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 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.
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.
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.
Random numbers are required for a variety of applications from secure communications to Monte Carlo simulation. Yet randomness is an asymptotic property, and no output string generated by a physical device can be strictly proven to be random. We report an experimental realization of a quantum random number generator (QRNG) with randomness certified by quantum contextuality and the Kochen-Specker theorem. The certification is not performed in a device-independent way but through a rigorous theoretical proof of each outcome being value indefinite even in the presence of experimental imperfections. The analysis of the generated data confirms the incomputable nature of our QRNG.
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.
Random Number Generators (RNGs) are used in various applications, including cryptography, scientific simulations, gaming, and virtual reality. The quality of these random numbers is crucial, especially for computational approaches that rely on unpredictability to produce accurate results. In this thesis, we perform a comprehensive assessment of the quality of a novel Quantum Random Number Generator (QRNG) based on single photon. QRNGs leverage quantum mechanical properties to produce truly random numbers. This ensures high entropy, unpredictability, and security, making QRNGs particularly useful in fields that require high-quality randomness.We evaluate the statistical properties of strings generated by our QRNG and then its practical application by comparing it to traditional Pseudo-Random Number Generators (PRNGs) in Monte Carlo Integration (MCI), a technique that estimates integrals using random sampling. In particular, we use two statistical randomness test suits, namely NIST and Dieharder, to assess the statistical properties of our QRNG. All tests within these tests were passed, confirming that the corresponding QRNG achieves true randomness with sufficient high entropy. Next, we use our QRNG-generated numbers to seed Monte Carlo Integration (MCI) and compare it against results obtained using classical PRNG. The QRNG-based MCI demonstrated more accurate and stable numerical estimates, showing lower error rates and improved convergence compared to PRNG-based simulations.These findings highlight the advantages of QRNGs in enhancing numerical accuracy and computational security. While QRNGs may have slower generation rates than PRNGs, their superior randomness quality allows for more reliable Monte Carlo-based simulations, suggesting that fewer samples may be needed to achieve the same level of accuracy compared to classical PRNGs. This research underscores the potential of QRNG-powered numerical methods, paving the way for their integration into more advanced applications, including scientific computing, financial modeling, and cryptographic security.
Random numbers are of vital importance in today’s world and used for example in many cryptographical protocols to secure the communication over the internet. The generators producing these numbers are Pseudo Random Number Generators (PRNGs) or True Random Number Generators (TRNGs). A subclass of TRNGs are the Quantum based Random Number Generators (QRNGs) whose generation processes are based on quantum phenomena. However, the achievable quality of the numbers generated from a practical implementation can differ from the theoretically possible. To ease this negative effect post-processing can be used, which contains the use of extractors. They extract as much entropy as possible from the original source and produce a new output with better properties. The quality and the different properties of a given output can be measured with the help of statistical tests. In our work we examined the effect of different extractors on two QRNG outputs and found that witg the right extractor we can improve their quality.