Unambiguous randomness from a quantum state

TL;DR

This paper introduces the concept of unambiguous quantum randomness, quantifies it for various states and measurements, and analyzes the impact of noise and eavesdroppers on quantum privacy and randomness.

Contribution

It defines and solves for the maximal unambiguous randomness in quantum states and measurements, extending to noisy scenarios and eavesdropper models.

Findings

Maximal unambiguous randomness proportional to the smallest eigenvalue of the state.

In noisy scenarios, joint eavesdroppers outperform independent ones beyond a critical noise threshold.

Identifies conditions where private randomness is completely compromised by eavesdroppers.

Abstract

Intrinsic randomness is generated when a quantum state is measured in any basis in which it is not diagonal. In an adversarial scenario, we quantify this randomness by the probability that a correlated eavesdropper could correctly guess the measurement outcomes. What if the eavesdropper is never wrong, but can sometimes return an inconclusive outcome? Inspired by analogous concepts in quantum state discrimination, we introduce the unambiguous randomness of a quantum state and measurement, and, relaxing the assumption of perfect accuracy, randomness with a fixed rate of inconclusive outcomes. We solve the maximal unambiguous randomness of any quantum state, optimised over all rank-one projective measurements, and find that it's proportional to the smallest eigenvalue of the state. We also solve these problems for any state and projective measurement in dimension two, as well as for an isotropically noisy state measured in an unbiased basis of any dimension. In the latter case, we find that, given a fixed amount of total noise, an eavesdropper correlated only to the noisy state is always outperformed by an eavesdropper with joint correlations to both a noisy state and a noisy measurement. In fact, we identify a critical error parameter beyond which the joint eavesdropper achieves perfect guessing probability, ruling out any possibility of private randomness.