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Chinese Optics Letters

Experimental randomness certification with a symmetric informationally complete positive operator-valued measurement

Chenxi Liu, Kun Liu, Xiaorun Wang, Luyan Wu, Jian Li, Qin Wang

Published Jan 1, 2020• 1 citation
Publisher

Abstract

Nonlocal correlations observed from entangled quantum particles imply the existence of intrinsic randomness. Normally, locally projective measurements performed on a two-qubit entangled state can only certify one-bit randomness at most, while non-projective measurement can certify more randomness with the same quantum resources. In this Letter, we carry out an experimental investigation on quantum randomness certification through a symmetric informationally complete positive operator-valued measurement, which in principle can certify the maximum randomness through an entangled qubit. We observe the quantum nonlocal correlations that are close to the theoretical values. In the future, this work can provide a valuable reference for the research on the limit of randomness certification.

DOI
10.3788/col202018.102701

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Nonlocal correlations observed from entangled quantum particles imply the existence of intrinsic randomness. Normally, locally projective measurements performed on a two-qubit entangled state can only certify one-bit randomness at most, while non-projective measurement can certify more randomness with the same quantum resources. In this Letter, we carry out an experimental investigation on quantum randomness certification through a symmetric informationally complete positive operator-valued measurement, which in principle can certify the maximum randomness through an entangled qubit. We observe the quantum nonlocal correlations that are close to the theoretical values. In the future, this work can provide a valuable reference for the research on the limit of randomness certification.
Abstract Quantum random numbers are fundamental resources ( Rev. Mod. Phys. 89 015004) for the fields of communications and information security. Usually, projective measurements can be applied on a two-qubit entangled state to generate quantum random numbers. However, it cannot certify more than one bit of randomness for one set of measurements. On contrast, non-projective measurements in-principle may be able to generate more random bits, and therefore show some merits among present methods. Here, we report the implementation of a quantum random number generator based on a symmetric informationally complete positive operator-valued measure. In our experiment, we prepare a maximally entangled state of two-qubits through spontaneous parametric down-conversion processes, and observe the violation of Gisin’s elegant Bell inequality with the value of 6.8138 ± 0.0822, exceeding the classical bound of 6. Through randomness extractors, including a Toeplitz extractor and a parallel linear feedback shift register based extractor, we finally extract a random string with an autocorrelation coefficient below 10 −3 and the string can pass all randomness tests of the National Institute of Science and Technology (NIST).
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.
Nonlocal correlations obtained from entangled quantum states are useful for device independent (DI) randomness certification [Nature (London) 464, 1021 (2010)]. The advantage of this DI protocol over the conventional quantum protocol is that even with untrusted experimental apparatus randomness can be certified. Although quantum entanglement is the necessary physical source for nonlocal correlation, these two are different concepts. There exist entangled states which produce no nonlocal correlation and hence are not useful for the DI randomness certification task. In this work we introduce measurement-device-independent randomness certification task and show that entangled states, having local description, can be useful resource in such task which otherwise are useless in corresponding DI scenario.
The nonlocal correlations exhibited when measuring entangled particles can be used to certify the presence of genuine randomness in Bell experiments. While nonlocality is necessary for randomness certification, it is unclear when and why nonlocality certifies maximal randomness. We provide a simple argument to certify the presence of maximal local and global randomness based on symmetries of a Bell inequality and the existence of a unique quantum probability distribution that maximally violates it. We prove the existence of NNN-party Bell tests attaining maximal global randomness by identifying those combinations of two-outcome measurements by each party providing NNN perfect random bits.
The outcomes of local measurements made on entangled systems can be certified to be random provided that the generated statistics violate a Bell inequality. This way of producing randomness relies only on a minimal set of assumptions because it is independent of the internal functioning of the devices generating the random outcomes. In this context it is crucial to understand both qualitatively and quantitatively how the three fundamental quantities -- entanglement, non-locality and randomness -- relate to each other. To explore these relationships, we consider the case where repeated (non projective) measurements are made on the physical systems, each measurement being made on the post-measurement state of the previous measurement. In this work, we focus on the following questions: For systems in a given entangled state, how many nonlocal correlations in a sequence can we obtain by measuring them repeatedly? And from this generated sequence of non-local correlations, how many random numbers is it possible to certify? In the standard scenario with a single measurement in the sequence, it is possible to generate non-local correlations between two distant observers only and the amount of random numbers is very limited. Here we show that we can overcome these limitations and obtain any amount of certified random numbers from an entangled pair of qubit in a pure state by making sequences of measurements on it. Moreover, the state can be arbitrarily weakly entangled. In addition, this certification is achieved by near-maximal violation of a particular Bell inequality for each measurement in the sequence. We also present numerical results giving insight on the resistance to imperfections and on the importance of the strength of the measurements in our scheme.