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arXiv:1402.1850 · Physical Review A

Device- and semi–device-independent random numbers based on noninequality paradox

Hong-Wei Li, Marcin Pawłowski, Ramij Rahaman, Guang‐Can Guo, Zheng‐Fu Han

Published Aug 11, 2015• 14 citations
arXivHTMLPDF

Abstract

In this work, we propose device-independent true random number expansion protocols based on noninequality paradoxes such as Hardy's and Cabello's nonlocality arguments, thus highlighting the noninequality paradox as an important resource for device-independent quantum-information processing, in particular for generating true randomness. As a byproduct, we find that the nonlocal bound of the Cabello argument with arbitrary dimension is the same as the one achieved in the qubits system. More interestingly, we propose a dimension witness paradox based on the Cabello argument which can be used for constructing a semi--device-independent true random number expansion protocol.

DOI
10.1103/physreva.92.022327

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In this work, we propose device-independent true random number expansion protocols based on noninequality paradoxes such as Hardy's and Cabello's nonlocality arguments, thus highlighting the noninequality paradox as an important resource for device-independent quantum-information processing, in particular for generating true randomness. As a byproduct, we find that the nonlocal bound of the Cabello argument with arbitrary dimension is the same as the one achieved in the qubits system. More interestingly, we propose a dimension witness paradox based on the Cabello argument which can be used for constructing a semi--device-independent true random number expansion protocol.
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.
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.
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.
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.
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.
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.