Unambiguous randomness from a quantum state

TL;DR

This work introduces two device-setting randomness notions—unambiguous randomness, where an eavesdropper makes no errors but may output an inconclusive result, and FRIO randomness, which allows a fixed inconclusive rate—and recasts them in a quantum state discrimination framework solved by semidefinite programs. The authors prove a general bound for any state: the maximal unambiguous randomness is $P_{UD}^{*}(\rho) = d\lambda_{\min}(\rho)$, with a constructive measurement achieving equality, and they derive exact results for qubits and for isotropically noisy states, including a closed-form FRIO expression. A key finding is that when both the state and the measurement are noisy, a joint-noise eavesdropper can outperform a single-noise one, and beyond a critical error, the joint eavesdropper can achieve perfect guessing, signaling a potential collapse of private randomness in realistic devices. These results illuminate the necessity of careful noise modelling in quantum randomness generation and suggest directions to extend the analysis to non-full-rank states and more general measurements.

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 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.

Figures (3)

• Figure 1: Unambiguous randomness. An eavesdropper's optimal decomposition of a qubit state $\vec{r}$, given a measurement in the basis $\{\pm \vec{m}\}$.
• Figure 2: FRIO randomness. An eavesdropper's optimal decomposition of a qubit state $\vec{r}$, given a measurement in the basis $\{\pm \vec{m}\}$ and FRIO parameter $Q$.
• Figure 3: Shared versus single noise. Eve's optimal guessing probability for the single noise case (dashed) and a lower bound on her guessing probability in the shared noise case (undashed).

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Corollary 2.1
• Theorem 3
• Theorem 4
• Corollary 4.1
• Theorem 5
• Theorem 6
• Theorem 1
• proof