restriction only constrains what particular probability distributions can
represent an agent’s knowledge of the physical state of the system and what
particular logical propositions about this physical state can correspond to
questions that can be asked.) The toy field theory can also be shown to
satisfy criterion (iii), that is, the principe of Leibnizianity (and hence
generalized noncontextuality as well) for the same reason that other
epistemically restricted classical statistical theories do Spekkens (2007,
2016a). Together with our demonstration that the toy field theory can
reproduce the TRAP phenomenology, this implies that the TRAP phenomenology is
classically explainable according to our preferred notion of
classicality.313131We pause here to comment on how our claim can be made to
square with Bohr’s pronouncement (echoed by many others) that we are forced in
quantum theory to acknowledge complementarity, that is, we are forced to
acknowledge that noncommuting observables cannot be represented by classical
random variables that are jointly well-defined. What we wish to emphasize is
that Bohr did not demonstrate that complementarity is forced upon us by the
operational phenomena of quantum theory. He merely asserted it. Bohr’s
pronouncements, in other words, did not meet the standards of the rigorous
methodology we are advocating here. It is only with the advent of certain
rigorous no-go theorems, such as the Kochen-Specker theorem Kochen and Specker
(1967); Bell (1966) and the no-go theorem for the generalized notion of
noncontextuality proposed in Ref. Spekkens (2005), that one is warranted in
concluding that it is impossible to represent certain sets of noncommuting
quantum observables by classical variables, at least not without violating the
assumption that the values must be assigned context-independently. The fact
that the TRAP phenomenology is not rich enough to support such a no-go theorem
is why a noncontextual model such as the toy field theory has the capacity to
reproduce this phenomenology.

Nonetheless, there will be other, more nuanced, aspects of interference
phenomena that are not classically explainable according to our preferred
notion of classicality. These aspects will arise as operational predictions
that are not realizable by a generalized-noncontextual ontological model. (The
notion of a locally causal model is likely not sufficient for this purpose,
because most interference experiments do not have the appropriate causal
structure for deriving a Bell-type theorem.) To determine to what extent
interference phenomena present a challenge to our preferred notion of
classicality, therefore, one should seek to identify those aspects of
interference phenomena that imply the failure of a generalized-noncontextual
ontological model.

Such a project has already been undertaken for other types of quantum
phenomena. For instance, the fact that nonorthogonal quantum states cannot be
discriminated with certainty is sometimes thought to be an intrinsically
quantum feature of state discrimination. However, if quantum states are
represented by probability distributions that overlap on the physical state
space, lack of perfect discriminability is to be expected. Nonetheless, it was
shown in Ref. Schmid and Spekkens (2018) that the particular tradeoff that
quantum theory predicts between the degree of nonorthogonality and the
probability of successful discrimination is inconsistent with the principle of
generalized noncontextuality. A similar analysis has also been done for state-
dependent cloning Lostaglio and Senno (2020). As another example, although
weak values in pre- and post-selected experiments can be given an account
within a generalized-noncontextual ontological model (see Ref. Karanjai  _et
al._ (2015) for instance), it was shown in Ref. Pusey (2014); Kunjwal  _et
al._ (2019) that anomalous weak values Aharonov  _et al._ (1988) cannot.

A natural follow-up work to this one, therefore, is to undertake the project
of determining what does resist being given an account within a generalized-
noncontextual model in the specific context of interference phenomena Catani
_et al._ .

####  V.1.4 Interpretational lessons

A motivation for studying classical statistical theories of the sort described
in this paper (and characterizing the principles of classicality that they
embody) is that the insights obtained by doing so are useful in guiding our
search for a more compelling interpretation of quantum theory.

In particular, Ref. Schmid  _et al._ (2021a) articulates a research program
that seeks to secure an interpretation of quantum theory that salvages the
spirit of locality and generalized noncontextuality by taking the quantum
state to describe knowledge, while going beyond the usual framework of
ontological models by allowing for intrinsically quantum notions of causation
and inference. It proposes a formalism that seeks to unscramble the omelette
of ontology and epistemology in quantum theory, to use the evocative imagery
of E.T. Jaynes Jaynes (1990). For various interference phenomena, we have here
shown that what was previously attributed to exotic types of causal influence
could be explained within the toy field theory by mere inference, i.e., merely
resolving some uncertainty based on the acquisition of new information. In
this sense, the toy field theory can also be understood as a small
contribution to the program Schmid  _et al._ (2021a, 2019); Schmid (2021);
Bartlett  _et al._ (2012); Fuchs (2002) of unscrambling the quantum omelette.

The toy field theory also suggests a novel take on the relationship between
the first-quantized and second-quantized descriptions in quantum theory. As we
noted in Sec. II.3, the first- and second-quantized descriptions are typically
understood as making different assumptions about the nature of the systems and
their properties: particles with motional degrees of freedom on the one hand,
or modes with excitational degrees of freedom on the other. Our analysis,
however, suggests that it is better to understand both descriptions as being
about modes with excitational degrees of freedom, with the first-quantized
description being merely a more coarse-grained description than the second-
quantized one, describing only a subset of the degrees of freedom that are
described in the latter.323232We provide some further details about how to
conceptualize a theory with an ontology of modes in Appendix C.5. We also
demonstrate that one can recast the toy field theory in terms of arrays of
spatially localized modes and a rule for free propagation of excitations in
Appendix C.6.

###  V.2 Further remarks

####  V.2.1 Why was the toy field theory not recognized earlier?

A natural question regarding the toy field theory is why it was not recognized
earlier that a local classical statistical theory of this type could reproduce
the basic phenomenology of quantum interference. One reason is that
interference phenomena have been studied predominantly in the first-quantized
rather than the second-quantized description, and this has exacerbated the
problem of seeing how they admit of a local explanation. The more significant
reason, however, is that a certain (largely unrecognized) interpretational
assumption—the reality of the quantum state—prevented researchers from
contemplating theories of this sort.

As has been emphasized in Refs. Spekkens (2007); Harrigan and Spekkens (2010)
most attempts to provide realist accounts of operational phenomena presume
from the outset that quantum states represent states of reality rather than
representing states of incomplete knowledge about reality. This assumption
implies, in particular, that the vacuum quantum state represents a state of
reality rather than representing a state of incomplete knowledge about
reality. It then follows that if a mode of an interferometer is described by
the vacuum quantum state, then its physical state is fixed. But if there is no
possibility of varying the physical state of that arm, then there is no
possibility of encoding information in its physical state. According to this
view, then, an arm of the interferometer that is in the vacuum quantum state
cannot propagate forward any information about devices on that arm because it
does not have the capacity to encode any such information.

One of the keys to the success of the toy field theory, therefore, is that the
analogue of a pure quantum state (and in particular, the analogue of the
vacuum quantum state) is a state of incomplete knowledge about reality, i.e.,
a probability distribution that has support on more than one physical
state.333333This is called a $\psi$-epistemic model in the foundations
literature Harrigan and Spekkens (2010). For further reading on the view that
quantum states are states of incomplete knowledge, see Refs. Spekkens (2007,
2016a). Note that the interpretational program of QBism Fuchs  _et al._
(2014); Fuchs and Schack (2013) also takes as its starting point the idea that
the quantum state does not represent reality. Indeed, early work on Quantum
Bayesianism laid some of the groundwork for Refs. Spekkens (2007, 2016a).
Nonetheless, there are ways of pursuing an epistemic view of quantum states
that are quite distinct from the QBist research program. See, e.g., Ref.
Schmid  _et al._ (2021a). Using concepts from statistical mechanics, one might
say that the vacuum quantum state is analogous to a macrostate and hence that
it is consistent with many microstates. Specifically, even if the occupation
number of a mode is 0, there are two possible values that its discrete phase
might take, and hence such a mode can still encode one bit of information.
This is what opens up the possibility that information about a device (e.g.,
whether it implements a which-way measurement or not) can be propagated to
other devices (such as the final detectors) through a mode which, in the
quantum account, is in the vacuum quantum state. In particular, in the case of
the Elitzur-Vaidman bomb-tester, it is what opens up the possibility that
information about whether the bomb is functional or faulty can be propagated
to the final detectors through the physical state of the $R$ mode even though
in the quantum account the $R$ mode is in the vacuum quantum state. 343434This
possibility is closely related to another feature of classical statistical
theories with an epistemic restriction, namely, that a transformation may
leave the probability distribution over physical states invariant even though
it modifies the physical state. This occurs whenever the transformation
implements a permutation within a set of physical states that are assigned
equal probability. In the toy field theory, for example, when the occupation
number is point-distributed, so that (by the epistemic restriction) the phase
is uniformly distributed, a phase flip transformation, $N\mapsto
N,\Phi\mapsto\Phi\oplus 1$, leaves the distribution invariant even though it
modifies the physical state. In summary, classical statistical theories with
an epistemic restriction render implausible what has been termed the ‘ontic
indifference principle’ in Ref. Hardy (2013), namely, that if a pure quantum
state is left invariant, then so too is any physical state that is assigned
nonzero probability by that quantum state.

The toy field theory consequently provides yet another example (in addition to
those presented in earlier work Spekkens (2007, 2016a); Bartlett  _et al._
(2012)) of how treating pure quantum states as states of incomplete knowledge
opens up new interpretational possibilities.

####  V.2.2 Inference, not influence

Another way in which our analysis has made use of the epistemic view of
quantum states is how the remote state update rule in quantum theory has come
out as analogous to the updating of a state of knowledge upon acquisition of
novel information. Because such updating is governed by Bayes’ rule, it is
typically termed Bayesian conditioning. This analogy has also been noted in
many previous works Spekkens (2007, 2016a); Leifer and Spekkens (2013); Fuchs
(2002).

Recall that the remote state update rule in quantum theory stipulates how the
quantum state of a system is updated when one learns the outcome of a
measurement (either destructive or nondestructive) on a different system that
is correlated with the first (by virtue of a common cause acting on both).
This is distinct from the usual state update rule in quantum theory, which
stipulates how the quantum state of a system is updated when one learns the
outcome of a direct (nondestructive) measurement on the system.

We noted that the counterpart in the toy field theory of the usual quantum
state update rule is a Bayesian conditioning followed by a disturbance. For a
nondestructive measurement of the occupation number of mode $R$, for instance,
there is a Bayesian conditioning based on the outcome of the measurement,
followed by a randomization of the phase of mode $R$.

On the other hand, the counterpart of the remote update rule involves only a
Bayesian conditioning, i.e., it is purely a resolution of some of the agent’s
uncertainty about the physical state (i.e., with no disturbance to the
latter). Specifically, given that the probability distribution prior to the
measurement is such that the occupation numbers of the pair of modes are
anticorrelated, it follows that learning the occupation number of mode $R$
causes one to update one’s knowledge of the occupation number of mode $L$.
Furthermore, it is only the agent’s knowledge about mode $L$ that changes as a
result of a measurement on the $R$ mode; the physical state of mode $L$ is not
changed. This clarifies how, in the toy field theory account of the TRAP
phenomenology, there is no need for a causal influence from the $R$ arm to the
$L$ arm of the interferometer.353535In Appendix C.1, we elaborate on the
importance of an interpretation of probabilities in terms of credences in our
approach. In Appendix C.2, we point out that one can nonetheless reframe the
predictions of the toy field theory in terms of an infinite ensemble of
repetitions of the experimental set-up. We show that in this reframing,
Bayesian updating about a single system is replaced by an updating of the
ensemble describing the set of copies of the system. In particular, we explain
the counterpart of the remote quantum state update rule in this ensemble
language, which provides another way of seeing why it is local.

In a similar manner, in the quantum eraser experiment, the remote state update
rule for $L$ and $R$ based on $A$ (i..e, the update rule for the composite
system comprising the pair of modes given a measurement on the auxiliary
system) corresponds, in the toy field theory, to simply updating one’s
knowledge of $L$ and $R$ based on what is learned about $A$ rather than any
nonlocal influence from $A$ to $L$ and $R$.

A final example of how the epistemic view of quantum states has been leveraged
in our analysis is how, in the toy field theory, phenomena that are sometimes
attributed to backwards-in-time causation can be instead attributed to merely
updating one’s knowledge about the past.

For those committed to the idea that a quantum state describes reality, the
distinction between the quantum state of a photon being an eigenstate of the
which-way observable (i.e., $\left|L\right>$ or $\left|R\right>$) and it being
a state from a complementary basis (i.e.,
$\tfrac{1}{\sqrt{2}}(\left|R\right>-\left|L\right>)$ or
$\tfrac{1}{\sqrt{2}}(\left|R\right>+\left|L\right>)$) is a distinction
concerning the physical state of the photon. The conventional discussions of
Wheeler’s delayed-choice experiment and the quantum eraser experiment
demonstrate that some researchers are tempted to endorse this idea even if the
difference in the quantum state assigned at some time is induced by learning
the outcome of a measurement implemented at a later time. In such a case, one
must either grant that there are backwards-in-time influences, or embrace a
kind of anti-realism (as Wheeler did).

In an epistemic interpretation of quantum states, by contrast, the fact that
learning the outcome of a measurement at one time can lead to an updating of
the quantum state for a system at an earlier time has a very natural
interpretation: it is simply an updating of one’s knowledge about the past.
Such updates are commonplace in science. For instance, we update our belief
about the existence of dinosaurs in the distant past based on observation of
the fossil record today.

From the perspective of the methodology we are advocating, it is fine if a
researcher wishes to assume that pure quantum states describe physical states
rather than states of incomplete knowledge (i.e., if, in the language of Ref.
Harrigan and Spekkens (2010), they wish to assume a $\psi$-ontic rather than a
$\psi$-epistemic ontological model), but this assumption must be made explicit
in the no-go theorem, so that it is evident that denying it is a way out of
the no-go result. The point, again, is that whatever one thinks of this
assumption, it is “not forced on us by experimental facts, but by deliberate
theoretical choice.”363636Many recent works, including the PBR theorem Pusey
_et al._ (2012), have sought to identify principles from which the assumption
of $\psi$-ontology can be derived (see Ref. Leifer (2014) for a review). The
naturalness of these principles is disputed by some of the authors of this
article (see, e.g., Ref. Spekkens (2012)). But regardless of what one thinks
about these principles, the important point is that all the standard no-go
theorems—including those of Bell and Kochen-Specker, but also that of
PBR—assume the framework of ontological models, and as we noted above, this is
the assumption whose rejection we believe holds the most promise for finding a
satisfactory realist interpretation of quantum theory.

####  V.2.3 Is the toy field theory ad hoc?

It is worth noting that the argument purporting to show that there is no local
causal explanation of the phenomenology of the Mach-Zehnder interferometer
never had the same force as the argument that there is no local causal
explanation of Bell inequality violations. This is because in the Mach-Zehnder
interferometer, one can always imagine some hypothetical system that
propagates from the location of the detector in the $R$ arm to the final
detectors (via the second beamsplitter). That is, one can always a imagine a
hypothetical system that mediates a causal influence from the former to the
latter without requiring such influences to be faster than the speed of light.
The argument that is typically presented against such a local causal
explanation proceeds along the following lines: “Yes, there might be
something—a ghost particle for example—that propagates information from the
region where the detector is (or is not) placed in the $R$ arm to the second
beamsplitter and that dictates the subsequent motion of the photon, but to
posit such a ghostly influence is ad hoc in the sense that it has nothing to
recommend it except the evidence that it is specifically designed to explain.”

Note that this response explicitly concedes that the inference from the TRAP
phenomenology to the failure of a local causal explanation is not logically
sound, but is rather only a plausibility argument. Nonetheless, an ad hoc
explanation is indeed an implausible one. Therefore, if our proposed account
of the TRAP phenomenology consisted of nothing more than to posit a ghostly
influence of the sort just described, then it would be appropriate to
criticize it as ad hoc and thus implausible. It is therefore worthwhile for us
to explicitly respond to this potential criticism.

To see why the toy field theory is not an ad hoc explanation, one must
consider its origin. It arose as a natural application of the ideas of Ref.
Spekkens (2007) to quantum interference phenomena.373737The basics of the toy
field theory and its implications for the interpretation of interference
phenomena were worked out in 2005. As we noted above, what is critical to
achieving a local causal explanation is the fact that the counterpart of the
vacuum quantum state in the toy field theory is a probability distribution
with support on many different physical states. But this fact is an
unavoidable consequence of the epistemic restriction, since the latter
dictates that every pure quantum state has such a probability distribution as
its counterpart. In other words, the possibility of sending information
through a mode which in the quantum account is described by the vacuum quantum
state is an inevitable consequence of the epistemic restriction. Given that
toy theories based on an epistemic restriction of this type can also reproduce
a long list of phenomena that are conventionally regarded as uniquely quantum
(see, e.g., Table II of Ref. Spekkens (2016a)), it follows that the epistemic
restriction underlying the toy field theory has much more to recommend it than
the fact that the toy field theory can reproduce the TRAP phenomenology of
quantum interference. Recalling the definition of an ad hoc explanation which
we presented above—having nothing to recommend it except the evidence that it
is specifically designed to explain—it is clear that the toy field theory
explanation of interference phenomena is not ad hoc.

####  V.2.4 Shifting the goal post

No doubt those researchers who are sympathetic to the view that interference
captures the essence of quantum theory will be tempted to respond to the
arguments of this article as follows:

> Sure, you have reproduced some of the phenomenology of quantum interference,
> but you haven’t reproduced all of it. What about all of the experiments
> involving beamsplitters that are not 50-50, or involving phase shifts other
> than $\phi=0$ and $\phi=\pi$? You can’t make sense of those in the toy field
> theory.383838In a similar vein, Yakir Aharonov’s response to a presentation
> on this topic Spekkens (2016b) was to concede that the toy field theory
> provides an adequate explanation of interference phenomena for bosons, but
> to insist that such an explanation will not be possible for fermions.

Our response is as follows. We agree that the full phenomenology of quantum
theory does not admit of an explanation in terms of a classical worldview. As
noted above, this is the lesson we take from the nonrealizability of certain
quantum correlations by a generalized-noncontextual ontological model or by a
locally causal model. As we noted in Section V.1.3, however, it remains an
open question which specific aspects of the full phenomenology of quantum
interference genuinely resist classical explanation.

If someone wishes to claim that aspects of interference beyond the TRAP
phenomenology demonstrate the impossibility of maintaining a classical
worldview, then not only must they specify precisely which aspects they have
in mind and how they propose to formalize the notion of classicality, they
must also back up their claim with a rigorous no-go theorem, following the
methodology we endorsed above. Until they do, the view that the phenomena in
question resist explanation in terms of a classical worldview is mere
speculation, and might only indicate a “lack of imagination”, to recall Bell’s
phrase.393939 In Appendices C.3 and C.4, we consider two examples of
additional interference phenomenology, based on minor modifications to the
Mach-Zehnder interferometer, that were speculated to resist classical
explanation in Ref. Hance and Hossenfelder (2022). We demonstrate that the
phenomenology of both examples can be easily accommodated within the toy field
theory, and therefore that they do not, in fact, resist classical explanation.

###  V.3 Send-off

In an article based on a 1981 conference presentation Feynman (1982), Feynman
returned to the question of what was the essential difficulty of quantum
theory. The article concerned whether quantum dynamics could be simulated
using a classical computer with local connections, and Feynman concluded that
it could not using an argument that is in essence the same as the one
appearing in Bell’s theorem. (In Feynman’s approach, the proof proceeds by
showing that the quantum correlations can only be reproduced using local
connections if the probabilities can go negative.) He notes furthermore that
the relevant experiment has been done and that it agrees with the quantum
predictions.404040Feynman did not cite Bell’s work explicitly, suggesting that
he might have rederived the result independently. However, as noted by
Whitaker Whitaker (2016), the fact that he was aware of the experimental tests
of Bell inequalities implies that he must at least have been aware of the
existence of Bell’s work. . In this article, therefore, Feynman takes the
essential difficulty to be the existence of a Bell-type no-go result. How does
one square this with his earlier claim, in the Lectures on Physics, that
“interference holds the essential difficulty”? (Especially given that, as we
have shown here, the TRAP phenomenology of interference is not associated to a
violation of a Bell inequality.) Did Feynman change his mind about what was
the essential difficulty?

No. It seems, rather, that he did not think that there was any significant
difference between Bell inequality violations and the phenomenology of
interference—both appeared to him to be classically inexplicable. Indeed, a
quote by Feynman regarding Bell’s theorem from a 1983 article reprinted in
Ref. Hey (2018) makes it clear that he in fact denied that the difficulty
uncovered by Bell was of a different nature than the difficulty associated
with understanding other quantum phenomena (including, presumably,
interference). Speaking of the ways in which quantum phenomena cannot be
accounted for in a classical worldview, he states:

> Bell’s Theorem is a contribution to [that set of ideas], which is to point
> out mathematically that it has to happen. People knew it had to happen
> before; all he did was to demonstrate it. It is not a theorem that anybody
> thinks is of any particular importance. We who use quantum mechanics have
> been using it all the time. It is not an important theorem. It is simply a
> statement of something we know is true—a mathematical proof of it.

The obvious criticism to make of Feynman here is that he is wrong in his
assessment of the degree of significance of Bell’s theorem. The criticism we
wish to emphasize, however, is not of this assessment, but of the reasons he
articulates in support of it. The cavalier attitude towards foundational
questions that is expressed in this quote is precisely the sort of
methodological looseness that we are advocating against. Interpretational
claims which practitioners of quantum theory believe to be true in the absence
of a mathematical proof are not “known to be true”. Indeed many such claims
are plainly mistaken. The claim that interference phenomenology captures the
essence of quantum theory is a prime example. A mathematically rigorous no-go
theorem is the bar that must be passed by any argument purporting to support
some interpretational claim. This is why Bell’s theorem is, in fact, a
singularly important result in twentieth-century physics. The absence of a
parallel no-go theorem establishing the classical nonexplainability of the
TRAP phenomenology of quantum interference is what left open the possibility
of the sort of classical account presented in this article. Feynman’s failure
to recognize this sort of possibility—which led to his mistaken claim
regarding interference phenomena being the essence of quantum
theory—illustrates why it is critical to back up interpretational claims with
mathematically rigorous no-go theorems.

## Acknowledgments

RWS would like to acknowledge the important contribution of Elliot Martin, who
worked as an undergraduate student on an early version of the ideas described
in this article as part of a research project in the summer and fall of 2005,
and to thank Jon Barrett for helpful discussions about the project at that
time. The authors also acknowledge discussions with Sabine Hossenfelder, Tim
Palmer, and Jonte Hance which served to motivate the inclusion of certain
discussions in the appendices. This research was supported by Perimeter
Institute for Theoretical Physics. Research at Perimeter Institute is
supported by the Government of Canada through the Department of Innovation,
Science and Economic Development Canada and by the Province of Ontario through
the Ministry of Research, Innovation and Science. ML was additionally
supported, in part, by grant number FQXi-RFP-IPW-1905 from the Foundational
Questions Institute and Fetzer Franklin Fund, a donor advised fund of Silicon
Valley Community Foundation. ML and LC were supported, in part, by the Fetzer
Franklin Fund of the John E. Fetzer Memorial Trust. LC was also supported by
the Army Research Office (ARO) (Grant No. W911NF-18-1-0178) and acknowledges
funding from the Einstein Research Unit ‘Perspectives of a Quantum Digital
Transformation’. DS also acknowledges support by the Foundation for Polish
Science (IRAP project, ICTQT, contract no.2018/MAB/5, co-financed by EU within
Smart Growth Operational Programme).

## References

  * Feynman  _et al._ (1963) R. P. Feynman, R. B. Leighton, and M. L. Sands,  _The Feynman Lectures on Physics_ (Addison-Wesley world student series, 1961-1963).
  * Bohr (1949) N. Bohr, in  _The Library of Living Philosophers, Volume 7. Albert Einstein: Philosopher-Scientist_ , edited by P. A. Schilpp (Open Court, 1949) pp. 199–241.
  * Elitzur and Vaidman (1993) A. C. Elitzur and L. Vaidman, Found. Phys. 23, 987 (1993).
  * Wheeler (1978) J. A. Wheeler,  _Mathematical Foundations of Quantum Theory_ , a. r. marlow ed. (Academic Press, 1978).
  * Mohrhoff (1996) U. Mohrhoff, Amer. J. Phys. 64, 1468 (1996).
  * Mohrhoff (1999) U. Mohrhoff, Amer. J. Phys. 67, 330 (1999).
  * Scully and Drühl (1982) M. O. Scully and K. Drühl, Phys. Rev. A 25, 2208 (1982).
  * Spekkens (2007) R. W. Spekkens, Phys. Rev. A 75, 032110 (2007).
  * Spekkens (2016a) R. W. Spekkens, in  _Quantum Theory: Informational Foundations and Foils_ , edited by G. Chiribella and R. W. Spekkens (Springer, Dordrecht, 2016) pp. 83–135.
  * Gottesman (1997) D. Gottesman, PhD thesis, California Institute of Technology (1997).
  * Catani and Browne (2017) L. Catani and D. E. Browne, New J. Phys. 96, 052112 (2017).
  * Catani and Browne (2018) L. Catani and D. E. Browne, Phys. Rev. A 98, 052108 (2018).
  * van Enk (2007) S. J. van Enk, Found. Phys. 37, 1447 (2007).
  * Pusey (2012) M. F. Pusey, Found. Phys. 42, 688 (2012).
  * Bartlett  _et al._ (2012) S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Phys. Rev. A 86, 012103 (2012).
  * Larsson (2012) J.-Å. Larsson, AIP Conf. Proc. 1424, 211 (2012).
  * Blasiak (2013) P. Blasiak, Phys. Lett. A 377, 847 (2013).
  * Disilvestro and Markham (2017) L. Disilvestro and D. Markham, Phys. Rev. A 95, 052324 (2017).
  * Hausmann  _et al._ (2021) L. Hausmann, N. Nurgalieva, and L. del Rio, arXiv:2105.03277 (2021).
  * Braasch and Wootters (2021) W. F. Braasch and W. K. Wootters, arXiv:2107.02728 (2021).
  * Bohm (1952a) D. Bohm, Phys. Rev. 85, 166 (1952a).
  * Blasiak (2015) P. Blasiak, New J. Phys. 17, 113043 (2015).
  * Spekkens (2016b) R. W. Spekkens, https://pirsa.org/16060102 (2016b).
  * Bell (1964a) J. S. Bell, Physics Physique Fizika 1, 195 (1964a).
  * Bell (1982) J. S. Bell, Found. Phys. 12, 989 (1982).
  * Bohm (1952b) D. Bohm, Phys. Rev. 85, 166 (1952b).
  * Bell (1964b) J. S. Bell,  _Speakable and Unspeakable in Quantum Mechanics_ , second edition ed. (Cambridge University Press, 1964).
  * Zehnder (1891) L. Zehnder, Zeitschrift für Instrumentenkunde 11, 275 (1891).
  * Mach (1892) L. Mach, Zeitschrift für Instrumentenkunde 12, 89 (1892).
  * Englert (1999) B.-G. Englert, Zeitschrift Naturforschung Teil A 54, 11 (1999).
  * Philippidis  _et al._ (1979) C. Philippidis, C. Dewdney, and B. J. Hiley, Il Nuovo Cimento B (1971-1996) 52, 15 (1979).
  * Penrose (1994) R. Penrose,  _Shadows of the Mind: A Search for the Missing Science of Consciousness_ (Vintage, 1994).
  * Scully  _et al._ (1991) M. O. Scully, B.-G. Englert, and H. Walther, Nature 351, 111 (1991).
  * Hillmer and Kwiat (2007) R. Hillmer and P. Kwiat, Scie. Amer. 296, 90 (2007).
  * Einstein  _et al._ (1935a) A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935a).
  * Herzog  _et al._ (1995) T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 75, 3034 (1995).
  * Schmid  _et al._ (2021a) D. Schmid, J. H. Selby, and R. W. Spekkens, arXiv:2009.03297 (2021a).
  * Spekkens (2019) R. W. Spekkens, arXiv:1909.04628 (2019).
  * Catani and Leifer (2020) L. Catani and M. Leifer, arXiv:2003.10050 (2020).
  * Kochen and Specker (1967) S. Kochen and E. Specker, J. Math. Mech. 17, 59 (1967).
  * Spekkens (2005) R. W. Spekkens, Phys. Rev. A 71, 052108 (2005).
  * Spekkens (2008) R. W. Spekkens, Phys. Rev. Lett. 101, 020401 (2008).
  * Schmid  _et al._ (2020) D. Schmid, J. H. Selby, M. F. Pusey, and R. W. Spekkens, arXiv preprint arXiv:2005.07161 (2020).
  * Schmid  _et al._ (2021b) D. Schmid, J. H. Selby, E. Wolfe, R. Kunjwal, and R. W. Spekkens, PRX Quantum 2, 010331 (2021b).
  * Shahandeh (2021) F. Shahandeh, PRX Quantum 2, 010330 (2021).
  * Kunjwal and Spekkens (2015) R. Kunjwal and R. W. Spekkens, Phys. Rev. Lett. 115, 110403 (2015).
  * Marvian (2020) I. Marvian, arXiv preprint arXiv:2003.05984 (2020).
  * Baldijão  _et al._ (2021) R. D. Baldijão, R. Wagner, C. Duarte, B. Amaral, and M. T. Cunha, P R X Quantum 2, 030351 (2021).
  * Mazurek  _et al._ (2016) M. D. Mazurek, M. F. Pusey, R. Kunjwal, K. J. Resch, and R. W. Spekkens, Nat. Comm. 7 (2016).
  * Mazurek  _et al._ (2017) M. D. Mazurek, M. F. Pusey, K. J. Resch, and R. W. Spekkens, arXiv:1710.05948 (2017).
  * Kunjwal and Spekkens (2018) R. Kunjwal and R. W. Spekkens, Phys. Rev. A 97, 052110 (2018).
  * Pusey (2018) M. F. Pusey, Phys. Rev. A 98, 022112 (2018).
  * Schmid  _et al._ (2018) D. Schmid, R. W. Spekkens, and E. Wolfe, Phys. Rev. A 97, 062103 (2018).
  * Spekkens  _et al._ (2009) R. W. Spekkens, D. H. Buzacott, A. J. Keehn, B. Toner, and G. J. Pryde, Phys. Rev. Lett. 102, 010401 (2009).
  * Chailloux  _et al._ (2016) A. Chailloux, I. Kerenidis, S. Kundu, and J. Sikora, New J. Phys. 18, 045003 (2016).
  * Ambainis  _et al._ (2019) A. Ambainis, M. Banik, A. Chaturvedi, D. Kravchenko, and A. Rai, Quantum Inf. Process. 18, 111 (2019).
  * Saha  _et al._ (2019) D. Saha, P. Horodecki, and M. Pawłowski, New J. Phys. 21, 093057 (2019).
  * Raussendorf (2013) R. Raussendorf, Phys. Rev. A 88, 022322 (2013).
  * Hoban  _et al._ (2011) M. J. Hoban, E. T. Campbell, K. Loukopoulos, and D. E. Browne, New J. Phys. 13, 023014 (2011).
  * Schmid  _et al._ (2021c) D. Schmid, H. Du, J. H. Selby, and M. F. Pusey, arXiv:2101.06263 (2021c).
  * Schmid and Spekkens (2018) D. Schmid and R. W. Spekkens, Phys. Rev. X 8, 011015 (2018).
  * Lostaglio and Senno (2020) M. Lostaglio and G. Senno, Quantum 4, 258 (2020).
  * Lostaglio (2020) M. Lostaglio, Phys. Rev. Lett. 125, 230603 (2020).
  * Pusey (2014) M. F. Pusey, Phys. Rev. Lett. 113, 200401 (2014).
  * Kunjwal  _et al._ (2019) R. Kunjwal, M. Lostaglio, and M. F. Pusey, Phys. Rev. A 100, 042116 (2019).
  * Yadavalli and Kunjwal (2020) S. A. Yadavalli and R. Kunjwal, arXiv:2006.00469 (2020).
  * Bell (1966) J. S. Bell, Rev. Mod. Phys. 38 (1966).
  * Karanjai  _et al._ (2015) A. Karanjai, E. Cavalcanti, S. Bartlett, and T. Rudolph, New J. Phys. 17, 073015 (2015).
  * Aharonov  _et al._ (1988) Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
  * (70) L. Catani, M. Leifer, G. Scala, D. Schmid, and R. W. Spekkens, in preparation .
  * Jaynes (1990) E. Jaynes, in Complexity, Entropy, and the Physics of Information edited by W. H. Zurek , 381 (1990).
  * Schmid  _et al._ (2019) D. Schmid, K. Ried, and R. W. Spekkens, Phys. Rev. A 100, 022112 (2019).
  * Schmid (2021) D. Schmid, PhD thesis, University of Waterloo (2021).
  * Fuchs (2002) C. A. Fuchs, arXiv preprint quant-ph/0205039 (2002).
  * Harrigan and Spekkens (2010) N. Harrigan and R. W. Spekkens, Found. Phys. 40, 125 (2010).
  * Fuchs  _et al._ (2014) C. A. Fuchs, N. D. Mermin, and R. Schack, American Journal of Physics 82, 749 (2014).
  * Fuchs and Schack (2013) C. A. Fuchs and R. Schack, Rev. Mod. Phys. 85, 1693 (2013).
  * Hardy (2013) L. Hardy, Int. J. Mod. Phys. B 27, 1345012 (2013).
  * Leifer and Spekkens (2013) M. S. Leifer and R. W. Spekkens, Phys. Rev. A 88, 052130 (2013).
  * Pusey  _et al._ (2012) M. F. Pusey, J. Barrett, and T. Rudolph, Nat. Phys. 8, 475 (2012).
  * Leifer (2014) M. Leifer, Quanta 3, 67 (2014).
  * Spekkens (2012) R. W. Spekkens, https://pirsa.org/12050021 (2012).
  * Hance and Hossenfelder (2022) J. R. Hance and S. Hossenfelder, arXiv preprint arXiv:2204.01768 (2022).
  * Feynman (1982) R. P. Feynman, International Journal of Theoretical Physics 21, 467 (1982).
  * Whitaker (2016) A. Whitaker, Amer. J. Phys. 84, 493 (2016).
  * Hey (2018) A. Hey,  _Feynman and computation_ (CRC Press, 2018).
  * Wigner (1995) E. P. Wigner, in  _Philosophical reflections and syntheses_ (Springer, 1995) pp. 247–260.
  * London and Bauer (2014) F. London and E. Bauer, in  _Quantum theory and measurement_ , Vol. 53, edited by J. A. Wheeler and W. H. Zurek (Princeton University Press, 2014).
  * Ghirardi  _et al._ (1986) G. C. Ghirardi, A. Rimini, and T. Weber, Phys. Rev. D 34, 470 (1986).
  * Pearle (1989) P. Pearle, Phys. Rev. A 39, 2277 (1989).
  * Bassi  _et al._ (2013) A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, Rev. Mod. Phys. 85, 471 (2013).
  * Einstein  _et al._ (1935b) A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935b).
  * Struyve and Westman (2007) W. Struyve and H. Westman, Proc. Roy. Soc. A 463, 3115 (2007).
  * Aharonov  _et al._ (2017) Y. Aharonov, E. Cohen, F. Colombo, T. Landsberger, I. Sabadini, D. C. Struppa, and J. Tollaksen, Proc. Nat. Acad. Scie. 114, 6480 (2017).
  * Everett (1957) H. Everett, Review of Modern Physics 20, 454 (1957).
  * Wallace (2012) D. Wallace,  _The emergent multiverse: Quantum theory according to the Everett interpretation_ (Oxford University Press, 2012).
  * Deutsch (1998) D. Deutsch,  _The Fabric of Reality_ (Penguin UK, 1998).
  * Hall  _et al._ (2014) M. J. W. Hall, D.-A. Deckert, and H. M. Wiseman, Phys. Rev. X 4, 041013 (2014).
  * De Finetti (2017) B. De Finetti,  _Theory of Probability: A critical introductory treatment_ (Wiley, 2017).
  * Caves  _et al._ (2002) C. M. Caves, C. A. Fuchs, and R. Schack, Journal of Mathematical Physics 43, 4537 (2002).
  * Boyer (1980) T. H. Boyer, in  _Foundations of Radiation Theory and Quantum Electrodynamics,_ (E. O. Barut, Plenum Press, New York, 1980).
  * Spekkens (2015) R. W. Spekkens, in  _Questioning the Foundations of Physics_ (Springer, 2015) pp. 5–16.

##  Appendix A A more formal account

###  A.1 The toy field theory

We begin by describing our diagrammatic representation, which follows the one
introduced in Ref. Spekkens (2007). The set of physical states of a mode is
represented diagrammatically as a set of 4 boxes, and a particular physical
state as a marked box. Thus, for instance, the physical state wherein $N=0$
and $\Phi=1$ is depicted as:

$\displaystyle\centering\includegraphics[width=82.38885pt]{OnticStatesSingleMode.pdf}.\@add@centering$
(23)

The set of physical states of a pair of modes is represented diagrammatically
by a $4\times 4$ grid of boxes. For instance, the physical state defined by
$N_{L}=1,\Phi_{L}=0,N_{R}=0,\Phi_{R}=1$ is represented as

$\displaystyle\centering\includegraphics[width=138.76157pt]{OnticStatesTwoModes.pdf}.\@add@centering$
(24)

An agent’s knowledge of a mode can be completely specified by a probability
distribution over $N$ and $\Phi$, denoted $p(N,\Phi)$. We will write
$p(X)=[x]$ to indicate that the distribution over the variable $X$ is a point
distribution with all weight on the value $x$. In particular, if $X$ is a
binary variable, then a distribution that assigns probability $w$ to $X=0$ can
be written as $p(X)=w[0]+(1-w)[1]$.

In the case where a mode is known to be unoccupied, so that $N=0$, while the
phase is unknown, the probability distribution is

$p(N,\Phi)=[0]\left(\tfrac{1}{2}[0]+\tfrac{1}{2}[1]\right),$

for which the marginal distribution on $N$ is a point distribution on the
value $0$, $p(N)=[0]$, and the marginal distribution on $\Phi$ is uniform,
$p(\Phi)=\tfrac{1}{2}[0]+\tfrac{1}{2}[1]$. (It follows that the marginal
distribution on the parity $N\oplus\Phi$ is also uniform,
$p(N\oplus\Phi)=\tfrac{1}{2}[0]+\tfrac{1}{2}[1]$, corresponding to the fact
that the parity is also unknown.)

This distribution is depicted diagrammatically as

$\displaystyle\centering\includegraphics[width=82.38885pt]{EpistemicStateSingleMode.pdf}.\@add@centering$
(25)

For the epistemic restriction that defines the toy field theory, the
probability distribution describing a valid state of knowledge is always a
flat distribution, so that one can get away with merely specifying which
physical states are known not to describe the actual physical state (depicted
as unfilled) and which are deemed to be possible options for the actual
physical state (depicted as blue).

Formally, therefore, the analogue in the toy field theory of the quantum state
$|1\rangle_{L}|0\rangle_{R}$ at the input of the Mach-Zehnder interferometer
is:

$\displaystyle p(N_{L},\Phi_{L},N_{R},\Phi_{R})$
$\displaystyle=[1]\left(\tfrac{1}{2}[0]+\tfrac{1}{2}[1]\right)[0]\left(\tfrac{1}{2}[0]+\tfrac{1}{2}[1]\right),$
(26)

which is depicted diagrammatically as:

$\displaystyle\centering\includegraphics[width=138.76157pt]{EpistemicStatesTwoModes.pdf}.\@add@centering$
(27)

The analogue in the toy field theory of the unitary associated to a 50-50
beamsplitter is the function described in Eq. (III.1.1), which we repeat here:

$\displaystyle N^{\rm out}_{L}=\Phi_{L}^{\rm in}\oplus\Phi_{R}^{\rm in}$
$\displaystyle N^{\rm out}_{R}=N_{L}^{\rm in}\oplus N_{R}^{\rm
in}\oplus\Phi_{L}^{\rm in}\oplus\Phi_{R}^{\rm in}$ $\displaystyle\Phi^{\rm
out}_{L}=N_{L}^{\rm in}\oplus\Phi_{R}^{\rm in}$ $\displaystyle\Phi^{\rm
out}_{R}=\Phi_{R}^{\rm in}.$ (28)

This corresponds to a permutation of the 16 possible physical states, depicted
diagrammatically as follows:

$\displaystyle\centering\includegraphics[width=138.76157pt]{ToyBeamSplitter.pdf}.\@add@centering$
(29)

We refer to this function as the beamsplitter dynamics.

If one changes coordinates from $N_{L},N_{R},\Phi_{L},\Phi_{R}$ to
$N_{L},\Delta N,\Phi_{L},\Delta\Phi$, where

$\displaystyle\Delta N\equiv N_{L}\oplus N_{R}$
$\displaystyle\Delta\Phi\equiv\Phi_{L}\oplus\Phi_{R},$ (30)

then the beamsplitter dynamics can be expressed as:

$\displaystyle N^{\rm out}_{L}=\Delta\Phi^{\rm in}$ $\displaystyle\Delta
N^{\rm out}=\Delta N^{\rm in}$ $\displaystyle\Delta\Phi^{\rm out}=N_{L}^{\rm
in}$ $\displaystyle\Phi^{\rm out}_{R}=\Phi_{R}^{\rm in},$ (31)

i.e., it acts as identity on $\Delta N$ and $\Phi_{R}$, and swaps the values
of $N_{L}$ and $\Delta\Phi$. In this form, the function is seen to be simply
the Swap Rule, articulated in subsection III.1. It is clear, therefore, that
the Swap Rule is simply another way of describing the function defined by Eq.
(III.1.1).

The impact of the first beamsplitter is determined by applying the function of
Eq. (III.1.1) to the distribution described in Eq. (A.1). One thereby finds
that the probability distribution after the first beamsplitter is

$\displaystyle p(N_{L},N_{R},\Phi_{L},\Phi_{R})$
$\displaystyle=\left(\tfrac{1}{2}[0][1]+\tfrac{1}{2}[1][0]\right)\left(\tfrac{1}{2}[0][1]+\tfrac{1}{2}[1][0]\right),$
(32)

which is depicted diagrammatically as:

$\displaystyle\centering\includegraphics[width=138.76157pt]{AntiDiagonalState.pdf}.\@add@centering$
(33)

This can also be inferred diagrammatically by working out the action of the
permutation depicted in Eq. (29) on the distribution represented in Eq. (27).

The case of the Mach-Zehnder interferometer with phase shifter. We now
consider the toy field theory account of a Mach-Zehnder interferometer with a
phase shifter in arm $R$, of the type depicted in Fig. 1.

A $\pi$ phase shift to a mode is represented by the function that flips the
value of the discrete phase of that mode, while preserving its occupation
number:

$\displaystyle N^{\rm out}=N^{\rm in}$ $\displaystyle\Phi^{\rm out}=\Phi^{\rm
in}\oplus 1,$ (34)

which is represented diagrammatically as:

$\displaystyle\centering\includegraphics[width=86.72267pt]{ToyPhaseShift.pdf}.\@add@centering$
(35)

Note that the presence of a phase shifter on arm $R$ is presumed to leave the
physical state of mode $L$ unchanged, so that it is the identity function that
acts on the latter.

It follows that, as a permutation on the physical states of the pair of modes,
a phase shifter on arm $R$ that implements a $\pi$ phase shift is represented
as:

$\displaystyle\includegraphics[width=138.76157pt]{PhaseShift2modes.pdf}.$ (36)

The phase flip dynamics of Eq. (A.1) maps the distribution of Eq. (A.1) to

$\displaystyle p(N_{L},N_{R},\Phi_{L},\Phi_{R})$
$\displaystyle=\left(\tfrac{1}{2}[0][1]+\tfrac{1}{2}[1][0]\right)\left(\tfrac{1}{2}[0][0]+\tfrac{1}{2}[1][1]\right),$
(37)

which is represented diagramatically as

$\displaystyle\centering\includegraphics[width=138.76157pt]{OffDiagonal.pdf}.\@add@centering$
(38)

This can also be inferred diagrammatically by applying the permutation of Eq.
(36) to Eq. (33).

The second beamsplitter then maps the two possible distributions, Eq. (A.1)
and Eq. (A.1), via the beamsplitter dynamics of Eq. (III.1.1), back to the
distribution of Eq. (A.1) (depicted diagrammatically in Eq. (27)) or to the
distribution

$\displaystyle p(N_{L},\Phi_{L},N_{R},\Phi_{R})$
$\displaystyle=[0]\left(\tfrac{1}{2}[0]+\tfrac{1}{2}[1]\right)[1]\left(\tfrac{1}{2}[0]+\tfrac{1}{2}[1]\right),$
(39)

which is depicted diagrammatically as:

$\displaystyle\centering\includegraphics[width=138.76157pt]{EpistemicStateROccupied.pdf}.\@add@centering$
(40)

These two cases correspond respectively to the $L$ mode or the $R$ mode being
occupied at the output of the second beamsplitter.

A diagrammatic account of the full evolutions of the probability distribution
for no phase shift and for a $\pi$ phase shift are summarized in the first two
rows of Fig. 4.

Figure 4:  The toy field theory account of the Mach-Zehnder interferometer
experiment with a phase shifter in arm $R$, implementing no phase shift or a
$\pi$ phase shift, and with a detector in arm $R$. The evolution of the state
of knowledge over the physical states of the two modes is depicted in each
case. We also provide the quantum evolution for direct comparison. Here,
$U_{\rm BS}$ denotes the unitary associated to the beamsplitter action,
described in Eq. (II.1), and $Z$ denotes the unitary associated to a $\pi$
phase shift, described in Eq. (3).

The detectors in the output ports of the Mach-Zehnder interferometer implement
a measurement of the occupation numbers of the $L$ and $R$ modes. In the toy
field theory, such measurements simply reveal the possessed values of $N_{L}$
and $N_{R}$, so we recover the prediction that when there is no phase shift,
it is the detector in the $L$ output port that always fires, while in the case
of a $\pi$ phase shift, it is the detector in the $R$ output port that always
fires.

The case of the Mach-Zehnder interferometer with which-way measurement. We now
turn to the case where there is a detector in the $R$ arm of the Mach-Zehnder
interferometer, as in Fig. 1b).

We begin by providing a formal account of the measurement update rule which
was described in subsection III.1.

If the initial probability distribution is $p(N,\Phi)$ and one obtains the
$N=0$ outcome in the measurement of occupation number (note that this can only
occur if $p(N,\Phi)$ assigns a nonzero probability to $N=0$), then the update
rule can be understood as consisting of two steps. Step 1 is the updating of
knowledge implied by the fact that learning the outcome allows one to resolve
some of one’s uncertainty regarding the physical state:

$\displaystyle p(N,\Phi)\mapsto
p^{\prime}(N,\Phi)=\frac{1}{\mathcal{N}}\delta_{N,0}p(N,\Phi)$ (41)

where $\mathcal{N}$ is a normalization constant that ensures that
$p^{\prime}(N,\Phi)$ is a normalized probability distribution. We depict this
diagrammatically as

$\displaystyle\centering\includegraphics[width=26.01613pt]{UpdateRuleSingleMode1.pdf},\@add@centering$
(42)

where the cross symbol on a physical state represents the fact that it has
been ruled out by the measurement and so its probability is set to zero. Step
2 is the updating of knowledge that is implied by the fact that the phase is
randomized,

$\displaystyle p^{\prime}(N,\Phi)\mapsto
p^{\prime\prime}(N,\Phi)\equiv\tfrac{1}{2}p^{\prime}(N,\Phi)+\tfrac{1}{2}p^{\prime}(N,\Phi\oplus
1),$ (43)

which we depict diagrammatically as

$\displaystyle\centering\includegraphics[width=82.38885pt]{UpdateRuleSingleMode2.pdf}.\@add@centering$
(44)

The toy field theory analogue of the quantum state update rule described in
Eq. (11) is obtained by composing these two steps:

$\displaystyle
p(N,\Phi)\mapsto[0]\left(\tfrac{1}{2}[0]+\tfrac{1}{2}[1]\right),$ (45)

which stipulates that regardless of the initial distribution $p(N,\Phi)$, the
final distribution is the one wherein $N$ is known to be 0 and $\Phi$ is
unknown.

Note that the measurement update rule implies that even if one has perfect
knowledge of the phase at the input of the measurement, one can infer nothing
about the phase at the output of the measurement. Also, even if one has
perfect knowledge of the phase at the output of the measurement (by knowing
the outcome of a subsequent measurement of phase for instance), then one can
infer nothing about the phase at the input of the measurement.

We now consider how this impacts a probability distribution over the physical
states for the pair of modes.

Again, consider a measurement of occupation number on mode $R$ that yields the
outcome 0. The probability distribution $p(N_{L},\Phi_{L},N_{R},\Phi_{R})$, is
then updated in two steps as before. The knowledge-updating step associated to
resolving uncertainty is:

$\displaystyle p(N_{L},\Phi_{L},N_{R},\Phi_{R})\mapsto$ $\displaystyle
p^{\prime}(N_{L},\Phi_{L},N_{R},\Phi_{R})$
$\displaystyle\equiv\frac{1}{\mathcal{N}}\delta_{N_{R},0}p(N_{L},\Phi_{L},N_{R},\Phi_{R}),$
(46)

where $\mathcal{N}$ is a normalization constant. It is depicted
diagrammatically as

$\displaystyle\centering\includegraphics[width=82.38885pt]{UpdateRuleTwoModes1.pdf}.\@add@centering$
(47)

The knowledge-updating step associated to phase randomization is

$\displaystyle p^{\prime}(N_{L},\Phi_{L},N_{R},\Phi_{R})\mapsto$
$\displaystyle\tfrac{1}{2}p^{\prime}(N_{L},\Phi_{L},N_{R},\Phi_{R})$
$\displaystyle+\tfrac{1}{2}p^{\prime}(N_{L},\Phi_{L},N_{R},\Phi_{R}\oplus 1),$
(48)

which is depicted diagrammatically as

$\displaystyle\centering\includegraphics[width=173.44534pt]{UpdateRuleTwoModes2.pdf}.\@add@centering$
(49)

The toy field theory analogue of the quantum state update rule, Eq. (12), is
obtained by composing these two steps,

$\displaystyle p(N_{L},\Phi_{L},N_{R},\Phi_{R})\mapsto
p^{\prime}(N_{L},\Phi_{L})[0]\left(\tfrac{1}{2}[0]+\tfrac{1}{2}[1]\right),$
(50)

where $p^{\prime}(N_{L},\Phi_{L})$ is the marginal over $N_{L}$ and $\Phi_{L}$
of $p^{\prime}(N_{L},\Phi_{L},N_{R},\Phi_{R})$.

We can now complete the analysis of the Mach-Zehnder inteferometer with a
detector in the $R$ arm. Consider the case where the detector does not fire.
We first apply the knowledge-updating step associated to resolving uncertainty
by virtue of learning that $N_{R}=0$, that is Eq. (A.1), to the distribution
in Eq. (A.1), obtaining

$\displaystyle
p(N_{L},N_{R},\Phi_{L},\Phi_{R})=[1][0]\left(\tfrac{1}{2}[0][1]+\tfrac{1}{2}[1][0]\right),$
(51)

which is depicted diagramatically as

$\displaystyle\centering\includegraphics[width=134.42113pt]{epistemicstateBayesianupdate.pdf}.\@add@centering$
(52)

We then apply the knowledge-updating step associated to randomization of the
local phase $\Phi_{R}$, that is, Eq. (A.1), to this distribution to obtain
(notice that we’ve reordered the variables)

$\displaystyle
p(N_{L},\Phi_{L},N_{R},\Phi_{R})=[1]\left(\tfrac{1}{2}[0]+\tfrac{1}{2}[1]\right)[0]\left(\tfrac{1}{2}[0]+\tfrac{1}{2}[1]\right),$
(53)

which is the same distribution as in Eq. (A.1), depicted diagrammatically in
Eq. (27).

A diagrammatic account of the full evolution of the probability distribution
when there is a which-way detector on arm $R$ which does not fire is given in
the third row of Fig. 4.

The case where the detector does fire, so that one implements the resolving of
uncertainty appropriate for learning that $N_{R}=1$ (rather than $N_{R}=0$) is
treated similarly.

###  A.2 The lesson of the toy field theory for the difference between first-
quantized and second-quantized descriptions

In Sec. III.3, we described an alternative to the toy field theory.
Specifically, we described a toy theory that stands to the first-quantized
description of the Mach-Zehnder interferometer as the toy field theory stands
to its second-quantized description.

This theory takes the system of interest to be a photon, and presumed that it
has two properties: a discrete which-way property describing whether the
photon is in the left or right arm, $W\in\\{L,R\\}$, and a discrete momentum
that is canonically conjugate to this which-way property,
$\Theta\in\\{0,1\\}$.

The physical state space can be depicted as a $2\times 2$ grid of boxes, and a
particular physical state as a marked box. Thus, for instance, the physical
state wherein $W=L$ and $\Theta=0$ is depicted as:

$\displaystyle\centering{\includegraphics[width=86.72267pt]{ToyStateSpaceFirstQuantized}}.\@add@centering$
(54)

The probability distributions $p(W,\Theta)$ that satisfy the epistemic
restriction are then denoted diagrammatically in the obvious way, by filling
in two of the four boxes.

In Sec. III.3, we noted that the translation from the toy field theory to the
first-quantized toy theory is given by the map of Eq. (III.3). This can be
represented diagrammatically as follows:

$\displaystyle\centering{\includegraphics[width=195.12767pt]{TranslationToy}}.\@add@centering$
(55)

This allows one to translate the second-quantized toy theory description of
evolution through the Mach-Zehnder interferometer, presented diagrammatically
in Fig. 4, into the first-quantized toy theory description of this evolution,
presented diagrammatically in Fig. 5.

Figure 5:  A first-quantized description of quantum interference phenomena and
how these are reproduced using the coarse-grained variables of the first-
quantized toy theory. At each time step, the state of knowledge over the
coarse-grained which-way variable $W$ and relative phase variable $\Theta$ is
depicted just above the corresponding quantum state. For the standard
interferometer (with and without the phase shift), the photon always ends up
in the same output port ($L$ and $R$, respectively). In the case where a
which-way measurement is carried out, the photon is equally likely to appear
in either output port. We also provide the quantum evolution for comparison.
Here, $H$ denotes the unitary associated to the beamsplitter action, described
in Eq. (II.1), and $Z$ denotes the unitary associated to a $\pi$ phase shift,
described in Eq. (3).

###  A.3 Formal toy-field-theoretic account of the quantum eraser experiment

In Sec. IV.3, we described the toy field theory account of the quantum eraser
experiment. Here, we present this account a bit more formally

The joint probability distribution of the pair of modes at the output ports of
the first beamsplitter is the one described in Eq. (A.1). The auxiliary system
$A$ is presumed to be prepared such that $Q_{A}$ is known, while $P_{A}$ is
unknown, described by the distribution
$p(Q_{A},P_{A})=[a_{0}](\tfrac{1}{2}[a_{+}]+\tfrac{1}{2}[a_{-}])$. It follows
that the joint probability distribution for the pair of modes together with
auxiliary system $A$ is

$\displaystyle p(N_{L},N_{R},Q_{A},\Phi_{L},\Phi_{R},P_{A})$
$\displaystyle=\left(\tfrac{1}{2}[0][1]+\tfrac{1}{2}[1][0]\right)[a_{0}]$
$\displaystyle\times\left(\tfrac{1}{2}[0][1]+\tfrac{1}{2}[1][0]\right)(\tfrac{1}{2}[a_{+}]+\tfrac{1}{2}[a_{-}]).$
(56)

The CNOT dynamics, described in Eq. (22), maps this to

$\displaystyle p(N_{L},N_{R},Q_{A},\Phi_{L},\Phi_{R},P_{A})$
$\displaystyle=\left(\tfrac{1}{2}[0][1][a_{1}]+\tfrac{1}{2}[1][0][a_{0}]\right)$
$\displaystyle\times\tfrac{1}{2}\left\\{\left(\tfrac{1}{2}[0][1]+\tfrac{1}{2}[1][0]\right)[a_{+}]+\left(\tfrac{1}{2}[0][0]+\tfrac{1}{2}[1][1]\right)[a_{-}]\right\\}.$
(57)

By considering the marginal $p(N_{R},Q_{A})$ of Eq. (A.3), it is clear that a
measurement of $Q_{A}$ allows one to infer the value of $N_{R}$, and so
constitutes an indirect measurement of $N_{R}$. Similarly, by considering the
marginal $p(\Phi_{L},\Phi_{R},P_{A})$ of Eq. (A.3), it is clear that a
measurement of $P_{A}$ allows one to infer the value of
$\Phi_{L}\oplus\Phi_{R}$ at the end of the measurement.

The argument for the reproduction of the phenomenology of the quantum eraser
then proceeds as follows. If one sorts the data by whether a measurement of
$Q_{A}$ yields $a_{0}$ or $a_{1}$, then one is sorting by whether $N_{R}$ has
value 0 or 1 and consequently in each subset one sees no interference. On the
other hand, if one sorts the data by whether a measurement of $P_{A}$ yields
$a_{+}$ or $a_{-}$, then one is sorting by whether $\Phi_{L}\oplus\Phi_{R}$
has value 0 or 1, and consequently in each subset one sees interference.

It is worth noting that one can also understand these predictions by means of
a graphical analysis. In order for the physical states of each of the three
systems (mode $L$, mode $R$, and auxiliary system $A$) to be represented on
different axes, the graphical representation of the joint distribution must be
three-dimensional. This is achievable but it is less intuitive to work with
than its two-dimensional counterpart and does not provide much additional
insight relative to the analysis just given, so we do not provide such an
analysis here.

##  Appendix B Contrasting the toy field theory with other realist accounts of
quantum interference

For the first two interpretational claims outlined in the introduction, there
are other ways of seeing that these are not forced on us by the phenomenology
of the Mach-Zehnder interferometer and its variants.

This is especially true with regards to the claim of the observer-dependence
of reality. Almost all researchers motivated to work on realist
interpretations of quantum theory are keen to eliminate any appeal to “psycho-
kinetic effects”. (The only prominent exceptions are certain proposals for
dynamical collapse models where the consciousness of the observer is
implicated in the mechanism that leads to collapse Wigner (1995); London and
Bauer (2014).)

Some realist interpretations also do away with the notion of wave-particle
complementarity. Bohmian mechanics Bohm (1952a), for instance, provides an
account of interference phenomenology where there is a particle and a wave
Philippidis  _et al._ (1979). Note, however, that only if the system of
interest is a single particle is it the case that the wave is defined over
familiar 3-dimensional physical space. For two particles, for instance, the
wave is defined on the 6-dimensional space of spatial configurations of the
pair (specifying the 3 spatial coordinates of each particle). Other
$\psi$-ontic models Harrigan and Spekkens (2010), such as the one proposed
recently by Blasiak Blasiak (2015), can also be understood as providing an
account of the single-particle phenomenology in terms of a particle and a wave
in physical space. Like Bohmian mechanics, such an account is not possible for
two or more particles.

Moreover, even in the case of a single particle, we noted earlier (in footnote
23) that having a particle and a wave is quite different from the way in which
wave-particle complementarity is rejected in the toy field theory, where there
are modes that have particle-like and wave-like properties which are
canonically conjugate to one another.

It is with the third interpretational claim, stating that the TRAP
phenomenology implies the failure of explainability in terms of local causes,
that the account provided by the toy field theory differs most from the other
accounts that deny the first two interpretational claims. The latter accounts
generally do require a rejection of the possibility of local causal
explanations for one or more of the interference phenomena discussed in this
article.

Consider in particular the quantum eraser experiment. The fact that the toy
field theory can account for the quantum eraser without nonlocality
demonstrates that nonlocality is not required to explain this phenomenology.
However, every $\psi$-ontic model, in particular, Bohmian mechanics, dynamical
collapse theories Ghirardi  _et al._ (1986); Pearle (1989); Bassi  _et al._
(2013), Blasiak’s model, etcetera, do require nonlocality to account for it.
This follows from the fact that the quantum eraser is simply a version of the
Einstein-Podolsky-Rosen (EPR) experiment Einstein  _et al._ (1935b) and every
$\psi$-ontic model is incapable of providing a local account of the
correlations obtained in the EPR experiment Harrigan and Spekkens (2010). By
contrast, $\psi$-epistemic models (such as the toy field theory and other
classical statistical theories with an epistemic restriction Spekkens (2007,
2016a)) are capable of doing so. This is possible because the correlations in
the EPR experiment (and those in the quantum eraser experiment) do not violate
any Bell inequalities. In other words, the reason that there are experimental
scenarios for which a local account is possible in the toy field theory but
not in any $\psi$-ontic model is that the latter sort of model posits strictly
more nonlocality than is required by Bell’s theorem.

It is worth noting a subtlety here. From the Bohmian perspective, the case of
the basic Mach-Zehnder interferometer is special among interference
experiments insofar as the Bohmian account in this case does not involve any
nonlocality. In the Bohmian account, the wavefunction of the photon branches,
with one branch of the wavefunction propagating through the $L$ arm and the
other through the $R$ arm. Meanwhile, the Bohmian particle takes one
particular arm, falling in the support of only one of these two branches of
the wavefunction. The branch of the wavefunction (or ‘wave’) containing the
Bohmian particle is termed ‘occupied’, while the other is termed ‘empty’.
Finally, a detector can have a local influence on the dynamics of the empty
wave, which in turn has an influence on the particle at the second
beamsplitter. In this sense, the empty wave plays a role analogous to the
phase of the unoccupied mode in the toy field theory. The analogy between
unoccupied modes and empty waves breaks down, however, in the quantum eraser
experiment. This is because the latter experiment involves not just the
photon, but the auxiliary 2-level system. The fact that these interact via a
CNOT gate implies that the pilot wave in the Bohmian account is the
wavefunction of this pair of systems, which unlike the wavefunction for the
photon cannot be understood as a wave over 3-dimensional physical space. Via
the dynamics of this wavefunction, the choice of what measurement to make on
the auxiliary system comes to determine the dynamics of the Bohmian particle
associated to the photon in a nonlocal fashion.

The model of Blasiak Blasiak (2015) is analogous to Bohmian mechanics insofar
as it avoids nonlocality in the case of the basic Mach-Zehnder interferometer,
but cannot provide a local account of the quantum eraser experiment.

There is another nuance worth noting concerning the possibility of securing
local causal explanations of interference phenomena in Bohmian mechanics.
Whether a given interference phenomena can be explained locally or not
actually depends on whether one is pursuing the Bohmian interpretational
methodology to quantum theory in its first-quantized description (which yields
standard Bohmian mechanics) or in its second-quantized description (which
yields a Bohmian version of quantum field theory Bohm (1952a); Struyve and
Westman (2007)). In the latter, rather than having a particle that has a
spatial location which is guided by a wave defined over physical space, one
has modes that have local field amplitudes which are guided by a wave defined
over the joint configuration space of those field amplitudes. The guiding wave
again fails to be a wave in 3-dimensional physical space and one can verify
that this sort of Bohmian account requires nonlocality in its account of the
phenomenology of even the basic Mach-Zehnder experiment. Unlike Bohmian
mechanics, the toy field theory manages to provide a local causal account of
the phenomenology of the Mach-Zehnder interferometer whether one uses the
first-quantized or second-quantized formalisms. The ontological account of the
experiment provided in the first-quantized toy theory is merely a coarse-
graining of the description of the experiment provided in the second-quantized
toy theory (the toy field theory). In the Bohmian approach, on the other hand,
the nature of the ontology and the explanation is completely different
depending on whether one develops a pilot-wave interpretation at the first-
quantized or second-quantized levels.

Aharonov et al. Aharonov  _et al._ (2017) also provide an account of
interference that does not endorse wave-particle complementarity but does
endorse nonlocality. In their account, a particle has a definite location
(therefore lies on one arm of the interferometer) but it also has a property
of modular momentum and the dynamics of the latter can depend on the situation
on the other arm of the interferometer, and is therefore nonlocal.

It is also useful to consider the extent to which the Everett interpretation
of quantum theory Everett (1957)(also known as the ‘many-worlds
interpretation’) satisfies or violates the different interpretational claims.
Whether wave-particle complementarity holds or not seems to us to depend on
which version of the Everett interpretation one endorses.

The Everett interpretation assumes that the wavefunction is an exhaustive
description of the ontology. In this sense, it is like Bohmian mechanics, but
without the Bohmian particle positions. The wavefunction is thought to define
branches, like in Bohmian mechanics, but one does not distinguish some
branches as actual and others as ‘empty’ based on the positions of the Bohmian
particles. Rather, one considers all branches to be simultaneously actual, but
to be describing parallel worlds. The way to define the branches (which in the
jargon of Everettians corresponds to solving the “preferred basis problem”) is
to select a preferred basis as one relative to which useful patterns arise
Wallace (2012), and this is typically picked out by decoherence. Relative to
this view, one could argue that in the Mach-Zehnder interferometer with the
detector in place, the preferred basis is such that the which-way property is
well-defined, whereas in the Mach-Zehnder interferometer without the detector,
the preferred basis is such that the relative phase property is well-defined.
In other words, if the properties obtaining in a given branch are just those
observables for which the wavefunction of that branch is an eigenstate, then
this version of the Everettian interpretation would seem to endorse wave-
particle complementarity.

However, the description of the Everett interpretation provided by Deutsch in
Ref. Deutsch (1998) suggests an ontology of particles alone, but wherein
particles can interact with their counterparts in parallel worlds. (In this
sense, it seems analogous to the “many interacting worlds” view of Ref. Hall
_et al._ (2014).) This version would seem to not endorse wave-particle
complementarity.

In either case, it is generally argued that the Everett interpretation
restores the possibility of local explanations. Indeed, it achieves locality
in a similar way to how, in Penrose’s account of the Elitzur-Vaidman bomb
tester, one can imagine restoring locality by allowing the counterfactual
world to influence the actual world. The difference is that in the Everettian
approach, all the worlds are actual.

The achievement of the toy field theory that is certainly not reproduced by
any version of the Everett interpretation is to account for the TRAP
phenomenology of interference while maintaining the classical worldview. In
other words, what we have shown here is that even if one grants that the
Everett interpretation salvages locality for the TRAP phenomenology, the price
it pays to do so—a radical deviation from a classical worldview— is not
necessitated by the phenomena.

Finally, we note that among all of the realist accounts of the TRAP
phenomenology of interference, only the one provided by the toy field theory
manages to preserve the classical paradigm of kinematics and dynamics (sets
for kinematics and functions for dynamics) while also preserving the principle
of Leibnizianity (described in Sec. V.1.3). This is because the alternative
realist accounts are all $\psi$-ontic, and it is known that every $\psi$-ontic
model violates the principle of generalized noncontextuality and thus also the
principle of Leibnizianity Spekkens (2005). In Sec. V.1.3, we noted that our
preferred notion of classicality includes assuming both the classical notions
of kinematics and dynamics as well as the principle of Leibnizianity, and so
only the account provided by the toy field theory manages to salvage our
preferred notion of classicality.

##  Appendix C Further elaborations

###  C.1 Probabilities as credences

In the epistemically restricted classical statistical theories of Refs.
Spekkens (2007); Bartlett  _et al._ (2012); Spekkens (2016a) and in the one
presented in this article (i.e., the toy field theory), probabilities are
conceptualized as quantifying the degrees of belief of an agent, that is, they
are conceptualized as credences. It is sometimes suggested that this means
that these theories are not really tested by the observation of relative
frequencies in the data. We here respond to this claim.

First, we note that it is unavoidable that there is some degree of uncertainty
about the parameters in a physical model. Therefore, the view that a theory
cannot be confirmed if it includes descriptions of an agent’s degrees of
beliefs implies a rejection of almost all scientific hypothesis-testing. In
other words, such a view about theory confirmation is clearly too extreme.

We therefore consider a less extreme view, namely, that assigning equal
credence to one measurement outcome or another does not imply that one should
believe that in an arbitrarily large number of repetitions of the experiment,
the relative frequency is likely to be one half.

The critic might argue as follows. To view probability assignments in these
theories as credences implies that they are viewed as degrees of belief about
a single system in a sequence of repetitions of an experiment, and such a
belief does not imply anything about the relative frequencies one expects to
see in the full sequence of repetitions. The following is a more formal
version of the claim. Let ${\bf p}$ be the probability distribution assigned
in a single run, say the $k$th run. For concreteness, imagine a sequence of
coin flips, so that ${\bf p}=(p_{0},p_{1})$ where $p_{0}$ is the probability
of heads and $p_{1}$ is the probability of tails. The critic’s view can be
formalized as the suggestion that the distribution ${\bf p}$ merely describes
the marginal on the $k$th system in the sequence (i.e., it arises from the
full distribution over all elements of the sequence by marginalization of all
but the $k$th system). Since the marginal on the $k$th system in a sequence
does not constrain the marginals for any other system, it is indeed the case
that, under such an interpretation of ${\bf p}$, nothing is implied about
which relative frequencies in the sequence are likely to be observed.

Our response is as follows. The view just articulated is not the
interpretation of the distribution ${\bf p}$ that is intended in our work.
Rather, we stipulate that the systems in the sequence of experiments are
identical and independently distributed (i.i.d.), so that to stipulate a
distribution ${\bf p}$ is to stipulate that the i.i.d. distribution is of the
form ${\bf p}\otimes{\bf p}\cdots\otimes{\bf p}$. In this case, the form of
${\bf p}$ does determine which relative frequencies in a sequence are most
likely. The law of large numbers implies that in the limit of an arbitrarily
large ensemble, the most likely relative frequencies are given by ${\bf p}$.
For instance, if an agent assigns equal credence to heads and tails coming up
for some coin and believes that a string of coin flips are i.i.d., then a run
of all heads is considered much less likely than one where the relative
frequency of heads is 1/2.

The observed relative frequencies can make an agent reject their hypothesized
probability distribution in this circumstance. The question one must ask is
simply whether the probability distribution hypothesized in the i.i.d. model
makes the observed relative frequencies likely or not. One can assess this,
for instance, by whether the hypothesis has a good $p$-value for the observed
finite-run relative frequencies.

Another claim one sometimes hears is that one must interpret probabilities as
objective chances rather than credences if one wishes to test these against
the observed data. For instance, a critic might say that 100 tosses of a coin
do serve to test the hypothesis of a certain weighting of the coin, where the
weighting is considered as a property of the coin that determines the
objective chance for each outcome, but they do not serve to test a hypothesis
about an agent’s credences regarding the coin.

Our response to this claim is the standard one that is made in favour of
interpretations of probability as credences, which appeals to the de Finetti
representation theorem De Finetti (2017); Caves  _et al._ (2002). de Finetti
showed that if an agent takes an arbitrarily large sequence of coin flips to
be exchangeable—i.e., the agent’s credences are invariant under any
permutation of elements of the sequence—then the agent’s credences about the
sequence must have the form $\int_{{\bf p}\in\Delta}\;{\bf p}\otimes{\bf
p}\cdots\otimes{\bf p}\;{\rm d}\mu({\bf p})$ where $\Delta$ is the simplex of
possibilities for the distribution ${\bf p}$ and $\mu$ is a probability
measure over $\Delta$.

The key thing to note about the form of the probability distribution on the
full sequence implied by the assumption of exchangeability is that it is
exactly the same as what one would write down if one assumed that ${\bf p}$
described objective chances, characterizing properties of the coin, and that
$\mu$ described the agent’s ignorance of these properties. The fact that the
same probability vector ${\bf p}$ is assigned to each coin flip in a sequence
does not, therefore, imply that there is some property of the coin (possibly
unknown) that ${\bf p}$ describes.

Note that in the i.i.d. scenario, the agent’s beliefs about the full sequence
of runs is ${\bf p}\otimes{\bf p}\cdots\otimes{\bf p}$, which is simply a
distribution of de Finetti form where the measure $\mu$ is a Dirac-delta
measure centred on ${\bf p}$. It corresponds to an exchangeable sequence
wherein there are no correlations among the different runs. Again, it is of
precisely the same form as the distribution one would assign if ${\bf p}$
described some property of the coin for which there was no uncertainty.

Figure 6:  A depiction of how the physical states describing the different
subensembles of the full ensemble of experimental runs transform in the Mach-
Zehnder interferometer with a phase-shifter in place. The first row of this
figure is the counterpart of the top row of the “phase shifter in place” row
of Fig. 4, while the second row is the counterpart of the bottom row of the
latter.

For an i.i.d. distrubtion ${\bf p}\otimes{\bf p}\cdots\otimes{\bf p}$, the law
of large numbers implies that in the limit of an infinitely large ensemble,
the relative frequencies converge to ${\bf p}$ with high likelihood.
Consequently, one can associate the distribution ${\bf p}$ to the likely
relative frequencies in this ensemble rather than to credences for single
systems in an i.i.d. sequence.

Although the concrete example we have used above is a sequence of coin flips,
the same can be said of probability distributions over the physical states of
a system in a sequence of repetitions of the same experiment. In the next
section, we show how to provide a description of the toy field theory account
of the TRAP phenomenology of the Mach-Zehnder interferometer in terms of an
ensemble of repetitions of the experiment.

###  C.2 Reframing our account in terms of an ensemble of repetitions of the
experiment

The discussion in the previous section implies that although we have up until
now conceptualized the probability distributions in the toy field theory as
states of incomplete knowledge about the physical state of a system in a
single run of an i.i.d. experiment, one can also conceptualize these
distributions as describing the likely relative frequencies of different
physical states in an infinite ensemble of runs of the experiment. Reframing
the predictions of the toy field theory in terms of relative frequencies can
be useful insofar as it provides a slightly different perspective on the
features of the theory. We will focus, in particular, on how our analysis of
locality appears in this reframing.

To begin with, consider the ensemble corresponding to the probability
distribution describing the pair of modes prior to the first beamsplitter,
depicted in the first column of Fig. 4 (this is the analogue of the initial
quantum state $\left|10\right>$). This ensemble includes four subensembles of
equal size wherein $N_{L}=1$ and $N_{R}=0$ in each of these while the local
phases take every possible pair of values: $(\Phi_{L},\Phi_{R})=(0,0)$ in one
subensemble, $(\Phi_{L},\Phi_{R})=(1,0)$ in another,
$(\Phi_{L},\Phi_{R})=(0,1)$ in another, and $(\Phi_{L},\Phi_{R})=(1,1)$ in the
fourth. In other words, the four subsensembles are associated to the four
physical states
$(N_{L},\Phi_{L},N_{R},\Phi_{R})=(1,0,0,0),(1,1,0,0),(1,0,0,1)$, and
$(1,1,0,1)$, which are depicted diagramatically in the first column of the
first four rows in the top half of Fig. 6. We have included a light blue
outline around the set of four physical states in the depictions of each
physical state to remind the reader of the full ensemble of which each
subensemble is a part.

Note that this decomposition of the ensemble into four subensembles occurs
also in the bottom half of Fig. 6 and in Fig. 7, since all three versions of
the Mach-Zehnder experiment start with the same ensemble.

For each of the four subsensembles, the first beamsplitter transforms the
associated physical state of the pair of modes according to the beamsplitter
transformation (which is depicted in (29) and repeated in Fig. 6). For each
physical state of the pair of modes at the input to the first beamsplitter
(depicted in the four rows of Fig. 6), its image under the beamsplitter
transformation is depicted in the second column of the same row. It follows
that the ensemble associated to the pair of modes at the output of the first
beamsplitter comprises four subensembles associated to the four physical
states depicted in that column. We have again indicated with a blue outline
this set of four physical states, thereby making evident the association
between this ensemble and the probability distribution depicted in the second
column of Fig. 4.

How the physical states associated to each of the four subensembles within the
full ensemble are transformed under the two possible phase shifts—no phase
shift or a $\pi$ phase shift—is depicted in the top and bottom halves of Fig.
6 respectively. In both cases, the set of four physical states associated to
the ensemble after the phase-shifter is depicted with a blue outline. In the
case of no phase shift, it corresponds to the probability distribution
depicted in the third column of the first row of Fig. 4, while in the case of
a $\pi$ phase shift, it corresponds to the one depicted in the third column of
the second row of Fig. 4.

Finally, within each of these two ensembles, how the physical states
associated to each of the four subensembles are transformed under the second
beamsplitter is also depicted in Fig. 6. Again, in both cases, the set of four
physical states associated to the final ensemble is depicted with a blue
outline and correspond to the probability distributions in the final column of
the first two rows of Fig. 4.

Now we turn to the case where a measurement of occupation number occurs in the
$R$ arm of the Mach-Zehnder interferometer.

Figure 7:  A depiction of how the physical states describing the different
subensembles of the full ensemble of experimental runs transform in the Mach-
Zehnder interferometer with a measurement of occupation number occuring in the
$R$ arm. The parts of this figure describing the two possible outcomes,
$N_{R}=0$ and $N_{R}=1$, are the counterparts of the two rows in the “Which-
way detector in place” part of Fig. 4.

Suppose one seeks to make predictions about what will occur at later times in
the experiment given that one has observed a particular outcome. Conditioning
on an outcome corresponds to updating the ensemble that is relevant for making
predictions. This updating can be understood as consisting of two steps.
First, one selects from the pre-existing ensemble the subensemble that is
consistent with the outcome that was observed. Second, the fact that a
measurement leads to a random disturbance—that is, one of several different
transformations to the physical state—implies that elements of the ensemble
selected in the first step get split into distinct elements (bifurcated in the
case of interest here), thereby leading to an increase in the number of
distinct subensembles.

Consider, in particular, the case wherein one conditions on obtaining the
$N_{R}=0$ outcome in the measurement of occupation number of the $R$ arm.

Recall that, after the first beam splitter, the ensemble of runs consists of
four subensembles, associated to physical states
$(N_{L},\Phi_{L},N_{R},\Phi_{R})=(1,1,0,0),(1,0,0,1),(0,1,1,0),(0,0,1,1)$
(depicted in the second column of Fig. 7). The first step of the ensemble
update associated to conditioning on $N_{R}=0$ is the selection step,
consisting of redefining the ensemble to include only the two subensembles
that are consistent with $N_{R}=0$, namely, the ones associated to the
physical states $(N_{L},\Phi_{L},N_{R},\Phi_{R})=(1,1,0,0),(1,0,0,1)$, which
are the top two cases in the second column of Fig. 7 (i.e., the two rows that
are labelled “outcome $N_{R}=0$”). This is the ensemble counterpart of the
knowledge-updating step associated to resolving uncertainty, depicted in Eq.
47, such that, prior to taking into account the disturbance to the physical
state, the ensemble is updated to the one associated to the probability
distribution depicted in (52).

With this description of the selection step of the ensemble update procedure
in hand, we can now explain why it does not involve any nonlocal influences.

When, based on the outcome of the measurement on the $R$ mode, one selects the
subensemble of physical states of the pair of modes that is consistent with
this outcome, it can happen that not only does the selection procedure exclude
certain physical states of the $R$ mode from the subensemble, it excludes
certain physical states of the $L$ mode as well. This is because the original
ensemble prior to the selection (the one describing the pair of modes at the
outputs of the first beamsplitter) embodies correlations between the
properties of the $L$ mode and those of the $R$ mode. Indeed, in the example
we are considering, where one selects the subensemble of physical states of
the pair of modes consistent with $N_{R}=0$, the fact that the original
ensemble has $N_{L}$ and $N_{R}$ taking opposite values ($N_{L}\oplus
N_{R}=1$) implies that by excluding from the subensemble all physical states
of the pair of modes having $N_{R}=1$, one thereby also excludes from the
subensemble all physical states of the pair of modes having $N_{L}=0$. The key
point to note is that this reduction in the set of properties of the $L$ mode
that are present in the selected subensemble (i.e., the elimination of any
physical states with $N_{L}=0$) is induced by the selection on the properties
of the $R$ mode and consequently does not imply a nonlocal influence from the
$R$ arm to the $L$ arm.

We now turn to the second step of the ensemble update procedure, the
bifurcation step, and explain why it also does not involve any nonlocal
influences.

As noted above, the subensemble after the selection process consists of two
physical states for the pair of modes. While both of these physical states
have the same occupation numbers for the modes, namely, $N_{L}=1$ and
$N_{R}=0$, they have different phases for the modes. Specifically, one has
$\Phi_{L}=0$ and $\Phi_{R}=1$ and the other has $\Phi_{L}=1$ and $\Phi_{R}=0$.
These two physical states are depicted in the second column of Fig. 7 in the
first and second rows respectively (i.e., the two rows that are labelled
“outcome $N_{R}=0$”). The fact that the phase randomization associated to the
measurement of occupation number is presumed to be local to the $R$ mode
implies that $\Phi_{R}$ is randomized while $\Phi_{L}$ remains undisturbed. It
follows that the element of the pre-randomization ensemble where the physical
state has $\Phi_{L}=1$ and $\Phi_{R}=0$ (the first row in Fig. 7) is mapped to
a pair of elements in the post-randomization ensemble, where both still have
$\Phi_{L}=1$, but where now one has $\Phi_{R}=1$ and the other has
$\Phi_{R}=0$. This bifurcation is depicted in the third column of the first
row of Fig. 7. Similarly, the other element of the pre-randomization ensemble,
where the physical state has $\Phi_{L}=0$ and $\Phi_{R}=1$ (the second row in
Fig. 7), is also mapped to a pair of elements in the post-randomization
ensemble. In this case, both still have $\Phi_{L}=0$, while one has
$\Phi_{R}=1$ and the other has $\Phi_{R}=0$. This bifurcation is depicted in
the third column of the second row of Fig. 7. The key point to note here is
that in both cases, the measurement on the $R$ mode has no effect on the
physical properties of the $L$ mode. Graphically, this is seen by the fact
that along the vertical axis of the discrete physical state space, which is
the axis that describes properties of the $L$ mode, the coordinate of the
physical state is unchanged. It is only along the horizontal axis of the
discrete physical state space, which is the axis that describes properties of
the $R$ mode, that the coordinate of the physical state changes.

The rest of the story regarding the evolution through the Mach-Zehnder
interferometer is relatively straightforward. We provide the details for
completeness.

The ensemble we are left with after these two steps of the updating procedure
is back to being comprised of four subensembles. These are associated to the
four physical states depicted in the third column of Fig. 7, which we have
indicated with a green outline. Clearly, this ensemble corresponds to the
probability distribution depicted in the third column of the third row of Fig.
4.

Fig. 7 also depicts how the physical states associated to each of the four
subensembles are transformed under the second beamsplitter. The ensemble
describing the output of the second beamsplitter is depicted in the final
column of Fig. 7, with the set of four physical states outlined in green. It
corresponds to the probability distribution depicted in the final column of
the third row of Fig. 4.

In the case wherein one obtains the outcome $N_{R}=1$ rather than $N_{R}=0$, a
similar analysis is possible: the ensembles describing the pair of modes
before and after the second beamsplitter are depicted with a red outline in
Fig. 7, and correspond to the probability distributions described in the third
and fourth columns of the fourth row of Fig. 4.

So we see that for both possible outcomes, we recover the TRAP phenomenology
of interference in the Mach-Zehnder interferometer without recourse to
nonlocal influences.

###  C.3 The case where the measurement is destructive

In the article thus far, we have considered only one kind of detector in
quantum theory, the type that implements a nondestructive measurement of the
occupation number of a mode. For these, the occupation number at its output
mode is presumed to be whatever the occupation number at the input mode was
found to be. But according to quantum theory, the TRAP phenomenology of
interference can be reproduced even if the detector implements a destructive
measurement of the occupation number, that is, one that absorbs the excitation
if it happens to be present, such that the occupation number of the input mode
may be found to be $1$, but the output mode is nonetheless left in a quantum
state with occupation number $0$.

Explicit in Feynman’s account (and to an even greater degree in Elitzur and
Vaidman’s account) is that the destruction of interference occurs even in the
case where the measurement does not detect the excitation on its arm. But in
this case, there is no difference in the quantum state update rule between
destructive and nondestructive measurements, as the output of the measurement
device is left in the quantum vacuum state in both cases. For this reason, the
distinction between destructive and nondestructive measurements is not
significant for discussions of the TRAP phenomenology. It follows that if one
can reproduce this phenomenology for either type of measurement in a classical
local model, one has undermined the claim that the phenomenology necessitates
a departure from the classical worldview. Hence, an explicit consideration of
destructive measurements is not required to establish our thesis.

Nonetheless, the question arises of whether the toy field theory can provide
an explicit local classical model for the case of a destructive measurement as
well. We here show that it can.

One way to implement a destructive measurement of occupation number on a given
arm of the interferometer is to place a photodetector there. (For a perfect
measurement of occupation number, one requires an ideal photodetector; it is
the latter that we have in mind for the purposes of this discussion.)

Another way to implement such a measurement on a given arm is to put a brick
or other absorbing medium there. Unlike a photodetector, the macroscopic
properties of a brick do not change in a noticeable way if it absorbs a single
excitation of the field mode. Nonetheless, because the microscopic properties
do register the absorption, the brick still implements a destructive
measurement of occupation number; it is just one for which the experimenter
does not learn the outcome. In this sense, having an arm blocked by a brick is
equivalent to having a photodetector in place and ignoring its outcome.
Moreover, the TRAP phenomenology concerns those cases where a detection is
registered at one of the output ports of the second beamsplitter, so that the
excitation is known not to be present in the $R$ arm, and for these cases it
is irrelevant whether it is a photodetector or a brick that implements the
destructive measurement.

We turn now to demonstrating how the toy field theory reproduces the TRAP
phenomenology of interference when the measurement of occupation number on the
$R$ arm is destructive.

First, we note explicitly why it is intuitively clear from the outset that the
toy field-theoretic model of a destructive measurement ought to be different
from its model of a nondestructive measurement in the second step of the
updating map, the step described as disturbance in the main text (see Eqs.
(43) and (44) and the surrounding discussion) and described as bifurcation in
the ensemble-reframing of Sec. C.2.

If $f$ denotes the function relating the phase of the output mode of a
measurement to the phase of its input mode, then one can describe the
disturbance/bifurcation step of the update map for a nondestructive
measurement, as follows: $f$ is sampled uniformly at random from the set
consisting of the identity function and the flip function and is applied to
the binary phase variable.

What would be the problem if a destructive measurement of occupation number
were also modelled in this fashion? The answer is that it is implausible that
such measurements implement the identity or flip functions on the phase. For
instance, if the measurement is implemented by placing a brick in an arm of
the interferometer, then it is implausible that the phase at the far side of
the brick (its “output port”) should be determined by the phase at the near
side of the brick (its “input port”).

It follows that the toy-field-theoretic model of a destructive measurement
should differ from that of a nondestructive measurement in such a way that an
implausible dependence of this sort is not required. We now show that this can
indeed be achieved, while still reproducing the TRAP phenomenology of
interference.

It is sufficient to take the disturbance/bifurcation step of the update rule
for a destructive measurement in the toy field theory to be defined as
follows. There are again two possibilities for the function $f$ that relates
the phase of the output mode of a measurement to the phase of its input mode.
But they are not the identity and flip functions. Rather, they correspond to
the other pair of functions from a binary variable to a binary variable (there
are only four such functions), namely, the function that is constant with
value 0 (the “reset-to-0” function) and the function that is constant with
value 1 (the “reset-to-1” function). As before, which of these functions is
applied is sampled uniformly at random.

Consider now what this implies for $\Phi_{R}$, the phase variable on the $R$
arm, at the output of the destructive measurement (for instance, the far side
of the brick). Because it is equally likely to be the image of the reset-to-0
map or the reset-to-1 map, it is simply sampled uniformly at random from the
values 0 and 1. What is key here is that its value is sampled independently of
its value at the measurement’s input (for instance, the near side of the
brick). There is consequently no dependence of the output phase on the input
phase. This harmonizes with the intuition that such a dependence would be
implausible for certain implementations of the destructive measurement, such
as placing a brick in the arm.414141It is useful to also consider the
occupation number on the $R$ arm, $N_{R}$, and the update rule for it. Recall
that in the case of a measurement that is nondestructive, $N_{R}$ at the
output is equal to $N_{R}$ at the input. By contrast, in the case of a
measurement that is destructive, $N_{R}$ at the output is always zero,
regardless of the value of $N_{R}$ at the input. So there is also no
dependence of the output occupation number on the input occupation number.

In summary, then, the disturbance/bifurcation step of the update map for a
destructive measurement is as follows: the physical state of the output mode
of the measurement is sampled uniformly at random from the pair of physical
states $(N,\Phi)=(0,0)$ and $(N,\Phi)=(0,1)$ completely independently of the
physical state of the input mode.

Finally, we can explain why, in spite of this difference in the
disturbance/bifurcation step of the update rule for the destructive case, one
can still reproduce the TRAP phenomenology.

The key is that the two different bifurcations result in the same rule for
updating the probability distribution over physical states, which is the only
thing that is relevant for the operational phenomenology.

To see this, consider the stochastic map on $\Phi_{R}$ associated with (i) an
equal probabilistic mixture of the identity function and the phase-flip
function, and (ii) an equal probabilistic mixture of the reset-to-0 function
and the reset-to-1 function. In both cases, it is the stochastic map
describing complete randomization of $\Phi_{R}$, that is, the map that
prepares the uniform probability distribution over the two values of
$\Phi_{R}$ regardless of the probability distribution fed into the stochastic
map. (For more details about this type of equivalence, see Sec. V.C of Ref.
Schmid  _et al._ (2021a).)

What does this imply? In the case where the occupation number on the $R$ arm
is found to be $N_{R}=0$, it implies that the probability distribution over
physical states at the measurement’s output is an equal mixture of the cases
where $\Phi_{R}=0$ and $\Phi_{R}=1$ independently of the value of $\Phi_{R}$
at the measurement’s input. And this is the case regardless of whether the
measurement is of the destructive or nondestructive variety. In short, despite
the differences in the bifurcation step of the update maps for destructive and
nondestructive measurements, both imply phase randomization.

As we explained in Sec. III.1.3, this phase randomization is what guarantees
that there is a randomization of where the detection is registered at the
output ports of the second beamsplitter and hence it is what guarantees that
the toy field theory reproduces the relevant phenomenology—that a measurement
of occupation number on an arm of the interferometer causes a loss of
interference. So, despite the differences in the disturbance/bifurcation step
of the update maps for destructive and nondestructive measurements, they both
reproduce the relevant TRAP phenomenology of interference.

Finally, the difference in the disturbance/bifurcation step does not change
any of the interpretational commitments of the toy field theory. In
particular, no nonlocal influences are required to reproduce the
phenomenology.

We end with a few more comments intended to head off possible confusions.

In the toy field theory, if a given field mode is known to be unoccupied, then
by the epistemic restriction, it is necessarily known to be in a uniform
mixture of the two possible phases, and it is also known that its physical
state is uncorrelated with the physical state of any other system. This is
true regardless of the reason for the mode being unoccupied, and it follows
that predictions that one makes about the outcomes of subsequent measurements
involving that mode are insensitive to these reasons. In particular, these
predictions are insensitive to whether a mode is known to be unoccupied by
virtue of the source in the experiment being turned off or by virtue of the
mode in question being at the output of a device that has absorbed the field
excitation that was produced at the source.

It follows in particular, that the output mode of a destructive measurement
device is described by such a probability distribution at all times. In other
words, it is the default situation for any mode that is unoccupied to have an
unknown phase that is not correlated with other systems. As such, this
situation does not need any special explanation within the toy field theory.
The far side of a brick, for instance, is always described by such a
probability distribution, regardless of whether it has just absorbed an
excitation on its near side or not.

The probability distribution just described—the uniform mixture of
$(N,\Phi)=(0,0)$ and $(N,\Phi)=(0,1)$ with no correlations with any other
system—is the counterpart within the toy field theory of the vacuum state in
quantum theory.424242In this sense, the toy field theory treats the quantum
vacuum state in a manner that is similar to how it is treated in stochastic
electrodynamics Boyer (1980). Furthermore, the features of the toy field
theory we have just outlined are in fact required if this probability
distribution is to emulate the analogous features of the quantum vacuum state.
Recall that in quantum theory, the reason that a given mode is in the quantum
vacuum state is irrelevant to the predictions about the outcomes of subsequent
measurements involving that mode, and the vacuum state is the default state of
any mode, appropriate for describing the far side of a brick regardless of
what is happening on its near side.

###  C.4 The case where a mirror is removed

Consider the case of the Mach Zehnder interferometer in the set-up of Fig.
1(a) or Fig. 1(b), but where the mirror on the right path has been removed.
This is depicted in Fig. 8. In this case, the $R$ mode at the input of the
second beamsplitter is no longer determined by the physical state of the $R$
mode at the output of the phase shifter or which-way detector, but rather it
tracks the physical state of another mode, the one aligned with the right
input of the second beamsplitter, but on the back side of the mirror, denoted
$R^{\prime}$. In other words, the transformation implemented in the case of no
mirror on the physical states of the $R$ mode and the $R^{\prime}$ mode is
simply a swap operation, i.e., the physical state of the $R$ output tracks
that of the $R^{\prime}$ input and the physical state of the $R^{\prime}$
output tracks that of the $R$ input. This is clearly analogous to the sort of
transformation one would use to describe the absence of the mirror in quantum
theory.

Figure 8: The Mach-Zehnder interferometers of Fig. 1, but with the mirror on
the $R$ mode removed.

Given that no source was shining onto the backside of the mirror in the setup
of the interferometer, the $R^{\prime}$ mode is known to be unoccupied. Hence,
by the epistemic restriction, the probability distribution over the physical
states of the $R^{\prime}$ mode is the one that is point distributed on
$N_{R^{\prime}}=0$ and uniformly distributed over values of the binary phase
$\Phi_{R^{\prime}}$. (If one traces the line made by the $R^{\prime}$ mode
back to its origin, it will meet some inert object, such as a wall of the
laboratory, and hence this case is really equivalent to the one wherein the
input to the second beamsplitter tracks the mode on the far side of a brick.)

It follows that the distribution over physical states of the mode $R$ at the
input of the second beamsplitter is the one that is point distributed on
$N_{R}=0$ and uniformly distributed over values of the binary phase
$\Phi_{R}$.

It is straightforward, therefore, to determine what our theory predicts in the
case where the mirror is removed. Recall that after the first beamsplitter,
the $L$ mode and the $R$ mode are equally likely to be the one that is
occupied. In those cases where it is the $R$ mode that is occupied, the
absence of the mirror on the right path implies that the excitation escapes
from the interferometer and thus neither detector at the output ports of the
second beamsplitter are found to fire. In those cases where it is the $L$ mode
that is occupied, the excitation remains in the interferometer and
consequently one of the detectors at the output ports of the second
beamsplitter fires. Which detector fires is determined (via the beamsplitter
update rule) by the relative phase at the input ports. But given that
$\Phi_{R}$ is uniformly distributed at this input, the relative phase
$\Phi_{L}\oplus\Phi_{R}$ is also uniformly distributed and consequently the
excitation is equally likely to be detected at either of the two output ports.
Thus, in the ensemble of cases where the excitation is detected, there is no
interference pattern.

Thus, the toy field theory reproduces the predictions of quantum theory for
the case of a Mach-Zehnder interferometer with one of the mirrors removed.

###  C.5 An ontology of field modes rather than an ontology of particles

The fundamental systems in the toy field theory are field modes, not
particles. We here elaborate a bit on the differences between modes and
particles in order to prevent possible confusions in interpreting the toy
field theory. As we already said in the main text, a particle is characterized
by having position and momentum as properties. These are aptly described as
motional degrees of freedom. A mode, on the other hand, is characterized by
having occupation number and phase as properties. These are best characterized
as excitational degrees of freedom. Spatially localized modes are parametrized
by position. As such, under a change in viewpoint from particles to modes, the
notion of position changes from being a dynamical variable whose value may
evolve over time as the particle evolves to being a parameter that indexes
which mode one is talking about, and it is the occupation numbers and the
phases of modes that are the dynamical variables. The notions of modes and
their excitational degrees of freedom are familiar and uncontroversial in the
study of field theories.

###  C.6 A description in terms of spatially localized modes

The fact that the ontology of the toy field theory is an ontology of modes
rather than particles is critical to providing a local causal explanation of
the TRAP phenomenology. To further elucidate this point, it is useful to
consider the analogy between a classical field theory and a cellular
automaton.

There are no particles in a cellular automaton; the fundamental systems are
the cells and the dynamics consists of changes of the internal states of these
cells (which are often taken to be binary but need not be). This is akin to
how, in a classical field theory, the systems are spatially localized modes
and the dynamics consists of changes of the internal states of these modes
(which are typically taken to be scalar or vector fields, but need not be).

In a cellular automaton, some set of cells may exhibit a distinctive pattern
that persists over time and propagates to adjacent sets of cells at every time
step. The motion of a glider in Conway’s Game of Life is the paradigmatic
example. Although this sort of motion may have the appearance of a particle
propagating through a discrete space, patterns of excitations (such as
Conway’s gliders) are not, strictly speaking, fundamental systems having
position as a property. Motion in a classical field theory is analogous, with
modes playing the role of cells.

A cellular automaton is said to invoke only local causal influences if the
update rule for the excitational state of a given cell (at a given time-step)
depends only on its state and that of its neighbours at the previous time-
step. The same is true of a classical field theory regarding the rule for
updating the physical states of a set of modes—the rule is local if the state
of a mode at one time depends only on its state at the earlier time and that
of spatially contiguous modes.

In the main text, we have emphasized that the toy field theory is local in the
sense that the $L$ and $R$ modes only exert causal influences on one another
when they become spatially contiguous, such as at a beamsplitter.434343Another
example is when a mode and another system (such as another mode) undergo the
CNOT dynamics, as in Eq. (22), which also requires them to be spatially
contiguous. However, there is one detail about our account of the experiment
that might still raise concerns regarding our claim of having a local model.
Specifically, we have conceptualized “the $L$ mode after the first
beamsplitter” as a single monolithic thing. We have not distinguished the
points on the left path that are at different distances from the first
beamsplitter. To claim that a field theory is truly local, it is reasonable to
require that one be able to provide an account in terms of modes that are
spatially localized, in addition to requiring that all interactions among
modes are local interactions. In other words, in addition to dynamical
locality, one must also have kinematical locality Spekkens (2015). As such, it
is useful to demonstrate explicitly that our toy field theory can be expressed
in terms of spatially localized modes. We now turn to that demonstration.

Given that our model involves a discrete space of physical states for each
mode, it makes sense to also discretize space and time in our model. Thus we
model the paths through the interferometer as a finite sequence of modes and
we model time as a finite sequence of time-steps.

Recall that it is common practice in physics to model a spatially localized
device in an experiment on some system by a local modification of the
Lagrangian that dictates the system’s evolution rather than trying to model
the device as a large number of fundamental systems (its atomic constituents)
in some particular configuration which then couple to the system of interest
by the universal Lagrangian. We now note that for models based on cellular
automata, there is a natural analogue of this practice, namely, to model the
device by a local modification of the cellular automaton’s update rule.

For simplicity, we will describe the experimental configuration as a circuit.
The cellular automaton model can then be described as follows: each wire in
the circuit is modelled as a 1-dimensional array of cells of the cellular
automaton, while each gate in the circuit is encoded as a modified update rule
for the cells that are the inputs and outputs of that gate.

We now redescribe the the Mach-Zehnder interferometer of Fig. 1 (with either
the phase shifter or which-way detector in place) in this manner. To begin
with, we recast the interferometer as a circuit, which we depict in Fig. 9(a).
Next, we decompose each wire in this circuit into a 1-dimensional array of
cells, as depicted in Fig. 9(b). We have numbered the cells on the left wire
by $L1,L2,\dots,L16$, and those on the right wire by $R1,R2,\dots,R16$.

Figure 9: (a) The circuit representation of the experimental setup of the
Mach-Zehnder interferometer of Fig. 1. BS denotes a beamsplitter. (b)
Modelling the circuit as a cellular automaton.

For the case of our toy field theory, a given cell $j$ has a physical state
described by a binary occupation number $N_{j}$ and a binary phase $\Phi_{j}$.
We denote the physical state of cell $j$ by
$\Lambda_{j}\equiv(N_{j},\Phi_{j})$.

The first update rule that needs to be stipulated for a circuit cellular
automaton is the one describing the time evolution of cells that are not the
input or output of any gate. We call this the free propagation update rule.

For the cellular automaton version of our toy field theory, we will use a
particular free propagation update rule which is an instance of what is called
the Margolus partitioning scheme, specialized to the case of a 1-dimensional
cellular automaton margolus1984physics. It is a rule that varies in its action
from one time-step to the next. Specifically, it is of the following form,
which is depicted schematically in Fig. 10,

$\displaystyle\textrm{for $t$ even}:$ $\displaystyle\;\;\textrm{for $j$
even}:\Lambda_{j}(t+1)=\Lambda_{j+1}(t),$ $\displaystyle\;\;\textrm{for $j$
odd}:\Lambda_{j}(t+1)=\Lambda_{j-1}(t).$ $\displaystyle\textrm{for $t$ odd}:$
$\displaystyle\;\;\textrm{for $j$ even}:\Lambda_{j}(t+1)=\Lambda_{j-1}(t),$
$\displaystyle\;\;\textrm{for $j$ odd}:\Lambda_{j}(t+1)=\Lambda_{j+1}(t).$
(58)

Figure 10: The update rule for free propagation.

The idea is that the internal state of a cell is swapped with one neighbour at
one timestep and the other neighbour at the next. In this way, an excitation
can move in either direction along a wire depending on the correlation between
the parity of its cell index and the parity of the time index. Such an update
rule for free propagation has the benefit of leading to dynamics that is time-
reversal symmetric.

The overall update rules for all cells in the circuit model of the Mach-
Zehnder interferometer are as depicted in Fig. 11.

Figure 11: The overall update rules for all cells.

Here, cell-groupings that are coloured green indicate places where the update
rule coincides with the free propagation update rule described above.
Consider, for instance, the cell $L8$. At even timesteps, it is grouped with
the cell $L9$ and the pair are subject to the update rule that simply swaps
their physical states:

$\displaystyle N_{L9}(t+1)=N_{L8}(t),$
$\displaystyle\Phi_{L9}(t+1)=\Phi_{L8}(t),$ and $\displaystyle
N_{L8}(t+1)=N_{L9}(t),$ $\displaystyle\Phi_{L8}(t+1)=\Phi_{L9}(t).$ (59)

At odd timesteps, by contrast, the cell $L8$ is grouped together with the cell
$L7$ and it is now this pair whose physical states are swapped:

$\displaystyle N_{L8}(t+1)=N_{L7}(t),$
$\displaystyle\Phi_{L8}(t+1)=\Phi_{L7}(t),$ and $\displaystyle
N_{L7}(t+1)=N_{L8}(t),$ $\displaystyle\Phi_{L7}(t+1)=\Phi_{L8}(t).$ (60)

Cell-groupings that are coloured blue in Fig. 11 indicate places where the
update rule is a modification of the free propagation rule to the functions
that describe the action of the associated gate. We have here adopted a
convention wherein the update rule for all gates differs from free propagation
only at even timesteps. The cells at the boundaries of the circuit of Fig. 11
interact at even timesteps with the internal degree of freedom of a source or
detector, and so there is just a single cell from the respective wires in the
blue-coloured cell grouping in these cases.

We can infer the modified update rules for gates from the functions that
describe the action of each gate and the fact that all of these functions are
time-reversal invariant.

For a gate on a single wire, for instance, time-reversal invariance means that
the function relating the state of cell $j+1$ at $t+1$ to the state of cell
$j$ at time $t$ is the same as the function that relates the state of cell $j$
at $t+1$ to the state of cell $j+1$ at time $t$. The analogous property holds
for the two-input two-output gates.

Consider the gate associated to the phase shifter and the pair of cells $R8$
and $R9$ that constitute its input and its output. From Eq. (A.1), one can
infer that the modified update rule for this pair of cells (at even timesteps)
is:

$\displaystyle N_{R9}(t+1)=N_{R8}(t),$
$\displaystyle\Phi_{R9}(t+1)=\Phi_{R8}(t)\oplus 1$ and $\displaystyle
N_{R8}(t+1)=N_{R9}(t),$ $\displaystyle\Phi_{R8}(t+1)=\Phi_{R9}(t)\oplus 1.$

Next, consider the gate associated to the first beam splitter and the
quadruple of cells $L4,R4,L5,R5$ constituting its inputs and outputs. From Eq.
(III.1.1), one can infer that the modified update rule for these cells (at
even timespets) is:

$\displaystyle N_{L5}(t+1)=\Phi_{L4}(t)\oplus\Phi_{R4}(t),$
$\displaystyle\Phi_{L5}(t+1)=N_{L4}(t)\oplus\Phi_{R4}(t),$ $\displaystyle
N_{R5}(t+1)=N_{R4}(t)\oplus N_{L4}(t)\oplus\Phi_{L4}(t)\oplus\Phi_{R4}(t),$
$\displaystyle\Phi_{R5}(t+1)=\Phi_{R4}(t),$ and $\displaystyle
N_{L4}(t+1)=\Phi_{L5}(t)\oplus\Phi_{R5}(t),$
$\displaystyle\Phi_{L4}(t+1)=N_{L5}(t)\oplus\Phi_{R5}(t),$ $\displaystyle
N_{R4}(t+1)=N_{R5}(t)\oplus N_{L5}(t)\oplus\Phi_{L5}(t)\oplus\Phi_{R5}(t),$
$\displaystyle\Phi_{R4}(t+1)=\Phi_{R5}(t).$

Now consider the cell $L1$, which is the output of the preparation device
associated to the source. This cell also has a modified update rule which (on
even timesteps), describing a nontrivial interaction with the internal state
of the device. For our purposes, it is irrelevant how the internal state of
such a device itself transforms when there are excitations travelling
backwards along the wires, since we are not considering these scenarios.
Consequently, it is sufficient to stipulate how $L1$ updates. If $t_{*}$
denotes an even timestep at which the source injects an excitation into the
interferometer, then

$\displaystyle N_{L1}(t_{*})=1,$ $\displaystyle\Phi_{L1}(t_{*})\;\textrm{is
sampled uniformly at random.}$

By contrast, if $t$ denotes an even timestep at which the source does not
inject an excitation into the interferometer, then

$\displaystyle N_{L1}(t)=0,$ $\displaystyle\Phi_{L1}(t)\;\textrm{is sampled
uniformly at random.}$

The cell $R1$, which is stipulated to correspond to a source of vacuum simply
never emits an excitation. Consequently, at all even timesteps $t$, we have

$\displaystyle N_{R1}(t)=0,$ $\displaystyle\Phi_{R1}(t)\;\textrm{is sampled
uniformly at random.}$

Now consider the gate associated to the which-way detector (labelled by $D$ in
Fig. 9(a)) and the pair of cells $R8$ and $R9$ that constitute its input and
its output. We consider the case where it implements a nondestructive
measurement of occupation number. Suppose that the detector fires at even
timestep $t_{*}$.

Recall that the knowledge-updating rule for such a measurement consisted of
two steps, one of which constituted a resolving of uncertainty regarding the
physical state and the other of which constituted a phase randomization.

We assume that the detector is of a type that can detect an excitation
travelling in either direction along the wire. (This is the case, for
instance, if a photon is detected by atomic absorption.) It follows that all
we can infer based on the fact that the detector fires at timestep $t_{*}$, is
that either $R8$ or $R9$ was excited at the end of the preceding timestep.
However, given that we assume that excitations are only injected into the
interferometer at $L1$, it follows that we can infer that it was in fact $R8$
that was excited.

Meanwhile, the phase randomization step can be described by the following
modified update rule for $R8$ and $R9$ (at even timesteps):

$\displaystyle N_{R9}(t+1)=N_{R8}(t),$
$\displaystyle\Phi_{R9}(t+1)\;\textrm{is sampled uniformly at random}$ and
$\displaystyle N_{R8}(t+1)=N_{R9}(t),$
$\displaystyle\Phi_{R8}(t+1)\;\textrm{is sampled uniformly at random.}$

The detectors at the output ports of the second beamsplitter act similarly to
the detector on the $R$ arm. Consider the cell $L16$, for instance. Whether
the detector fires at even timestep $t$ or not is simply a function of whether
$N_{L16}=1$ or not at that timestep. Regardless of whether it fires or not,
this detector acts as a source of vacuum for $L16$. Hence, the modified update
rule for $L16$ at all even timesteps is exactly that of Eq. (C.6), namely,

$\displaystyle N_{L16}(t)=0,$ $\displaystyle\Phi_{L16}(t)\;\textrm{is sampled
uniformly at random.}$

This concludes our demonstration that the toy field theory can be described in
terms of spatially localized modes interacting locally.