A new definition of random events is considered in the context of quantum random bit generators. It generalizes the definition of randomness by von Mises and overcomes its deficiencies. We proposed to eliminate the rule for choosing an acceptable place using the characteristic matrix of a random experiment. According to the new model of randomness random events forms an atomic, complete and fully distributive Boolean algebra. It is shown that the probability distribution generated by random variables in this model is not a measure, but only a finite-additive function of events in the case of continuous rational or real random variables. Numerical experiments with quantum random number generators were carried out. Their results have shown that the model is consistent with experimental observations.
A new definition of random events is considered in the context of quantum random bit generators. It generalizes the definition of randomness by von Mises and overcomes its deficiencies. We proposed to eliminate the rule for choosing an acceptable place using the characteristic matrix of a random experiment. According to the new model of randomness random events forms an atomic, complete and fully distributive Boolean algebra. It is shown that the probability distribution generated by random variables in this model is not a measure, but only a finite-additive function of events in the case of continuous rational or real random variables. Numerical experiments with quantum random number generators were carried out. Their results have shown that the model is consistent with experimental observations.
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.
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.
The influence of “Grundbegriffe” by A. N. Kolmogorov (published in 1933) on education in the area of probability and its impact on research in stochastics cannot be overestimated. We would like to point out three aspects of the classical probability theory “calling for” an upgrade: (i) classical random events are black-and-white (Boolean); (ii) classical random variables do not model quantum phenomena; (iii) basic maps (probability measures and observables { dual maps to random variables) have very different “mathematical nature”. Accordingly, we propose an upgraded probability theory based on Łukasiewicz operations (multivalued logic) on events, elementary category theory, and covering the classical probability theory as a special case. The upgrade can be compared to replacing calculations with integers by calculations with rational (and real) numbers. Namely, to avoid the three objections, we embed the classical (Boolean) random events (represented by the f0; 1g-valued indicator functions of sets) into upgraded random events (represented by measurable {0; 1}-valued functions), the minimal domain of probability containing “fractions” of classical random events, and we upgrade the notions of probability measure and random variable.
Abstract The nature of randomness and constructive and provable methods to obtain (extract) it from observations of physical systems are discussed. True randomness, which exists only in a microcosm in the quantum-mechanical description of physical systems, is a fundamental property of quantum systems, which manifests itself in the outcomes of measurements upon quantum systems. The classical description of physical systems does not include any randomness and, in fact, it is introduced ‘manually’ by means of uncertainty — unknown initial conditions. Methods to really ‘feel’ quantum randomness are discussed using the example of a quantum device, a random number generator. Issues related to the ‘proof’ of randomness — testing of numerical sequences — are reviewed, and logical constructions that underlie such testing are analyzed. A mathematical apparatus is used to this end, which does not require special academic training, so standard knowledge from university courses on quantum mechanics and probability theory is sufficient. The authors aim to track a unified logical path from the origin of randomness in the quantum domain to its extraction, physical implementation, and testing.
The last few years have been characterized by a tremendous development of quantum information and probability and their applications, including quantum computing, quantum cryptography, and quantum random generators. In spite of the successful development of quantum technology, its foundational basis is still not concrete and contains a few sandy and shaky slices. Quantum random generators are one of the most promising outputs of the recent quantum information revolution. Therefore, it is very important to reconsider the foundational basis of this project, starting with the notion of irreducible quantum randomness. Quantum probabilities present a powerful tool to model uncertainty. Interpretations of quantum probability and foundational meaning of its basic tools, starting with the Born rule, are among the topics which will be covered by this issue. Recently, quantum probability has started to play an important role in a few areas of research outside quantum physics—in particular, quantum probabilistic treatment of problems of theory of decision making under uncertainty. Such studies are also among the topics of this issue.
In 1933, Kolmogorov synthesized the basic concepts of probability that were in general use at the time into concepts and deductions from a simple set of axioms that said probability was a σ-additive function from a boolean algebra of events into [0, 1]. In 1932, von Neumann realized that the use of probability in quantum mechanics required a different concept that he formulated as a σ-additive function from the closed subspaces of a Hilbert space onto [0,1]. In 1935, Birkhoff & von Neumann replaced Hilbert space with an algebraic generalization. Today, a slight modification of the Birkhoff-von Neumann generalization is called “quantum logic”. A central problem in the philosophy of probability is the justification of the definition of probability used in a given application. This is usually done by arguing for the rationality of that approach to the situation under consideration. A version of the Dutch book argument given by de Finetti in 1972 is often used to justify the Kolmogorov theory, especially in scientific applications. As von Neumann in 1955 noted, and his criticisms still hold, there is no acceptable foundation for quantum logic. While it is not argued here that a rational approach has been carried out for quantum physics, it is argued that (1) for many important situations found in behavioral science that quantum probability theory is a reasonable choice, and (2) that it has an arguably rational foundation to certain areas of behavioral science, for example, the behavioral paradigm of Between Subjects experiments.