All star-incompatible measurements can certify steering-based randomness

Abstract

Certifying that quantum randomness generated by untrusted devices is unpredictable to an attacker (say, Eve) is crucial for device-independent security. Bipartite protocols where only one of the parties is trusted are termed one-sided device-independent (1SDI) or steering-based protocols, where the untrusted party (say, Alice) performs measurements on her part of a bipartite entangled state to steer the subsystem of the trusted party (say, Bob) into different ensembles (collectively, an assemblage) of quantum states. Recent work has shown that an assemblage has certified randomness if and only if it is realizable by a set of measurements that are star-incompatible, i.e., the measurement setting of interest for the guessing probability of Eve is incompatible with at least one of the remaining measurement settings of Alice. However, it remains conceivable that there exist star-incompatible measurements that cannot certify steering-based randomness, just like there exist incompatible measurements that cannot certify bipartite Bell nonlocality. Here we prove that any set of star-incompatible measurements can generate steering-based randomness, thereby establishing an equivalence between the two notions. We further introduce a weight-based measure of star-incompatibility and lower bound the amount required to certify a given randomness, capturing the qualitative and quantitative interplay between the quantum resources of star-incompatibility and steering-based randomness.

Figures (3)

• Figure 1: One-sided device-independent (1SDI) randomness certification. The single arrows represent quantum information, and the double arrows represent classical information. Alice chooses the $x$-th measurement from her set of measurements, and gets an outcome $a_x\in\mathcal{A}$. Bob is in a trusted lab, and can fully characterize the post-measurement state by state tomography paris2004quantum, obtaining an assemblage $\{\sigma^{a_x|x}_\mathrm{B}\}_{a_x\in\mathcal{A},x\in\mathcal{X}}$. An attacker named Eve, space-like separated from Alice and Bob, guesses Alice's measurement outcomes. The guess is successful if $e=a_x$.
• Figure 2: Equivalence between star-incompatibility, star-steerability, and certified randomness. The dashed arrows indicate implications shown in Proposition 1 in Ref. li2025necessary, the solid arrows are our results. The double left and right arrows indicate equivalence.
• Figure 3: Star-incompatibility weight $W^1(\mathbb{M}^{\mathcal{A}|\mathcal{I}}_\mathrm{A})$ (dashed), and $2(1-p^1_\mathrm{guess}(\sigma(\mathbb{M}^{\mathcal{A}|\mathcal{I}}_\mathrm{A},\rho_{\mathrm{A}\mathrm{B}}))$ (solid) given in Theorem \ref{['theorem:weight']} for a PMD $\mathbb{M}^{\mathcal{A}|\mathcal{I}}_{\mathrm{A}}:=\{\mathsf{M}^{\mathcal{A}|i}_\mathrm{A}\}_{i=1}^3$ where $\mathsf{M}^{\mathcal{A}|i}_\mathrm{A}$ is given by Eq. \ref{['eq:pauli']}. $\eta$ is in increments of $0.01$ from $\eta=0.65$. As expected from analytic results heinosaari2008notesliang2011specker, wherever $\eta$ is below $0.707$, the star-incompatibility weight is numerically $0$ and $p^1_\mathrm{guess}(\sigma(\mathbb{M}^{\mathcal{A}|\mathcal{I}}_\mathrm{A},\rho_{\mathrm{A}\mathrm{B}}))$ is $1$.

Theorems & Definitions (16)

• Definition 1
• Theorem 1
• Definition 2
• Lemma 1
• Lemma 2
• proof : Proof of Theorem \ref{['theorem:incompatibility-randomness']}
• Corollary 1
• Remark 1
• Remark 2
• Definition 3