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.
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.
In this Thesis, we develop the only class of QRNGs, to date, that has been proved to be better than any PRNG. This type of QRNG is based on a located form of the Kochen-Specker Theorem. This certification is unique because i) the assumptions used have been experimentally validated, ii) the robustness of measurements was proved theoretically, iii) the quality of quantum random digits generated with this class of QRNG was experimentally shown to be better than that of the best PRNGs, demonstrating for the first time the strong effects of incomputability on the outputs of a quantum random number generator. Finally, we propose a method using high-quality randomness for inverting a class of one-way functions in a pre-given time.
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 paper we propose a quantum random number generator (QRNG) that uses an entangled photon pair in a Bell singlet state and is certified explicitly by value indefiniteness. While ‘true randomness’ is a mathematical impossibility, the certification by value indefiniteness ensures that the quantum random bits are incomputable in the strongest sense. This is the first QRNG setup in which a physical principle (Kochen–Specker value indefiniteness) guarantees that no single quantum bit that is produced can be classically computed (reproduced and validated), which is the mathematical form of bitwise physical unpredictability. We discuss the effects of various experimental imperfections in detail: in particular, those related to detector efficiencies, context alignment and temporal correlations between bits. The analysis is very relevant for the construction of any QRNG based on beam-splitters. By measuring the two entangled photons in maximally misaligned contexts and using the fact that two bitstrings, rather than just one, are obtained, more efficient and robust unbiasing techniques can be applied. We propose a robust and efficient procedure based onXORing the bitstrings together – essentially using one as a one-time-pad for the other – to extract random bits in the presence of experimental imperfections, as well as a more efficient modification of the von Neumann procedure for the same task. We also discuss some open problems.
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.
This paper presents the design and theoretical framework for a quantum random number generator (QRNG) rooted in the principles of quantum mechanics. The core innovation lies in leveraging the inherent randomness associated with quantum measurement, specifically the uncertainty principle, to generate truly random numbers, a significant departure from the limitations of classical random number generators. The proposed system utilizes a measurement process involving a quantum system, such as a photon polarization or a quantum bit, to produce a sequence of random bits. The generated bits are then processed to eliminate bias and ensure high uniformity. The theoretical analysis focuses on quantifying the randomness inherent in the measurement process and demonstrating the potential for generating high-quality random numbers. This work addresses the growing demand for secure communication, cryptography, and simulation applications that require truly random numbers, offering a fundamentally different approach compared to traditional methods.
Quantum random number generators can provide genuine randomness by appealing to the fundamental principles of quantum mechanics.In general, a physical generator contains two parts-a randomness source and its readout.The source is essential to the quality of the resulting random numbers; hence, it needs to be carefully calibrated and modeled to achieve information-theoretical provable randomness.However, in practice, the source is a complicated physical system, such as a light source or an atomic ensemble, and any deviations in the real-life implementation from the theoretical model may affect the randomness of the output.To close this gap, we propose a source-independent scheme for quantum random number generation in which output randomness can be certified, even when the source is uncharacterized and untrusted.In our randomness analysis, we make no assumptions about the dimension of the source.For instance, multiphoton emissions are allowed in optical implementations.Our analysis takes into account the finite-key effect with the composable security definition.In the limit of large data size, the length of the input random seed is exponentially small compared to that of the output random bit.In addition, by modifying a quantum key distribution system, we experimentally demonstrate our scheme and achieve a randomness generation rate of over 5 × 10 3 bit=s.