Sequential Semi-Device-Independent Quantum Randomness Certification

TL;DR

The paper tackles certifying quantum randomness in sequential prepare-and-measure experiments under semi-device-independent assumptions. It develops a general SDP-based framework for bounding Eve’s guessing probability and hence the min-entropy, first under fully trusted state preparations and then in a semi-DI setting with bounded state overlaps, enabling randomness certification despite untrusted devices and adaptive attacks. By employing dual SDP formulations and entropy accumulation via affine min-tradeoff functions derived from Gauss-Radau quadrature, the work accommodates both i.i.d. and non-i.i.d. rounds, providing practical finite-size bounds. The results show that maximum-confidence measurements can distribute certifiable randomness across sequential observers, with explicit criteria and thresholds for two-state, two-measurement scenarios and extensions to $N$ sequential measurements. This approach has practical relevance for semi-DI quantum randomness generation and could inform future multipartite cryptographic protocols and semi-DI QKD implementations.

Abstract

Quantum measurements under realistic conditions reveal only partial information about a system. Yet, by performing sequential measurements on the same system, additional information can be accessed. We investigate this problem in the context of semi-device-independent randomness certification using sequential maximum confidence measurements. We develop a general framework and versatile numerical methods to bound the amount of certifiable randomness in such scenarios. We further introduce a technique to compute min-tradeoff functions via semidefinite programming duality, thus making the framework suitable for bounding the certifiable randomness against adaptive attacking strategies through entropy accumulation. Our results establish sufficient criteria showing that maximum confidence measurements enable the distribution and certification of randomness across a sequential measurement chain.

Figures (4)

• Figure 1: Sequential prepare-and-measure framework. Alice (A) encodes a symbol $x$ into a quantum state $\rho_x$. This is sent to Bob (B), observing a measurement outcome $b$. Lastly, Charlie (C) measures Bob's post-measurement state $\sigma_x$ and produces an outcome $c$.
• Figure 2: Semi-device-independent framework. We assume that Alice's preparations are pure with bounded pair-wise overlaps. All operations performed after the preparation process are completely uncharacterised.
• Figure 3: Shannon and min-entropies certified in two sequential maximum confidence measurements. Alice prepares two pure states with $\left|\braket{\psi_0|\psi_{1}}\right|\geq \delta$. Bob and Charlie perform maximum confidence measurement with white noise component $1-r$, and rates of inconclusive outcomes $Q$ and $r\delta/Q$ respectively. We also show the certifiable min-entropy in Charlie's device in the measurement-device independent setting (Charlie$^\ast$).
• Figure 4: Numerical search of the bound $r\delta \leq \Delta$ at which randomness can be simultaneously certified in all $N$ sequential maximum confidence measurements.