Psi $-Epistemic Interpretations of Quantum Theory

ψ-epistemic interpretations of quantum theory have a measurement problem

Joshua B. Ruebeck1, Piers Lillystone1, and Joseph Emerson2,3

1Institute for Quantum Computing and Department of Physics and Astronomy University of Waterloo, Waterloo, Ontario N2L 3G1, Canada 2 Institute for Quantum Computing and Department of Applied Math University of Waterloo, Waterloo, Ontario N2L 3G1, Canada 3 Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada February 11, 2020

ψ-epistemic interpretations of quantum theory tum state is merely a state of knowledge about the real maintain that quantum states only represent in- state of the system. A very thorough review of these complete information about the physical states two stances can be found in [24]. A third recently ar- of the world. A major motivation for this view ticulated category of ψ-doxastic interpretations [25–31] is the promise to provide a reasonable account argue that the quantum state is a state of belief, and of state update under measurement by assert- are distinguished from ψ-epistemic interpretations by ing that it is simply a natural feature of updat- the fact that they deny that a system has some ‘real ing incomplete statistical information. Here we state.’ While not all interpretations conform to these demonstrate that all known ψ-epistemic onto- three descriptors, they are useful categories insofar as logical models of quantum theory in dimension they allow us to qualitatively discuss certain features d ≥ 3, including those designed to evade the con- separately from the particular interpretation in which clusion of the PBR theorem, cannot represent they are embedded. state update correctly. Conversely, interpreta- The ψ-epistemic class has garnered attention as a tions for which the wavefunction is real evade view which provides very appealing explanations of such restrictions despite remaining subject to otherwise paradoxical features of quantum theory like long-standing criticism regarding physical dis- the state update rule [17], the classical limit under continuity, indeterminism and the ambiguity of quantum chaos [20], no cloning [32], and entangle- the Heisenberg cut. This revives the possibil- ment [33]. For example, through this viewpoint state ity of a no-go theorem with no additional as- update is not a physical ‘collapse’ process and therefore sumptions, and demonstrates that what is usu- not subject to paradoxes, indeterminism, and disconti- ally thought of as a strength of epistemic inter- nuity; rather it is understood as analogous to the non- pretations may in fact be a weakness. pardoxical ‘collapse’ of a subjective probability distribution via Bayes’ rule upon consideration of new information. While several ψ-epistemic models have been pro- 1 Introduction posed [16, 22, 34–38], they generally have undesirable features or are restricted to a subtheory of full quan- There are many interpretations of quantum theory1. arXiv:1812.08218v3 [quant-ph] 9 Feb 2020 tum theory. None achieve all of the features that an Among the many diﬀerences between these interpreta- optimistic ψ-epistemicist would expect. tions, one that often takes center stage is the stance that This suggests the possibility that a fully satisfactory they take towards the wavefunction or quantum state. ψ-epistemic interpretation cannot actually explain all Three broad categories have been identiﬁed which cap- of quantum theory despite the qualitatively compelling ture a number of interpretations. Two of these cate- features of such a view2. This suspicion has led to a gories are more commonly juxtaposed: ψ-ontic inter- number of no-go theorems in recent years which estab- pretations [1–14] posit that the quantum state is a part of the real (physical) state of a system, whereas ψ- 2 epistemic interpretations [15–23] argue that the quan- As is well-known, the work of Bell [39] has shown that no ψ- epistemic interpretation can evade the non-locality that manifests Joshua B. Ruebeck: [email protected] trivially in ψ-ontic interpretations; this trivial manifestation of non-locality in ψ-ontic interpretations is an oft-forgotten insight 1This is an understatement. from Einstein [15, 18, 23].

Accepted in Quantum 2019-09-27, click title to verify 1 lish that, given at least one additional assumption, any ontic distinction4. Here we take some preliminary steps consistent interpretation of quantum theory cannot be in both of these directions, and argue that ψ-epistemic ψ-epistemic [36, 40–43]. These no-go theorems are gen- models are the natural arena in which to investigate in- erally proven within the ontological models formalism, teresting behavior of state update under measurement. which describes a large class of existing interpretations In Section2, we describe the ontological models for- of quantum theory [21, 44, 45]. malism, adding a description of state update under Within the ontological models formalism3, ψ- measurement and motivating its importance. Despite epistemic models can be given a precise mathematical this motivation, one might still argue that many dis- definition called the ψ-epistemic criterion [22]. This cri- tinctly quantum phenomena (e.g. Bell inequality vi- terion allows the possibility of conclusively ruling out olations) can be described without reference to state this type of model. Outside of this framework, it is un- update; thus, from an operationalist point of view, we likely that ψ-epistemic models can be precluded with shouldn’t need to consider state update in order to in- any kind of certainty; ψ-doxastic interpretations, for vestigate these phenomena. However, we show in Sec- example, do not fit neatly into the ontological models tion3 that the consideration of state update actually framework and thus are not necessarily ruled out by places nontrivial restrictions on how one can represent these no-go theorems. This is despite the fact that they even a prepare-and-measure-once experiment. Thus our share many of the features which make ψ-epistemic in- results are directly applicable to models which have only terpretations appealing. In the present paper we restrict specified behavior for a single measurement. Section4 our attention to the ontological models framework. reviews a number of examples of ontological models The fact that an extra assumption is required to rule from the literature; in each case we either specify its out ψ-epistemic theories has purportedly been demon- state update rule (in dimension d = 2, for ψ-ontic mod- strated by the existence of ψ-epistemic models which, els, and for some models of subtheories) or prove its while being individually unsatisfactory for various rea- impossibility (for all known ψ-epistemic models in di- sons, do satisfy at least the bare minimum requirements mension d ≥ 3). Finally, we discuss the implications of a ψ-epistemic theory [35–37]. All of these models of our results and describe some open questions in Sec- were specified within a prepare-measure framework, so tion5. they have been proven to reproduce quantum statistics for all experiments that involve preparing a state and then measuring it once. In this paper we show that, 2 Defining measurement update in on- if we allow sequential measurements in the operational description, these models cannot reproduce operational tological models statistics. Our main contribution in this work is thus to demon- 2.1 The ontological models formalism strate that the state update rule imposes severe con- In the standard treatment, an operational theory [44] is straints on ψ-epistemic models. This is in contrast to described by a set of preparations P, a set of transfor- the prevailing view that, as articulated by Leifer, “a mations T , and a set of measurements M along with a straightforward resolution of the collapse of the wave- probability distribution function, the measurement problem, Schrödingerscat and friends is one of the main advantages of ψ-epistemic Pr(k|M,T,P ). (1) interpretations” [24]. As a consequence, we revive the possibility of a general no-go theorem for ψ-epistemic This quantity describes the probability of some mea- models that doesn’t rely on an additional assumption surement outcome k ∈ given an experimenter’s choice such as the locality assumption required in [40] which Z of P ∈ P, T ∈ T , and M ∈ M. When consider- conflicts with the non-locality that is implied by Bell’s ing transformations this is called the prepare-transform- theorem [43,46]. See AppendixA for further discussion measure operational framework, and when we omit on this point. transformations it is the prepare-measure framework. Although state update under measurement has been Often we take P to be the set of pure quantum state described in a few specific models [22, 34, 47, 48] and preparations, T to be the full set unitary maps on a discussed with regards to contextuality [49], it has yet to Hilbert space, and M to be all projective measurements be treated generally or in relation to the ψ-epistemic/ψ- on this Hilbert space. In this case, we say we are de- 3A note on potentially confusing terminology—an ontological model assumes the reality of some kind of state (hence “onto- 4Although the Leggett-Garg inequalities [50, 51] might be con- logical”), but does not assume the reality of the quantum state strued as a general treatment of state update, it is more accurate specifically (and so is not necessarily ψ-ontic). to say that they are about the absence of state update.

Accepted in Quantum 2019-09-27, click title to verify 2 scribing the full quantum theory5; in contrast, a sub- given the previous state λ and the choice T of trans- theory is described by taking subsets of P, T , M for the formation. Finally, measurements are represented by a full quantum theory. For example, in quantum infor- response function ξ(k|λ, M) which describes the prob- mation settings we often consider only measurements ability of an outcome k given the ontic state λ and the in the standard basis. choice of measurement M. We say that an ontological Note that this standard deﬁnition involves a single model successfully reproduces quantum theory if measurement and a single measurement outcome de- Z Z spite the fact many important quantum experiments dλ0 dλ ξ(k|λ0,M)Γ(λ0|λ, T )µ(λ|P ) (e.g. Stern-Gerlach, double slit [52]) and quantum algo- Λ Λ rithms (e.g. measurement-based error correction [53]) = PrQ(k|M,T,P ) (4) involve multiple measurements. Thus we will refer to ∀P ∈ P,T ∈ T ,M ∈ M, the usual deﬁnition of prepare-measure as prepare-and- measure-once experiments. In this paper, we are con- or, in a prepare-and-measure-once experiment, cerned with multiple measurements, so we will also have Z to describe probabilities like dλ ξ(k|λ, M)µ(λ|P ) = PrQ(k|M,P ) (5) Λ ∀P ∈ P,M ∈ M. Pr(k2, k1|M2,M1,P ). (2)

Additionally, we note that positive operator valued In both of these cases, PrQ indicates the outcome proba- measures (POVMs) do not fully specify how a mea- bility calculated by operational quantum theory for the surement updates a state. Although one can obtain particular experiment under consideration. Again, we must supplement this definition in order to a POVM {Ek} from a set of generalized measurement operators {M } by the relation E = M †M , the de- model sequential measurement. In textbook quantum k k k k theory, the state updates during a measurement in a composition of {Ek} into {Mk} is not unique. Thus although M is often described by POVMs, considera- way that depends on the previous state, the measure- tion of state update requires that we specify generalized ment procedure, and the measurement outcome: For a measurement operators instead. As an example of when measurement outcome corresponding to a projector Πk, this is important, consider a coarse-graining of the mea- a quantum state |ψi after measurement will be surement {M } = {[0], [1], [2]}, where we denote the k 0 Πk |ψi projector onto a state |ψ i = . (6) hψ|Πk|ψi [ψ] = |ψihψ| (3) We allow for dependence on all of these things by choosing to represent this via a state update rule as in [24]. We can either coarse-grain coherently, i.e. 0 0 η(λ |k, λ, M). As with µ, Γ, and ξ, we require that η is measure {Mk} = {[0] + [1], [2]} or we can coarse-grain normalized. Although this object looks very similar to decoherently by measuring {Mk} and then combining the transition matrix for transformations, it is distin- outcomes 0 and 1 into a single measurement result guished by two important features which we emphasize and ‘forgetting’ which one actually occurred. While by choosing a new symbol to represent it. these two processes are represented by the same POVM The first distinction is simple, in that η depends on a {Ek} = {[0] + [1], [2]}, their state update behavior is measurement outcome k, while Γ does not; this is anal- different: if the state √1 (|0i + |1i) is measured, it will 2 ogous to the fact that generally in quantum theory we stay the same in the coherent case or update to the can only implement measurement update maps proba- 1 mixed state 2 ([0] + [1]) in the decoherent case. bilistically (i.e. by post-selecting on a not-necessarily- An ontological model [21, 44, 45] supplements this op- deterministic measurement outcome). erational point of view by asserting that a system has The second distinction is the fact that η(λ0|k, λ, M) is a state λ, called an ontic state. To specify an onto- not defined for all λ ∈ Λ. Roughly speaking, it doesn’t logical model, we first choose an ontic state space Λ. make sense to ask “What is the new state λ0 after mea- Then, preparations are described by a preparation dis- suring state λ and obtaining outcome k?” if the out- tribution µ(λ|P ), which is the probability of preparing come k could not have occurred given the previous state some state λ ∈ Λ given the preparation P . Transforma- λ. To express this formally, we define the support of a 0 tions are described by a transition matrix Γ(λ |λ, T ), distribution which is the probability of preparing a new state λ0 Supp(ξ(k|·)) = {λ ∈ Λ: ξ(k|λ) > 0} (7) 5Larger sets can be considered (e.g. including mixed states, CPTP maps, or non-projective measurements), but all of the as the set of ontic states on which it is nonzero. Us- models studied in this paper fit the given definition. ing this, we can say that η(λ0|k, λ, M) is well-defined

Accepted in Quantum 2019-09-27, click title to verify 3 only for λ ∈ Supp(ξ(k|·)). This property of η is anal- state |ψi, to distinguish it from the support of a particu- ogous to the fact that Eq.6 is only well-defined for lar preparation Pψ. Then a pair of states |ψi , |φi is on- hψ|Πk|ψi= 6 0; i.e. only when the measurement Πk tologically distinct in a particular model if ∆φ ∩∆ψ = ∅, could have responded to the previous quantum state ψ. and ontologically indistinct otherwise7. This leads us This second distinction is central to the main result to the standard definition of a ψ-epistemic ontological of this paper; by showing that η is non-normalizable on model [23, 24]: some domain, we are able to conclude that that domain cannot be part of the support of ξ. Definition 1 (ψ-epistemic). An ontological model is ψ- The consistency condition with quantum theory is epistemic if there exists a pair of states |ψi , |φi that are given by equations similar to Eqs.4 and5; see Ap- ontologically indistinct; i.e. pendixB for more formal treatments of the properties of ∃ |ψi , |φi : ∆ ∩ ∆ 6= ∅. (9) η and other extensions of the ontological models formal- φ ψ ism discussed so far. The rigorous treatment in the ap- As noted in [24], this definition is highly permissive in pendix shows that, given the pre-existing assumptions the sense that, if an ontological model were to contain of the ontological models formalism, this is the only only a single pair of quantum states that are ontolog- way to represent state update under sequential mea- ically indistinct, then it would not achieve the full ex- surement; this is why we refer to our results as using planatory power expected of the ψ-epistemic viewpoint; ‘no additional assumptions.’ this is exactly the case with the ABCL model discussed This paper is not explicitly concerned with contex- 0 in Section 4.2.4. tuality [16, 44], but we are careful to ensure that we There are, however, proposals to strengthen the no- do not assume noncontextuality. Under the definition tion of ψ-epistemicity, two of which are relevant to our of generalized contextuality given in [44], an ontologi- discussion [24, 54]. cal model is noncontextual if two operationally equivalent preparation, transformation, or measurement pro- Definition 2 (Pairwise ψ-epistemic). An ontological cedures are always represented by equivalent prepara- model is pairwise ψ-epistemic if, for all pairs |ψi , |φi tion distributions, transition matrices, or response func- of nonorthogonal quantum states, |ψi and |φi are onto- 6 tions, respectively . It is contextual otherwise. While logically indistinct. this generalized definition may be too broad, account- ing for contextuality under this definition also accounts Definition 3 (Never ψ-ontic). An ontological model is for the traditional definition [16], and is therefore more never ψ-ontic if every ontic state λ ∈ Λ is in the support inclusive. Although, as stated above, we do not assume of at least two quantum states: noncontextuality of any kind, most of our results hold for a projector independent of its full measurement con- ∀λ ∈ Λ: ∃ψ, φ : λ ∈ ∆ψ ∩ ∆φ. (10) text. We are explicit about this when it is the case, and write ξ(Π|λ) rather than ξ(k = 0|λ, M = {Π,...}) for Note that both of these definitions imply the weaker notational convenience; η(λ0|λ, Π) is defined similarly. notion of ψ-epistemicity, but are independent from one another. 2.2 ψ-epistemic models 2.3 Some easy cases of state update rules Here we focus on a set of precise criteria for ψ-epistemic interpretations within the ontological models formal- There are two cases in which, given a prepare-and- ism. Consider first the ψ-epistemic criterion, proposed measure-once ontological model for quantum theory, we in [23] as a test for whether an interpretation admits at can always augment it with a state update rule for mea- least some quantum states that are not uniquely deter- surement. First, if we only include rank-1 projective mined by the underlying state of reality. Following [49], measurements in the subtheory we’re modeling, we can we account for potential preparation contextuality by simply re-prepare in the measured (unique, pure) state: defining 0 0 [ η(λ |k, λ, M{Πi}) = µ(λ |PΠk ) for tr(Πk) = 1. (11) ∆ψ = Supp(µ(·|Pψ)) (8) Pψ ∈Pψ 7For this purposes of this paper we assume there exists a mea- where P ⊆ P is the set consisting of every possible sure that is absolutely continuous with respect to all other mea- ψ sures in the ontological model. Therefore, we can work with prob- preparation of |ψi. We refer to ∆ψ as the support of a ability densities, rather than the full measure-theoretic treatment. While this assumption is not strictly true in all of our models, it 6This definition is actually complicated slightly by the inclu- does not affect our results and significantly simplifies our presen- sion of sequential measurements, but we do not discuss this here. tation.

Accepted in Quantum 2019-09-27, click title to verify 4 This is normalized for all λ since µ is normalized, and get outcome k, where Π is the kth projector in M. Then faithfully reproduces quantum statistics since µ does. let PM,k,Pφ ∈ Pψ be the preparation procedure associ- It is independent of the previous state λ, which, besides ated with post-selection of this measurement outcome being unsatisfying, is also not possible in general (this after a particular preparation Pφ. It must be normal- follows from Section3). ized on ∆ψ: Second, it is quick to prove, again by construction, Z that ψ-ontic models can always be given a state up- 0 0 1 = dλ µ(λ |PM,k,Pφ ) date rule. Since there is a unique quantum state |ψλi ∆ψ associated with every ontic state λ, we can deﬁne Z Z 0 0 = dλ dλ η(λ |k, λ, M)µ(λ|Pφ)   † ∆ψ ∆φ 0 0 Mk[ψλ]Mk Z Z η(λ |k, λ, M) = µλ P = . (12) 0 0 † = dλ µ(λ|Pφ) dλ η(λ |k, λ, M) tr Mk[ψλ]M k ∆φ ∆ψ

Again, normalization and faithfulness follow because µ Since η is always positive, normalization of µ(λ|Pφ) then has these properties. Note that this construction works implies that for any kind of measurement, not just projective mea- Z surements. 0 0 dλ η(λ |k, λ, M) = 1 ∀λ ∈ ∆φ. These observations together suggest that in order ∆ψ to ﬁnd anything interesting involving state update, we ought to examine higher-rank measurements in ψ- If η is normalized on a region, its support must be con- epistemic models. This suspicion will be conﬁrmed by tained in that region. Thus for all λ that are consistent the main result of this paper, which applies to exactly with some preparation |φi that could result in the post- these types of measurements and models. measurement state |ψi,

Supp(η(·|k, λ, M)) ⊆ ∆ψ. 3 The consequences of state update The fact that this is true for all |ψi that could result We now prove our central claim that consideration of from the measurement leads to Eq. 42. a rule for state update under measurement has consequences for the response function of an ontological We note that Lemma1 is easy to account for in ψ- model, so that consistent state update puts restrictions ontic theories and for rank-1 measurements, since in on how one may represent even a prepare-and-measure- both cases Sλ,Π has a single element. This is why we once experiment. We begin with a lemma that articu- were able to write down update rules for these situations lates a general property of the update rule η, and then in Section 2.3. Outside of these trivial cases, Eq. (42) examine its consequences for response functions ξ. is a very restrictive condition; depending on the structure of Sλ,Π, the intersection may be a very small set. Lemma 1. Suppose we have an ontological model with In particular, if any two of the post-selected quantum ontic space Λ, preparation distributions µ(λ|P ), in- states are orthogonal, then Sλ,Π is empty. dicator functions ξ(k|λ, M), and state update maps η(λ0|k, λ, M). For a particular ontic state λ and mea- Theorem 1 (Main theorem). Suppose that a projec- surement projector Π, we deﬁne the set Sλ,Π of quantum tor Π maps two states |αi , |βi to ontologically distinct states that one could obtain after measurement of any states Π |αi , Π |βi. Then the response function for Π quantum state consistent with λ: cannot have support on the overlap of |αi , |βi for any ( ) measurement context of Π; i.e. Π |φi

|ψi∈Sλ,Π ξ(Π|λ) = 0.

Proof. Suppose that measuring a state |φi with a mea- Both the lemma and the theorem hold for non- surement M results in the updated state |ψi when we projective measurements as well. We emphasize that

Accepted in Quantum 2019-09-27, click title to verify 5 this result does not say anything directly about the over- rule: lap of the supports of quantum states, just their overlap within the support of a particular response func- Λ = S2 tion. In the following section, we deploy this theorem 1 by showing that, in every known ψ-epistemic model for µ(~λ|P ) = Θ(ψ~ · ~λ)ψ~ · ~λ ψ π d ≥ 3, measurements of this type exist and have sup- ~ 0 ~ ~ 0 ~ port on the relevant overlaps, leading to contradiction Γ(λ |λ, TU ) = δ(λ − RU λ) ~ ~ ~ and demonstrating that these models cannot reproduce ξ(k|λ, Mφ) = Θ(kφ · λ) state update under measurement. 1 η(~λ0|k, ~λ, M ) = Θ(kφ~ · ~λ0)kφ~ · ~λ0 (16) φ π

4 Examples of state update under mea- Here Θ is the Heaviside step function, RU is the rotation of the Bloch sphere corresponding to a unitary U, and surement (or its impossibility) k ∈ {+1, −1}. This particular state update rule is not particularly satisfactory in an explanatory sense, since We provide a number of examples of ontological models it is independent of the previous ontic state. from the literature, illustrating some of the properties described in the previous sections. For each model, we either specify its state update rule or prove that it can- 4.1.2 Montina model not reproduce state update. Although many of these models can be easily defined for arbitrary types of mea- In [34], Montina introduces an ontological model based surements, we only consider projective measurements on the Kochen-Specker model. The model was con- for simplicity and notational consistency. structed to show that state update in a qubit can be successfully modeled by only updating a finite amount of information in the ontic state. To do so, Montina 4.1 ψ-epistemic models of a qubit extends the ontic space of the Kochen-Specker model by taking two vectors ~x+1, ~x−1 on the Bloch sphere and The only ψ-epistemic models that are able to represent adding two bits, labeled r and s, such that the vectors state update are those that restrict to modeling a qubit. on the Bloch sphere are dynamic under transformations This corresponds to the fact that the only nontrivial but remain static under state update (Fig. 1b). The bit projective measurements in a qubit are rank-1, which, r ∈ {−1, +1} acts as an index which decides which of as described earlier, do not result in restrictions from the two Bloch vectors is ‘active;’ s ∈ {−1, +1} stores the state update. result of a hypothetical standard basis measurement on the state. We take the standard basis to be defined by a special vector ~n pointing along the z-axis. 4.1.1 Kochen-Specker model As in the Kochen-Specker model, unitary transformations act by rotating the Bloch vectors; additionally, The Kochen-Specker model of a qubit [16, 24] is an ex- if the vector ~xr (i.e. the ‘active’ Bloch vector) crosses emplar of what we look for in a ψ-epistemic theory, with the horizontal equator of the sphere during this trans- the unfortunate feature that it only works in d = 2 di- formation, then the bit s flips to −s. r does not change mensions. It is both pairwise ψ-epistemic and never during a transformation. ψ-ontic, and provides a very intuitively pleasing inter- A measurement in the standard basis simply reveals pretation of the statistical nature of quantum theory. the value of s, and then updates r so that the active We take the ontic space to be the unit sphere S2, and vector is the one which was more closely aligned with denote by ψ~ the Bloch vector corresponding to |ψi un- ~n at the time of measurement. For any other basis, we der the usual mapping. Preparations and measurement apply the unitary that maps our desired measurement outcomes are represented by distributions over hemi- basis to the standard basis, measure, and then rotate spheres, with response functions uniform and prepara- back—this whole process has been wrapped into our tion distributions peaked towards the center (Fig. 1a). definitions of η and ξ below. In either case, the vectors Unitary transformations are represented by rotations of ~x , ~x do not change during measurement. the sphere. Since the only nontrivial measurements on +1 −1 a qubit are rank-1 measurements, this is a case where Finally, we prepare a state ψ~ by measuring in the ba- we can use the state update rule described in Eq. 11 sis {ψ,~ −ψ~} and applying a rotation that maps −ψ~ → ψ~ and just re-prepare the measured state for our update if we measured −ψ~. Summarizing these constructions,

Accepted in Quantum 2019-09-27, click title to verify 6 (a) Kochen-Specker model (b) Montina model (c) Beltrametti-Bugajski model (d) Bell Model

Figure 1: Visualizations of the state space, preparation distributions, and response functions for (a) the Kochen-Specker model, (b) Montina’s model, (c) the Beltrametti-Bugajski model for d = 2, and (d) Bell’s model for d = 2. Blue represents the support of preparations, and green the support of the response functions, where possible. Black objects are generic elements of the state space. we can write update (Eq. 11) that we used for the Kochen-Specker

2 2 model is not the only possibility; even though all mea- Λ = S × S × {−1, +1} × {−1, +1} surements in this model are rank-1, η has nontrivial λ = (~x+1, ~x−1, r, s) dependence on the previous ontic state λ. Thus just 1 h i because we can construct a trivial update rule in some µ(λ|P ) = Θ s ~x · ψ~ (~x · ~n) ψ (4π)2 r r cases does not mean that there is then nothing interest- 2 2 ing to investigate. It also includes these features while ~ ~ · Θ r ~x+1 · ψ − ~x−1 · ψ remaining pairwise ψ-epistemic and never ψ-ontic.

0 0 0 Γ(λ |λ, TU ) = δ ~x+1 − RU ~x+1 δ ~x−1 − RU ~x−1 0 0 0 4.2 Models of full quantum theory for arbitrary · Θ[ss (~xr · ~n)(~xr · ~n)] Θ[rr ] dimension h i ξ(k|λ, M ) = Θ ks (~x · ~n) ~x · φ~ φ r r 4.2.1 Beltrametti-Bugajski model 0 0 0 η(λ |k, λ, Mφ) = δ ~x+1 − ~x+1 δ ~x−1 − ~x−1 The Beltrametti-Bugajski model [9, 24] is perhaps the h 0 ~ ~i simplest ontological model that describes a system of ar- · Θ ss (~xr · ~n) ~xr · φ (~xr0 · ~n) ~xr0 · φ bitrary dimension. Although it is ψ-ontic, it is the start- 2 2 0 ~ ~ ing point for the construction of the next three models · Θ r ~x+1 · φ − ~x−1 · φ in this section. For a d-dimensional quantum system, (17) we take the ontic space to be the quantum state space, which we denote PHd−1 (the projective Hilbert space of The original presentation is not stated in terms of the dimension d − 1). Preparations, transformations, mea- ontological models formalism, and only explicitly mod- surements, and state update rules then follow directly els measurements in the standard basis. This lead to from the usual quantum rules: the claim that state update under measurement is ac- counted for by updating a single bit, but it is clear from Λ = PHd−1 the form of η above that by including all measurements µ(λ|Pψ) = δ(|λi − |ψi) in our subtheory we have caused both bits to be up- 0 0 dated during measurement. Γ(λ |λ, TU ) = δ(|λ i − U |λi)

We include this model here for two reasons. First, it ξ(k|λ, M{Πi}) = hλ| Πk |λi ! is one of the few ontological models in the literature that Π |λi η(λ0|k, λ, M ) = δ |λ0i − k (18) has explicitly considered state update under measure- {Πi} p ment. Second, it demonstrates that the generic rank-1 hλ| Πk |λi

Accepted in Quantum 2019-09-27, click title to verify 7 This provides an example of the generic update-rule for refer the reader to [35] for a more thorough construction ψ-ontic models (Eq. 12), and is depicted in Fig. 1c. and motivation. The LJBR model has the same ontic space as the Bell 4.2.2 Bell’s model model, so we again write ontic states as λ = (|λi , pλ). It is constructed in a preferred basis {|ji}, which we use Lewis et al. [35] extended a model of a qubit orignally in defining two helper functions. First, proposed by Bell [39] to arbitrary dimension, which can be seen as a modification of the Beltrametti-Bugajski zj(|λi) = inf tr([λ][φ]). (20) model [24]. The ontic space is the Cartesian product of |φi:tr([j][φ])≥1/d the projective Hilbert space with the unit interval [0, 1]. d−1 Note that zj(|λi) > 0 if and only if tr([j][λ]) > , Now we write λ as an ordered pair λ = (|λi , pλ) where d |λi ∈ PHd−1, as in the Beltrametti-Bugajski model, so zj(|λi) is nonzero for at most a single element of the preferred basis; we denote this unique vector as |jλi. and pλ ∈ [0, 1]. Preparations remain essentially the same, becoming a product distribution of a delta func- Second, we define a permutation πM,λ for each mea- tion on the quantum state space with a uniform dis- surement M and ontic state λ: tribution over the unit interval. The response functions trM [j ] ≥ trM [j ] ≥ · · · divide up the unit interval into lengths corresponding to πM,λ(0) λ πM,λ(1) λ probabilities of measuring each outcome, and respond · · · ≥ tr MπM,λ(|M|−1)[jλ] . (21) with outcome k when p is in the corresponding inter- λ If there is no |j i, i.e. z (|λi) = 0 for all j, then we take val (Fig. 1d). This has the effect of making the model λ j π to be the identity permutation. The final element outcome deterministic. M,λ we need before defining the model itself is a set Λ = PHd−1 × [0, 1] Ej = {λ : zj(|λi) > 0} (22) µ(λ|Pψ) = δ(|λi − |ψi) 0 0 Γ(λ |λ, TU ) = δ(|λ i − U |λi) defined for each basis vector. Without further ado, the  k−1  full specification of the model: X ξ(k|λ, M ) = Θ p − tr(Π [λ]) {Πi}  λ j  Λ = PHd−1 × [0, 1] j=0 λ = (|λi , pλ)  k  X Y · Θ−pλ + tr(Πj[λ]) µ(λ|Pψ) = δ(|λi − |ψi) Θ[pλ − zj(|ψi)] j=0 j ! X Π |λi + zj(|ψi)µEj (λ) η(λ0|k, λ, M ) = δ |λ0i − k (19) {Πi} p j hλ| Πk |λi " k−1 # X Since this model is still ψ-ontic, we once again use the ξ(k|λ, M{Πi}) = Θ pλ − tr ΠπM,λ(l)[λ] generic state update rule for ψ-ontic models. In this l=0 case, we can also see that this works because of the " k # X structure of the preparations as product distributions. · Θ −pλ + tr ΠπM,λ(l)[λ] (23) Since every state has a uniform distribution over pλ, l=0 and this is uncorrelated with |λi, we can just update where µ (λ) is the uniform distribution over E . |λi according to the Beltrametti-Bugajski update rule Ej j Roughly, all quantum states |ψi with tr([j][ψ]) > d−1 and then re-randomize uniformly over pλ. d will have support on Ej, and so will all overlap with each other. The permutation included in the definition of the 4.2.3 LJBR model measurements is constructed so that this shared support In [35], Lewis et al. define a ψ-epistemic model based does not affect the prepare-and-measure-once statistics: on their generalization of Bell’s model. Referred to here the measurement ordered first by the permutation has as the LJBR model, it is motivated by the observation a support which entirely contains Ej. This model is not that the order of segments in the response function of pairwise ψ-epistemic, nor is it never ψ-ontic. Note ad- Bell’s model does not matter: a re-ordering of these ditionally that it was originally only defined for rank-1 segments allows arbitrary modification of preparation projective measurements, but it works just as well for distributions within a subset of the ontic space, so they higher-rank projective measurements without modifica- can be made to overlap. We present here a brief description. tion of the ‘most epistemic’ version of this model, and This model is the first to fall to Theorem1:

Accepted in Quantum 2019-09-27, click title to verify 8 Theorem 2. The LJBR model cannot represent state and is intended to demonstrate the possibility of a pair- update under measurement in dimension d ≥ 3. wise ψ-epistemic model. This is the only known example of a pairwise ψ-epistemic model in d ≥ 3, but it still Proof. The general idea of the proof is to find a is not never ψ-ontic [24, 54]. These models have come measurement which maps any two nonidentical states under criticism for their “unnaturalness,” but we show |αi , |βi to two again nonidentical states Π |αi , Π |βi here that their problems go deeper due to an inability which both have no support on any of the E , and so j to represent state-update. are ontologically distinct. We begin with ABCL , defining a couple of helper Consider the preferred basis |ji of the LJBR model 0 functions like in the LJBR model. Rather than ordering and the generalized x-basis defined by measurements with respect to traces with a preferred d−1 basis, we use the defining states |αi , |βi and a function 1 X jk 2πi/d |Xki = √ ω |ji , ω = e . (24) gαβ(Π) = min{tr(Π[α]), tr(Π[β])}. (27) d j=0 We now define a new permutation σ 8 for each mea- These x-basis states have the property tr([X ][k]) = 1 M j d surement M [24]: for all j, k. There must exist two elements |Xk1 i , |Xk2 i of the x-basis such that |αi , |βi differ on that two- gαβ(MσM (0)) ≥ gαβ(MσM (1)) ≥ · · · ≥ gαβ(MσM (|M|−1)). dimensional subspace or else |αi , |βi would be identi- (28) cal. Pick two such elements, and consider the projector With this, we can specify the ABCL0 model. Π = [X ] + [X ]. The quantum overlap of the post- k1 k2 Λ = PHd−1 × [0, 1] measurement state Π |αi with any basis vector |ji is, using the submultiplicativity of the trace, λ = (|λi , pλ)  tr([j]Π[α]Π) 2 d − 1  Θ(p − ε)δ(|λi − |ψi)  λ ≤ tr([j]Π) = ≤ (25)  tr(Π[α]) d d  1  + Θ(ε − p )[δ(|λi − |αi) + δ(|λi − |βi)]  2 λ and the same is true for Π |βi. As described above, µ(λ|Pψ) = d−1 only states with tr([j][ψ]) > have overlap with  if |ψi = |αi , |βi d  any other states in the LJBR model, so the post-   measurement states are ontologically distinct; by The- δ(|λi − |ψi) otherwise orem1, ξ(Π|λ) = 0 for all λ ∈ Ej for all j since |αi and  k−1  |βi were arbitrary. X ξ(k|λ, M ) = Θ p − trΠ [λ] However, when measured in the context of the rest of {Πi}  λ σM (j)  the rank-1 x-basis projectors, Π will be ordered first by j=0 π for all λ since  k  M,λ X · Θ −p + trΠ [λ] (29) 2 1  λ σM (j)  tr(Π[j]) = > (26) j=0 d d |hα|βi| for all j. Thus, by the construction of the LJBR model, for ε ≤ d . Now the preparation distributions for ξ(Π|λ) = 1 for all λ ∈ Ej, resulting in a contradiction. |αi and |βi overlap on {|αi , |βi}×[0, ε]; as in the LJBR model, the permutation in ξ ensures that the preparation change doesn’t affect the prepare-and-measure- 4.2.4 ABCL models once statistics by making sure measurements whose support must contain this overlap region are ordered first. In [36], Aaronson et. al. construct two ψ-epistemic Once again, this model fails to meet the conditions re- models. The first, which we will call ABCL0 , is very quired in order to faithfully represent state update: closely related to the LJBR model but is not identical; Theorem 3. The ABCL model cannot represent state rather than continuous regions of quantum states which 0 update under measurement in dimension d ≥ 3. overlap, this model has exactly one pair of quantum states which are ontologically indistinct. However, it Proof. Call the two states defining the model |αi , |βi. gains the feature that any two nonorthogonal quantum Let Π = [α] + [γ], where |γi is some state such that states can be chosen as the single pair that overlaps. hα|γi = 0 and 0 < |hγ|βi|2 < 1 − |hα|βi|2. (30) With malice aforethought, we will call this defining pair |αi , |βi. 8To be precise, we should label this with α, β as well to em- The second, ABCL1 , is a convex mixture (to be de- phasize that it belongs to the model defined by that particular pair of states. fined) of the ABCL0 model constructed for all |αi , |βi

Accepted in Quantum 2019-09-27, click title to verify 9 Under this measurement, |αi maps to |αi and |βi does |hψ|φi| = |hψ0|φ0i| implies that |ψi , |φi are ontologi- not get mapped to either |αi or |βi. Thus the post- cally distinct if and only if |ψ0i , |φ0i are ontologically measurement states are ontologically distinct, so by distinct. If the model also includes all CPTP maps, 0 0 0 0 Theorem1, ξ(Π|λ) = 0 for all λ ∈ ∆α ∩ ∆β. then |hψ|φi| ≥ |hψ |φ i| implies that |ψ i , |φ i are onto- For the other half of the contradiction, note that logically distinct if |ψi , |φi are. tr(Π[α]) = 1 means gαβ(Π) = tr(Π[β]) > 0 (since It immediately follows that the LJBR and ABCL0 |αi , |βi are nonorthogonal) and gαβ(I − Π) = 1 − models cannot faithfully represent unitary transforma- tr(Π[α]) = 0, so Π is ordered first by σM when mea- tions. In each model there exist quantum states which sured in the context M = {Π, I − Π}. Thus ξ(Π|λ) = 1 are ontologically distinct from every other state; pick for all λ ∈ ∆α ∩ ∆β, resulting in a contradiction. one of these states, and it is easy to find examples of pairs of ontologically distinct states with any inner We outline the ABCL1 model schematically and refer product. the reader to [24, 36] for details. Given two ontological However, transformations cannot necessarily rule out models specified by Λ1, µ1, ξ1 and Λ2, µ2, ξ2 respectively, the ABCL1 model: since it is pairwise ψ-epistemic, the authors define a convex combination of these models the unitary condition could in principle be satisfied. as Λ3, µ3, ξ3 such that That said, the transformation rule would be compli-

Λ3 = Λ1 ⊕ Λ2 cated because it would have to map between models that are mixed together, so it is certainly an open ques- bµ = pµ + (1 − p)µ 3 1 2 tion whether this is actually possible. ξ3 = ξ1 + ξ2 (31) Here p ∈ (0, 1) is some mixing parameter. If there’s 4.3 Models of subtheories overlap between two states in either of models 1 or 2, Although we have dealt so far with models that in- then model 3 has overlap on these states. The ABCL1 model is then defined essentially as a convex mixture clude the full quantum set of preparations, transformations, and measurements, there is the possibility that of the ABCL0 models for all pairs |αi , |βi, taking care with respect to the uncountable size of this set. we can retain ψ-epistemic models of subtheories. It In order to include state update in a convex com- turns out that although the stabilizer subtheory can be bination of ontological models, the most obvious (and represented by a ψ-epistemic model, the more general perhaps only) option is to specify Kitchen Sink model which models any finite subtheory cannot in general represent state update under mea- η3 = η1 + η2. (32) surement. The failure of the ABCL model to reproduce state 1 4.3.1 Kitchen Sink model update follows directly from the failure of the ABCL0 model. The Kitchen Sink model is a ψ-epistemic ontological model for any finite subtheory of quantum the- Theorem 4. The ABCL1 model cannot represent state update under measurement in dimension d ≥ 3 ory [37, Section IIIC]. Given a finite set of projective measurements M = {M (i)}, we choose our on- Proof. When we take a convex combination of models, tic states to be a list of measurement outcomes λ = we see that (λ1, λ2, . . . , λ|M|). That is, λi = k means that if (i) (i) Supp(ξ3(k|·,M)) =Supp(ξ1(k|·,M)) ∪ Supp(ξ2(k|·,M)) M = {Πj } is measured on the ontic state λ, the outcome Πk will occur with certainty. For a system of Supp(η3(·|k, λ, M)) = Supp(η1(·|k, λ, M)) dimension d, the maximum number of projectors in any ∪ Supp(η2(·|k, λ, M)) given measurement is d, so we pad all of our measure- Thus if either of models 1 or 2 violates Theorem1, ments with 0s until they have d elements. The Kitchen model 3 must violate it as well. Since all of the ABCL0 Sink model is then defined by models being mixed violate Theorem1, so must ABCL 1 Λ = |M| . Zd |M| Y (i) µ(λ|ψ) = tr Π [ψ] 4.2.5 A note on transformations λi i=1 As Leifer notes, transformations also play a role in re- (i) ξ(k|λ, M ) = δ(k, λi) (33) stricting the structure of ψ-epistemic ontological models [24, Section 8.1]. If an ontological model suc- The Kitchen Sink is pairwise ψ-epistemic for any sub- cessfully represents all unitary transformations, then theory, and also never ψ-ontic if we include all pure

Accepted in Quantum 2019-09-27, click title to verify 10 states in our subtheory. Transformations can addition- This final expression will be zero if and only if at least ally be modeled under the assumption of a closed sub- one of its factors is 0. The first two factors are nonzero theory. by Eqs. 35 and 36. The rest of the factors are nonzero We can only rule out the Kitchen Sink model for cer- due to Eq. 34 and the completeness condition on the tain subtheories, as it is easy to construct subtheories measurements. Thus Eq. 39 is nonzero and we have with trivial update rules (e.g. by only including rank- a contradiction, so state update cannot be represented 1 measurements). That said, our requirements are few faithfully. and are satisfied by the multi-qupit stabilizer subtheory, arguably the most important subtheory of quantum This demonstrates that Theorem1 can create trou- theory. Specifically, we only need to include two states ble even in subtheories. In particular, the stabilizer subtheory satisfies the requirements in Eqs. 34–37, so it |αi , |βi and two measurements M (1) = {Π, I−Π},M (2) satisfying cannot be modeled by the Kitchen Sink. That said, we can show that the stabilizer subtheory still supports a hα|βi= 6 0 (34) ψ-epistemic interpretation using other models. hα|Π|αi= 6 0 (35) hβ|Π|βi= 6 0 (36) 4.3.2 Qupit stabilizer subtheory M (2) distinguishes Π |αi and Π |βi (37) We begin with the straightforward case of n p- dimensional systems, for p an odd prime. In this The first is required because we don’t expect orthog- case, the stabilizer subtheory has an ontological model onal states to be ontologically indistinguishable. The given by the discrete Wigner function [33, 55] (see Ap- next two stipulate that there is a nonzero chance of ob- pendixC for definitions of the stabilizer subtheory and taining an outcome Π when measuring |αi and |βi, so the phase-point operaters A ): that its support overlaps with their support. The last λ condition implies that the post-measurement states are n n Λ = Zp × Zp ontologically distinct [24]. 1 µ(λ|P ) = tr(A [ψ]) Theorem 5. For any finite subtheory containing states ψ pn λ and measurements satisfying the conditions given in ξ(k|λ, M ) = tr(Π A ) (40) Eqs. 34–37, the Kitchen sink model cannot model state {Πi} k λ update under measurement. As the Wigner function is a quasi-probability distribu- Proof. Theorem1 implies that, if Π maps |αi , |βi to tion [56], it necessarily takes on negative values if we ontologically distinct states, then try to model the full quantum theory. If, however, we Z restrict to modeling preparations, transformations, and dλ ξ(Π|λ)µ(λ|α)µ(λ|β) = 0. (38) measurements in the qupit stabilizer subtheory, then the Λ representation is positive and it forms a well-defined on- We evaluate this quantity for the Kitchen Sink’s re- tological model [33, 57]. It is both pairwise ψ-epistemic sponse functions and preparation distributions, using and never ψ-ontic. the states and measurement M (1) satisfying Eqs. 34– The stabilizer subtheory presents a challenge in that 37: measuring a single qupit is described by a rank-pn−1 Z measurement, which may run into trouble due to The- dλ ξ(k = 0|λ, M (1))µ(λ|α)µ(λ|β) Λ orem1. In particular, there are many examples in the |M| stabilizer subtheory of the type of measurements that X Y (j) (j) = δ(0, λ ) tr M [α] tr M [β] broke the Kitchen Sink model. Nonetheless, we can 1 λj λj |M| j=1 specify the update rule λ∈Zr X (1) (1) 0 1 tr(AλΠkAλ0 Πk) = δ(0, λ1) tr M [α] tr M [β] η(λ |k, λ, M ) = . (41) λ1 λ1 {Πi} n p tr(ΠkAλ) λ1∈Zr |M| Note the normalization factor in η which makes clear Y X (j) (j) · tr Ml [α] tr Ml [β] that η is only defined in the support of ξ. The fact j=2 l∈Zr that this successfully reproduces quantum statistics fol- = tr M (1)[α] tr M (1)[β] lows from the fact that post-selected measurement is 0 0 a completely positive map, and this is how completely |M| positive maps are represented in quasi-probability rep- Y X (j) (j) · tr Ml [α] tr Ml [β] (39) resentations [56]. η is always positive for the stabilizer j=2 l∈Zr subtheory, though we do not include the proof here.

Accepted in Quantum 2019-09-27, click title to verify 11 For the case p = 2, Lillystone and Emerson construct challenges to achieving this conclusion. In particular, a ψ-epistemic model of the n-qubit stabilizer formalism we note that all of the ψ-epistemic models that we that successfully represents state update under mea- considered share the property of outcome determinism, surement [48]. This model starts from the Kitchen Sink which means that Theorem1 may be less trouble in model and augments Λ so that the problematic over- non-outcome-deterministic models. At the very least, laps of the Kitchen sink are removed. This model is not any no-go theorem will have to include measurements, pairwise ψ-epistemic, but a modified version (see ap- states, and/or transformations from outside the stabi- pendix of [48]) is never-ψ-ontic. We don’t present the lizer subtheory since we have shown that ψ-epistemic construction here because it is significantly more convo- models for this subtheory exist. luted than the model above for p ≥ 3. This reflects the What import does our result have for the general in- often-observed ill-behaved nature of the qubit stabilizer terpretational project of quantum theory? First of all, subtheory. we have demonstrated that ψ-epistemicists have yet an- Tangentially, if we extend the Wigner function to the other challenge to overcome: a successful explanation of full quantum theory, we get negatively represented state state-update. This is in contrast with the usual claim update, as expected. One consequence of this is that that this arena is one where epistemic interpretations some state updates can’t be normalized, so the Wigner have an advantage over ontic interpretations. As we function state update must include a renormalization emphasized in the introduction, our results only strictly step not allowed in ontological models or quasiproba- apply to interpretations that can be described by the on- bility representations. Although further discussion of tological models formalism, but there may be a qualita- state update under measurement in quasi-probability tive message for epistemic and doxastic interpretations representations is beyond the scope of this paper, we that are outside this formalism as well. note that Theorem1 does not hold for quasi-probability representations so this could be one potential direction for related future work. 6 Acknowledgements

This research was supported by the Canada First Re- 5 Discussion search Excellence Fund and the NSERC Discovery pro- We have demonstrated that state update under mea- gram. J.B.R. acknowledges funding from an Ontario surement poses a serious challenge to ψ-epistemic inter- Graduate Scholarship. The authors would like to thank pretations of quantum theory in the ontological models Matt Leifer and David Schmid for productive discus- framework: all currently known ψ-epistemic models for sions. We would also like to thank two anonymous full quantum theory in d ≥ 3 cannot faithfully represent reviewers and our editor at Quantum, whose remarks state update. This runs in direct contrast to the prevail- motivated the discussion in AppendixA. ing view that ψ-epistemic models provide a compelling explanation of state update. There are a number of remaining open questions. References Most pressingly, we have re-opened the possibility of proving a general ψ-onticity result without additional [1] David Bohm. A Suggested Interpretation of the assumptions—will the methods of this paper be useful Quantum Theory in Terms of ”Hidden” Variables. in doing so? I. Physical Review, 85(2):166–179, January 1952. On the one hand, the proofs above do not rule out ISSN 0031-899X. DOI: 10.1103/PhysRev.85.166. the possibility of extending the ontology of the broken [2] Hugh Everett. ”Relative State” Formulation of models in order to represent state update under mea- Quantum Mechanics. Reviews of Modern Physics, surement while still retaining the epistemicity of the 29(3):454–462, July 1957. ISSN 0034-6861. DOI: model. This is exactly the route taken in [48] for the n- 10.1103/RevModPhys.29.454. qubit stabilizer subtheory. Granted, the n-qubit stabi- [3] Paul Dirac. The Principles of Quantum Mechanics. lizer subtheory was brought within an inch of ψ-onticity Oxford University Press, Oxford, 1958. by this process, so it seems unlikely that a similar tech- [4] G. C. Ghirardi, A. Rimini, and T. Weber. Uni- nique will work for the full quantum theory. ﬁed dynamics for microscopic and macroscopic sys- In the other direction, we’ve shown that consideration tems. Physical Review D, 34(2):470–491, July 1986. of state update puts powerful constraints on the struc- ISSN 0556-2821. DOI: 10.1103/PhysRevD.34.470. ture of ψ-epistemic models. These restrictions would [5] Philip Pearle. Combining stochastic dynamical ideally lead to a categorical statement like “ψ-epistemic state-vector reduction with spontaneous localiza- models cannot represent state-update,” but there are tion. Physical Review A, 39(5):2277–2289, March

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Accepted in Quantum 2019-09-27, click title to verify 13 Physics, 47(12):122107, December 2006. ISSN [45] Nicholas Harrigan and Terry Rudolph. Ontologi- 0022-2488, 1089-7658. DOI: 10.1063/1.2393152. cal models and the interpretation of contextuality. [34] Alberto Montina. Dynamics of a qubit as a classical September 2007. arXiv:0709.4266. stochastic process with time-correlated noise: Min- [46] Joseph Emerson, Dmitry Serbin, Chris Sutherland, imal measurement invasiveness. Physical Review and Victor Veitch. The whole is greater than the Letters, 108(16), April 2012. ISSN 0031-9007, 1079- sum of the parts: On the possibility of purely sta- 7114. DOI: 10.1103/PhysRevLett.108.160501. tistical interpretations of quantum theory. Decem- [35] Peter G. Lewis, David Jennings, Jonathan Bar- ber 2013. arXiv:1312.1345. rett, and Terry Rudolph. Distinct Quantum States [47] Lorenzo Catani and Dan E. Browne. Spekkens’ Can Be Compatible with a Single State of Reality. toy model in all dimensions and its relationship Physical Review Letters, 109(15), October 2012. with stabilizer quantum mechanics. New Journal ISSN 0031-9007, 1079-7114. DOI: 10.1103/Phys- of Physics, 19(7):073035, July 2017. ISSN 1367- RevLett.109.150404. 2630. DOI: 10.1088/1367-2630/aa781c. [36] Scott Aaronson, Adam Bouland, Lynn Chua, and [48] Piers Lillystone and Joseph Emerson. A Contex- George Lowther. ψ-epistemic theories: The role tual ψ-Epistemic Model of the n-Qubit Stabilizer of symmetry. Physical Review A, 88(3), Septem- Formalism. April 2019. arXiv:1904.04268. ber 2013. ISSN 1050-2947, 1094-1622. DOI: [49] Angela Karanjai, Joel J. Wallman, and Stephen D. 10.1103/PhysRevA.88.032111. Bartlett. Contextuality bounds the efficiency of [37] Nicholas Harrigan, Terry Rudolph, and Scott classical simulation of quantum processes. Febru- Aaronson. Representing probabilistic data ary 2018. arXiv:1802.07744. via ontological models. September 2007. [50] A. J. Leggett and Anupam Garg. Quantum me- arXiv:0709.1149. chanics versus macroscopic realism: Is the flux there when nobody looks? Physical Review Letters, [38] Joel J. Wallman and Stephen D. Bartlett. Non- 54(9):857–860, March 1985. DOI: 10.1103/Phys- negative subtheories and quasiprobability repre- RevLett.54.857. sentations of qubits. Physical Review A, 85(6), [51] Clive Emary, Neill Lambert, and Franco Nori. June 2012. ISSN 1050-2947, 1094-1622. DOI: Leggett-Garg Inequalities. Reports on Progress 10.1103/PhysRevA.85.062121. in Physics, 77(1):016001, January 2014. ISSN [39] John S. Bell. On the Problem of Hidden Vari- 0034-4885, 1361-6633. DOI: 10.1088/0034- ables in Quantum Mechanics. Reviews of Mod- 4885/77/1/016001. ern Physics, 38(3):447–452, July 1966. ISSN 0034- [52] J. J. Sakurai and Jim Napolitano. Modern Quan- 6861. DOI: 10.1103/RevModPhys.38.447. tum Mechanics. Addison-Wesley, Boston, 2nd edi- [40] Matthew F. Pusey, Jonathan Barrett, and Terry tion, 2011. ISBN 978-0-8053-8291-4. Rudolph. On the reality of the quantum state. Na- [53] Daniel Gottesman. An Introduction to Quan- ture Physics, 8(6):476–479, May 2012. ISSN 1745- tum Error Correction and Fault-Tolerant Quantum 2473, 1745-2481. DOI: 10.1038/nphys2309. Computation. April 2009. arXiv:0904.2557. [41] Roger Colbeck and Renato Renner. A system’s [54] Alberto Montina. Comment on math overflow wave function is uniquely determined by its under- question: ‘psi-epistemic theories’ in 3 or more di- lying physical state. New Journal of Physics, 19 mensions, December 2012. (1):013016, January 2017. ISSN 1367-2630. DOI: [55] Victor Veitch, Seyed Ali Hamed Mousavian, Daniel 10.1088/1367-2630/aa515c. Gottesman, and Joseph Emerson. The Resource [42] Lucien Hardy. Are quantum states real? Inter- Theory of Stabilizer Computation. New Journal of national Journal of Modern Physics B, 27(01n03): Physics, 16(1):013009, January 2014. ISSN 1367- 1345012, January 2013. ISSN 0217-9792, 1793- 2630. DOI: 10.1088/1367-2630/16/1/013009. 6578. DOI: 10.1142/S0217979213450124. [56] Christopher Ferrie. Quasi-probability represen- [43] Shane Mansfield. Reality of the quantum state: tations of quantum theory with applications Towards a stronger ψ -ontology theorem. Physical to quantum information science. Reports on Review A, 94(4), October 2016. ISSN 2469-9926, Progress in Physics, 74(11):116001, November 2469-9934. DOI: 10.1103/PhysRevA.94.042124. 2011. ISSN 0034-4885, 1361-6633. DOI: [44] R. W. Spekkens. Contextuality for prepara- 10.1088/0034-4885/74/11/116001. tions, transformations, and unsharp measure- [57] David Gross. Non-negative Wigner functions in ments. Physical Review A, 71(5), May 2005. prime dimensions. Applied Physics B, 86(3):367– ISSN 1050-2947, 1094-1622. DOI: 10.1103/Phys- 370, February 2007. ISSN 0946-2171, 1432-0649. RevA.71.052108. DOI: 10.1007/s00340-006-2510-9.

Accepted in Quantum 2019-09-27, click title to verify 14 [58] Nix Barnett and James P. Crutchﬁeld. Com- putational Mechanics of Input-Output Processes: Structured transformations and the -transducer. Journal of Statistical Physics, 161(2):404–451, Oc- tober 2015. ISSN 0022-4715, 1572-9613. DOI: 10.1007/s10955-015-1327-5. [59] Matthew S. Leifer and Matthew F. Pusey. Is a time symmetric interpretation of quantum theory possible without retrocausality? Proc. R. Soc. A, 473(2202):20160607, June 2017. ISSN 1364-5021, 1471-2946. DOI: 10.1098/rspa.2016.0607. [60] Ronald A. Howard and James E. Matheson. In- ﬂuence Diagrams. Decision Analysis, 2(3):127– 143, September 2005. ISSN 1545-8490. DOI: 10.1287/deca.1050.0020.

Accepted in Quantum 2019-09-27, click title to verify 15 A Is state update an additional assump- ontological model that is not required by the bare for- tion? malism. As another example, the Colbeck-Renner argument [41] claims to not make any assumptions of the Throughout this paper, we have used the phrase ‘no kind described in the PIP; they do, however, assume additional assumptions’ in order to contrast our results a spacetime structure which is, again, additional to with those of existing no-go theorems. By this we don’t the ontological models formalism. Spacetime and lo- mean to raise an argument over the semantics of exactly cality arguments are also what motivate the PIP. These what one means by ‘assumption,’ but rather we mean relativistically-motivated considerations are interesting to say that the consideration of state update, which in their own right, but since Bell’s theorem already rules we propose may be able to categorically rule out ψ- out ψ-epistemic models based on reasonable spacetime epistemic models, is very different from the ‘additional structure (i.e. the locality assumption), the goal of our assumptions’ used in existing no-go theorems. paper is to arrive at a similar conclusion without impos- To make this precise, we can break up our considering spacetime structure. In particular, a ψ-epistemicity ation of state update into two parts. The first part is no-go theorem would directly imply the results of Bell’s simply the fact that we are considering a broader set of theorem and is thus a stronger result [24]. empirical situations than considered previously: exper- Whether operational state update, i.e. the Lüders iments involving sequential measurements. The onto- rule, is considered an assumption of some kind or a logical models formalism, as originally defined, does not consequence of other operational axioms is irrelevant have the tools to model this empirical situation, so we to our analysis. The key point for our analysis is that are forced to extend the formalism to do so. This leads the state-update rule is an experimentally validated op- to the second part, which is an assumption about how erational feature/prediction of QM theory, much like we ought to model this empirical situation. While, in entanglement or unitary dynamics, and therefore is an principle, there are many ways one can do this, we show empirical feature of the QM framework that the frame- in AppendixB that there is only one way to do this that work of ontological models ought to explain. So just is consistent with the broader conceptual underpinnings like the discovery from Bell that entangled states lead (i.e. pre-existing assumptions) of the ontological mod- to new constraints/insights into the necessary features els formalism. By examining the core assumptions of of (ψ-epistemic) ontological models, similarly our in- the ontological models formalism rather than its strict sight is that the dynamics associated with state-update mathematical formulation, we show that all of the struc- also leads to new constraints/insights into the necessary ture associated with state update follows directly from features of (ψ-epistemic) ontological models. the pre-existing assumptions of the ontological models formalism. This is in contrast to the assumptions used in previ- B Justifying the form of the state up- ously published no-go theorems. First of all, they start date rule: ontological models as hidden from empirical situations that can be described by the vanilla ontological models formalism. Then, they im- Markov models of stochastic channels pose a particular restriction on the form of the ontological models that happens directly at the level of the ontic In the context of state update under measurement, it is state; for example, the PBR theorem [40] assumes the illuminating to motivate the definition of an ontologi- preparation independence postulate (PIP) which says cal model from the point of view of the hidden Markov that says that, for two subsystems A and B, if the models (HMM) literature. We do this in order to (a) quantum state is a product state, then the preparation provide a rigorous treatment of multiple-measurement distribution µ is a product distribution: scenarios presented informally in Section 2.1 and (b) clarify the assumptions that define the ontological mod-

ΛAB = ΛA × ΛB (42) els framework. In the process, we use these assumptions to show that the form of our state update rule is the only and one consistent with the already-deﬁned components of ΨAB = ΨA ⊗ ΨB the ontological models formalism. =⇒ µ(λA, λB|ΨAB) = µ(λA|ΨA) · µ(λB|ΨB). (43) We picture a quantum circuit as a memoryful stochastic channel (Fig.2). The channel that we often dis- We can see from this symbolic expression that this is cuss with regards to a quantum circuit is the (quantum) indeed a direct assumption about the structure of the channel that takes the input quantum state and maps ontological model, rather than an empirical considera- it to the output quantum state. For present purposes, tion. More to the point, it imposes structure on the we will instead think of it as a channel from the ex-

Accepted in Quantum 2019-09-27, click title to verify 16 k k0 Note that this does not imply a Markov process, since it ←− ←− M depends in general on the entire histories a0 and k0. All U we mean by single-symbol is that we are not specifying P U 00 the probabilities over the whole future, just a single M 0 U 0 output symbol. We now construct an HMM as follows:

Definition 6 (Hidden Markov Model). A hidden Markov model (HMM) of a stationary, causal channel is specified by an additional random variable λ taking 0 k0 = 0 k1 = 0 k2 = 0 k3 = k k4 = 0 k5 = k values in a state space Λ. It is given a joint probabil- ←→ ←→ ←→ ity distribution over λ0 , a0 , k0 so that the recurrence relation above (Eq. 46) becomes channel 0 00 0 a0 = P a1 = U a2 = U a3 = M a4 = U a5 = M ←− ←− ←− Pr(k0, λ1|a0, λ0, a 0, k 0, λ 0) = Pr(k0, λ1|a0, λ0). Figure 2: A quantum circuit can be pictured as a stochastic (47) channel, as described in the text. The inputs to the channel at are the choice of operation, and the outputs kt report the In other words, the state λ renders the future condi- results of measurements. tionally independent of the past and induces a Markov process over the state space Λ that mediates the channel statistics. This is the property of the ontological perimenter to individual measurement outcomes used models formalism called “λ-mediation” in [59]. An in- repeatedly at each time step. Pictorially, one might fluence diagram [60] of a stationary, causal channel is think of this as ‘rotating the channel ninety degrees’ in shown in Figure3 before and after the specification of a circuit diagram. an HMM. The input string ←→a = . . . a a a a a ... of the 0 −2 −1 0 1 2 To see that specification of an HMM as in Eq. 47 channel is the experimenter’s choice of action, which is equivalent to the definition of an ontological model we take to be a sequence of operational preparation, given in Section 2.1, we first note that generally we transformation, or measurement procedures. The out- ←→ don’t think of preparations and transformations having put string k0 of the channel reports either the results of output; to account for this, we stipulate that they give a measurements or a trivial output for preparations and trivial, deterministic output k0 = 0. We then factor the transformations. The subscript 0 indicates in both cases probability distribution from Eq. 47 and look separately the time that we take as an origin/reference point. The at the cases where a0 is a preparation, transformation, channel is then described by the conditional probability or measurement: ←→ ←→ distribution Pr(k0 |a0 ). Following [58], we denote sub- strings with a = a a . . . a , and also define t:t+L t t+1 t+L−1 Pr(k0, λ1|λ0, a0) the past ←−a = a and future −→a = a . We can now t −∞:t t t:∞ = Pr(λ |k , λ , a )Pr(k |λ , a ) define two properties of stochastic channels: 1 0 0 0 0 0 0  µ(λ |P )δ a = P ∈ P Definition 4 (Stationary). A stationary channel is one  1 k0,0 0 = Γ(λ |λ ,T )δ a = T ∈ T that has time-translation symmetry, so statistics are not 1 0 k0,0 0  affected by our choice of time-origin: η(λ1|k0, λ0,M)ξ(k0|λ0,M) a0 = M ∈ M ←→ ←→ (48) Pr(kt:t+L| at ) = Pr(k0:L|ao ) and ←→ ←→ ←→ ←→ ←→ Pr( kt | at ) = Pr(k0 |a0 ) ∀t, L, a . (44) where δ is the Kronecker delta. This can be seen as the rigorous definition of η which emerges naturally Definition 5 (Causal). A causal channel is one for from this recognition that ontological models are equiv- which a finite output substring depends only on input alent to hidden Markov models. In particular, this symbols in its past: gives a mathematical reason why η is only defined in ←→ ←− ←− the support of ξ. If we take the joint distribution Pr(kt:t+L| a ) = Pr(kt:t+L| a t+L). ∀t, L, a (45) Pr(k0, λ1|λ0, a0) to be the more fundamental object, It is shown in [58] that a channel satisfying these two then it is clear we can obtain ξ directly by marginal- properties can be specified entirely by the single-symbol ization recurrence relation Z ←− ←− ξ(k0|λ0,M) = dλ1 Pr(k0, λ1|λ0,M) (49) Pr(k0|a0, a0, k0). (46) Λ

Accepted in Quantum 2019-09-27, click title to verify 17 refer the reader to [33, 55].

Mathematical objects The generalizations of the X k−1 k0 k1 k2 and Z operators to a single qupit are deﬁned by their ...... action on the standard basis {|ji} for j = 0, 1, . . . , p−1: X |ji = |j + 1i (51) a a a a −1 0 1 2 Z |ji = ωj |ji (52) (a) A causal and stationary channel ω = e2πi/p (53)

k−1 k0 k1 k2 All integer arithmetic is done modp. Then the full set of generalized Pauli operators on n qupits is given by λ λ λ ... 0 1 2 ... ( Nn−1 xj zj j=0 X Z p = 2 T(x,z) = (54) Nn−1 xj zj /2 xj zj a−1 a0 a1 a2 j=0 ω X Z p > 2 for x = (x , x , . . . , x ) ∈ n (b) A hidden Markov model of the above channel 0 1 n−1 Zp n and z = (z0, z1, . . . , zn−1) ∈ Zp Figure 3: Influence diagrams [60] for stochastic channels. The boxes represent choices made by the experimenter, circles rep- We also define the symplectic inner product as resent random variables, and arrows represent a possible causal influence. Note that no arrows point backwards in time, and [(x, z), (x0, z0)] = z · x0 − x · z0. (55) that in the hidden Markov model the state λt mediates all causal influences through time. Finally, the phase-point operators Aλ, for λ = (x, z) ∈ n n Zp ×Zp , are a symplectic Fourier transform of the Pauli operators: which is always well defined, and then find η by rear- ranging Eq. 48: 1 X [λ,λ0] 0 Aλ = n ω Tλ (56) p 0 n n Pr(k0, λ1|λ0,M) λ ∈Zp ×Zp η(λ1|k0, λ0,M) = (50) ξ(k0|λ0,M) The stabilizer subtheory A stabilizer group S is a Thus clearly η(λ |k , λ ,M) is only well-defined when 1 0 0 set of pn mutually commuting Pauli operators, which ξ(k |λ ,M) 6= 0, which is how we defined its support. 0 0 can be specified by a set of n generators. There is a Definitions4–6 constitute an equivalent formulation unique state (up to global phase) which is an eigenvector of the ontological models formalism. The assumptions of all of these operators with eigenvalue +1; we say that of this construction can be broken down as follows: (a) S stabilizes this state. For example, for two qubits, the quantum theory is described by a stochastic channel, Bell state (b) this channel is stationary, (c) it is causal, and (d) we |Ψi = |00i + |11i (57) assign the system a state which acts as an HMM of the channel. The authors of [59] identify (c) and (d), calling is stabilized by them non-retrocausality and λ-mediation, respectively. Assumptions (a) and (b) were implicitly present but not SΨ = hZ1Z2,X1X2i (58) explicitly identified. Since our formulation of the state = {I,Z1Z2,X1X2, −Y1Y2}. (59) update rule follows directly from these assumptions, we see that it is not ‘additional’ to the ontological models Here a subscript indicates on which qubit the operator formalism, but a unique extension to describe a more is acting, e.g. Z1 = Z ⊗ I describes Z acting on the general empirical scenario. first qubit. A stabilizer state, then, is a state which is stabilized by a group of pn Pauli operators. Stabilizer measurements are simply measurements of C A brief introduction to the stabilizer the Pauli operators. Note that since each Pauli opera- subtheory tor has p eigenvalues, these amount to a measurement of p projectors, each with rank pn−1. Lower-rank pro- We focus here on the stabilizer subtheory for n qupits, jectors can be constructed by performing commuting where p is a prime. For a more detailed exposition, we Pauli measurements sequentially.

Accepted in Quantum 2019-09-27, click title to verify 18 Finally, the transformations of the stabilizer subtheory are called Cliﬀord transformations. These are the transformations that map the set of Pauli operators to itself, up to a global phase. In other words, it is the normalizer of the Pauli group. The stabilizer subtheory thus consists of preparations corresponding the set of stabilizer states, measurements of Pauli observables, and Cliﬀord transformations, along with convex combinations thereof.

Accepted in Quantum 2019-09-27, click title to verify 19