Randomness plays a central role in the quantum mechanical description of our interactions. We review the relationship between the violation of Bell inequalities, non signaling and randomness. We discuss the challenge in defining a random string, and show that algorithmic information theory provides a necessary condition for randomness using Borel normality. We close with a view on incomputablity and its implications in physics.
Randomness plays a central role in the quantum mechanical description of our interactions. We review the relationship between the violation of Bell inequalities, non signaling and randomness. We discuss the challenge in defining a random string, and show that algorithmic information theory provides a necessary condition for randomness using Borel normality. We close with a view on incomputablity and its implications in physics.
Abstract This is a review of the issue of randomness in quantum mechanics, with special emphasis on its ambiguity; for example, randomness has different antipodal relationships to determinism, computability, and compressibility. Following a (Wittgensteinian) philosophical discussion of randomness in general, I argue that deterministic interpretations of quantum mechanics (like Bohmian mechanics or ’t Hooft’s Cellular Automaton interpretation) are strictly speaking incompatible with the Born rule. I also stress the role of outliers, i.e. measurement outcomes that are not 1-random. Although these occur with low (or even zero) probability, their very existence implies that the no-signaling principle used in proofs of randomness of outcomes of quantum-mechanical measurements (and of the safety of quantum cryptography) should be reinterpreted statistically, like the second law of thermodynamics. In three appendices I discuss the Born rule and its status in both single and repeated experiments, review the notion of 1-randomness (or algorithmic randomness) that in various guises was investigated by Kolmogorov and others and treat Bell’s (Physics 1:195–200, 1964) Theorem and the Free Will Theorem with their implications for randomness.
Randomness is fundamental in quantum theory, with many philosophical and practical implications. In this paper we discuss the concept of algorithmic randomness, which provides a quantitative method to assess the Borel normality of a given sequence of numbers, a necessary condition for it to be considered random. We use Borel normality as a tool to investigate the randomness of ten sequences of bits generated from the differences between detection times of photon pairs generated by spontaneous parametric downconversion. These sequences are shown to fulfil the randomness criteria without difficulties. As deviations from Borel normality for photon-generated random number sequences have been reported in previous work, a strategy to understand these diverging findings is outlined.
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.
Contextuality means non-existence of a joint distribution for random variables recorded under mutually incompatible conditions, subject to certain constraints imposed on how the identity of these variables may change across these conditions. In simple quantum systems contextuality is indicated by violations of Bell-type or Leggett-Garg-type inequalities. These inequalities, however, are predicated on the assumption of no-signaling, defined as invariance of the distributions of measurement results with respect to other (e.g., earlier in time) measurements' settings. Signaling makes the inequalities inapplicable: a non-signaling system with any degree of contextuality, however high, loses any relation to this concept as soon as it exhibits any degree of signaling, however small. This is unsatisfactory. We describe a principled way of defining and measuring contextuality in arbitrary systems with random outputs, whether signaling is absent or present.
This is a review of the issue of randomness in quantum mechanics, with special emphasis on its ambiguity; for example, randomness has different antipodal relationships to determinism, computability, and compressibility. Following a (Wittgensteinian) philosophical discussion of randomness in general, I argue that deterministic interpretations of quantum mechanics (like Bohmian mechanics or 't Hooft's Cellular Automaton interpretation) are strictly speaking incompatible with the Born rule. I also stress the role of outliers, i.e. measurement outcomes that are not 1-random. Although these occur with low (or even zero) probability, their very existence implies that the no-signaling principle used in proofs of randomness of outcomes of quantum-mechanical measurements (and of the safety of quantum cryptography) should be reinterpreted statistically, like the second law of thermodynamics. In appendices I discuss the Born rule and its status in both single and repeated experiments, and review the notion of 1-randomness \nintroduced by Kolmogorov, Chaitin, Martin-Lo"f, Schnorr, and others.
Einstein was never satisfied with quantum mechanics. He argued that quantum mechanics was incomplete for two main reasons; it violated the locality principle and the separability principle. The violation of separability is an unavoidable consequence of quantum interactions. Non-separability can be seen in quantum entanglement. Non-locality, however, is more controversial. Einstein and his associates published the EPR paper in order to argue for the incompleteness of quantum mechanics. Years later, John Bell formulated what became known as the Bell Inequalities in response to the EPR paper. The Bell Inequalities are seen as a major obstacle for quantum locality. I will argue that non-locality is not a necessary implication of the Bell Inequalities. The Bell Inequalities were developed using Bell’s locality requirement as a major premise. Bell’s locality requirement can be described in terms of two conditions, parameter independence and outcome independence. A violation of either condition will lead to a violation of the Bell inequalities. Parameter independence is not violated by the results of experimental quantum physics. So, it can be argued that violations of the Bell inequalities are caused by the violation of outcome independence. Such a violation of outcome independence does not imply non-locality if we accept some form of holism or non-separability. Thus, by including some form of holism or non-separability into our picture of the quantum realm we can develop a theory that does not conflict with locality. This paper will discuss different types of holism and non-separability and how they can be used to help understand quantum phenomena.