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 $QK_C$, 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.
Quantum Martin-Löf randomness (q-MLR) for infinite qubit sequences was introduced by Nies and Scholz. We define a notion of quantum Solovay randomness which is equivalent to q-MLR. The proof of this goes through a purely linear algebraic result about approximating density matrices by subspaces. We then show that random states form a convex set. Martin-Löf absolute continuity is shown to be a special case of q-MLR. Quantum Schnorr randomness is 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 $QK$. Many properties enjoyed by prefix-free Kolmogorov complexity ($K$) have analogous versions for $QK$; notably a counting condition. Several connections between Solovay randomness and $K$, including the Chaitin type characterization of Solovay randomness, carry over to those between weak Solovay randomness and $QK$. We work towards …
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 compres…
The education system for students in physics suffers (worldwide) from the absence of a deep course in probability and randomness. This is the real problem for students interested in quantum information theory, quantum optics, and quantum foundations. Here the primitive treatment of probability and randomness may lead to deep misunderstandings of theory and wrong interpretations of experimental results. Since during my visits (in 2013 and 2014) to IQOQI a number of students (experimenters!) asked me permanently about foundational problems of probability and randomness, especially inter-relation…
Ideal quantum random number generators (QRNGs) can produce algorithmically random and thus incomputable sequences, in contrast to pseudo-random number generators. However, the verification of the presence of algorithmic randomness and incomputability is a nontrivial task. We present the results of a search for algorithmic randomness and incomputability in the output from two different QRNGs, performed by applying tests based on the Solovay-Strassen test of primality and the Chaitin-Schwartz theorem. The first QRNG uses measurements of quantum vacuum fluctuations. The second QRNG is based on po…
Quantum systems are the ultimate touchstone for the production of random sequences of numbers. Spatially spread entangled systems allow the generation of identical random sequences in remote locations. The impossibility of observing a quantum system, without disturbing it, ensures that the messages encoded using these sequences cannot be eavesdropped. This is the basis of Quantum Key Distribution. It is then of crucial importance knowing whether the sequences generated in the practice by spatially spread entangled states are truly random, or not. Yet, that knowledge is not immediate. One of th…
In the mentioned paper we presented results of the estimation of Kolmogorov complexity of sequences of random numbers generated in a famous Bell's experiment, aimed to study the security of QKD. We focused on series of time differences between successive detections of coincidences, and found that randomness cannot be taken for granted. It was then criticized that the theorems that demonstrate the randomness of series produced in Bell's experiments involve series of measurement outcomes, not of measurement times. Here we reply to this objection and present data of series of outcomes, showing th…
The notion of probability plays a crucial role in quantum mechanics. It appears in quantum mechanics as the Born rule. In modern mathematics which describes quantum mechanics, however, probability theory means nothing other than measure theory, and therefore any operational characterization of the notion of probability is still missing in quantum mechanics. In this paper, based on the toolkit of algorithmic randomness, we present a refinement of the Born rule, as an alternative rule to it, for specifying the property of the results of quantum measurements in an operational way. Algorithmic ran…
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 …
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 …
University of California, Irvine（加州大学尔湾分校）；Halıcıoğlu Data Science Institute and Department of Philosophy, University of California, San Diego（加州大学圣地亚哥分校哈奇奥卢数据科学研究所与哲学系）；Department of Logic and Philosophy of Science, University of California, Irvine（加州大学尔湾分校逻辑与科学哲学系）
Algorithmic Randomness, Exchangeability, and the Principal Principle
Halıcıoğlu Data Science Institute and Department of Philosophy, University of California, San Diego（加州大学圣地亚哥分校哈利奇奥格鲁数据科学研究所与哲学系）；Department of Logic and Philosophy of Science, University of California, Irvine（加州大学尔湾分校逻辑与科学哲学系）
In contrast with software-generated randomness (called pseudo-randomness), quantum randomness is provable incomputable, i.e.\ it is not exactly reproducible by any algorithm. We provide experimental evidence of incomputability --- an asymptotic property --- of quantum randomness by performing finite tests of randomness inspired by algorithmic information theory.
Different observers do not have to agree on how they identify a quantum system. We explore a condition based on algorithmic complexity that allows a system to be described as an objective "element of reality". We also suggest an experimental test of the hypothesis that any system, even much smaller than a human being, can be a quantum mechanical observer.
Randomness plays a central rol 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.
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 w…
本文给出了量子Kolmogorov复杂性的定义。在经典设置中，字符串的柯尔莫哥洛夫复杂度是可以产生该字符串作为其输出的最短程序的长度。它是字符串中包含的固有随机性（或信息）的度量。我们将量子比特串的量子柯尔莫哥洛夫复杂度定义为通用量子图灵机的最短量子输入的长度，该通用量子图灵机以高保真度产生初始量子比特串。Vitanyi的定义（Proceedings of the 15th IEEE Annual Conference on Computational Complexity，2000）测量经典信息的量，而我们考虑量子比特串中的量子信息的量。我们认为，我们的定义是自然的，是一个准确的表示量的量子信息包含在一个量子态。
The generation of series of random numbers is an important and difficult problem. Appropriate measurements on entangled states have been proposed as the definitive solution, based on the impossibility of exploiting quantum non locality to get faster than light signaling. There is a controversy regarding what is preferable to produce series with utilizable randomness in practice, high or low entanglement. We prepare biphotons with three different levels of entanglement, easy entangled, marginally entangled and no entangled. Randomness is evaluated, independently of the quantum non locality argu…
