The inferential system interpretation of quantum mechanics

John O. Campbell
May 2017

Carlo Rovelli’s proposal for basing a relational quantum mechanics on a set of information theoretic postulates (1) appears to have recently been realized in the research of Phillipp Hoehn (2).  This program holds great promise and after eighty years of misconception we may finally be making a start in formulating a comprehensible interpretation of quantum theory. 

Rovelli's initial sketch of a relational quantum theory was posted as a preprint in 1994 (3) and has since been updated, most recently in 1997. Although much of his subsequent research has focused on other topics, he has further developed his views on relational quantum theory in two subsequent papers (4; 5). 

It is remarkable that in the conclusion of his 1997 outline of relational quantum theory he writes that it had just come to his attention that Wojciech Zurek’s had precedence (1982) in developing ‘conclusions that are identical to the ones developed here.’ (1) . Although Rovelli acknowledges that Zurek arrived at these conclusions fifteen years earlier, he does not mention, in this or his subsequent papers, Zurek’s on-going research program advancing the subject. We might well pay close attention to Zurek’s research not only because he had a fifteen-year head start on Rovelli but also because he has focused on the development of this topic throughout his career and has published dozens of papers on the subject.

Hoehn claims that his axiomatic formulation of quantum theory supports Rovelli’s interpretation and that Hoehn’s ‘successful reconstruction can be viewed as a completion of these ideas for qubit systems’ (2).  Neither Hoehn’s nor Rovelli’s subsequent papers mention or cite Zurek’s research.

Hoehn’s development of quantum theory is most instructive as it is in terms of information theory. Hoehn considers an observer who entails a probabilistic model of the state of some phenomena and is able to ask questions and receive answers regarding the phenomena. He specifies some constraints on the model: the questions and the answers in the form of 5 rules or postulates. The answers are used to update the probabilistic state model according to the principles of Bayesian inference and in this manner the observer’s model becomes a ‘catalogue of knowledge’ regarding the observed phenomena. 

Importantly, the observer, its questions, answers and the Bayesian process used to update its model form a system that will accumulate evidence-based knowledge. I have used the term ‘inferential systems’ (6; 7) to describe similar systems that operate to accumulate knowledge in non-quantum or classical reality. The Rovelli/Hoehn (RH) paradigm posits this same mechanism at the basis of quantum theory and thus suggests that inferential systems may be a unifying paradigm across the entire scope of reality. This suggestion is important because it removes quantum phenomena from the ‘weird’ interpretations traditionally applied to it and instead places it firmly within the same paradigm that is used to describe many other natural systems.

As Hoehn demonstrates, the model, or ‘catalogue of knowledge’ which the observer will evolve by following the postulates is quantum theory itself. In other words, the model that the observer will infer to describe and predict the phenomena under consideration is quantum theory. 

Hoehn thereby demonstrates that quantum theory is the logical product of an inferential system which processes information in a manner consistent with his postulates. Rovelli stresses that the ‘observer’ in this paradigm may be any physical object (1) and hence provides an explanation for the universal applicability of quantum theory. It is most natural to place the observer at the level of the quantum system itself. All quantum systems are associated with a state function which receives and processes information. The HR paradigm explains why this state function is quantum mechanical and thus offers an explanation for quantum mechanics: quantum mechanics is the body of knowledge which any system will infer when constrained to exchange and process information in accord with Hoehn’s postulates. 

The startling conclusion is that reality at the micro level may be described by quantum theory because its information acquisition and processing is constrained in accord with Hoehn’s postulates.

Zurek’s 1982 paper which contains ‘conclusions that are identical to the ones developed’ in Rovelli’s 1997 paper, came near the beginnings of Zurek’s research program into decoherence or the nature of quantum interactions. Through this research Zurek attempts to answer a central question posed by quantum theory: why do we never experience weird quantum phenomena such as superpositions as described by theory, why do we instead experience definite classical outcomes? The answer he provides is described on his FQXi page (8): 

Zurek hopes that the answer may lie at the intersection of three well-developed ideas in quantum theory. The first is decoherence, a theory Zurek has been instrumental in advancing since 1981. Decoherence describes what happens when a system in superposition interacts with its environment: It becomes entangled with particles in the outside world that record its location, causing the superposition to fall apart (decohere) into a probabilistic mixture of definite states at specific locations. Only systems in perfect isolation can remain in superposition. A single errant photon can destroy the superposition, like the faintest breeze knocking down a house of cards.

Decoherence explains how familiar, classical physics emerges from the haze of quantum mechanics, and clears up vexing paradoxes like that of Schrödinger’s famous cat, locked in a box with a poisonous substance that will be released if a radioactive atom decays. While the box is closed, the usual story goes, the radioactive atom is in a superposition state in which it has both decayed and not decayed, leaving the trapped cat in limbo too. Decoherence helps resolve this, however: We don’t need to worry about the cat being alive and dead at once because a macroscopic feline can’t maintain the perfect isolation required to hold the superposition.

Zurek has derived his research results starting from the traditional quantum axioms of the Copenhagen interpretation. In the process, he has reduced the number of required axioms and has largely eliminate the problematic axioms which lead to the ‘measurement problem’ that has plagued quantum theory since its inception. However, as Hoehn’s set of axioms describing an inferential system is an equivalent starting point for the derivation of quantum theory, Zurek could have equally well performed his research starting from Hoehn’s and Rovelli’s framework.

This, as we will see, has some important advantages as Zurek’s more recent research has led him to conclusions that resonate very well with Rovelli’s interpretation. In the meantime Zurek’s research program has moved beyond decoherence (8). 

But decoherence does not explain why all observers share the same classical reality—that is, why reality is objective rather than subjective. For that, Zurek turns to the second piece of the puzzle, the theory of quantum Darwinism, which Zurek and his colleagues and collaborators have been developing over the past dozen years. Quantum Darwinism specifies how the environment selects and disseminates information about favored states. These favored states emerge through a process dubbed "einselection," or environment induced superselection, and are recorded and copied by the environment during decoherence. "This proliferation of "copies" of the states allows many observers to find out independently about the system, without disturbing it by measurements—hence, without getting in each others’ way," explains Zurek. "This consensus defines "objectivity" of the classical everyday reality."

To be clear, Zurek has named this theory ‘Quantum Darwinism’ not out of whimsy but rather because he considers the process he is describing as a true Darwinian process (9):

In the end one might ask: “How Darwinian is Quantum Darwinism?” Clearly, there is survival of the fittest, and fitness is defined as in natural selection. 

Zurek’s understanding of quantum systems focuses on the environment surrounding the quantum system, an environment which is itself composed of quantum systems, and the information which those systems forming the environment may share with a particular quantum system that they ‘witness’. He has found that most of the information necessary to fully describe a quantum system is not able to survive in its environment. Only a very small subset of the ‘fittest’ information can survive and proliferate leaving many redundant copies. This subset of information is ‘classical’ or Holevo information as distinct from purely quantum information known as ‘quantum discord’. Thus, most of the information which fully describes the state of a quantum system cannot be communicated to any other entity or observer (10; 11). The small subset of information which can survive the transfer and be communicated to other entities forms the basis of objective classical reality.

Observers seeking information regarding the quantum system may sample the environment and each observer will find the same information leading to their experience of an objective, classical reality. We should remember that it is common for observers to gather information concerning a quantum system by sampling its environment. For example, everything we ‘see’ is the result of photons in the environment of the quantum system. At a basic level, as classical information is the only information which may be communicated, it is this information that forms an objective classical reality and Zurek characterizes this process as the emergence of classical reality from its quantum substrate.

The name ‘quantum Darwinism’ is descriptive as classical reality is composed from the fittest information, in relation to its environment, the information which can survive and proliferate.

Zurek’s and HR’s paradigms describe quantum systems from two separate viewpoints. Zurek maintains a focus on the environment surrounding the quantum system and the information describing the quantum system which may be transferred to this environment. He describes this process as a Darwinian process in the sense that only ‘fit’ quantum information can survive in its environment just as in biology where only ‘fit’ genetic information may survive in its environment. Fit genetic information is capable of constructing reproductively successful phenotypes, fit quantum information is capable of constructing reproductively successful classical phenomena. In the quantum case the selection of fit information results in the existence of classical reality in the biological case the selection of fit information results in the existence of the biosphere.

On the other hand, HR maintain an ‘outward facing’ focus on the quantum system itself and the information it may contain describing the other quantum systems composing its environment. Through an inferential procedure detailed in the postulates, the quantum system elicits, receives and processes information and as a result infers a quantum worldview or, in HR terms, a quantum catalogue of knowledge. 

It may appear that although HR and Zurek once shared a common view that this has diverged significantly since. How can we reconcile their apparent divergent focuses on the interior and exterior aspects of quantum systems? The answer may lie in the ‘repeatability’ postulate of quantum theory: an immediately repeated measurement yields the same outcome. Although the measured outcome may have initially been predicted by the quantum state with a probability less than one, following the initial measurement the quantum state becomes homologous with the outcome so that a repeated measurement is predicted to produce the same outcome with probability 1. The interior quantum system becomes homologous with the information regarding it that is recorded in the environment. The interior and exterior views are synchronized so that interior and exterior descriptions are equivalent.

Even given this equivalency how can we reconcile the interior description in terms of an inferential process with the exterior description in terms of a Darwinian process? Recent results show this to be straight forward; the mathematics of Darwinian processes are the mathematics of Bayesian inference. The ‘relative fitness’ of a Darwinian description is the likelihood of a Bayesian description. 

For example, the Darwinian change in the frequency of biological alleles between generations has long been described by population biologists as (12):

where ‘p’ is the probability of the particular allele in the latter generation, p is the probability of the particular allele in the former generation, R_A is the fitness of the particular allele and  is the average fitness of all competing alleles. This equation describes a Bayesian update where the frequency in the previous generation is updated by the ratio of two other probabilities, the ratio of the fitness of the particular allele to the average fitness of all alleles for that characteristic.

The Price equation, which is the mathematics of generalized Darwinian processes, has been shown to be equivalent to the mathematics of Bayesian inference and so we might understand Darwinian processes as physically instantiated instances of Bayesian inference (13; 6). In the quantum realm, for example, we may understand Zurek’s quantum Darwinism as equivalent to the inferential process described by HR.

This may be considered as a conceptual breakthrough because inferential systems are found throughout nature wherever knowledge is accumulated. Specifically, knowledge stores or ‘catalogues’ are accumulated via inferential systems in biology, neural based behaviour and culture (6; 7). Equivalently, in each instance, these accumulations of knowledge evolve through a Darwinian process. The inclusion of quantum theory within this framework hints at a truly universal mechanism at the root of existence.

References

1. Relational Quantum Mechanics. Rovelli, Carlo. s.l. : International Journal of Theoretical Physics, 1996, Vols. 35 (1996) pp. 1637-78.

2. Quantum theory from rules on information acquisition. Hoehn, Philipp Andres. s.l. : Entropy, 2017, Vols. 19(3), 98;.

3. Wikipedia. Relational quantum mechanics. Wikipedia. [Online] [Cited: 5 24, 2017.] https://en.wikipedia.org/wiki/Relational_quantum_mechanics.

4. Relative information at the foundation of physics. Rovelli, Carlo. s.l. : ArXiv preprint:1311.0054, 2013.

5. Meaning = Information + Evolution. Rovelli, Carlo. s.l. : arXiv:1611.02420 [physics.hist-ph], 2016.

6. Universal Darwinism as a process of Bayesian inference. Campbell, John O. s.l. : Front. Syst. Neurosci., 2016, System Neuroscience. doi: 10.3389/fnsys.2016.00049.

7. Campbell, John O. Einstein's Enlightenment. s.l. : Createspace, 2017. ASIN: B06XNZDGCS.

8. Becker, Kate. Realities NeverEnding Story. FQXi Community. [Online] April 17, 2014. http://fqxi.org/community/articles/display/189.

9. Quantum Darwinism. Zurek, Wojciech H. s.l. : http://www.nature.com/nphys/journal/v5/n3/abs/nphys1202.html, 2009, Nature Physics, vol. 5, pp. 181-188.

10. Complementarity of quantum discord and classically accessible information. Zurek, Wojciech and Zwolak, Michael. s.l. : Scientific Reports 3, Article number: 1729, 2013. doi:10.1038/srep01729.

11. Quantum discord cannot be shared. Streltsov, Alexander and Zurek, Wojciech. 4, s.l. : American Physical Society - Physical review letters, 2013, Vol. 111.

12. Ricklefs, Robert E. Ecology. Concord, Massachusetts : Chiron Press, 1979. p. 448.

13. Natural selection. V. How to read the fundamental equations of evolutionary change in terms of information theory. Frank, Steven, A. 2012, Journal of Evolutionary Biology, Vols. 25:2377-2396.

Quantum theory as an inferential system

John O. Campbell

April 2017

In a number of previous blog posts, books (1; 2) and papers (3; 4) I have developed the notion that due to a number of new findings a wonderfully unifying scientific interpretation may now be possible. This interpretation focuses on ‘inferential systems’ which may operate throughout nature and which accumulate the knowledge required for the existence of complex systems. Such inferential systems are typified by internal probabilistic models which are updated by evidence. The internal models code an executable strategy for existence which manifests as type of generalized phenotype or, in Dawkins’ terms, vehicles. In turn these phenotypes or vehicles collect evidence concerning the success of the coded strategy, evidence that is used to update the internal model in a Bayesian manner and accumulate a catalogue of knowledge that specifies a strategy for existence.

This paradigm is non-controversial in its application to many complex systems which emerge from physical reality including biology (5), neural-based behaviour (6; 7) and cultural evolution (8). Indeed, the consensus understanding within each of these fields is consistent with the inferential system model (9). It is however a greater challenge for this paradigm to explain physical systems. While some physical theories or interpretations conform to the paradigm (10; 11), they are, yet, far from consensus.

However, a recent research program conducted by Philipp Hohn derives quantum theories, our most fundamental physical theories, from informational or Bayesian postulates (12; 13) and thus demonstrates how quantum theory may arise through the actions of an inferential system. His papers develop quantum theory within the context of an ‘observer’ who interrogates natural systems with binary questions that may be answered using experimental evidence. The statistics over all possible answers to these experimental questions forms the ‘state space’ of the system. 

The culmination of Hohn’s program is the demonstration, given some reasonable constraints on the observer’s ability to acquire information, that the internal model or ‘catalogue of knowledge’ (13) that the observer will develop by evolving their model using the principles of Bayesian inference is quantum theory.  In other words, the evidence-based inferential system he describes will infer quantum theory from the evidence it receives. 

Unfortunately, quantum theory is only now emerging from over a century of conceptual confusion whose lingering effects tend to place Hohn’s findings in an ambiguous context. Some of the key scientists who developed quantum theory, including Niels Bohr, interpreted this theory as inconsistent with our usual understanding of scientific theories. They made two key speculative interpretations which have long haunted the theory: 

1) The words ‘measurement’ in the quantum postulates refers to human activities and therefore the fundamental theory involves humans and/or human consciousness.

2) Quantum theory does not describe the actual world but is rather a kind of abstract or Platonic description which at best only indirectly describes the real world.

  

The first of these casts a shadow on Hohn’s findings as his conclusion may appear somewhat trivial, in the sense that it is historically obvious that science has inferred its understanding of quantum phenomena from the evidence and that this inferential process has resulted in the catalogue of knowledge known as quantum theory. From this perspective Hohn’s conclusions appear little more than an account of how quantum theory was inferred by scientists. This perspective hinges on Hohn’s ‘observers’ being interpreted as human, scientific observers.

On the other hand, Hohn’s conclusion may be interpreted in a more profound light where ‘observers’ are not constrained to scientific observers but rather may be any entity operationally capable of gathering and processing empirical evidence. In this sense, an observer is any entity which acts as a quantum phenomenon and thus extends the appropriate title ‘observer’ to all quantum entities. 

This ambiguity between the anthropocentric status of ‘observer’ or ‘measurement’ has dogged quantum theory from its beginnings. However modern developments seem to have come down in favor of the broadly-based understanding that the use of these words in quantum theory does not constrain them to human ‘observers’ or only to ‘measurements’ performed by humans. As Wojciech Zurek notes (14):

The dividing line between what is and what is known to be has been blurred forever. While abolishing this boundary, quantum theory has simultaneously deprived the “conscious observer” of a monopoly on acquiring and storing information: Any correlation is a registration, any quantum state is a record of some other quantum state. 

A human presence or consciousness is not required for the world to operate in a quantum manner. All quantum states may be considered observers.

Hohn however, appears to endorse Bohr’s anthropocentric interpretation of this issue. He quotes approvingly Bohr’s statement that (15): 

There is no quantum world. There is only an abstract quantum physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature...

Perhaps the best that can be said is that Bohr’s statement contradicts principles considered central to science in its denial that science is fundamentally a description of actual reality. Bohr was entirely spot on in the sense that so far science has only developed an abstract theory concerning quantum phenomena. Science has not yet discovered the details of the actual reality which quantum theory describes but we should expect this to eventually become known. Claims of the completeness of quantum theory are premature; as Einstein noted, quantum theory is obviously incomplete. What physics says about Nature has value only to the extent that its descriptions share mutual information with how Nature is and the purpose of physics or any other science is to maximize this mutual information.

It is evident that there is a very long way yet to go on this path towards maximization. Our ignorance is immense. This path may even be of an infinite length. Any claim of complete understanding is hopelessly premature and only presents obstacles to further understanding which we can expect to be developed a little further along the path.

We should understand that ‘measurement’ of quantum phenomena is not an experience unique to humans. Quantum entities ‘measure’ each other all the time. Measurements conducted by humans are merely set-ups for us to view naturally-occurring quantum interactions; interactions which occur all the time, whether humans are watching or not.

This confusion may be at least partially resolved by an understanding that models of phenomena occur at many different levels within nature. Those models which are constructed by humans participating in science attempt to model other aspects of nature and many of these ‘aspects of nature’ involve models of their own. Thus, scientific models often describe other models. For example, the science of genetics describes the genetic models found in organisms and due to the centrality of genetics within biology this model is crucial to our understanding of most aspects of biology. The point I would like to stress is that the actual genetic models are not the creation of scientists but rather are models coded in DNA and existing within organisms. They are what nature is. On the other hand, the scientifically constructed model of genetics is a description of nature’s models written in DNA; the scientific models are models of models and have value only to the extent that they accurately describe or share mutual information with nature’s actual models.

The same relationship may be found in neuroscience; mental models are not the product of scientists rather they are models coded in neurons within brains. The scientifically constructed models which attempt to model mental models are likewise models of models. As the great neuroscientist, Karl Friston noted (6):

Our capacity to construct conceptual and mathematical models is central to scientific explanations of the world around us. Neuroscience is unique because it entails models of this model making procedure itself. There is something quite remarkable about the fact that our inferences about the world, both perceptual and scientific, can be applied to the very process of making those inferences: Many people now regard the brain as an inference machine that conforms to the same principles that govern the interrogation of scientific data.

During the decade since Friston wrote the above he has expanded this paradigm to biology and perhaps to existence in general (16).

If we take Hohn’s demonstration at face value and accept that his ‘observer’ may be any entity capable of receiving and processing quantum information then we may extend this paradigm to quantum physics and view scientifically constructed models of quantum phenomena as scientific models of nature’s models. 

The second lingering speculation concerning quantum theory, that it does not describe what nature actually is, also cast a shadow on Hohn’s findings. Since the inception of quantum theory, a debate has raged between those who view quantum theory as ‘epistemology’ (a description of what we can know about reality) and those who view it as ‘ontology’ or how nature actually is. 

Einstein championed the view that science describes ontology and that the ultimate aim of science is to describe what nature is.  

Bohr was less constrained by this traditional view of science as naturalism. For example, he promoted the idea of vitalism (the belief that life contains non-physical phenomena) in biology long after almost all biologists had firmly rejected that notion. As the biologist, Ernst Mayr wrote (17): 

we might note in passing a rather peculiar twentieth-century phenomenon-the development of vitalistic beliefs among physicists. Niels Bohr was apparently the first to suggest that special laws not found in inanimate nature might operate in organisms. He thought of these laws as analogous to the laws of physics except for their being restricted to organisms.

The development of quantum theory was deeply tainted with non-naturalistic explanations, leading E.T. Jaynes to quip that the theory’s accepted norm was ‘A standard of logic that would be considered a psychiatric disorder in other fields’ (18) . As the historian of science, Juan Miguel Marin, observes (19):

Not only was consciousness introduced hypothetically at the birth of quantum physics, but the term ‘mystical’ was also used by its founders to argue in favour and against such an introduction. In private conversations, at least as early as the 1927 Solvay Congress, the founders discussed ideas about quantum theory, ‘mysticism’ and consciousness. It was also around this time that Einstein accused Bohr of introducing ‘mysticism’ into physics.

This debate may be mitigated by an insistence that scientific theories are models of nature which strive to maximize the mutual information they share with nature. Scientific theories are what we can say about how nature is. This ‘ontic’ or naturalistic position gains support from some recent papers (20; 21) which claim to decide conclusively that quantum theory is a description of what nature is (21):

This means that we can deduce the quantum state from a knowledge of the ontic state. Hence, if these assumptions are correct, we can claim that the quantum state is a real thing (it is written into the underlying variables that describe reality).

If we reject mysticism and accept the position that the quantum state describes how nature actually is, then we can interpret Hohn’s paradigm in a more significant manner. His ‘observers’ may be interpreted as any quantum entity that can interact or acquire information at the quantum level.  This information acquisition involves a probabilistic model or state function of the information expected to be received. As the quantum entity acquires information or evidence it updates its probabilistic model in a Bayesian manner. As a result of this evidence-based evolution the wave function may be seen as a knowledge repository or catalogue which contains knowledge capable of making highly accurate predictions. This knowledge catalogue is the quantum entity’s ‘worldview’ and is equivalent to quantum theory. It is in this sense that our scientific quantum theory shares mutual information with nature operating at the quantum level.

In this view quantum entities are but another instance of nature’s many inferential systems and scientific quantum theory is but a human model which encapsulates one of nature’s many models. We may understand quantum phenomena within a naturalistic framework where it forms a level of existence within a nested hierarchy of levels that include biology, neural based behaviour and culture.  Each level is engaged in a common process which provides a unified view of existence over many levels of scientific subject matter. This common process is the inference of knowledge from information, a process by which knowledge evolves to explore the many strategies for existence found in nature.

References

1. Campbell, John O. Universal Darwinism: The path of knowledge. s.l. : CreateSpace, 2011.

2. —. Darwin does physics. s.l. : CreateSpace, 2015.

3. Bayesian Methods and Universal Darwinism. Campbell, John O. s.l. : http://arxiv.org/abs/1001.0068, 2009. AIP Conf. Proc. 1193, 40 (2009), DOI:10.1063/1.3275642. pp. 40-47.

4. Universal Darwinism as a process of Bayesian inference. Campbell, John O. s.l. : Front. Syst. Neurosci., 2016, System Neuroscience. doi: 10.3389/fnsys.2016.00049.

5. Darwin, Charles. The Origin of Species. sixth edition. New York : The New American Library - 1958, 1872. pp. 391 -392.

6. Free Energy and the brain. Friston, Karl and Klass, Stephan. 2007, Synthese, 159, pp. 417-458.

7. The visual system’s internal model of the world. Lee, Tai Sing. Proceedings of the IEEE. Institute of Electrical and Electronics Engineers, Vols. 103(8), 1359–1378.

8. A framework for the unification of the behavioral sciences. Gintis, Herbert. 2007, BEHAVIORAL AND BRAIN SCIENCES.

9. Campbell, John O. Einstein's Enlightenment. s.l. : Createspace, 2017. ASIN: B06XNZDGCS.

10. Smolin, Lee. The life of the cosmos. s.l. : Oxford University Press, 1998.

11. Quantum Darwinism. Zurek, Wojciech H. s.l. : http://www.nature.com/nphys/journal/v5/n3/abs/nphys1202.html, 2009, Nature Physics, vol. 5, pp. 181-188.

12. Quantum theory from rules on information acquisition. Hohn, Philipp Andres. s.l. : Entropy, 2017, Vols. 19(3), 98; .

13. Quantum theory from questions. Hohn, Philipp Andres and Wever, Christopher S.P. s.l. : PhysRevA.95.012102, 2017.

14. Decoherence and the Transition form Quantum to Classical - Revisited. Zurek, Wojciech H. s.l. : http://arxiv.org/ftp/quant-ph/papers/0306/0306072.pdf, 2003.

15. Pais, Abraham. The genius of science: a portrait gallery. s.l. : Oxford University Press, 2000. ISBN-10: 0198506147.

16. Life as we know it. Friston, Karl. s.l. : Journal of the Royal Society Interface, 2013, Vol. 10: 20130475.

17. Mayr, Ernst. This is Biology: The science of the living world. s.l. : Harvard University Press, 1998. ISBN 9780674884694.

18. Jaynes, Edwin T. Clearing up the mysteries - the original goal. [book auth.] John Skillings. Maximum Entropy and Bayesian Methods. s.l. : http://bayes.wustl.edu/etj/articles/cmystery.pdf, 1989.

19. 'Mysticism' in quantum mechanics: the forgotten controversy. Marin, Juan Miguel. 2009, European Journal of Physics, pp. 807 - 822.

20. On the reality of quantum states. Pusey, Matthew F., Barrett, Jonathan and Randolph, Terry. 2012, Nature Physics 8 , pp. 475 - 478.

21. Are quantum states real? Hardy, Lucien. s.l. : http://arxiv.org/abs/1205.1439, 2013, International Journal of Modern Physics B.