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arXiv:1503.03691 · Physical Review A

Semi-device-independent randomness expansion with partially free random sources

Yuqian Zhou, Hongwei Li, Yukun Wang, Dandan Li, Fei Gao, Qiaoyan Wen

Published Aug 13, 2015• 16 citations
arXivHTMLPDF

Abstract

By proposing device-independent protocols, Pironio et al. [Nature (London) 464, 1021 (2010)] and Colbeck et al. [Nat. Phys. 8, 450 (2012)] proved that new randomness can be generated by using perfectly free random sources or partially free ones as seed. Subsequently, Li et al. [Phys. Rev. A 84, 034301 (2011)] studied this topic in the framework of semi-device-independent and proved that new randomness can be obtained from perfectly free random sources. Here we discuss whether and how partially free random sources bring us new randomness in a semi-device-independent scenario. We propose a semi-device-independent randomness expansion protocol with partially free random sources and obtain the condition that the partially free random sources should satisfy to generate new randomness. In the process of analysis, we acquire a two-dimensional quantum witness. Furthermore, we get the analytic relationship between the generated randomness and the two-dimensional quantum witness violation.

DOI
10.1103/physreva.92.022331

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By proposing device-independent protocols, Pironio et al. [Nature (London) 464, 1021 (2010)] and Colbeck et al. [Nat. Phys. 8, 450 (2012)] proved that new randomness can be generated by using perfectly free random sources or partially free ones as seed. Subsequently, Li et al. [Phys. Rev. A 84, 034301 (2011)] studied this topic in the framework of semi-device-independent and proved that new randomness can be obtained from perfectly free random sources. Here we discuss whether and how partially free random sources bring us new randomness in a semi-device-independent scenario. We propose a semi-device-independent randomness expansion protocol with partially free random sources and obtain the condition that the partially free random sources should satisfy to generate new randomness. In the process of analysis, we acquire a two-dimensional quantum witness. Furthermore, we get the analytic relationship between the generated randomness and the two-dimensional quantum witness violation.
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.
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.
Semi-device-independent randomness expansion is the protocol which can generate and certify perfect free random numbers, without making detailed assumptions on the internal working of quantum devices, except the bound on the dimension of the Hilbert space. The two-dimensional (2D) quantum witness violation has been used to generate guaranteed fresh randomness from untrusted devices in the prepare-and-measure configuration. In this paper we first show that the 2D quantum witness violation can be simulated by a classical deterministic process by relaxing the assumption of the dimension. A deterministic process to simulate the generation of random numbers is given, under the hidden variables model, with just the high dimensional measurement device and one qubit communication. This makes the generated number essentially deterministic, which is said to be pseudorandom. Then we also show that when we relax the freedom of choice, there is the exact region for which quantum correlations can be simulated by the classical process in the task of randomness expansion.
By testing the classical correlation violation between two systems, true random numbers can be generated and certified without applying classical statistical method. In this work, we propose a true random-number expansion protocol without entanglement, where the randomness can be guaranteed only by the two-dimensional quantum witness violation. Furthermore, we only assume that the dimensionality of the system used in the protocol has a tight bound, and the whole protocol can be regarded as a semi-device-independent black-box scenario. Compared with the device-independent random-number expansion protocol based on entanglement, our protocol is much easier to implement and test.
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.
It is a well-known fact in classical information theory that no deterministic procedure can extract close-to-ideal randomness from an arbitrary entropy source. On the other hand, if additional knowledge about the source is available -- e.g., that it is a sequence of independent Bernoulli trials -- then deterministic extractors do exist. For quantum entropy sources, where in addition to classical random variables we consider quantum side information, the use of extra knowledge about their structure was pioneered in a recent publication [C. Foreman and L. Masanes, Quantum 9, 1654 (2025)]. In that work, the authors provide deterministic extractors for device-independent randomness generation with memoryless devices achieving a sufficiently high CHSH score. In this work, we port their construction to the prepare-and-measure scenario. Specifically, we prove that the considered functions are also extractors for memoryless devices in settings with partial trust, either in the state preparation or in the measurement, as well as in a semi-device-independent setting under an overlap assumption on the prepared quantum states. Within this last setting, we simulate the resulting randomness generation protocol on a novel and experimentally relevant family of behaviors, observing positive key rates already for 7×1037\times 10^37×103 rounds.