A Metaphor between Modeling Relations

One of the fundamental questions in quantum mechanics concerns the interpretation of the QM formalism: what does QM “say” about the material world?  Do the probability distributions in QM reflect corresponding innate properties in the material quantum world, or do they reflect only uncertainties in our state of knowledge about the material quantum world? In a paper[1] entitled “In defense of the epistemic view of quantum states: a toy theory”  by Robert Spekkens, the case for interpreting QM as in the latter manner is put forth. To refer to these two interpretations, the terms “ontic” and “epistemc” are used. The meanings of these terms in this specific context are described by Spekkens:

     We begin by clarifying the dichotomy between states of reality and states of knowledge. To be able to refer to this distinction conveniently, we introduce the qualifiers ontic, (from the Greek ontos, meaning “to be”) and epistemic (from the Greek episteme, meaning “knowledge”). An ontic state is a state of reality, whereas an epistemic state is a state of knowledge. To understand the content of the distinction, it is useful to study how it arises in an uncontroversial context: that of classical physics.
The first notion of state that a student typically encounters in their study of classical physics is the one associated with a point in phase space. This state provides a complete specification of all the properties of the system (in particle mechanics, such a state is sometimes called a “Newtonian state”). It is an ontic state. On the other hand, when a student learns classical statistical mechanics, a new kind of state is introduced, corresponding to a probability distribution over the phase space (sometimes called a “Liouville state”). This is an epistemic state. The critical difference between a point in phase space and a probability distribution over phase space is not that the latter is a function. An electromagnetic field configuration is a function over three-dimensional space, but is nonetheless an ontic state. What is critical about a probability distribution is that the relative height of the function at two different points is not a property of the system (unlike the relative height of an electromagnetic field at two points in space). Rather this relative height represents the relative likelihood that some agent assigns to the two ontic states associated with those points of the phase space. The distribution describes only what this agent knows about the system. [bold added]

The intent of this post is to cast the issue in terms of the Rosen Modeling Relation, in order to better grasp the overall situation. Thus, we begin with the following Modeling Relation:


Since in both interpretations the material quantum system under study is the same, and the measurements are the same, and obviously the formal model is the same, the distinction between the epistemic and ontic interpretations cannot be in any of these elements of the Modeling Relation. Moreover, the QM formalism appears to decode back to the same observables (in the sense that the predictions the QM machinery makes are correct) in both cases. But the Decoding is also where the difference resides: in the ontic interpretation, the semantics of the decoding specify that the material correlates of the quantum probability distributions are actual properties of the system, whereas in the epistemic interpretation, the semantics of the decoding specify that the probability distributions are to be decoded into a property of the modeler, namely the state of knowledge in the modeler about the material quantum system. From this perspective, the claim that QM is an incomplete theory (such as Einstein’s assertion in the 1935 EPR paper) is a claim about the Decoding in that Modeling Relation. (The interested reader may also want to read the paper by Dress[2], which discusses EPR in light of the Rosen Modeling Relation.) As we know, the Encoding and Decoding are not entailed from within the Modeling Relation[3]; that is, the Encodings/Decodings do not derive entirely from either the model or the natural system under study. Instead, they must be supplied by the modeler from outside the Modeling Relation — they are “free creations of the mind”.  How, then, is it possible to discern whether the ontic or epistemic view is correct? What Spekkens proposes:

            We shall argue for the superiority of the epistemic view over the ontic view by demonstrating how a great number of quantum phenomena that are mysterious from the ontic viewpoint, appear natural from the epistemic viewpoint. These phenomena include interference, noncommutativity, entanglement, no cloning, teleportation, and many others. Note that the distinction we are emphasizing is whether the phenomena can be understood conceptually, not whether they can be understood as mathematical consequences of the formalism, since the latter type of understanding is possible regardless of one’s interpretation of the formalism. The greater the number of phenomena that appear mysterious from an ontic perspective but natural from an epistemic perspective, the more convincing the latter viewpoint becomes. For this reason, the article devotes much space to elaborating on such phenomena.

           Of course, a proponent of the ontic view might argue that the phenomena in question are not mysterious if one abandons certain preconceived notions about physical reality. The challenge we offer to such a person is to present a few simple physical principles by the light of which all of these phenomena become conceptually intuitive (and not merely mathematical consequences of the formalism). Our impression is that this challenge cannot be met. By contrast, a single information-theoretic principle, which imposes a constraint on the amount of knowledge one can have about any system, is sufficient to derive all of these phenomena in the context of a simple toy theory, as we shall demonstrate.

The situation Spekkens proposes consists of a “toy model”, which will sit on the right-hand side of the Modeling Relation. Because this is a toy theory, the intent is not to attempt to have the toy model commute with a material quantum system. Indeed, since what is being debated is precisely what can be imputed back to a material system from the QM formalism, this would be unhelpful. Instead, the left-hand side will consist of a surrogate, a formal system which is intended to stand-in for the material quantum system in this scenario. Spekkens proceeds by asserting minimal properties to this surrogate, since the aim is not to create or recreate a quantum theory but, rather, to demonstrate “how a great number of quantum phenomena that are mysterious from the ontic viewpoint, appear natural from the epistemic viewpoint.” Specifically, his investigation rests on providing his surrogate with what he calls “The Knowledge Balance Principle”: If one has maximal knowledge, then for every system, at every time, the amount of knowledge one possesses about the ontic state of the system at that time must equal the amount of knowledge one lacks. The resulting situation is this Modeling Relation:


He then proceeds to examine the consequences of such a Modeling Relation with the following result:

            We have considered the consequences of a principle of equality between knowledge and ignorance to the structure of the set of possible states of knowledge. We have examined the manner in which such states of knowledge may be decomposed into convex sums, decomposed into “coherent” sums, transformed, inverted, updated, remotely “steered”, cloned, broadcast, teleported, and so forth. In all of these respects we have found that they resemble quantum states. This is strongly suggestive that quantum states should be interpreted as states of incomplete knowledge.

     The toy theory contains almost no physics. The motional degree of freedom was assumed classical, and there were no masses or charges or forces or fields or Hamiltonians anywhere in the theory. Although this is a shortcoming from the perspective of obtaining an empirically adequate theory, it helps make the case for the epistemic view. Specifically, it supports the idea that a great number of quantum phenomena, and in particular all the phenomena that the toy theory reproduces, have nothing to do with physics, but rather concern only the manipulation of our information about the world.

This in itself is intriguing. A second paper by S. J. van Erik[4], extends and enhances the toy model presented by Spekkens. Erik’s variation on Spekkens’ toy theory modifies the Knowledge Balance Principle in such a way that his enhanced toy model additionally exhibits “the correct maximum violation of the Bell-CHSH inequalities”. Although these results may appear compelling, does this prove, in any sense, that the epistemic interpretation is the correct one? In Rosennean terminology, does the Modeling Relation between a formal system (the surrogate) with a specified property (the Knowledge Balance Principle) and a toy model entail a particular encoding/decoding in the original Modeling Relation between a material quantum system and the QM formalism? As Spekkens points out, the toy theory contains almost no physics, nor can one derive QM from it. Plainly, then, the answer is no, and Spekkens is appropriately cautious when he concludes his results are only “strongly suggestive” that “quantum states should be interpreted as states of incomplete knowledge”.

The term “suggestive”, though, does help us identify the nature of the relationship between these two Modeling Relations. What we can see is that there are behaviors in the toy model Modeling Relation that correspond to behaviors in the QM Modeling Relation. These behaviors are as dependent upon the properties of the systems on the left-hand sides of their respective Modeling Relations as they are on their respective models. (In Rosennean terms, if we were to fractionate the Modeling Relations, we would lose these behaviors.) Therefore, the “suggestive” relationship is between entire Modeling Relations. What Spekkens would like to do is to impute the epistemic properties which underly the toy theory Modeling Relation to the original QM Modeling Relation. Such an imputation would be a decoding from the former Modeling Relation to the latter. Thus, the situation would appear to be that of metaphor, where there is a decoding but no encoding [5], as shown below:


What is interesting here is that the systems which sit on either side of this Modeling Relation are themselves Modeling Relations! Although one does not derive proof from such a scenario, this use of metaphor has been and can be a powerful tool. When metaphor is applied to entire Modeling Relations, as in this case, it provides a means by which an epistemological/ontological relationship can be probed.  Thus, we come back around to the quote from Spekkens that “the distinction we are emphasizing is whether the phenomena can be understood conceptually, not whether they can be understood as mathematical consequences of the formalism”. I would suggest that “understood conceptually” in this context cannot merely mean the toy theory or the toy theory Modeling Relation; but rather, the metaphorical relationship as described above,  since it is only once this metaphor has been established that there is any potential consequence for quantum mechanics.

Note: A useful background paper added as [6].


[1] Spekkens, R. 2005. “In defense of the epistemic view of quantum states: a toy theory”. arXiv:quant-ph/040152v2.

[2] Dress, W. 1999. “Epistemology and the Rosen Modeling Relation”. PDF

[3] Rosen, R. 2000. Essays on Life Itself. Chapter 10.

[4] van Erik, S. 2007. “A toy model for quantum mechanics”. arXiv:0705.2742v1

[5] Rosen, R. 1991. Life Itself. Chapter 3.

[6] Fuchs, C.  2002. “Quantum Mechanics as Quantum Information (and only a little more)”. arXiv:quant-ph/0205039v1


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