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Physical Review A

Randomness expansion with a one-sided trusted device

Hong Wang, Kai Chen, Ming Gao, Zhi Ma, Weilong Wang, Chaobo Li, Chuangui Ma

Published May 8, 2015• 1 citation
Publisher

Abstract

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.

DOI
10.1103/physreva.91.052308

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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.
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.
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.
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.
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 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.
In this paper we develop a method for investigating semi-device-independent randomness expansion protocols that was introduced in Li et al. [H.-W. Li, P. Mironowicz, M. Paw\l{}owski, Z.-Q. Yin, Y.-C. Wu, S. Wang, W. Chen, H.-G. Hu, G.-C. Guo, and Z.-F. Han, Phys. Rev. A 87, 020302(R) (2013)]. This method allows us to lower bound, with semi-definite programming, the randomness obtained from random number generators based on dimension witnesses. We also investigate the robustness of some randomness expanders using this method. We show the role of an assumption about the trace of the measurement operators and a way to avoid it. The method is also generalized to systems of arbitrary dimension and for a more general form of dimension witnesses than in our previous paper. Finally, we introduce a procedure of dimension witness reduction, which can be used to obtain from an existing witness a new one with a higher amount of certifiable randomness. The presented methods find an application for experiments [J. Ahrens, P. Badziag, M. Pawlowski, M. Zukowski, and M. Bourennane, Phys. Rev. Lett. 112, 140401 (2014)].