

The unpredictability of quantum physics gives rise to intrinsic randomness. In an adversarial scenario, any additional degrees of freedom must be attributed to an eavesdropper with correlations to the measurement set-up. The true randomness is then quantified by the probability that she correctly guesses the measurement outcomes, optimised over all possible strategies. Extremal measurements are appealing here, since they do not allow information to leak to such an eavesdropper. Beyond projective measurements, however, a simple question remains open: how much intrinsic randomness can be generated by a given extremal measurement? In a step towards solving it, we characterise the randomness generated by any unbiased extremal rank-one measurement acting on any state, solving the problem explicitly in dimension two. Four-outcome qubit measurements of this type are tomographic, so these results hold for fully source-device-dependent randomness too. The tetrahedral symmetric informationally complete (SIC) measurement, we find, has the least intrinsic randomness within this class. We also present the skewed SIC family of measurements, and use them to partially solve an open problem: we prove that $2 \log d$ bits of randomness, the maximal amount, can be generated device-dependently (or source-device-independently) in any dimension in which there exists a SIC measurement.