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arXiv:1101.4711 · Computability

Von Neumann Normalisation of a Quantum Random Number Generator

Alastair A. Abbott, Cristian S. Calude

Published Jan 1, 2012• 6 citations
arXivHTMLPDF

Abstract

In this paper we study von Neumann un-biasing normalisation for ideal and real quantum random number generators, operating on finite strings or infinite bit sequences.In the ideal cases one can obtain the desired un-biasing.This relies critically on the independence of the source, a notion we rigorously define for our model.In real cases, affected by imperfections in measurement and hardware, one cannot achieve a true un-biasing, but, if the bias "drifts sufficiently slowly", the result can be arbitrarily close to un-biasing.For infinite sequences, normalisation can both increase or decrease the (algorithmic) randomness of the generated sequences.A successful application of von Neumann normalisation-in fact, any un-biasing transformation-does exactly what it promises, un-biasing, one (among infinitely many) symptoms of randomness; it will not produce "true" randomness.

DOI
10.3233/com-2012-001

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In this paper we study von Neumann un-biasing normalisation for ideal and real quantum random number generators, operating on finite strings or infinite bit sequences.In the ideal cases one can obtain the desired un-biasing.This relies critically on the independence of the source, a notion we rigorously define for our model.In real cases, affected by imperfections in measurement and hardware, one cannot achieve a true un-biasing, but, if the bias "drifts sufficiently slowly", the result can be arbitrarily close to un-biasing.For infinite sequences, normalisation can both increase or decrease the (algorithmic) randomness of the generated sequences.A successful application of von Neumann normalisation-in fact, any un-biasing transformation-does exactly what it promises, un-biasing, one (among infinitely many) symptoms of randomness; it will not produce "true" randomness.
In this thesis we study the generation of randomness from quantum mechanical sources: quantum random number generators (QRNGs). Such devices are thought to provide a better quality of randomness than is possible with classical devices, but the issue is in need of more rigorous study. In Chapter 1 we provide the necessary background for the thesis from both computability and probability theory. We then present and extend some recent results providing a more theoretical grounding for the indeterminism in quantum mechanics. In particular, we show that sequences of quantum random bits are incomputable in the strongest sense: no bit can be provably computed in advance. In Chapter 2 we study in detail the use of von Neumann normalisation for QRNGs producing both finite strings and infinite sequences of bits; such normalisation methods are necessary in order to counter for experimental imperfections. We show that, in the case of a constantly biased source, this normalisation gives the desired uniform distribution. The effect of this normalisation on the (in)computability of the generated sequences is studied, and it is shown that algorithmic randomness and Borel normality are preserved, but e-randomness (and thus incomputability) is, in general, not. Experimental bounds for the extent of departure from the uniform distribution in the non-ideal case of a slowly drifting bias are derived. In Chapter 3 we propose a new QRNG which uses entangled photon pairs and is certified to produce incomputable bits by value indefiniteness; this is the first QRNG with explicit certification by a physical principle. The effects of various experimental imperfections are discussed in detail, and care is taken to make the proposed QRNG as robust as is possible to these. A robust method based on XORing the produced bitstrings together is proposed to further improve the quality of the distribution produced by the QRNG. Various improvements and optimisations to this scheme are discussed.
Our aim is to experimentally study the possibility of distinguishing between quantum sources of randomness--recently proved to be theoretically incomputable--and some well-known computable sources of pseudo-randomness. Incomputability is a necessary, but not sufficient "symptom" of "true randomness". We base our experimental approach on algorithmic information theory which provides characterizations of algorithmic random sequences in terms of the degrees of incompressibility of their finite prefixes. Algorithmic random sequences are incomputable, but the converse implication is false. We have performed tests of randomness on pseudo-random strings (finite sequences) of length 2322^{32}232 generated with software (Mathematica, Maple), which are cyclic (so, strongly computable), the bits of πππ, which is computable, but not cyclic, and strings produced by quantum measurements (with the commercial device Quantis and by the Vienna IQOQI group). Our empirical tests indicate quantitative differences, some statistically significant, between computable and incomputable sources of "randomness".
We propose what we believe to be a new approach to nondeterministic random-number generation. The randomness originated from the uncorrelated nature of consecutive laser pulses with Poissonian photon statistics and that of photon number detections is used to generate random bit, and the von Neumann correction method is used to extract the final random bit. This method is proved to be bias free in randomness generation, provided that the single photon detections are mutually independent. Further, it has the advantage in fast random bit generation, since no postprocessing is needed. A true random-number generator based on this method is realized, and its randomness is tested and guaranteed using three statistical test batteries.
Randomness is widely used in various fields, including encryption, statistical analysis, and numerical simulations. Randomness is also a fundamental resource in science and engineering. For such applications, we usually need to provide unbiased and independent random bits. This raises the issue of where to get these supposed random bits. Quantum random number generators (QRNGs) generate real random numbers based on the inherent randomness of quantum measurements. In practice, unfortunately, quantum randomness is inevitably mixed with classical randomness due to classical noise. Also, randomness is often correlated and biased. It is necessary to process the resulting raw bits sequence and convert them to good-quality output values that are as close to uniform distribution as possible. Random extractors are required for this. We analyze the randomness obtained by quantum random number generators as well as various examples of post-processing. We discuss the types of randomness extractors. Also, we use information-theoretically secure extractors and present improved novel randomness extraction.
Generation of random numbers is a central problem for many applications in the field of information processing, including, e.g., cryptography, in classical and quantum regime, but also mathematical modeling, Monte Carlo methods, gambling and many others. Both, the quality of the randomness and efficiency of the random numbers generation process are crucial for the most of these applications. Software produced pseudorandom bit sequences, though sufficiently quick, do not fulfill required randomness quality demands. Hence, the physical hardware methods are intensively developed to generate truly random number sequences for information processing and electronic security application. In the present paper we discuss the idea of the quantum random number generators. We also present a variety of tests utilized to assess the quality of randomness of generated bit sequences. In the experimental part we apply such tests to assess and compare two quantum random number generators, PQ4000KSI (of company ComScire US) and JUR01 (constructed in Wroclaw University of Science and Technology upon the project of The National Center for Research and Development) as well as a pseudorandom generator from the Mathematica Wolfram package. Finally, we present our new prototype of fully operative miniaturized quantum random generator JUR02 producing a random bit sequence with velocity of 1 Mb/s, which successfully passed standard tests of randomness quality (like NIST and Dieharder tests). We also shortly discuss our former concept of an entanglement-based quantum random number generator protocol with unconditionally secure public randomness verification.
Nies and Scholz introduced the notion of a state to describe an infinite sequence of qubits and defined quantum-Martin-Lof randomness for states, analogously to the well known concept of Martin-L\"of randomness for elements of Cantor space (the space of infinite sequences of bits). We formalize how 'measurement' of a state in a basis induces a probability measure on Cantor space. A state is 'measurement random' (mR) if the measure induced by it, under any computable basis, assigns probability one to the set of Martin-L\"of randoms. Equivalently, a state is mR if and only if measuring it in any computable basis yields a Martin-L\"of random with probability one. While quantum-Martin-L\"of random states are mR, the converse fails: there is a mR state, x which is not quantum-Martin-L\"of random. In fact, something stronger is true. While x is computable and can be easily constructed, measuring it in any computable basis yields an arithmetically random sequence with probability one. I.e., classical arithmetic randomness can be generated from a computable, non-quantum random sequence of qubits.