Quantum superpositions cannot be epistemic
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Abstract
Quantum superposition states are behind many of the curious phenomena exhibited by quantum systems, including Bell nonlocality, quantum interference, quantum computational speedup, and the measurement problem. At the same time, many qualitative properties of quantum superpositions can also be observed in classical probability distributions leading to a suspicion that superpositions may be explicable as probability distributions over less problematic states; that is, a suspicion that superpositions are epistemic. Here, it is proved that, for any quantum system of dimension \(d>3\), this cannot be the case for almost all superpositions. Equivalently, any underlying ontology must contain ontic superposition states. A related question concerns the more general possibility that some pairs of nonorthogonal quantum states \(\psi \rangle ,\phi \rangle \) could be ontologically indistinct (there are ontological states which fail to distinguish between these quantum states). A similar method proves that if \(\langle \phi \psi \rangle ^{2}\in (0,\frac{1}{4})\), then \(\psi \rangle ,\phi \rangle \) must approach ontological distinctness as \(d\rightarrow \infty \). The robustness of these results to small experimental error is also discussed.
Keywords
Quantum Superposition Foundations Ontic Epistemic Ontology1 Introduction
Is the quantum state ontic (a state of reality) or epistemic (a state of knowledge)? This, rather old, question is the subject of the nowfamous PBR theorem [15], which proves that the quantum state of a system is ontic given reasonable assumptions about the ontic structure of multipartite systems. Whilst these assumptions appear weak and wellmotivated, they have also been frequently challenged and, as a result, many recent papers have sought to address the onticity of the quantum state using only singlesystem arguments [1, 2, 5, 11, 13, 14]. These theorems and discussions are reviewed in Ref. [10].
All of this work addresses the epistemic realist, who assumes that a physical system is always in some definite ontic state (realist) and hopes that uncertainty about the ontic state might explain certain features of quantum systems (epistemic). The features that the epistemic realist might like to explain in this way include indistinguishability of nonorthogonal states, nocloning, stochasticity of measurement outcomes, and the exponential increase in state complexity with increasing system size [18]. Preparing some quantum state \(\psi \rangle \) must result in some ontic state \(\lambda \) obtaining, so some probability distribution, called a preparation distribution, must describe the probabilities with which each \(\lambda \) obtains in that preparation. In general, preparation distributions for some pair of nonorthogonal quantum states might overlap—there might be ontic states accessible by preparing either of those quantum states. The main strategy of the singlesystem ontology arguments is to prove that, in order to preserve quantum predictions, these overlaps must be unreasonably small—too small to explain any quantum features.
This paper initially concentrates on quantum superposition states defined with respect to some specified orthonormal basis (ONB). Superpositions are behind quantum interference, the uncertainty principle, waveparticle duality, entanglement, Bell nonlocality [4], and the probable increased computational power of quantum theory [9]. Perhaps most alarmingly, superpositions give rise to the measurement problem, so captivatingly illustrated by the “Schrödinger’s cat” thought experiment.
Schrödinger’s cat is set up to be in a superposition of \(\mathrm{dead}\rangle \) and \(\mathrm{alive}\rangle \) quantum states. The epistemic realist (and probably the cat) would ideally prefer the ontic state of the cat to only ever be one of “dead” or “alive” (viz., only in ontic states accessible to either the \(\mathrm{dead}\rangle \) or \(\mathrm{alive}\rangle \) quantum states). In that case, the cat’s apparent quantum superposition would be epistemic—there would be nothing ontic about the superposition state. Conversely, if there are ontic states which can only obtain when the cat is in a quantum superposition (and never when the cat is in either quantum \(\mathrm{alive}\rangle \) or \(\mathrm{dead}\rangle \) states) then the superposition is unambiguously ontic: there are ontological features which correspond to that superposition but not to nonsuperpositions so that superposition is real.
Obviously quantum superpositions are different from proper mixtures of basis states. The question here is rather whether quantum superpositions over basis states can be understood as probability distributions over some subset of underlying ontic states, where each such ontic state is also accessible by preparing some basis state.
The epistemic realist perspective on the foundations of quantum theory is not only philosophically attractive but also appears to be tenable. Theories in which the quantum state is explained in an epistemically realist manner have been demonstrated to reproduce interesting subsets of quantum theory which include characteristically quantum features [3, 12, 18, 19]. Moreover, they include theories where superpositions are not ontic in the sense described above. The question of the reality of superpositions in quantum theory is, therefore, very much open.
For example, in Spekkens’ toy theory [18] the “toybit” reproduces a subset of qubit behaviour. A toy bit consists of four ontic states, a , b , c , d , and four possible preparations, \(0),1),+),)\), which are analogous to the correspondingly named qubit states. Each preparation corresponds to a uniform probabilistic distribution over exactly two ontic states: 0) is a distribution over a and b; 1) a distribution over c and d; \(+)\) over a, c; and \()\) over b, d. Full details of how these states behave and how they reproduce qubit phenomena is described in Ref. [18]. For the purposes here, it suffices to notice that all ontic states corresponding to the superpositions states \(+)\) and \()\) are also ontic states corresponding to either 0) or 1)—this toy theory has nothing on the ontological level which can be identified as a superposition so the superpositions are epistemic. Such models, therefore, lend credibility to the idea that quantum superpositions themselves might, in a similar way, fail to have an ontological basis.
Previous singlesystem theorems that bound ontic overlaps to argue for the onticity of the quantum state [2, 5, 11, 13, 14] share at least these shortcomings: (i) they prove that there exists some pair of quantum states (taken from a specific set) with bounded overlap, rather than bounding overlaps between arbitrary quantum states and (ii) when the overlaps are proved to approach zero in some limit, the quantum states involved also approach orthogonality in that same limit [10].
In this paper it is proved that, for a \(d>3\) dimensional quantum system, almost all quantum superpositions with respect to any given ONB must be ontic. A very similar argument can be used to obtain a general bound on ontic overlaps for \(d>3\), which addresses the above shortcomings. Finally, the noise tolerance of these results is discussed.
2 Ontological models
The appropriate framework for discussing epistemic realism is that of ontological models [7, 8, 10]. It is flexible enough for most realist approaches to quantum ontology to be cast as ontological models [1] including, but not limited to, Bohmian theories, spontaneous collapse theories, and naïve wavefunctionrealist theories.^{1}
An ontological model of a system has a set \(\Lambda \) of ontic states \(\lambda \in \Lambda \). The ontic state which the system occupies dictates the properties and behaviour of the system, regardless of any other theory (such as quantum theory) which may be used to describe it.
An ontological model for a quantum system is constrained by the fact that it must reproduce the predictions of quantum theory (at least where they are empirically verifiable). Recall that a quantum system is described with a ddimensional complex Hilbert space \(\mathcal {H}\) with \(\mathcal {P}(\mathcal {H})\mathop = \limits ^\mathrm{def}\{\psi \rangle \in \mathcal {H}\,:\,\left\ \psi \right\ =1,\psi \rangle \sim \mathrm{e}^{i\theta }\psi \rangle \}\) as the set of distinct pure quantum states.^{2} Quantum superpositions are defined with respect to some ONB \(\mathcal {B}\) of \(\mathcal {H}\) and are simply those \(\psi \rangle \in \mathcal {P}(\mathcal {H})\) for which \(\psi \rangle \not \in \mathcal {B}\).
The preparation distributions^{3} \(\mu (\lambda )\) for some state \(\psi \rangle \in \mathcal {P}(\mathcal {H})\) form a set \(\Delta _{\psi \rangle }\) since different ways of preparing the same \(\psi \rangle \) may result in different distributions \(\mu \in \Delta _{\psi \rangle }\). If \(\Delta _{\psi \rangle }\) is a singleton for every \(\psi \rangle \in \mathcal {P}(\mathcal {H})\), then the ontological model is preparation noncontextual ^{4} for pure states (otherwise, it is preparation contextual). Let \(\Lambda _{\mu }\mathop = \limits ^\mathrm{def}\{\lambda \in \Lambda \,:\,\mu (\lambda )>0\}\) be the support of the distribution \(\mu \).
3 Measuring overlaps
4 Quantum superpositions are real
From Eqs. (5) and (11), a superposition \(\psi \rangle \not \in \mathcal {B}\) can only be epistemic if the asymmetric overlap \(\varpi (i\rangle \,\,\mu )\) is maximal for every \(\mu \in \Delta _{\psi \rangle }\) and all \(i\rangle \in \mathcal {B}\). Therefore, the statement that “not every quantum superposition can be epistemic” is rather weak. A more interesting question is whether an individual superposition state \(\psi \rangle \in \mathcal {B}\) can be epistemic.
Theorem 1
Consider a quantum system of dimension \(d>3\) and define superpositions with respect to some ONB \(\mathcal {B}\). Almost all quantum superposition states \(\psi \rangle \not \in \mathcal {B}\) are ontic.
Proof
Let \(\psi \rangle \) be an arbitrary superposition state \(\psi \rangle \not \in \mathcal {B}\) and assume only that \(\psi \rangle \) is not an exact 50:50 superposition of two states in \(\mathcal {B}\). This is true for almost all superpositions and guarantees that there exists some \(0\rangle \in \mathcal {B}\) such that \(\langle 0\psi \rangle ^{2}\in (0,\frac{1}{2})\).

\(\langle 0\psi \rangle ^{2}=\alpha ^{2}=\langle \phi \psi \rangle ^{2}\) so there exists a unitary \(U\in \mathcal {S}_{\psi \rangle }\) for which \(U0\rangle =\phi \rangle \);

and the inner products \(\langle 0\psi \rangle ^{2}\), \(\langle \phi \psi \rangle ^{2}\), \(\langle 0\phi \rangle ^{2}\) satisfy Eq. (9) and, therefore, the triple \(\{\psi \rangle ,\phi \rangle ,0\rangle \}\) is antidistinguishable.
5 Bounds on general overlaps
Theorem 1 establishes the reality of almost all superpositions in \(d>3\) by bounding an asymmetric overlap. This suggests that a similar method may be used to prove a general bound on ontic overlaps.
Recall shortcomings (i) and (ii) of the previous singlesystem ontology arguments as mentioned in Sect. 1. Shortcoming (i) leaves open the possibility that many pairs of quantum states could have significant ontic overlaps, while (ii) casts doubt on the significance of those zerooverlap limits (as orthogonal states are distinguishable and, therefore, must be trivially ontologically distinct).
The following theorem address these shortcomings:
Theorem 2
The proof, in “Appendix 1”, closely follows that of Theorem 1. The assumption of pure state preparation noncontextuality with respect to stabiliser unitaries is required to replace the assumption used in Theorem 1 that \(\psi \rangle \) is an epistemic superposition with respect to \(0\rangle \).
6 Noise tolerance
Thus far Eq. (3) has been assumed, demanding that quantum statistics are exactly reproduced by valid ontological models. However, it is impossible to verify this. At most, experiments demonstrate quantum probabilities hold to within some finite additive error \(\epsilon \in (0,1]\). It is, therefore, necessary to consider noise tolerant versions of the above theorems.
Unfortunately, the asymmetric overlap is a noise intolerant quantity—there exist simple ontological models in which every pair of quantum states have unit asymmetric overlap and still reproduce quantum probabilities to within any given \(\epsilon \in (0,1]\). However, an alternative overlap measure, the symmetric overlap \(\omega (\psi \rangle ,\phi \rangle )\) [2, 5, 10, 11, 14], is robust to small errors and Theorem 2 can be modified to bound the symmetric overlap in a noise tolerant way.
Theorem 3
The proof is provided in “Appendix 1”. This theorem makes Theorem 2 noise tolerant at the expense of weakening the bound (and only applying for \(d>5\)). This is because the simple bound on symmetric overlap [Eq. (23)] is lower than that for the asymmetric overlap [Eq. (5)] and, therefore, more difficult to improve upon.
Note that this theorem does not immediately imply that almost all superpositions are real. However, by demonstrating that Theorem 2’s arguments can be made robust against error, it suggests that a noisetolerant version of Theorem 1 should also be possible. Even so, a noisetolerant version of Theorem 1 would require the definition of “epistemic superposition” to be modified since it is currently defined in terms of the noise intolerant asymmetric overlap and is therefore noise intolerant.
7 Discussion
Assuming that quantum statistics are exactly correct, Theorem 1 proves that, for \(d>3\), almost all superpositions defined with respect to any given basis \(\mathcal {B}\) must be real. Therefore, any epistemic realist account of quantum theory must include ontic features corresponding to superposition states. The unfortunate cat cannot be put out of its misery.
A similar method and construction is used in Theorem 2 to prove that, for arbitrary states satisfying \(\langle \phi \psi \rangle ^{2}\in (0,\frac{1}{4})\), ontic overlap must approach zero as d increases for fixed \(\langle \phi \psi \rangle ^{2}\). Theorem 3 makes this robust against small errors in quantum probabilities, at the expense of weakening the bound. Both theorems require an extra assumption: pure state preparation noncontextuality with respect to stabiliser unitaries. Pure state preparation contextuality is often implicitly assumed wholesale, so this assumption should not be very controversial. Moreover, “Appendix 2” provides a heuristic argument to the effect that this type of contextuality is a natural assumption in practice.
These results are damaging to any epistemic approach to quantum theory compatible with the ontological models formalism that reproduces quantum statistics exactly. Such a programme can never hope to epistemically explain superpositions, including macroscopic superpositions. Moreover, for any moderately large system, a large number of pairs of nonorthogonal states cannot overlap significantly, making it unlikely that such overlaps can satisfactorily explain quantum features.
As a result tolerant to small errors, it is possible that Theorem 3 could be experimentally tested. Such a test would require demonstration of small errors in probabilities for a wide range of measurements on a \(d>5\) dimensional system.
The methodology of Theorems 1 and 2 is tightly linked to the asymmetric overlap as a probability, making noisetolerant versions a challenge to extract. If the conclusion from Theorems 1 and 2 could be obtained though an operational methodology (closer to that of Bell’s theorem [4] or the PBR theorem [15]) this would likely lead to better noisetolerant extensions and better opportunities for experimental investigation. Such an operational version may also make it easier to discover any information theoretic implications of these results.
Footnotes
 1.
Conversely, ontological models are irrelevant for any “antirealist”, “instrumentalist”, “positivist”, or “Copenhagenlike” theories denying the existence of an underlying ontology. For example, quantumBayesian theories are exempt from ontological model analysis.
 2.
For simplicity, take \(d<\infty \).
 3.
In fact, this treatment of ontological models is not as general as it should be. Reference [10] notes that, instead of probability distributions, one should consider general probability measures \(\mu \) over a measurable space \((\Lambda ,\Sigma )\) and ontological models can be reformulated measuretheoretically. The presentation here implicitly, and problematically, assumes some canonical measure \(\mathrm {d}\lambda \) over \(\Lambda \) with respect to which all of the probability distributions can be defined. It is possible to derive the results presented here in the more rigorous formulation, but doing so would be at the expense of conceptual clarity. In light of this simplification some of the proofs presented here will also lack in mathematical rigour at certain steps, though more thorough versions of the same results can be derived.
 4.
 5.
 6.
A more mathematically rigorous treatment would fully consider this step in the light of \(\Delta _{\phi \rangle }\) being uncountable in the general case. Such a discussion is omitted for the sake of conceptual clarity and since a more rigorous treatment would also have to account for the issues raised in footnote 3.
 7.
Similarly to the previous footnote, a fully rigorous treatment would include a proof of this step, which is omitted for conceptual clarity and since mathematical rigour has already been sacrificed for conceptual clarity earlier in the paper.
 8.
Once again, such a more rigorous formulation of the problem would require a full justification of this step.
Notes
Acknowledgments
I would like to thank Jonathan Barrett, Owen Maroney, Dominic C. Horsman, and Matty Hoban for insightful discussions as well as an anonymous referee for thorough and insightful comments. This work is supported by the Engineering and Physical Sciences Research Council (EPSRC); the European Coordinated Research on Longterm Challenges in Information and Communication Sciences and Technologies (CHISTERA) project on Device Independent Quantum Information Processing (DIQIP); and the Foundational Questions Institute (FQXi) Large Grant “Thermodynamic vs information theoretic entropies in probabilistic theories”.
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