Quantum Superpositions Cannot be Epistemic

Quantum superposition states are behind many of the curious phenomena exhibited by quantum systems, including Bell non-locality, quantum interference, quantum computational speed-up, 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 \emph{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 non-orthogonal 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.

Is the quantum state ontic (a state of reality) or epistemic (a state of knowledge)? This, rather old, question is the subject of the now-famous PBR theorem [1], which proves the onticity of the quantum state under reasonable assumptions on the ontic structure of multipartite quantum systems. However, these assumptions have been openly challenged and, as a result, many recent papers have addressed the same question using only single-system arguments [2][3][4][5][6][7]. These theorems and discussions are reviewed in Ref. [8].
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 non-orthogonal states, no-cloning, stochasticity of measurement, and exponential state complexity with increasing system size [9]. Preparing some quantum state |ψ must result in some ontic state λ obtaining, so some probability distribution, called a preparation distribution, must describe the probabilities with which each λ obtains in that preparation. In general, preparation distributions for non-orthogonal 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 letter 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 non-locality [10], and the probable increased computational power of quantum theory [11]. Perhaps most alarmingly, superpositions give rise to the measurement problem, so captivatingly illus-trated by the "Schrödinger's cat" thought experiment.
Schrödinger's cat is set up to be in a superposition of |dead and |alive 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 |dead or |alive 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 |alive or |dead states) then the superposition is unambiguously ontic: there are ontological features which correspond to that superposition but not to non-superpositions, so that superposition is real.
Obviously quantum superpositions are different from proper mixtures of basis states, the question here is rather whether quantum superpositions can be understood as probability distributions over some underlying ontic states, each of which is also accessible by preparing some basis state. Previous single-system theorems that bound ontic overlaps in order to argue for the onticity of the quantum state [2][3][4][5][6] share at least these shortcomings: (i) they prove the existence of a pair of quantum states from a specific set with a bounded overlap, rather than bounding overlaps between arbitrary quantum states and (ii) the limits in which they prove that overlaps approach zero are limits in which the quantum states approach orthogonality [8].
In this letter it is proved that, for a d > 3 dimensional quantum system, almost all quantum superpositions w.r.t. 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.
Ontological models. The appropriate framework for discussing epistemic realism is that of ontological models [8,12,13]. It is flexible enough for most realist approaches to quantum ontology to be cast as an ontological models [7] including, but not limited to, Bohmian theories, spontaneous collapse theories, and naïve wavefunction-realist theories [14].
An ontological model of a system has a set Λ of ontic states λ ∈ Λ. 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 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 d-dimensional complex Hilbert space H with P(H) def = {|ψ ∈ H : ψ = 1, |ψ ∼ e iθ |ψ } as the set of distinct pure quantum states [15]. Quantum superpositions are defined w.r.t. some ONB B of H, and are simply those |ψ ∈ P(H) for which |ψ ∈ B.
A measurement M of a quantum system can be represented as a set of outcomes: either vectors of some ONB B ′ (for an ONB measurement) or POVM elements (for a general POVM measurement). An ontological model assigns a set Ξ M of conditional probability distributions, called response functions P M ∈ Ξ M , to M . A method for performing measurement M selects some P M ∈ Ξ M which gives the probability of obtaining outcome E ∈ M conditional on the ontic state of the system. A preparation of |ψ via µ ∈ ∆ |ψ followed by a measurement M via P M ∈ Ξ M therefore returns outcome E ∈ M with probability An ontological model assigns a set Γ U of stochastic maps γ to each unitary transformation U over H. A method for performing U selects some γ ∈ Γ U which, given that the system is in ontic state λ ′ , describes a probability distribution γ(·|λ ′ ) of states which λ ′ may be mapped to. A preparation of |ψ via µ ∈ ∆ |ψ followed by U via γ ∈ Γ U results in an ontic state distributed according to It is required that ν ∈ ∆ U|ψ , since this an example of a procedure preparing the quantum state U |ψ .
For now, assume that measurement statistics predicted by quantum theory are exactly correct, so valid ontological models must reproduce them. Therefore, for every |ψ ∈ P(H), every unitary U over H, and every measurement M , any choices of preparation µ ∈ ∆ |ψ , stochastic map γ ∈ Γ U , and response function P M ∈ Ξ M , must satisfy (3) It shall be useful to consider the stabiliser subgroups of unitaries S |ψ def = {U : U |ψ = |ψ } for each |ψ ∈ P(H).
In particular, an ontological model is preparation noncontextual with respect to stabiliser unitaries of |ψ if and only if for every µ ∈ ∆ |ψ , U ∈ S |ψ , and γ ∈ Γ U the action of γ, according to Eq. (2), leaves the preparation distribution µ unaffected.
Quantum superpositions are real. Define quantum superpositions w.r.t. some ONB B and consider any superposition state |ψ ∈ B. If every ontic state accessible by preparing any µ ∈ ∆ |ψ is also accessible by preparing some |i ∈ B, then |ψ has no ontology independent of B in the ontological model. Such a |ψ is called an epistemic or statistical superposition and must satisfy |i ∈B ̟(|i | µ) = 1, ∀µ ∈ ∆ |ψ , or equivalently, (10) The alternative occurs when there exists some subset of ontic states λ ∈ Λ B ψ ⊂ Λ for which µ(λ) > 0 for some µ ∈ ∆ |ψ , but ν(λ) = 0 for every ν ∈ ∆ |i ∈B . That is, the ontic states λ ∈ Λ B ψ are accessible from preparations of |ψ but not from any preparations of any |i ∈ B, making |ψ an ontic or real superposition.
From Eqs. (5,11), a superposition |ψ ∈ B can only be epistemic if the asymmetric overlap ̟(|i | µ) is maximal for every µ ∈ ∆ |ψ and all |i ∈ B. Therefore, the statement that "not every quantum superposition can be epistemic" is very weak. A more interesting question is whether an individual superposition state |ψ can be epistemic.
Theorem 1. Consider a quantum system of dimension d > 3 and define superpositions w.r.t. some ONB B. Almost all quantum superposition states |ψ ∈ B are ontic.
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 previous singlesystem ontology arguments mentioned above. (i) leaves open the possibility of significant ontic overlaps for many pairs of quantum states. (ii) casts doubt on the significance of those zero-overlap limits, as orthogonal states are distinguishable and therefore ontologically distinct anyway.
The following theorem address these shortcomings.
The proof, in appendix , closely follows that of Thm. 1. Assuming pure state preparation non-contextuality w.r.t. stabiliser unitaries is required to replace the assumption that |ψ is an epistemic superposition w.r.t. |0 .
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 ǫ ∈ (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 ǫ ∈ (0, 1]. However, an alternative overlap measure, the symmetric overlap ω(|ψ , |φ ) [2,[4][5][6]8], is robust to small errors and Thm. 2 can be modified to bound the symmetric overlap in a noise tolerant way.
The proof is provided in appendix . It makes Thm. 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. (21)] 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 Thm. 2's arguments can be made robust against error, it suggests that a noise tolerant version of Thm. 1 should also be possible.
Discussion. Assuming that quantum statistics are exactly correct, Thm. 1 proves that, for d > 3, almost all superpositions defined w.r.t. any given basis 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 Thm. 2 to prove that, for arbitrary states satisfying | φ|ψ | 2 ∈ (0, 1 4 ), ontic overlap must approach zero as d increases for fixed | φ|ψ | 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 non-contextuality w.r.t. stabiliser unitaries. Pure state preparation contextuality is often implicitly assumed wholesale, so this assumption should not be too controversial. Moreover, appendix 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 non-orthogonal 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 Thm. 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 Thms. 1,2 is tightly linked to the asymmetric overlap as a probability, making noise tolerant versions a challenge to extract. If the conclusion from Thms. 1,2 could be obtained though an operational methodology (closer to that of Bell's theorem [10] or the PBR theorem [1]) this would likely lead to better noise tolerant extensions and better opportunities for experimental investigation. Such an operational version may also make any information theoretic implications of these results clearer.
I would like to thank Jonathan Barrett, Owen Maroney, Clare Horsman, and Matty Hoban for insightful discussions. This work is supported by the Engineering and Physical Science Research Council, the CHIST-ERA DIQIP project, and the FQXi Large Grant "Thermodynamic vs information theoretic entropies in probabilistic theories".

Theorem 3: A noise tolerant bound on the symmetric overlap
This proof uses the the assumptions, notation, and constructions from Thm. 2, except this time it is only assumed that the ontological model reproduces quantum probabilities to within some additive error ǫ ∈ (0, 1]. It will also be necessary to define the tri-partite symmetric overlap between three probability distributions µ, ν, χ [8] ω(µ, ν, χ) Consider any pair of preparation distributions µ ∈ ∆ |ψ , ν ∈ ∆ |0 . From Thm. 2 it is known that there exist U i ∈ S |ψ such that U i |0 = |φ i . For each U i consider any corresponding stochastic map γ i ∈ Γ Ui . By assumption, preparations of |ψ are non-contextual with respect to stabiliser unitaries so each γ i maps µ to itself. Let each γ i map ν to some χ i ∈ ∆ |φi . For notational convenience, let ν = χ 0 and define the setsĨ Consider a preparation of |ψ via µ, followed by a measurement M in the basis B ′ . Similarly to Thm. 2, the aim is to bound ω(µ, ν) by considering the probability of obtaining either of the measurement outcomes |0 or |1 ′ , given by The trick is to do this in such a way that all possible errors are accounted for. In order to link this quantum probability to symmetric overlaps, consider the following subsets of Λ.
Roughly, this is the region of the overlap between µ and χ i for which χ i is greater than all other χ j =i , but smaller than µ.
• Similarly, for each i > j ∈ I consider Θ j Note that these sets are defined to be disjoint, for i = j: The point of these subsets is the way in which they relate to symmetric overlaps. From the definitions of symmetric overlaps it is not difficult to verify that Proceed by separating the probability Eq. (43) according to subsets in which λ may obtain: The final line follows simply because P M (|0 ∨ |1 ′ , λ ∈ Ω ij |µ) ≤ P M (λ ∈ Ω ij |µ) = Ωij dλµ(λ).
For the i = 0 term in the first line of Eq. (47), define the function ξ(λ) This can be simplified by noting that, for any Ω ⊆ Λ, = ǫ (54) so that The i ∈Ĩ terms of the first line of Eq. (47) follow in a similar way. Define ζ i (λ) Together, Eqs. (47,55,59) produce The i = 0 term of the second line of Eq. (60) can be bounded as follows The i ∈Ĩ terms of the second line of Eq. (60) can be similarly bounded So now combining Eqs. (60,64,67) it is found that Equation (68) can be further reduced by adding any negative quantity. For example, consider Boole's inequality Therefore, Eq. (68) reduces to This can be further simplified by noting so that Eq. (70) becomes having used Eqs.(44,45). As a final step, consider how the tripartite symmetric overlaps are bounded by ǫ. Consider the measurement Conservation of probability requires that for all λ ∈ Λ. Consider the tripartite symmetric overlap, Eqs. (41,45), for µ, χ i , χ j . Then Applying Eq. (81) to Eq. (74), one finds that having used Eq. (42). From Eq. (20) it is seen that there always exist choices for µ ∈ ∆ |ψ , ν ∈ ∆ |0 such that ω(|ψ , |0 ) ≤ ω(µ, ν) and so by combining Eqs. (43,83) This completes the proof. A note is in order about the tightness of this bound, assuming arbitrarily small ǫ. At d = 4, this bound cannot improve upon that of Eq. (21) for any α 2 ∈ (0, 1 4 ). At d = 5, an improvement is possible for some values of α. It is only for d > 5 that this bound is capable of improving upon Eq. (21) for all values of α ∈ (0, 1 4 ). This is because the theorem extends the methods of Thms. 1,2, which are closely linked to the asymmetric overlap, to the symmetric overlap.
It may be possible to improve on the error term above by more carefully using higher Bonferroni inequalities [20], rather than just Boole's inequality as used above. Considering quad-partite and higher-order overlaps (rather than stopping at the tripartite overlap, as done here) may also improve the error. Doing so may improve the scaling with d.

Justifying Preparation Non-contextuality with respect to Stabiliser Unitaries
The ontological models formalism combines fundamental objective ontology and operational notions. The fundamental ontology is reflected in the idea that ontic states represent actual states of affairs, independently of any other theories an observer might use to describe the same system. On the other hand, the only way to reason about this largely-unspecified ontological level is operationally: how does it respond to preparations, transformations, and measurements that we can actually perform.
An assumption of non-contextuality is an assumption about these operational bridges between our capabilities and the ontology. The following argument is designed to defend the idea that operational preparations of some pure quantum state |ψ ∈ P(H) can reasonably be assumed to be non-contextual with respect to stabiliser unitaries S |ψ of the same state.
Any specific operational method for preparing some state |ψ ∈ P(H) may be thought of as a black box which the system is fed into. When the system is fed out of the box, it is promised that the box has prepared the system in state |ψ according to some specific method. In terms of ontological models, any preparation distribution µ ∈ ∆ |ψ can be considers in terms of such a box.
Suppose you design some experiment, which involves preparing |ψ via µ. Scientists implementing that experiment would obtain the corresponding black box to be sure that the method corresponding to µ is indeed used. Once prepared, the system will the need to be presented to other pieces of apparatus. However, there will always be variation in how the system is treated between preparation and the action of any other apparatus, any amount of motion or passage of time or other (seemingly innocuous) treatment amounts to applying some unitary U to the system. Each scientist will, no doubt, be careful to ensure that the system is not disturbed from its preparation state, so it can be safely assumed that any such U is a stabiliser unitary S |ψ . However, the point remains that some unknown U ∈ S |ψ is inevitably applied to the system after preparation via µ, and this can never be perfectly accounted for. Therefore, to analyse the result of the experiment, one has to allow for some unknown U ∈ S |ψ to by applied (via some unknown γ ∈ Γ U ) after preparation of |ψ via µ. On this minimally realistic operational level, an arbitrary preparation distribution µ can never be prepared unscathed, one has to account for the inevitable, unknown, subsequent stabiliser unitary. The effective preparation distribution that one must therefore use to describe the experiment has to be one that is non-contextual with respect to such transformations, allowing the experiment to still be analysed despite the application of an unknown U ∈ S |ψ .
One must be careful to only consider operational features that are not, even in principle, impossible to reliably perform. Since the sets of preparation distributions for any given quantum states are, in the end, operational in character, one may safely restrict to preparation distributions that satisfy certain sensible realistic requirements. The above heuristic argument aims to establish pure state preparation non-contextuality with respect to stabiliser unitaries as such a realistic requirement.