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Physical Review A

Reduced gap between observed and certified randomness for semi-device-independent protocols

Yukun Wang, Su‐Juan Qin, Xia Wu, Fei Gao, Qiaoyan Wen

Published Nov 19, 2015• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.92.052321

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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.
The prepare-and-measure framework based on energy constraints offers a practical middle ground between fully device-dependent and device-independent quantum cryptography. The only assumption on an otherwise uncharacterized prepare-and-measure device is that the energy of the prepared states is bounded. Existing security analyses of this framework assume that the preparation and measurement devices share at most classical correlations, and under this assumption certified lower bounds on the extractable randomness have been established. Recent work has shown that an adversary who pre-distributes entanglement between the devices can mount attacks that are strictly more powerful than those available when the devices share only classical correlations, reducing the extractable randomness below the previously certified rates. This leaves open the fundamental question of whether randomness can be certified at all in this scenario. We address this open question by constructing semidefinite programming relaxations of the guessing probability by adapting the Navascués-Pironio-Acín hierarchy to the energy-constrained setting where shared entanglement between the devices is permitted. Our relaxations yield certified lower bounds on the extractable randomness without enforcing any restrictions on the dimensions of the quantum state shared between the preparation and measurement devices. We show that these certified lower bounds are strictly positive for a range of energy values, thereby answering the open question affirmatively: certified randomness generation is theoretically possible in the energy-constrained SDI framework even in the presence of a fully quantum adversary.
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 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.
Semi-device-independent random number expansion (SDI-RNE) protocols require some truly random numbers to generate fresh ones, with making no assumptions on the internal working of quantum devices except for the dimension of the Hilbert space. The generated randomness is certified by non-classical correlation in the prepare-and-measure test. Until now, the analytical relations between the amount of the generated randomness and the degree of non-classical correlation, which are crucial for evaluating the security of SDI-RNE protocols, are not clear under both the ideal condition and the practical one. In the paper, first, we give the analytical relation between the above two factors under the ideal condition. As well, we derive the analytical relation under the practical conditions, where devices' behavior is not independent and identical in each round and there exists deviation in estimating the non-classical behavior of devices. Furthermore, we choose a different randomness extractor (i.e., two-universal random function) and give the security proof.
The majority of recent works investigating the link between non-locality and randomness, e.g. in the context of device-independent cryptography, do so with respect to some specific Bell inequality, usually the CHSH inequality. However, the joint probabilities characterizing the measurement outcomes of a Bell test are richer than just the degree of violation of a single Bell inequality. In this work we show how to take this extra information into account in a systematic manner in order to optimally evaluate the randomness that can be certified from non-local correlations. We further show that taking into account the complete set of outcome probabilities is equivalent to optimizing over all possible Bell inequalities, thereby allowing us to determine the optimal Bell inequality for certifying the maximal amount of randomness from a given set of non-local correlations.
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.