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Quantum superpositions cannot be epistemic

  • John-Mark A. Allen
Open Access
Regular Paper

Abstract

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 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.

Keywords

Quantum Superposition Foundations Ontic  Epistemic Ontology 

1 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 now-famous PBR theorem [15], which proves that the quantum state of a system is ontic given reasonable assumptions about the ontic structure of multi-partite systems. Whilst these assumptions appear weak and well-motivated, 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 single-system 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 non-orthogonal states, no-cloning, 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 non-orthogonal quantum states might overlap—there might be ontic states accessible by preparing either of those quantum states. The main strategy of the single-system 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, wave-particle duality, entanglement, Bell non-locality [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 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 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 “toy-bit” 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 ac; and \(|-)\) over bd. 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 single-system 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 wave-function-realist 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 d-dimensional 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 distributions3 \(\mu (\lambda )\) 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 non-contextual 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 \(\mu \).

A measurement M of a quantum system can be represented as a set of outcomes: either vectors of some ONB \(\mathcal {B}^{\prime }\) (for an ONB measurement) or POVM elements (for a general POVM measurement). An ontological model assigns a set \(\Xi _{M}\) of conditional probability distributions, called response functions \(\mathbb {P}_{M}\in \Xi _{M}\), to M. A method for performing measurement M selects some \(\mathbb {P}_{M}\in \Xi _{M}\) which gives the probability of obtaining outcome \(E\in M\) conditional on the ontic state of the system. A preparation of \(|\psi \rangle \) via \(\mu \in \Delta _{|\psi \rangle }\) followed by a measurement M via \(\mathbb {P}_{M}\in \Xi _{M}\), therefore, returns outcome \(E\in M\) with probability
$$\begin{aligned} \mathbb {P}_{M}(E\,|\,\mu )=\int _{\Lambda }\mathrm {d}\lambda \,\mu (\lambda )\mathbb {P}_{M}(E\,|\,\lambda ). \end{aligned}$$
(1)
Transformations acting on a system must correspond to stochastic maps on its space of ontic states \(\Lambda \). An ontological model assigns a set \(\Gamma _{U}\) of stochastic maps \(\gamma \) to each unitary transformation U over \(\mathcal {H}\). A method for performing U selects some \(\gamma \in \Gamma _{U}\) which, given that the system is in ontic state \(\lambda ^{\prime }\), describes a probability distribution \(\gamma (\cdot |\lambda ^{\prime })\), so that the probably that \(\lambda ^{\prime }\) is mapped to \(\lambda \) under this operation is \(\gamma (\lambda |\lambda ^{\prime })\). A preparation of \(|\psi \rangle \) via \(\mu \in \Delta _{|\psi \rangle }\) followed by a transformation U via \(\gamma \in \Gamma _{U}\) results in an ontic state distributed according to the distribution \(\nu \), given by
$$\begin{aligned} \nu (\lambda )=\int _{\Lambda }\mathrm {d}\lambda ^{\prime }\mu (\lambda ^{\prime })\gamma (\lambda \,|\,\lambda ^{\prime }),\quad \forall \lambda \in \Lambda . \end{aligned}$$
(2)
It is required that \(\nu \in \Delta _{U|\psi \rangle }\), since this an example of a procedure preparing the quantum state \(U|\psi \rangle \).
For now, assume that measurement statistics predicted by quantum theory are exactly correct, so valid ontological models for quantum systems must reproduce them. Therefore, for every \(|\psi \rangle \in \mathcal {P}(\mathcal {H})\), every unitary U over \(\mathcal {H}\), and every measurement M, any choices of preparation \(\mu \in \Delta _{|\psi \rangle }\), stochastic map \(\gamma \in \Gamma _{U}\), and response function \(\mathbb {P}_{M}\in \Xi _{M}\), must satisfy
$$\begin{aligned} \langle \psi |U^{\dagger }EU|\psi \rangle =\int _{\Lambda }\mathrm {d}\lambda \int _{\Lambda }\mathrm {d}\lambda ^{\prime }\mu (\lambda ^{\prime })\gamma (\lambda |\lambda ^{\prime })\mathbb {P}_{M}(E\,|\,\lambda ),\quad \forall E\in M. \end{aligned}$$
(3)
It shall be useful to consider the stabiliser subgroups of unitaries \(\mathcal {S}_{|\psi \rangle }\mathop = \limits ^\mathrm{def}\{U\,:\,U|\psi \rangle =|\psi \rangle \}\) for each \(|\psi \rangle \in \mathcal {P}(\mathcal {H})\). In particular, an ontological model is preparation non-contextual with respect to stabiliser unitaries of \(|\psi \rangle \) if and only if for every \(\mu \in \Delta _{|\psi \rangle }\), \(U\in \mathcal {S}_{|\psi \rangle }\), and \(\gamma \in \Gamma _{U}\) the action of \(\gamma \), according to Eq. (2), leaves the preparation distribution \(\mu \) unaffected (that is, \(\nu \) in Eq. (2) would be equal to \(\mu \)).

3 Measuring overlaps

One way to quantify the overlap between preparation distributions is the asymmetric overlap \(\varpi (|\phi \rangle |\mu )\) [1, 13, 14], defined as the probability of obtaining an ontic state \(\lambda \) accessible from some preparation of \(|\phi \rangle \) when sampling from \(\mu \). Formally,
$$\begin{aligned} \varpi (|\phi \rangle \,|\,\mu )\mathop = \limits ^\mathrm{def}\int _{\Lambda _{|\phi \rangle }}\mathrm {d}\lambda \,\mu (\lambda ), \end{aligned}$$
(4)
where \(\Lambda _{|\phi \rangle }\mathop = \limits ^\mathrm{def}\cup _{\nu \in \Delta _{|\phi \rangle }}\Lambda _{\nu }\) is the total support of all possible preparations of \(|\phi \rangle \). By Eq. (3), the asymmetric overlap must be upper bounded by the Born rule measurement probability (proof in “Appendix 1”)
$$\begin{aligned} \varpi (|\phi \rangle \,|\,\mu )\le |\langle \phi |\psi \rangle |^{2},\quad \forall \mu \in \Delta _{|\psi \rangle }. \end{aligned}$$
(5)
That is, the probability of obtaining a \(\lambda \) compatible with \(|\phi \rangle \) when preparing \(|\psi \rangle \) cannot exceed the probability of getting the measurement outcome \(|\phi \rangle \) having prepared \(|\psi \rangle \).
This quantifies overlaps between pairs of quantum states, but what of multi-partite overlaps? The asymmetric multi-partite overlap \(\varpi (|0\rangle ,|\phi \rangle ,...\,|\,\mu )\) acts like the union of the bipartite overlaps \(\varpi (|0\rangle |\mu )\), \(\varpi (|\phi \rangle |\mu )\), etc. It is defined as the probability of obtaining a \(\lambda \in \Lambda _{|0\rangle }\cup \Lambda _{|\phi \rangle }\cup ...\) when sampling from \(\mu \). Formally,
$$\begin{aligned} \varpi (|0\rangle ,|\phi \rangle ,...\,|\,\mu )\mathop = \limits ^\mathrm{def}\int _{\Lambda _{|0\rangle }\cup \Lambda _{|\phi \rangle }\cup ...}\mathrm {d}\lambda \,\mu (\lambda ). \end{aligned}$$
(6)
From Eqs. (4) and (6) and Boole’s inequality, it is clear that
$$\begin{aligned} \varpi (|0\rangle ,|\phi \rangle ,...\,|\,\mu )\le \varpi (|0\rangle \,|\,\mu )+\varpi (|\phi \rangle \,|\,\mu )+ \cdots \end{aligned}$$
(7)
Quantum states are only perfectly distinguishable if they are mutually orthogonal. Distinguishable states must be ontologically distinct (their preparation distributions cannot overlap) to satisfy Eq. (3). The opposite concept of anti-distinguishability is much more useful in discussions of ontic overlaps [10]. A set \(\{|\psi \rangle ,|\phi \rangle ,...\}\subset \mathcal {P}(\mathcal {H})\) is anti-distinguishable if and only if there exists a measurement \(M=\{E_{\lnot \psi },E_{\lnot \phi },...\}\) such that
$$\begin{aligned} \langle \psi |E_{\lnot \psi }|\psi \rangle =\langle \phi |E_{\lnot \phi }|\phi \rangle =\cdots =0, \end{aligned}$$
(8)
i.e. the measurement can tell, with certainty, one state from the set that was not prepared. It has been proven [2, 6] that if some inner products \(a=|\langle \phi |\psi \rangle |^{2}\), \(b=|\langle 0|\psi \rangle |^{2}\), \(c=|\langle 0|\phi \rangle |^{2}\) satisfy
$$\begin{aligned} a+b+c<1,\quad (1-a-b-c)^{2}\ge 4abc, \end{aligned}$$
(9)
then the triple \(\{|\psi \rangle ,|\phi \rangle ,|0\rangle \}\) must be anti-distinguishable. Anti-distinguishable triples \(\{|\psi \rangle ,|\phi \rangle ,|0\rangle \}\) are useful because \(\Lambda _{|\psi \rangle }\cap \Lambda _{|\phi \rangle }\cap \Lambda _{|0\rangle }=\emptyset \) and, therefore, \(\varpi (|0\rangle ,|\phi \rangle |\mu )=\varpi (|0\rangle |\mu )+\varpi (|\phi \rangle |\mu )\) for all \(\mu \in \Delta _{|\psi \rangle }\), as proved in “Appendix 1”.

4 Quantum superpositions are real

Define quantum superpositions with respect to some ONB \(\mathcal {B}\) and consider any superposition state \(|\psi \rangle \not \in \mathcal {B}\). If every ontic state accessible by preparing any \(\mu \in \Delta _{|\psi \rangle }\) is also accessible by preparing some \(|i\rangle \in \mathcal {B}\), then \(|\psi \rangle \) has no ontology independent of \(\mathcal {B}\) in the ontological model. Such a \(|\psi \rangle \) is called an epistemic or statistical superposition and must satisfy
$$\begin{aligned} \sum _{|i\rangle \in \mathcal {B}}\varpi (|i\rangle \,|\,\mu )= & {} 1,\;\forall \mu \in \Delta _{|\psi \rangle },\;\text {or equivalently,}\end{aligned}$$
(10)
$$\begin{aligned} \varpi (|i\rangle \,|\,\mu )= & {} |\langle i|\psi \rangle |^{2},\quad \forall |i\rangle \in \mathcal {B},\mu \in \Delta _{|\psi \rangle }. \end{aligned}$$
(11)
The alternative occurs when there exists some subset of ontic states \(\lambda \in \Lambda _{\psi }^{\mathcal {B}}\subset \Lambda \) for which \(\mu (\lambda )>0\) for some \(\mu \in \Delta _{|\psi \rangle }\), but \(\nu (\lambda )=0\) for every \(\nu \in \Delta _{|i\rangle \in \mathcal {B}}\). That is, the ontic states in \(\Lambda _{\psi }^{\mathcal {B}}\) are accessible by preparing \(|\psi \rangle \) but not by preparing any \(|i\rangle \in \mathcal {B}\), making \(|\psi \rangle \) an ontic or real superposition.

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 \(|\langle 0|\psi \rangle |^{2}\in (0,\frac{1}{2})\).

Define an ONB \(\mathcal {B}^{\prime }=\{|0\rangle \}\cup \{|i^{\prime }\rangle \}_{i=1}^{d-1}\) containing this \(|0\rangle \) such that
$$\begin{aligned} |\psi \rangle =\alpha |0\rangle +\beta |1^{\prime }\rangle +\gamma |2^{\prime }\rangle , \end{aligned}$$
(12)
where \(\alpha \in \mathbb {R}\), \(\alpha \in (0,\,1/\sqrt{2})\), and \(\beta \mathop = \limits ^\mathrm{def}\sqrt{2}\alpha ^{2}\). Such bases always exists since \(|\langle 0|\psi \rangle |^{2}=\alpha ^{2}\) and \(|\alpha |^{2}+|\beta |^{2}=\alpha ^{2}(1+2\alpha ^{2})<1\). With respect to the same \(\mathcal {B}^{\prime }\), define
$$\begin{aligned} |\phi \rangle \mathop = \limits ^\mathrm{def}\delta |0\rangle +\eta |1^{\prime }\rangle +\kappa |3^{\prime }\rangle , \end{aligned}$$
(13)
where \(\delta \mathop = \limits ^\mathrm{def}1-2\alpha ^{2}\), \(\eta \mathop = \limits ^\mathrm{def}\sqrt{2}\alpha \). This can always be normalised because \(|\delta |^{2}+|\eta |^{2}=(1-2\alpha ^{2})^{2}+2\alpha ^{2}<1\).
The above construction has been chosen such that
  • \(|\langle 0|\psi \rangle |^{2}=\alpha ^{2}=|\langle \phi |\psi \rangle |^{2}\) so there exists a unitary \(U\in \mathcal {S}_{|\psi \rangle }\) for which \(U|0\rangle =|\phi \rangle \);

  • 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 anti-distinguishable.

For any preparation distribution \(\mu ^{\prime }\in \Delta _{|\psi \rangle }\) of \(|\psi \rangle \), consider \(\varpi (|0\rangle |\mu ^{\prime })\). For any unitary V and any corresponding \(\gamma \in \Gamma _{V}\), \(\mu ^{\prime }\) is evolved to some \(\mu \in \Delta _{V|\psi \rangle }\) as in Eq. (2). This operation cannot decrease the asymmetric overlap \(\varpi (V|0\rangle |\mu )\ge \varpi (|0\rangle |\mu ^{\prime })\) and, in particular, letting \(V=U\) one finds
$$\begin{aligned} \varpi (|\phi \rangle |\mu )\ge \varpi (|0\rangle |\mu ^{\prime }). \end{aligned}$$
(14)
A proof of this is provided in “Appendix 1”. Therefore, there must exist preparation distributions \(\mu ,\mu ^{\prime }\in \Delta _{|\psi \rangle }\) satisfying Eq. (14).
Assume towards a contradiction that \(|\psi \rangle \) is an epistemic superposition so that Eq. (11) holds and, in particular, \(\varpi (|0\rangle |\mu )=\varpi (|0\rangle |\mu ^{\prime })=\alpha ^{2}\). By Eq. (14) it is, therefore, found that
$$\begin{aligned} \varpi (|\phi \rangle \,|\,\mu )\ge \varpi (|0\rangle \,|\,\mu ). \end{aligned}$$
(15)
Consider, then, a preparation of the state \(|\psi \rangle \) via \(\mu \) followed by an ONB measurement M in the \(\mathcal {B}^{\prime }\) basis. Since \(|\psi \rangle \) was prepared, \(\lambda \in \Lambda _{|\psi \rangle }\) and the only possible measurement outcomes are \(|0\rangle \), \(|1^{\prime }\rangle \), and \(|2^{\prime }\rangle \). By Eq. (3), almost all \(\lambda \in \Lambda _{|0\rangle }\) must return the outcome \(|0\rangle \) with certainty. Similarly, almost all \(\lambda \in \Lambda _{|\phi \rangle }\) can only return \(|0\rangle \), \(|1^{\prime }\rangle \), or \(|3^{\prime }\rangle \) as the measurement outcome. Therefore, the probability of obtaining outcomes \(|0\rangle \) or \(|1^{\prime }\rangle \) must be lower bounded by the probability of obtaining a \(\lambda \in \Lambda _{|0\rangle }\cup \Lambda _{|\phi \rangle }\); formally,
$$\begin{aligned} \mathbb {P}_{M}(|0\rangle \vee |1^{\prime }\rangle \,|\,\mu )\ge & {} \varpi (|0\rangle ,|\phi \rangle \,|\,\mu )=\varpi (|0\rangle \,|\,\mu )+\varpi (|\phi \rangle \,|\,\mu )\nonumber \\\ge & {} 2\varpi (|0\rangle \,|\,\mu ), \end{aligned}$$
(16)
where the equality follows because \(\{|0\rangle ,|\psi \rangle ,|\phi \rangle \}\) is anti-distinguishable and the final line follows from Eq. (15), which is found by assuming that \(|\psi \rangle \) is an epistemic superposition.
In order to satisfy Eq. (3)
$$\begin{aligned} \mathbb {P}_{M}(|0\rangle \vee |1^{\prime }\rangle \,|\,\mu )=|\langle 0|\psi \rangle |^{2}+|\langle 1^{\prime }|\psi \rangle |^{2}=\alpha ^{2}+2\alpha ^{4}. \end{aligned}$$
(17)
Combining Eqs. (16) and (17) it is found that
$$\begin{aligned} \varpi (|0\rangle \,|\,\mu )\le \alpha ^{2}\left( \frac{1}{2}+\alpha ^{2}\right) <\alpha ^{2}. \end{aligned}$$
(18)
But, this contradicts the assumption that \(|\psi \rangle \) is an epistemic superposition which implies \(\varpi (|0\rangle |\mu )=\alpha ^{2}\) by Eq. (11). Therefore, if the predictions of quantum theory are to be exactly reproduced, any such \(|\psi \rangle \) must be an ontic, rather than epistemic, superposition.\(\square \)

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 single-system 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 zero-overlap limits (as orthogonal states are distinguishable and, therefore, must be trivially ontologically distinct).

The following theorem address these shortcomings:

Theorem 2

Consider a \(d>3\) dimensional quantum system and any pair \(|\psi \rangle ,|0\rangle \in \mathcal {P}(\mathcal {H})\) such that \(|\langle 0|\psi \rangle |^{2}\mathop = \limits ^\mathrm{def}\alpha ^{2}\in (0,\frac{1}{4})\). Assume that pure state preparations of \(|\psi \rangle \) are non-contextual with respect to stabiliser unitaries of \(|\psi \rangle \). For any preparation distribution \(\mu \in \Delta _{|\psi \rangle }\), the asymmetric overlap must satisfy
$$\begin{aligned}&\varpi (|0\rangle \,|\,\mu ) \le \alpha ^{2}\left( \frac{1+2\alpha }{d-2}\right) \end{aligned}$$
(19)
$$\begin{aligned}&\lim _{d\rightarrow \infty }\varpi (|0\rangle \,|\,\mu ) = 0 \end{aligned}$$
(20)
and so becomes arbitrarily small as d increases, independently of \(\alpha \).

The proof, in “Appendix 1”, closely follows that of Theorem 1. The assumption of pure state preparation non-contextuality 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.

Suppose you are given some \(\lambda \in \Lambda \) obtained by sampling from either \(\mu \) or \(\nu \) (each with equal a priori probability). If you try to guess which of \(\mu ,\nu \) was used, then \(\omega (\mu ,\nu )/2\) is defined to be the average probability of error when using the optimal strategy. This is known to correspond to [2, 14]
$$\begin{aligned} \omega (\mu ,\nu )\mathop = \limits ^\mathrm{def}\int _{\Lambda }\mathrm {d}\lambda \min \{\mu (\lambda ),\nu (\lambda )\}. \end{aligned}$$
(21)
Extending this to quantum states themselves, rather than to preparation distributions, gives the symmetric overlap
$$\begin{aligned} \omega (|\psi \rangle ,|\phi \rangle )\mathop = \limits ^\mathrm{def}\sup _{\mu \in \Delta _{|\psi \rangle },\nu \in \Delta _{|\phi \rangle }}\omega (\mu ,\nu ). \end{aligned}$$
(22)
Quantum theory provides an upper bound on the symmetric overlap, since any quantum procedure for distinguishing \(|\psi \rangle ,|\phi \rangle \) is also a method for distinguishing \(\mu \in \Delta _{|\psi \rangle },\nu \in \Delta _{|\phi \rangle }\) in an ontological model. As \(\frac{1}{2}\left( 1 - \sqrt{1 - |\langle \phi | \psi \rangle |^2} \right) \) is the minimum average error probability when distinguishing \(|\psi \rangle ,|\phi \rangle \) within quantum theory5 it follows that \(\omega (\mu ,\nu )\le 1-\sqrt{1-|\langle \phi |\psi \rangle |^{2}}\) holds for every \(\mu \in \Delta _{|\psi \rangle },\nu \in \Delta _{|\phi \rangle }\) and so
$$\begin{aligned} \omega (|\psi \rangle ,|\phi \rangle )\le 1-\sqrt{1-|\langle \phi |\psi \rangle |^{2}}. \end{aligned}$$
(23)

Theorem 3

Consider the assumptions of Theorem 2, but only assume that the probabilities predicted by quantum theory are accurate to within \(\pm \epsilon \), for some \(\epsilon \in (0,1]\). The symmetric overlap must satisfy
$$\begin{aligned} \omega (|0\rangle ,|\psi \rangle )\le & {} \alpha ^{2}\left( \frac{1+2\alpha }{d-2}\right) +\frac{(3d^{2}-7d)}{2(d-2)}\epsilon . \end{aligned}$$
(24)
This bound is tighter than Eq. (23) for \(d>5\) for small \(\epsilon \).

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 noise-tolerant version of Theorem 1 should also be possible. Even so, a noise-tolerant 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 non-contextuality 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 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 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 noise-tolerant 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 noise-tolerant 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. 1.

    Conversely, ontological models are irrelevant for any “anti-realist”, “instrumentalist”, “positivist”, or “Copenhagen-like” theories denying the existence of an underlying ontology. For example, quantum-Bayesian theories are exempt from ontological model analysis.

  2. 2.

    For simplicity, take \(d<\infty \).

  3. 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 re-formulated measure-theoretically. 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. 4.

    Preparation non-contextuality for pure states is often implicitly assumed because it rarely affects arguments [10]. Rather, preparation contextuality for mixed quantum states is more often discussed [17]. However, explicit preparation contextuality for pure states will be needed here.

  5. 5.

    By using the Helstrom measurement [2, 20].

  6. 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. 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. 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 Long-term Challenges in Information and Communication Sciences and Technologies (CHIST-ERA) 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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Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of OxfordOxfordUK

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