This paper is a comment on the paper "Quantum Mechanics and Algorithmic Randomness" was written by Ulvi Yurtsever \cite{Yurtsever} and the briefly explanation of the algorithmic randomness of quantum measurements results. There are differences between the computability of probability sources, ( which means there is an algorithm that can define the way that random process or probability source generates the numbers ) and the algorithmic randomness of the sequences or strings which are produced by a source. We may have the source without a computable algorithm for that but it can produce compressible or incompressible strings. For example, so far there is no computable algorithm that can define the abstract meaning of randomness even the easiest one, Bernoulli probability distribution. Historically and philosophically there many scientist believe the existence of the algorithm for a random process is a contradiction because in their opinion, in the definition of a random variable, implicitly assumed that there is no reason for the happening of an event and we just know the probabilities. There is however no need to enter into this matter here. As in the paper mentioned, all the algorithms for simulating a random process try to pass the statistical tests and be close to the abstract meaning of it.
This paper is a comment on the paper "Quantum Mechanics and Algorithmic Randomness" was written by Ulvi Yurtsever \cite{Yurtsever} and the briefly explanation of the algorithmic randomness of quantum measurements results. There are differences between the computability of probability sources, ( which means there is an algorithm that can define the way that random process or probability source generates the numbers ) and the algorithmic randomness of the sequences or strings which are produced by a source. We may have the source without a computable algorithm for that but it can produce compressible or incompressible strings. For example, so far there is no computable algorithm that can define the abstract meaning of randomness even the easiest one, Bernoulli probability distribution. Historically and philosophically there many scientist believe the existence of the algorithm for a random process is a contradiction because in their opinion, in the definition of a random variable, implicitly assumed that there is no reason for the happening of an event and we just know the probabilities. There is however no need to enter into this matter here. As in the paper mentioned, all the algorithms for simulating a random process try to pass the statistical tests and be close to the abstract meaning of it.
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 232 generated with software (Mathematica, Maple), which are cyclic (so, strongly computable), the bits of π, 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".
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.
The generation of high-quality random numbers is a cornerstone of numerous fields, including cryptography, simulation, and statistical modeling. Classical methods for random number generation often rely on deterministic algorithms that, despite their complexity, ultimately produce predictable sequences. This paper proposes a novel approach to random number generation that harnesses the unique properties of quantum mechanics, specifically quantum measurement. We argue that classical algorithms are fundamentally limited in their ability to produce true randomness. Our approach utilizes quantum measurement processes, projecting a qubit into a superposition state, to generate random numbers. The resulting sequences exhibit statistical properties superior to those attainable through classical methods, offering a potentially significant advancement in the field of randomness generation. We detail the theoretical underpinnings, outlining the core mechanism and highlighting the key advantages. This work establishes a framework for leveraging quantum phenomena to achieve truly random number generation. ---
Nies and Scholz defined quantum Martin-Löf randomness (q-MLR) for states (infinite qubitstrings). We define a notion of quantum Solovay randomness and show it to be equivalent to q-MLR using purely linear algebraic methods. Quantum Schnorr randomness is then introduced. A quantum analogue of the law of large numbers is shown to hold for quantum Schnorr random states. We introduce quantum-K, (QK) a measure of the descriptive complexity of density matrices using classical prefix-free Turing machines and show that the initial segments of weak Solovay random and quantum Schnorr random states are incompressible in the sense of QK. Several connections between Solovay randomness and K carry over to those between weak Solovay randomness and QK. We then define QKC, using computable measure machines and connect it to quantum Schnorr randomness. We then explore a notion of `measuring' a state. We formalize how `measurement' of a state induces a probability measure on the space of infinite bitstrings. A state is `measurement random' (mR) if the measure induced by it, under any computable basis, assigns probability one to the set of Martin-Löf randoms. I.e., measuring a mR state produces a Martin-Löf random bitstring almost surely. While quantum-Martin-Löf random states are mR, the converse fails: there is a mR state, ρ which is not quantum-Martin-Löf random. In fact, something stronger is true. While ρ is computable and can be easily constructed, measuring it in any computable basis yields an arithmetically random sequence with probability one. So, classical randomness can be generated from a computable state which is not quantum random. We conclude by studying the asymptotic von Neumann entropy of computable states.
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.
Randomness is a phenomena which we are confronted with all the time. Will it rain today? Will the train be on time? What present(s) will I receive at Christmas? But are such phenomena truly random?Good randomness is essential for many applications. For instance, cryptography, the art of hiding information from malicious parties, is only as good as the sources of randomness that underlie it. Quantum mechanics, the theory of microscopic phenomena, can only predict the probability of events. For instance quantum theory can only predict the probability that a radioactive nucleus will decay, not when the nucleus will decay. Does this mean that microscopic phenomena are truly random?By studying systems of two entangled particles, it can be shown both theoretically and experimentally that events at the microscopic scale are truly random, truly unpredictable. Beyond its philosophical implications, this result also has important potential applications. Indeed it implies that one can build random number generators whose output is random and certifies that the device works correctly. That is, if the random number generator malfunctions in some way, if the numbers it produces cease to be random, this will automatically be detected. By extending this idea, one could also build quantum cryptographic systems and quantum computers that certify that they work correctly. We discuss the perspectives for practical implementations.