Tuesday, 30 May 2017

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, RA 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.


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.