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Springer eBooks

Using Information Entropy to Measure Stationary Randomness of Quantum Random Sequences

Weizhong Yang, Yamin Luo, Zhefei Li, Jeffrey Zheng

Published Dec 17, 2018• 1 citation
Publisher

Abstract

Abstract Different statistical measurements can be used to determine stationary randomness for random sequences. This chapter proposes a testing scheme for random sequences using information entropy as measurements. Datasets are collected from University of Science & Technology of China (USTC), three quantum random sequences are selected for testing. Multiple results are created on three maps, entropy curves, and quantitative measurements of stationary randomness are compared. Three differences of Max-Min entropy variation ratios are bounded in $$[0.08,0.09]\%$$ [ 0.08 , 0.09 ] % region. The whole structure has measurable stationary properties.

DOI
10.1007/978-981-13-2282-2_21

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Abstract Different statistical measurements can be used to determine stationary randomness for random sequences. This chapter proposes a testing scheme for random sequences using information entropy as measurements. Datasets are collected from University of Science & Technology of China (USTC), three quantum random sequences are selected for testing. Multiple results are created on three maps, entropy curves, and quantitative measurements of stationary randomness are compared. Three differences of Max-Min entropy variation ratios are bounded in [0.08,0.09]%[0.08,0.09]\%[0.08,0.09]% [ 0.08 , 0.09 ] % region. The whole structure has measurable stationary properties.
Abstract In this chapter, a testing model is used to apply statistical probability in multiple distributions on three maps for a selected sequence to check refined stationary randomness on quantum sequences. Three random data sequences are collected from two quantum random resources: one from Australian National University (ANU) and two (initial and secure) from University of Science and Technology of China (USTC). Multiple results are created on three maps, and measurements of stationary randomness are illustrated and compared. Three samples show distinct stationary properties.
Randomness is a valuable resource in science, cryptography, engineering, and information technology. Quantum-mechanical sources of randomness are attractive because of the indeterminism of individual quantum processes. Here, we consider the production of random bits from polarization measurements on photons. We first present a pedagogical discussion of how the quantum randomness inherent in such measurements is connected to quantum coherence, and how it can be quantified in terms of the quantum state and an associated entropy value known as min-entropy. We then explore these concepts by performing a series of single-photon experiments that are suitable for the undergraduate laboratory. We prepare photons in different nonentangled and entangled states, and measure these states tomographically. We use the information about the quantum state to determine, in terms of the min-entropy, the minimum amount of randomness produced from a given photon state by different bit-generating measurements. This is helpful in assessing the presence of quantum randomness and in ensuring the quality and security of the random-bit source.
We present an efficient method to extract the amount of true randomness that can be obtained by a quantum random number generator (QRNG). By repeating the measurements of a quantum system and by swapping between two mutually unbiased bases, a lower bound of the achievable true randomness can be evaluated. The bound is obtained thanks to the uncertainty principle of complementary measurements applied to min-entropy and max-entropy. We tested our method with two different QRNGs by using a train of qubits or ququart and demonstrated the scalability toward practical applications.
Abstract The advantages of quantum random number generators (QRNGs) over pseudo-random number generators (PRNGs) are normally attributed to the nature of quantum measurements. This is often seen as implying the superiority of the sequences of bits themselves generated by QRNGs, despite the absence of empirical tests supporting this. Nonetheless, one may expect sequences of bits generated by QRNGs to have properties that pseudo-random sequences do not; indeed, pseudo-random sequences are necessarily computable, a highly nontypical property of sequences. In this paper, we discuss the differences between QRNGs and PRNGs and the challenges involved in certifying the quality of QRNGs theoretically and testing their output experimentally. While QRNGs are often tested with standard suites of statistical tests, such tests are designed for PRNGs and only verify statistical properties of a QRNG, but are insensitive to many supposed advantages of QRNGs. We discuss the ability to test the incomputability and algorithmic complexity of QRNGs. While such properties cannot be directly verified with certainty, we show how one can construct indirect tests that may provide evidence for the incomputability of QRNGs. We use these tests to compare various PRNGs to a QRNG, based on superconducting transmon qutrits and certified by the Kochen–Specker theorem, to see whether such evidence can be found in practice. While our tests fail to observe a strong advantage of the quantum random sequences due to algorithmic properties, the results are nonetheless informative: some of the test results are ambiguous and require further study, while others highlight difficulties that can guide the development of future tests of algorithmic randomness and incomputability.
Random numbers generated via quantum process is indeterministic, which are of essential to the cryptographic communication and the large-scale computer modeling. However, in realistic scenarios, classical noises can inevitably contaminate the raw sequences of a quantum randomness number generator (QRNG), and then compromise the security of the QRNG. Min-entropy is a useful approach that can quantify the randomness independent of side-information. To enhance the extractable randomness of the raw sequences arising from the QRNG, we propose a new method which exploits non-uniform quantization methods instead of uniform sampling methods and effectively enhances the extractable randomness from the QRNG at a high quantum-to-classical-noise ratio (QCNR). Given a QCNR as 50 dB and a 16-bit analog-to-digital converter (ADC), the worst-case conditional min-entropy of the non-uniform quantization scheme is improved by nearly 11% compared with that of the uniform sampling scheme.
We develop a quantum version of the probability estimation framework [arXiv:1709.06159] for randomness generation with quantum side information. We show that most of the properties of probability estimation hold for quantum probability estimation (QPE). This includes asymptotic optimality at constant error and randomness expansion with logarithmic input entropy. QPE is implemented by constructing model-dependent quantum estimation factors (QEFs), which yield statistical confidence upper bounds on data-conditional normalized Rényi powers. This leads to conditional min-entropy estimates for randomness generation. The bounds are valid for relevant models of sequences of experimental trials without requiring independent and identical or stationary behavior. QEFs may be adapted to changing conditions during the sequence and trials can be stopped any time, such as when the results so far are satisfactory. QEFs can be constructed from entropy estimators to improve the bounds for conditional min-entropy of classical-quantum states from the entropy accumulation framework [Dupuis, Fawzi and Renner, arXiv:1607.01796]. QEFs are applicable to a larger class of models, including models permitting experimental devices with super-quantum but non-signaling behaviors and semi-device dependent models. The improved bounds are relevant for finite data or error bounds of the form e−κse^{-κs}e−κs, where sss is the number of random bits produced. We give a general construction of entropy estimators based on maximum probability estimators, which exist for many configurations. For the class of (k,2,2)(k,2,2)(k,2,2) Bell-test configurations we provide schemas for directly optimizing QEFs to overcome the limitations of entropy-estimator-based constructions. We obtain and apply QEFs for examples involving the (2,2,2)(2,2,2)(2,2,2) Bell-test configuration to demonstrate substantial improvements in finite-data efficiency.