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arXiv:1109.5259 · Physical Review A

Semi-device-independent randomness certification using n→1 quantum random access codes

Hong-Wei Li, Marcin Pawłowski, Zhen−Qiang Yin, Guang‐Can Guo, Zheng‐Fu Han

Published May 11, 2012• 111 citations
arXivHTMLPDF

Abstract

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 $n$ and find it optimal for $n=3$, and provide the explanation for this fact.

DOI
10.1103/physreva.85.052308

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We study random-number-expansion protocols based on the n\ensuremath→1n\ensuremath{\rightarrow}1n\ensuremath→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→1n\ensuremath{\rightarrow}1n\ensuremath→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.
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 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.
One of the striking properties of quantum mechanics is the occurrence of the Bell-type non-locality. They are a fundamental feature of the theory that allows two parties that share an entangled quantum system to observe correlations stronger than possible in classical physics. In addition to their theoretical significance, non-local correlations have practical applications, such as device-independent randomness generation, providing private unpredictable numbers even when obtained using devices delivered by an untrusted vendor. Thus, determining the quantity of certifiable randomness that can be produced using a specific set of non-local correlations is of significant interest. First, we present an experimental realization of recent Bell-type operators designed to provide private random numbers that are secure against adversaries with quantum resources. We use semi-definite programming to give lower bounds on the generated randomness in terms of both min-entropy and von Neumann entropy in a device-independent scenario. Our results demonstrate the first experiment that certifies nearly two bits of randomness from binary measurements of two parties [1], [2]. Apart from single-round certification, we provide an analysis of finite-key protocol for quantum randomness expansion using the Entropy Accumulation Theorem. Second, we present an efficient and practical method for certifying quantum randomness using generalized measurements. Indeed, we have derived a method for self-testing the presence of POVMs. We present the certification of more than one bit of min-entropy from a single POVM on one of the qubits of an entangled state using a variant of the elegant Bell inequality. We also obtain more than one bit of randomness in a prepare and measure scenario of a similar structure with a POVM on a single qubit [3]. We provide numerical simulations to demonstrate the effectiveness of our randomness certification and expansion.
Quantum communication with systems of dimension larger than two provides advantages in information processing tasks. Examples include higher rates of key distribution and random number generation. The main disadvantage of using such multi-dimensional quantum systems is the increased complexity of the experimental setup. Here, we analyze a not-so-obvious problem: the relation between randomness certification and computational requirements of the post-processing of experimental data. In particular, we consider semi-device independent randomness certification from an experiment using a four dimensional quantum system to violate the classical bound of a random access code. Using state-of-the-art techniques, a smaller quantum violation requires more computational power to demonstrate randomness, which at some point becomes impossible with today's computers although the randomness is (probably) still there. We show that by dedicating more input settings of the experiment to randomness certification, then by more computational postprocessing of the experimental data which corresponds to a quantum violation, one may increase the amount of certified randomness. Furthermore, we introduce a method that significantly lowers the computational complexity of randomness certification. Our results show how more randomness can be generated without altering the hardware and indicate a path for future semi-device independent protocols to follow.
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.