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Entropy

Semi-Device-Independent Randomness Expansion Using n→1 Parity-Oblivious Quantum Random Access Codes

Xunan Wang, Xu Chen, Mengke Xu, Wanglei Mi, Xiao Chen

Published Jun 28, 2025• 0 citations
Publisher

Abstract

Quantum mechanics enables the generation of genuine randomness through its intrinsic indeterminacy. In device-independent (DI) and semi-device-independent (SDI) frameworks, randomness generation protocols can further ensure that the output remains secure and unaffected by internal device imperfections, with certification grounded in violations of generalized Bell inequalities. In this work, we propose an SDI randomness expansion protocol using n→1 parity-oblivious quantum random access code (PO-QRAC), where the presence of true quantum randomness is certified through the violation of a two-dimensional quantum witness. For various values of n, we derive the corresponding maximal expected success probabilities. Notably, for n=4, the expected success probability obtained under our protocol exceeds the upper bound reported in prior work. Furthermore, we establish an analytic relationship between the certifiable min-entropy and the quantum witness value, and demonstrate that, for a fixed witness value, PO-QRAC–based protocols certify more randomness than those based on standard QRACs. Among all configurations satisfying the parity-obliviousness constraint, the protocol based on the 3→1 PO-QRAC achieves optimal randomness expansion performance.

DOI
10.3390/e27070696

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Quantum mechanics enables the generation of genuine randomness through its intrinsic indeterminacy. In device-independent (DI) and semi-device-independent (SDI) frameworks, randomness generation protocols can further ensure that the output remains secure and unaffected by internal device imperfections, with certification grounded in violations of generalized Bell inequalities. In this work, we propose an SDI randomness expansion protocol using n→1 parity-oblivious quantum random access code (PO-QRAC), where the presence of true quantum randomness is certified through the violation of a two-dimensional quantum witness. For various values of n, we derive the corresponding maximal expected success probabilities. Notably, for n=4, the expected success probability obtained under our protocol exceeds the upper bound reported in prior work. Furthermore, we establish an analytic relationship between the certifiable min-entropy and the quantum witness value, and demonstrate that, for a fixed witness value, PO-QRAC–based protocols certify more randomness than those based on standard QRACs. Among all configurations satisfying the parity-obliviousness constraint, the protocol based on the 3→1 PO-QRAC achieves optimal randomness expansion performance.
We have proved that new randomness can be certified by partially free sources using $2\ensuremath{\rightarrow}1$ quantum random access code (QRAC) in the framework of semi-device-independent (SDI) protocols [Y.-Q. Zhou, H.-W. Li, Y.-K. Wang, D.-D. Li, F. Gao, and Q.-Y. Wen, Phys. Rev. A 92, 022331 (2015)]. To improve the effectiveness of the randomness generation, here we propose the SDI randomness expansion using $3\ensuremath{\rightarrow}1$ QRAC and obtain the corresponding classical and quantum bounds of the two-dimensional quantum witness. Moreover, we get the condition which should be satisfied by the partially free sources to successfully certify new randomness, and the analytic relationship between the certified randomness and the two-dimensional quantum witness violation.
We study random-number-expansion protocols based on the $n\ensuremath{\rightarrow}1$ quantum random access codes (QRACs). We consider them in the semi-device-independent scenario where the inner workings of the devices are unknown to us but we can certify the dimensions of the systems being communicated. This approach does not require the use of entanglement and makes the physical realization of these protocols much easier than in the standard device-independent scenario. In our work, we propose a protocol for randomness expansion, compute min-entropy for the semi-device-independent protocol, and investigate $n\ensuremath{\rightarrow}1$ QRACs with a view to their use in randomness-expansion protocols. We also calculate the dependence of the effectiveness of the randomness generation on nnn and find it optimal for n=3n=3n=3, and provide the explanation for this fact.
Random number generation is vital for cryptography. The classical world faces inherent limitations in generating provably true random numbers. Quantum mechanics, however, provides a natural foundation for genuinely unpredictable outcomes. In random number generation protocols, source-independent protocols ensure security even with untrusted source devices. Hence, we propose a sourceindependent expansion protocol with partially free sources to balance security and feasibility. In our proposed protocol, an untrusted source is interrogated by trusted measurement devices, utilizing a partially free source for randomness expansion. Based on Quantum random access codes (QRACs) in a three-dimensional Hilbert space with partially free source, we construct a quantum witness, derive the relationship between randomness generation efficiency and the violation of this witness. This demonstrates the advantage of utilizing higher-dimensional systems to generate more certified randomness.
The randomness expansion capabilities of semi-device-independent (SDI) prepare and measure protocols are analyzed under the sole assumption that the Hilbert state dimension is known. It is explicitly proved that the maximum certifiable entropy that can be obtained through this set of protocols is $\ensuremath{-}{log}_{2}[\frac{1}{2}(1+\frac{1}{\sqrt{3}})]$ and the same is independent of the dimension witnesses used to certify the protocol. The minimum number of preparation and measurement settings required to achieve this entropy is also proven. An SDI protocol that generates the maximum output entropy with the least amount of input setting is provided. An analytical relationship between the entropy generated and the witness value is obtained. It is also established that certifiable entropy can be generated as soon as the dimension witness crosses the classical bound, making the protocol noise-robust and useful in practical applications.
Quantum systems can be used to certify the existence of true randomness in nature, and semi-device-independent (SDI) randomness expansion has been used to achieve this goal under the prepare-measure configuration. Usually, randomness is certified with the degree of violation of some inequalities. However, the observed statistics characterizing SDI protocols contain much richer information than just a single value of the inequality. Thus, there might be a gap between the randomness in observed statistics and the randomness we certify. The purpose of this paper is to optimize the use of the full observed statistics to reduce that gap. It shows that certifying randomness directly from the full observed statistics but not some inequality is available in the SDI protocols. Besides, average of guessing probabilities should be considered as randomness quantitative standards, since it certifies more randomness than the previous maximal guessing probability method. In the analysis, we take into account the assumption that the devices and the potential adversary are classically correlated.
We study a semi-device-independent (SDI) randomness expansion protocol with a trusted source or detector. It is not difficult to characterize the source or the detector when their dimensions are limited in the SDI scenario. By adding the reasonable assumption, our protocol offers higher min-entropy and, thus, can be used to certify more randomness than the standard SDI protocol of H.-W. Li et al. [Phys. Rev. A 84, 034301 (2011)]. Moreover, randomness expansion can be realized with a lower detection efficiency (about 85.4%) in the trusted source case than the standard one (about 93.4%). Meanwhile, this advantage is robust against parameter deviations of the source. But on the other hand, the sharp measurement cannot improve the performance much in this case. So, there is no need to characterize the measurement accurately. Furthermore, an analytical formula describing the relationship between the dimension witness value and the min-entropy of the output string is derived directly.