ψ-epistemic interpretations of quantum theory have a mea- surement 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 distribu- tion via Bayes' rule upon consideration of new informa- tion. 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, Schr¨odingerscat 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.