Julian Schwinger provided to physics a mathematical reconstruction of quantum mechanics on the basis of the characteristics of sequences of measurements occurring at the atomic level of physical structure. The central component of this reconstruction is an algebra of symbols corresponding to quantum measurements, conceived of as discrete processes, which serve to relate experience to theory; collections of outcomes of identically circumscribed such measurements are attributed expectation values, which constitute the predictive content of the theory. The outcomes correspond to certain phase parameters appearing in the corresponding symbols, which are complex numbers, the algebra of which he finds by a process he refers to as 'induction'. Schwinger assumed these (individually unpredictable) phase parameters to take random, uniformly distributed definite values within a natural range. I have previously suggested that the 'principle of plenitude' may serve as a basis in principle for the occurrence of the definite measured values that are those members of the collections of measurement outcomes from which the corresponding observed statistics derive (Jaeger 2015Found. Phys.45, 806-819. (doi:10.1007/s10701-015-9893-6)). Here, I evaluate Schwinger's assumption in the context of recent critiques of the notion of randomness and explicitly relate the randomness of these phases with the principle of plenitude and, in this way, provide a fundamental grounding for the objective, physically irreducible probabilities, conceived of as graded possibilities, that are attributed to measurement outcomes by quantum mechanics.
Julian Schwinger provided to physics a mathematical reconstruction of quantum mechanics on the basis of the characteristics of sequences of measurements occurring at the atomic level of physical structure. The central component of this reconstruction is an algebra of symbols corresponding to quantum measurements, conceived of as discrete processes, which serve to relate experience to theory; collections of outcomes of identically circumscribed such measurements are attributed expectation values, which constitute the predictive content of the theory. The outcomes correspond to certain phase parameters appearing in the corresponding symbols, which are complex numbers, the algebra of which he finds by a process he refers to as 'induction'. Schwinger assumed these (individually unpredictable) phase parameters to take random, uniformly distributed definite values within a natural range. I have previously suggested that the 'principle of plenitude' may serve as a basis in principle for the occurrence of the definite measured values that are those members of the collections of measurement outcomes from which the corresponding observed statistics derive (Jaeger 2015Found. Phys.45, 806-819. (doi:10.1007/s10701-015-9893-6)). Here, I evaluate Schwinger's assumption in the context of recent critiques of the notion of randomness and explicitly relate the randomness of these phases with the principle of plenitude and, in this way, provide a fundamental grounding for the objective, physically irreducible probabilities, conceived of as graded possibilities, that are attributed to measurement outcomes by quantum mechanics.
Quantum mechanics is a physical theory supplying probabilities corresponding to expectation values for measurement outcomes. Indeed, its formalism can be constructed with measurement as a fundamental process, as was done by Schwinger, provided that individual measurements outcomes occur in a random way. The randomness appearing in quantum mechanics, as with other forms of randomness, has often been considered equivalent to a form of indeterminism. Here, it is argued that quantum randomness should instead be understood as a form of unpredictability because, amongst other things, indeterminism is not a necessary condition for randomness. For concreteness, an explication of the randomness of quantum mechanics as the unpredictability of quantum measurement outcomes is provided. Finally, it is shown how this view can be combined with the recently introduced view that the very appearance of individual quantum measurement outcomes can be grounded in the Plenitude principle of Leibniz, a principle variants of which have been utilized in physics by Dirac and Gell‐Mann in relation to the fundamental processes. This move provides further support to Schwinger's “symbolic” derivation of quantum mechanics from measurement.
The outcomes of quantum measurements are generally considered to be random, but despite the fact that this randomness is an important element in quantum information theory, its nature is not well understood. In this thesis, we study several issues relating to the origin and certification of quantum randomness and unpredictability. One of the key results in forming our understanding of quantum mechanics as an intrinsically indeterministic theory is the Kochen-Specker theorem, which shows the impossibility to consistently assign simultaneous noncontextual definite values to all quantum mechanical observables prior to measurement. However, the theorem, under the assumption that any definite values must be noncontextual, only strictly shows that some observables must be value indefinite. We strengthen this result, proving a stronger variant of the Kochen-Specker theorem showing that, under the same assumption, if a system is prepared in an arbitrary state j i, then every observable A is value indefinite unless j i is an eigenstate of A. The indeterministic nature of quantum measurements does little to explain how the quality of quantum randomness differs from classical randomness. We show that, subject to certain physical assumptions, a sequence of bits generated by the measurement of value indefinite observables is guaranteed, in the infinite limit, to be strongly incomputable. We further discuss how this can be used to build a quantum random number generator certified by value indefiniteness. Next, we study the notion of unpredictability, which is central to the concept of (quantum) randomness. In doing so, we propose a formal model of prediction that can be used to asses the predictability of arbitrary physical experiments. We investigate how the quantum features of value indefiniteness and complementarity can be used to certify different levels of unpredictability, and show that the outcome of a single measurement of a value indefinite quantum observable is formally unpredictable. Finally, we study the relation between this notion of unpredictability and the computability-theoretic certification of quantum randomness.
The ability to transfer information securely is essential in the modern world we live in. While the field of cryptography has long researched ways to do this, events over the last couple of years have brought the issues of secure information transfer squarely into the public eye. As a consequence, security and cryptography have quickly become issues of social and political importance, sparking debates about the values of privacy and the role of government. An essential ingredient of modern cryptography is the generation of randomness: cryptographic techniques are built on the premise that one has access to random bits. It is well known, however, that computers cannot produce algorithmically random sequences – that is, sequences with maximal algorithmic information content – but are doomed to produce ‘pseudorandomness’. This lack of randomness can be, and has been, exploited to compromise security, see [4]. The active field of quantum information theory has proposed approaches to provide supposedly ‘unbreakable’ security by exploiting various quantum phenomena. This security unfortunately relies on assumptions about the nature of quantum measurements and their ability to generate random bits. Anton Zeilinger summarises this by postulating that the simplest quantum systems, qubit, can hold only one bit of classical information [6]. This foundational principle is in line with a wider paradigm shift to view quantum information as an extension of classical information, but it is nonetheless unsatisfying to simply postulate this principle, given its importance in determining the practical advantages of quantum information theory. In order to understand better how quantum mechanics can help generate meaningful information, we instead look to relate the outcomes of quantum measurements to formal properties of the system based on more fundamental assumptions. Indeed, we have shown that a) some of the postulated properties of quantum information follow from the formal structure of the theory and b) a purely formal notion of information within a quantum world can generate, via measurement, meaningful and useful information in the macroscopic world (for example, in cryptography). The indeterminism of quantum measurements can be formalised via the notion of value indefiniteness. To explain this concept, let us consider an arbitrary quantum system and ask whether the outcome of a measurement of any observable quantity A (such as the energy of the system, its angular momentum (spin), etc.) is determined prior to the measurement. If this is the case, then we say the observable is value definite with value v(A); otherwise, the observable is value indefinite and v(A) is undefined. Mathematically, one can reduce all observable quantities to so-called projection observables, which can only take the values 0 or 1. Thus, the question of whether the outcome of several measurements can be simultaneously determined in advance can be rephrased in terms of the information ‘carried’ by a particular system in a definite quantum state. Classically, one expects that all quantities are determined in advance, and hence all observables are value definite. Quantum mechanically, however, the belief is that this is not the case, and the information content of quantum systems is limited. Formulating carefully the notion of value indefiniteness allows us to formalise the notion of (quantum) indeterminism; however, this doesn’t help clarify whether quantum systems are indeed value indefinite or not. Staying in this formal framework, the Kochen-Specker theorem [5] provides a first positive result, showing that at least some observables must be value indefinite if one makes the assumption known as non-contextuality, which states that any definite values that exist must be independent of other compatible measurements that may or may not be performed on the system. Under the same assumption, this theorem can be strengthened to show that only one single observable can have the definite value 1 (see [2]). Furthermore, only observables that can be measured simultaneously with this single one can have the definite value 0; the rest must all be value indefinite. Since the preparation of a system involves precisely ensuring that, usually via measurement, the system is in a definite state with respect to some desired observable, this result shows that no other incompatible observable can be value definite. That is, preparing a system in a definite state by making the ‘preparation’ observable value definite specifies completely the information content of the quantum system. This is an example of syntactical quantum information acquiring meaning at the level of the quantum itself. Can the syntactical information at the level of the quantum generate meaning outside the quantum, that is, at the macroscopic level? The results cited above hold in the Hilbert-space framework of quantum mechanics and are formulated only in terms of a syntactical notion of information. Their real importance becomes evident when one interprets them in the context of quantum measurements. Specifically, if we prepare a quantum system in a known state, they allow us to ‘locate’ observables which we can measure, but which are value indefinite; that is, observables whose measurement results are not specified by any pre-existing property of the quantum system. Furthermore, with respect to a mathematical model of unpredictability which we developed in [3], the results of these measurements can be shown to be absolutely unpredictable. In this way we get a mathematical explanation and justification of the largely accepted intuition that quantum mechanics is inherently unpredictable, and that this unpredictability arises from the phenomenon of indefiniteness within the quantum world. Furthermore, if one considers a hypothetical infinite sequence generated by the repeated measurement of a quantum value indefinite observable, one can prove that these sequences must be strongly incomputable, technically ‘bi-immune’ [1]. Such sequences cannot be generated by any Turing machine or classical computer, showing that value indefiniteness leads to a clear classical/quantum split in a purely algorithmic context. Importantly, bi-immunity is a property of observed, macroscopic quantities, quite separate from the quantum framework in which the value indefiniteness is formalised. This form of macroscopic meaning created from the lack of syntactic information is precisely the scenario that quantum random number generators try to create, and which is essential for the development and certification of quantum cryptographic systems. The fact that the macroscopic information created goes beyond anything classically obtainable serves as a valuable practical resource, outside of and removed from the quantum formalism. To conclude, quantum information creates meaning within the quantum and via measure- ment, the lack of information within the quantum, creates meaning and valuable information at the macroscopic level. Acknowledgments The second author thanks Prof. S. Marcus for useful conversations on quantum information theory. This work was supported in part by Marie Curie FP7-PEOPLE-2010-IRSES Grant RANPHYS. References and Notes Abbott, A.A.; Calude, C.S.; Svozil, K. Strong Kochen-Specker theorem and incomputability of quantum randomness. Physical Review A 2012, 86, 062109. Abbott, A.A.; Calude, C.S.; Svozil, K. Value-indefinite observables are almost everywhere. Physical Review A 2013, 89, 032109. Abbott, A.A.; Calude, C.S.; Svozil, K. On the unpredictability of individual quantum measurement outcomes. CDMTCS Research Report 2014, 458. Calude, C.S. Quantum randomness & cryptology. CyberTalk 2013, 3, 42–43. Kochen, S.; Specker, E. The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics 1967, 17, 59–87. Zeilinger, A. A foundational principle for quantum mechanics. Foundations of Physics 1999, 29(4),631–643.
When investigating various interpretations of quantum mechanics one notices that each interpretation contains an element which escapes a complete and full description. This element is always associated with the stochasticity of the individual event in the quantum measurement process. It appears that the implications of this limit to any description of the world has not been sufficiently appreciated with notable exceptions of, for example, Heisenberg, Pauli and Wheeler. If we assume that a deeper foundation of quantum mechanics is possible, the question arises which features such a philosophical foundation might have. It is suggested that the objective randomness of the individual quantum event is a necessity of a description of the world in view of the significant influence the observer in quantum mechanics has. It is also suggested that the austerity of the Copenhagen interpretation should serve as a guiding principle in a search for deeper understanding. In recent years, a significant increase of interest in the foundations of quantum mechanics can be observed. This increase is certainly related to the immense progress made during the last two decades in experiments with individual quantum systems. This technological progress has made it possible to really perform more and more of the famous gedanken experiments which were so important in the early days of the theory. Not only did these experiments confirm the theory in every detail, they also opened up avenues for new experimental directions which might even lead some day to new technology. While such applications are certainly still far away, areas like quantum cryptography[1], quantum computation[2] and interaction-free measurement[3], just to name a few examples, certainly deserve the attention they receive. All this activity has also drawn more attention to the problems of understanding and interpreting quantum mechanics. It seems that there, a definitive consensus has not yet been reached. As witnesses quote authorities like Feynman as saying[4]: I think can safely say that nobody today understands quantum mechanics, whereby he apparently included himself, or like Roger Penrose[5], who describes his opinion remarking that while the theory agrees incredibly well with experiment and while it is of profound mathematical beauty, it makes absolutely no sense. Why did even physicists who have contributed so significantly to quantum theory Feynman was even rewarded the Nobel Prize for one of its mathematical formulations choose such emphatic and strong formulations? Why is it that the average physicist hears very little about such problems during his/her education? Why is the comprehension of the theory very often focused on formalism while questions that probe for deeper meaning are usually not tackled? To take a closer look at the situation, let us turn to a brief analysis of the interpretation of quantum mechanics. We note that there are at least two levels of interpreting a formalism, a physical theory. The first, basic, level supplies the rules that determine which element of the formalism corresponds to which measurable quantity or to which observable fact in a concrete experimental situation. Theses rules are a large, mostly not explicated but only implicit, set of instructions. These instructions concern the manner of working with the statements of the theory in order to obtain predictions for future experiments or events, and they concern the instructions on how to proceed in experiment in order to demonstrate or test a theoretical
Quantum mechanics has several deficiencies as a complete theoretical description of the measurement process. Among them is the fact that the quantum mechanical description of correlations between the single measurements of a sequence is quite problematic. A single measurement is defined to be a preparation followed by an observation. In particular, one feels that an infinite sequence of such single measurements which corresponds to the measurement of a question O on a state η where η does not lie entirely in an eigenspace of O should generate a random output sequence. However, quantum mechanics seems to say nothing about this. In this paper, physical theories are defined in such a manner that correlations between single measurements are explicitly included. In particular, a physical theory is considered to be a mapping U, with domain in the set [Qsτ] of infinite instruction strings for carrying out infinite sequences of single measurements and range in the set of probability measures defined on A, the usual σ algebra of subsets of Ω. Ω is the set of all infinite sequences of natural numbers. A fundamental property which any valid physical theory must satisfy is that it agrees with experiment. It is proposed and discussed here that much of the intuitive meaning of agreement between a theory U and experiment with respect to H is given by the statement ∀Qsτ [U(Qsτ) defined⇒E(H,U(Qsτ),ψQsτ)],where U(Qsτ) is the probability measure U associates with the infinite instruction string Qsτ and ψQsτ is the outcome sequence obtained by carrying out Qsτ. E(H, U(Qsτ),ψQsτ) is the statement that all formulas in H with one free sequence variable which are true on Ω almost everywhere with respect to U(Qsτ) are true for ψQsτ. H is a subclass of the class of all formulas in a formal language L. A theorem is proved which states that, if U(Qsτ) corresponds to a nontrivial product probability measure and U H-agrees with experiment, then the outcome sequence ψQsτ is H-random. H-randomness is defined here in terms of the statement E(H, P, φ). Another property of a valid physical theory, which is defined here, is that, for some Qsτ, U(Qsτ) must be determinable on much of AH from ψQsτ. Sufficient conditions for this property to hold are given. AH is the class of all H definable subsets of Ω. Some properties of the statement E(H, P, φ) are given. Among other things, it is proved that, if E(H, P, φ) holds and P is a nontrivial probability measure on A, then φ is not definable in H.
As physics searches for invariants in observations, this paper looks for invariants of probabilistic observation without assuming physical structure. Structure emerges from the basic assumption of science that new information shall lead to more accurate knowledge of the invariants. This leads to statistically unique random variables for expressing observed information: Complex probability amplitudes. Predictions are also just random variables computed from observed data, and must become more accurate with more observations as input. This singles out the quantum mechanical superposition principle. The external conditions of a probabilistic experiment can themselves be monitored at the most detailed level, resulting in observation of coicidence probabilities. The invariants of any multi-coincidence experiment are the same as those of a one-event experiment with the same number of possible outcomes. An observable probability turns out to be controllable by two independent experimental conditions, naturally parametrized as a direction in a 3-dimensional space. In summary, the probabilistic paradigm itself defines a unique method of forming concepts and making predictions. The method appears irrefutable within probability, because, whenever a prediction turns out wrong the existence of an as yet unmonitored condition is postulated, and a formal way to incoroporate it is shown. The Hilbert space formalism of quantum theory seems to be isomorphic to this method.