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
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 is usually defined in terms of some loosely connected axioms and rules. Such a foundation is far from the beauty of, e.g., the `principles' underlying classical mechanics. Motivated, in addition, by notorious interpretation problems, there have been numerous attempts to modify or `complete' quantum mechanics. A first attempt was based on so-called hidden variables; its proponents essentially tried to expel the non-classical nature of quantum mechanics. More recent proposals intend to complete quantum mechanics not within mechanics proper but on a `higher (synthetic) level'; by means of a combination with gravitation theory (R Penrose), with quantum information theory (C M Caves, C A Fuchs) or with psychology and brain science (H P Stapp). I think it is fair to say that in each case the combination is with a subject that, per se , suffers from a very limited understanding that is even more severe than that of quantum mechanics. This was acceptable, though, if it could convincingly be argued that scientific progress desperately needs to join forces. Quantum mechanics of a closed system was a beautiful and well understood theory with its respective state being presented as a point on a deterministic trajectory in Liouville space---not unlike the motion of a classical N -particle system in its 6N -dimensional phase-space. Unfortunately, we need an inside and an outside view, we need an external reference frame, we need an observer. This unavoidable partition is the origin of most of the troubles we have with quantum mechanics. A pragmatic solution is introduced in the form of so-called measurement postulates: one of the various incompatible properties of the system under consideration is supposed to be realized (i.e. to become a fact, to be defined without fundamental dispersion) based on `instantaneous' projections within some externally selected measurement basis. As a result, the theory becomes essentially statistical rather than deterministic; furthermore there is an asymmetry between the observed and the observing. This is the point where consciousness may come in. Complemented by an introduction and several appendices, Henry Stapp's book consists essentially of three parts: theory, implications, and new developments. The theory part gives a very readable account of the Copenhagen interpretation, some aspects of a psychophysical theory, and, eventually, hints towards a quantum foundation of the brain--mind connection. The next part, `implications', summarizes some previous attempts to bridge the gap between the working rules of quantum mechanics and their possible consequences for our understanding of this world (Pauli, Everett, Bohm, Heisenberg). The last section, `new developments', dwells on some ideas about the conscious brain and its possible foundation on quantum mechanics. The book is an interesting and, in part, fascinating contribution to a field that continues to be a companion to `practical' quantum mechanics since its very beginning. It is doubtful whether such types of `quantum ontologies' will ever become (empirically) testable; right now one can hardly expect more than to be offered some consistent `grand picture', which the reader may find more or less acceptable or even rewarding. Many practicing quantum physicists, though, will remain unimpressed. The shift from synthetic ontology to analytic ontology is the foundation of the present work. This means that fundamental wholes are being partitioned into their ontologically subordinate components by means of `events'. The actual event, in turn, is an abrupt change in the Heisenberg state describing the quantum universe. The new state then defines the tendencies associated with the next actual event. To avoid infinite regression in terms of going from one state of tendencies to the next, consciousness is there to give these events a special `feel', to provide a status of `intrinsic actuality'. The brain of an alert human observer is similar in an important way to a quantum detection device: it can amplify small signals to large macroscopic effects. On the other hand, actual events are not postulated to occur exclusively in brains. They are more generally associated with the formation of records. Records are necessarily part of the total state of the universe: it is obvious that the state of the universe cannot undergo a Schrödinger dynamics and at the same time record its own history. `The full universe consists therefore of an exceedingly thin veneer of relatively sluggish, directly observable properties resting on a vast ocean or rapidly fluctuating unobservable ones.' The present ideas also bear on how the world should be seen to develop. While conventional cosmology encounters problems as to how to define the intial conditions, which would enter the governing equations of motion, here `the boundary conditions are set not at some initial time, but gradually by a sequence of acts that imposes a sequence of constraints. After any sequence of acts there remains a collection of possible worlds, some of which will be eliminated by the next act.' Connected with those acts is `meaning': there has always been some speculation about the special significance of local properties in our understanding of the world. One could argue that correlations (even the quantum correlations found, e.g., in the EPR-experiments) were as real as anything else. But also Stapp stresses the special role of locality: the `local observable properties, or properties similar to them are the natural, and perhaps exclusive, carriers of meaning in the quantum universe. From this point of view the quantum universe tends to create meaning.' This sounds like an absolute concept: meaning not with respect to something else, but defined intrinsically---not easy to digest. The role of consciousness in the developing quantum universe requires more attention. `The causal irrelevance of our thoughts within classical physics constitutes a serious deficiency of that theory, construed as a description of reality.' This is taken to be entirely different within quantum mechanics. `The core idea of quantum mechanics is to describe our activities as knowledge-seeking and knowledge-using agents.' `21st century science does not reduce human beings to mechanical automata. Rather it elevates human beings to agents whose free choices can, according to the known laws, actually influence their behaviour.' An example with respect to perception is discussed: `Why, when we look at a triangle, do we experience three lines joined at three points and not some pattern of neuron firings?' The brain `does not convert an actual whole triangle into some jumbled set of particle motions; rather it converts a concatenation of separate external events into the actualization of some single integrated pattern of neural activity that is congruent to the perceived whole triangle.' How convincing is this proposal? It is hard to tell. I think Henry Stapp did a good job, but there are tight limitations to any such endeavour. Quantum mechanics is often strange indeed, but it also gives rise to our classical world around us. For the emergence of classicality jumps and measurement projections (the basic phenomena connected with those fundamental events of choice) are not needed. Therefore, I doubt whether the explanation of the evolution of our world really allows (or requires) that much free choice. On the other hand, most scientist will agree that empirical science was not possible without free will: we could not ask independent questions if this asking was part of a deterministic trajectory. The fact that the result of a quantum measurement is indeterminate (within given probabilities) does certainly not explain free will. How about the type of measurment? The experimentalist will have to assume that he can select the pertinent observable within some limits. But given a certain design the so-called pointer basis (producing stable measurement results) is no longer a matter of free choice. `The main theme of classical physics is that we live in a clocklike universe.' Today it is often assumed that the universe was a big (quantum-) computer or a cellular automaton. Many would be all too happy to leave that rather restrictive picture behind. But where to go? Stapp suggests giving consciousness a prominent role: `The most profound alteration of the fundamental principles was to bring consciousness of human beings into the basic structure of the physical theory.' How far we are able to go in this direction will depend on the amount of concrete research results becoming available to support this view.
The dominance of the Copenhagen interpretation is the result of a historical accident (Cushing, 1994). At the Solvay conference in 1927, Louis de Broglie put forward the deterministic pilot wave theory, described later. Never slow to criticize, Pauli jumped up and demolished de Broglie’s proposal, and de Broglie was so discouraged by this that he did not develop his interpretation any further. The Copenhagen interpretation was then developed by Bohr, Heisenberg, Pauli and many others, as described in the previous chapter. It was strongly supported by von Neumann’s proof of the impossibility of hidden variables. Years later, in 1952, Bohm published a deterministic hidden variable theory, showing that there must be something wrong with von Neumann’s proof. The error was found by John Bell in 1966, as described in the last chapter, and he also showed how Pauli’s objections to de Broglie’s original idea can be answered. If this had been done at the conference in 1927, quantum mechanics could have been interpreted deterministically from the beginning. The essential mistake in the Copenhagen interpretation is to treat it as a complete account of the behaviour of each individual system. It is frequently claimed that it enables us to calculate, at least in principle, everything that can be measured. Thus Hooft (1997, p. 11) said that ‘The laws of quantum mechanics have been formulated very accurately. We know exactly how to compute anything we would like to know.’ Also Peierls (1997, p. 25) has said that quantum mechanics ‘had become a complete and consistent scheme capable of giving a unique answer to any questions relating to actual or possible observations’. Even if this were true, it would not imply that it is the final complete theory, for it is always possible that some new phenomenon might be found that cannot be calculated quantum-mechanically. However, it is not true: there are many phenomena such as the time of decay of a radioactive nucleus, or the direction a particle is scattered by a nucleus, that cannot be calculated, even in principle. Quantum mechanics is therefore an incomplete theory. We can only calculate the statistical properties of these systems, such as the half-life of a radioactive decay or the differential scattering cross-section that gives the probabilities that a particle is scattered through various angles. All measurements of quantum systems are of this statistical character. It might be thought that the possibility of making measurements on a single electron provides an exception to this. However, even in such cases the electron is continually bathed in a fluctuating background radiation from nearby atoms. Since this is variable, that electron is a member of an ensemble of electrons subject to different fluctuations. All these examples clearly demonstrate the statistical character of quantum mechanics. ‘I am rather firmly convinced’, Einstein (in Schilpp, 1949, pp. 666, 671) remarked, ‘that the essentially statistical character of contemporary quantum theory is solely to be ascribed to fact that this theory operates with an incomplete description of physical systems’. As a result, ‘the c-function is to be understood as the description not of a single system but of an ensemble of systems’. So, ‘if the statistical quantum theory does not pretend to describe the individual system completely, it appears unavoidable to look elsewhere for a complete description of the individual systems’. Thus ‘the difficulties of theoretical interpretation disappear, if one views the quantum-mechanical description as a description of an ensemble of systems’. If this is achieved, ‘the statistical theory would, within the framework of future physics, take an approximately analogous position to statistical mechanics within the framework of classical mechanics’. ‘Thus, essentially, nothing has changed since Galileo or Newton or Faraday concerning the status of the ‘‘observer’’ or our ‘‘consciousness’’ or of our ‘‘information’’ in physics’ (Popper, 1982, p. 46). Once this is accepted, the quantum paradoxes that plagued the Copenhagen interpretation are easily resolved, as described below. In any textbook of quantum mechanics problems such as the calculation of the energy levels of the hydrogen atom are solved as if that hydrogen atom is alone in the universe. As Feynman (1972) remarked, ‘when we solve a quantum-mechanical problem, what we really do is to divide the universe into two parts – the system in which we are interested and the rest of the universe. We then usually act as if the system in which we are interested comprised the whole universe.’ In fact, however, every hydrogen atom on which measurements are made is surrounded by other atoms and exposed to their radiations. Quantum mechanics somehow takes this into account and gives the average behaviour of an ensemble of hydrogen atoms. A more detailed theory takes these influences into account. One attempt to do this is stochastic electrodynamics, described in a later section.
Long ago, and almost in parallel with the “orthodox” Copenhagen/standard interpretation, other interpretations of quantum mechanics were proposed. Giving an exhaustive discussion of all points of view that have been put forward since then would probably be an impossible task. Moreover, while one can distinguish big families among the interpretations, it is also possible to combine them in many ways, with an almost infinite number of nuances. The Copenhagen/standard interpretation itself, as we have seen, is certainly not a monolithic construction, but can be declined in various forms. In this chapter, we will therefore limit ourselves to a general description of the major families of interpretations. We will begin with a brief description of some frequent attitudes observed among scientists in laboratories, who do not necessarily pay extreme attention to the foundations of quantum mechanics, even when they do experiments in quantum physics. In practice, they often use pragmatic rules, which are sufficient to interpret all their experiments, and prefer to avoid difficult questions about the very nature of the measurement process. For instance, one popular point of view is the “correlation interpretation”, which can be considered as a “minimal interpretation” – minimal but sufficient for all practical purposes; it is accepted as a valid rule by a large majority of physicists, even those who prefer to supplement it with other elements in order to reach a more precise interpretation for the whole theory. We will then proceed to discuss various families of interpretations that are less common, including additional variables, modified Schrödinger dynamics, consistent histories, Everett interpretation, etc.
Since its inception, the many counter-intuitive aspects and results of quantum mechanics have provoked strong philosophical debates and many interpretations. Even fundamental issues, such as Max Born's basic rules concerning probability amplitudes and probability distributions, took decades to be appreciated by society and many leading scientists. Richard Feynman once said, think I can safely say that nobody understands quantum mechanics.[48] According to Steven Weinberg, There is now in my opinion no entirely satisfactory interpretation of quantum mechanics. The Copenhagen interpretation - due largely to the Danish theoretical physicist Niels Bohr - remains the quantum mechanical formalism that is currently most widely accepted amongst physicists, some 75 years after its enunciation. According to this interpretation, the probabilistic nature of quantum mechanics is not a temporary feature which will eventually be replaced by a deterministic theory, but instead must be considered a final renunciation of the classical idea of causality. The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-307447-311734.pdf
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
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.