# What aspects of the phenomenology of interference witness nonclassicality?

Lorenzo Catani<EMAIL_ADDRESS>Electrical Engineering and
Computer Science Department, Technische Universität Berlin, 10587 Berlin,
Germany  Matthew Leifer  Institute for Quantum Studies and Schmid College of
Science and Technology, Chapman University, One University Drive, Orange, CA,
92866, USA  Giovanni Scala  International Centre for Theory of Quantum
Technologies, University of Gdansk, 80-308 Gdansk, Poland  David Schmid
International Centre for Theory of Quantum Technologies, University of Gdansk,
80-308 Gdansk, Poland  Robert W. Spekkens  Perimeter Institute for Theoretical
Physics, 31 Caroline Street North, Waterloo, Ontario Canada N2L 2Y5

###### Abstract

Interference phenomena are often claimed to resist classical explanation.
However, such claims are undermined by the fact that the specific aspects of
the phenomenology upon which they are based can in fact be reproduced in a
noncontextual ontological model [Catani et al. arXiv:2111.13727]. This raises
the question of what other aspects of the phenomenology of interference do in
fact resist classical explanation. We answer this question by demonstrating
that the most basic quantum wave-particle duality relation, which expresses
the precise trade-off between path distinguishability and fringe visibility,
cannot be reproduced in any noncontextual model. We do this by showing that it
is a specific type of uncertainty relation, and then leveraging a recent
result establishing that noncontextuality restricts the functional form of
this uncertainty relation [Catani et al. arXiv:2207.11779]. Finally, we
discuss what sorts of interferometric experiment can demonstrate contextuality
via the wave-particle duality relation.

##  I Introduction

The canonical thought experiment concerning wave-particle duality, as with so
many thought experiments in the foundations of quantum theory, originates with
Einstein. Bohr attributes it to him in Ref. Bohr (1949), and describes it
thus:

> If a semi-reflecting mirror is placed in the way of a photon, having two
> possibilities for its direction of propagation, the photon may either be
> recorded on one, and only one, of two photographic plates […] in the two
> directions in question, or else we may, by replacing the plates by mirrors,
> observe effects exhibiting an interference between the two reflected wave-
> trains.

This pair of possibilities is depicted in Figs. 1 (a) and 1 (b), respectively.
The semi-reflecting mirror can be presumed to be a 50-50 beamsplitter, and in
the case where the mirrors are in place, the interference is observed by
recombining the beams at a second 50-50 beamsplitter. This set-up, of course,
is the famous Mach-Zehnder interferometer Zehnder (1891); Mach (1892).

Figure 1: Einstein’s thought experiment concerning wave-particle duality. a) A
single photon source at the left input of a 50-50 beamsplitter leads to
detection in one and only one arm. b) If the detectors on the arms are not
present and the two beams are recombined at a second 50-50 beamsplitter,
interference is observed.

In modern versions of the thought experiment, it is usually further noted that
(i) if the photographic plates in Fig. 1 (a) are replaced by nondestructive
detectors and the beams are again recombined at the second 50-50 beamsplitter
(so that one has a Mach-Zehnder interferometer wherein a nondestructive which-
way measurement is implemented), then one still does not see any interference,
and (ii) as it is assumed that the source generates only a single photon, it
suffices to have a detector in just one of the arms to implement a which-way
measurement and destroy the interference. Consequently, the modern version of
the thought experiment differs from that depicted in Fig. 1 in that it
typically considers two configurations of a Mach-Zehnder interferometer: the
first, which replaces the set-up depicted in Fig. 1 (a), is an interferometer
wherein there is a (nondestructive) detector on one of the arms between the
pair of beamsplitters; the second is a variant of Fig. 1 (b) in which there is
a phase shifter on one arm, and where it is sufficient to consider two choices
of the local phase shift that allow one to toggle the relative phase shift
between 0 and $\pi$. In the second configuration, one sees perfect
interference, with the dark port (the one corresponding to destructive
interference) toggling between left and right as one toggles the choice of the
local phase shift. In the first configuration, the presence of the detector
destroys this interference.

Some version of this thought experiment is familiar to all students of quantum
theory, since certain aspects of its phenomenology have been heralded as
capturing the essence of quantum theory, with Feynman being one of the most
prominent advocates of this view Feynman  _et al._ (1961-1963). (Note that
Feynman considers the case of the double-slit experiment, but the aspects of
the phenomenology that he makes use of are features that the double-slit and
the Mach-Zehnder interferometer have in common.) The relevant aspects of the
phenomenology include those in the previous paragraph (and some extensions
thereof); see Ref. Catani  _et al._ (2021) for a complete discussion. In that
article, the collection of these aspects is called the TRAP phenomenology,
where TRAP stands for “traditionally regarded as problematic”.

The claim that the TRAP phenomenology captures the essence of quantum theory
is unlike similar claims that focus on Bell inequality violations Bell (1964,
1995); Clauser  _et al._ (1969); Brunner  _et al._ (2014) or violations of
inequalities derived from generalized noncontextuality Spekkens (2005);
Spekkens  _et al._ (2009); Kunjwal and Spekkens (2015, 2018); Schmid  _et al._
(2018); Chaturvedi  _et al._ (2021) because it is not supported by a no-go
theorem that establishes rigorously the inconsistency between the TRAP
phenomenology and certain stipulated principles that formally define a notion
of classical explainability (such as the principles of locality or
noncontextuality within the framework of ontological models Harrigan and
Spekkens (2010)). This leaves open the possibility that a classical model of
this interference phenomenology could be constructed. Indeed, such a model
does exist, as was shown in Ref. Catani  _et al._ (2021). This model is
classical in the sense that its kinematical state space is represented by a
set and its dynamics are represented by functions on this set Schmid  _et al._
(2021a), while also respecting the principles of locality and generalized
noncontextuality.

We here take a phenomenon to be classically explainable if and only if it can
be reproduced in a generalized-noncontextual ontological model Spekkens
(2005). Generalized noncontextuality can be motivated by a methodological
version of Leibniz’s principle of the ontological identity of empirical
indiscernibles Spekkens (2019); Schmid  _et al._ (2021a), or by the principle
of operational no fine-tuning Catani and Leifer (2020). Realizability by a
generalized-noncontextual ontological model implies the existence of a locally
causal model Bell (1964, 1995), the existence of a Kochen-Specker-
noncontextual model Kochen and Specker (1967) for sharp measurements Spekkens
(2014); Kunjwal and Spekkens (2015), the existence of a simplex-embedding for
a generalized probabilistic theory Schmid  _et al._ (2021b); Selby  _et al._
(2021), and the existence of a nonnegative quasiprobability representation
Schmid  _et al._ (2020); Spekkens (2008), and is implied by the existence of a
macrorealist explanation Schmid (2022). Under appropriate conditions, various
other indicators of nonclassicality (including anomalous weak values Pusey
(2014), violations of the Leggett-Garg inequality Schmid  _et al._ , and
universal quantum computation Howard  _et al._ (2014); Schmid  _et al._
(2022)) imply the failure of generalized noncontextuality. The failure of
generalized noncontextuality is also a resource for various forms of
information processing Spekkens  _et al._ (2009); Chailloux  _et al._ (2016);
Ambainis  _et al._ (2019); Saha  _et al._ (2019); Schmid and Spekkens (2018);
Lostaglio and Senno (2020); Saha and Chaturvedi (2019); Hameedi  _et al._
(2017); Yadavalli and Kunjwal (2022); Roch i Carceller  _et al._ (2022); Flatt
_et al._ (2022).

Having a precise and well-motivated boundary between classicality and
nonclassicality is important because it informs our attempts to solve various
problems: (i) the project of extending quantum theory into new domains, such
as gravitational physics, and of proposing quantum versions of various
classical theoretical frameworks, such as the frameworks of causal inference
Pearl (2009) or algorithmic information theoryLi and Vitányi (2019); Chaitin
(2004); (ii) the project of uncovering novel quantum-over-classical advantages
for various tasks in computation, communication, and cryptography Nielsen and
Chuang (2010); (iii) the project of determining the correct interpretation of
the quantum formalism.

Given that the TRAP phenomenology of interference can be reproduced in a
generalized-noncontextual ontological model, and hence is classically
explainable, it is natural to ask: what aspects of interference phenomena, if
any, are genuinely nonclassical? That is: what aspects of interference
phenomena cannot be reproduced by any generalized-noncontextual ontological
model?

An easy but relatively uninformative way to answer this question is to take
any known proof of the failure of generalized noncontextuality and recast it
into an interferometric scenario. For instance, a proof involving one or two
abstract qubits can be instantiated interferometrically by associating each
qubit with the two-dimensional Hilbert space corresponding to the which-path
degree of freedom of the system, termed a dual-rail qubit. This can always be
done, but one learns very little in the process, since it is merely a
restatement of known facts about the Hilbert space structure of quantum theory
in the language of interferometers. The more difficult but more informative
way to answer the question is to find some operational phenomenology that was
previously thought to be significant to the study of quantum interference and
show that this phenomenology witnesses nonclassicality, in the sense of being
inconsistent with a generalized-noncontextual or local ontological model.

We show herein that the functional form of the most basic wave-particle
duality relation in quantum theory Greenberger and Yasin (1988) is such an
aspect.

A wave-particle duality relation is a tradeoff relation between two quantities
that are meant to represent wave-like and particle-like behaviours
respectively. For the relation of interest to us here Wootters and Zurek
(1979); Greenberger and Yasin (1988), these quantities are the fringe
visibility $\mathcal{V}$ and the path distinguishability $\mathcal{P}$. The
path distinguishability is defined in terms of the statistics of a measurement
of the type depicted in Fig. 2 (a), which we term a which-way measurement,
while the fringe visibility is defined in terms of the statistics of a
measurement of the type depicted in Fig. 2 (b), which we term a which-phase
measurement. (Their precise definitions are given further on.) If we
conceptualize the two experiments depicted in Fig. 1 as involving a
preparation stage and a measurement stage for the dual-rail qubit (the
division between the stages being indicated by the blue dashed line), then we
see that the two possibilities for the measurement stage are what we have
picked out in Figs. 2 (a) and (b).

The tradeoff relation we will consider between the fringe visibility
$\mathcal{V}$ and path distinguishability $\mathcal{P}$ holds for all states
of the dual-rail qubit. One of the ways of preparing an arbitrary state of the
dual-rail qubit is depicted in Fig. 2 (c), where the beamsplitter is allowed
to have any reflectivity $r$ in the range $[0,1]$ and there is a phase shifter
on one of the arms that can implement any phase shift $\phi$ in the range
$[0,2\pi)$. Thus, for each preparation of Fig. 2 (c), we can imagine following
it up with the measurement in Fig. 2 (a) or the measurement in Fig. 2 (b), and
it is the statistics of these two counterfactual possibilities for the
measurement that are constrained by the wave-particle duality relation of
interest to us.

Figure 2:  Measurement and preparation stages of the interferometric
experiment. a) Which-way and b) which-phase measurements. c) A preparation
procedure with beamsplitter reflectivity $r$ and phase shift $\phi$ allowing
for the implementation of any quantum state of the dual-rail qubit.

We can now summarize our main result. Under a particular symmetry condition
(which we discuss below and which always holds in quantum theory),
noncontextuality implies the constraint

$\mathcal{V}+\mathcal{P}\leq 1.$ (1)

This constitutes a noncontextuality inequality that is noise-robust in the
sense of Refs. Mazurek  _et al._ (2016a); Kunjwal and Spekkens (2015); Pusey
(2018). Meanwhile, the quantum tradeoff between visibility $\mathcal{V}$ and
distinguishability $\mathcal{P}$ is described by the relation Wootters and
Zurek (1979); Greenberger and Yasin (1988)

$\mathcal{V}^{2}+\mathcal{P}^{2}\leq 1.$ (2)

In Fig. 3, this tradeoff is plotted alongside the noncontextual bound of Eq.
(1). It is clear from this plot that the noncontextual bound can be violated
in quantum theory, demonstrating the impossibility of explaining this aspect
of quantum phenomenology in terms of a noncontextual ontological model.

Figure 3: Plots of fringe visibility $\mathcal{V}$ versus path
distinguishability $\mathcal{P}$. The blue curve is the quantum tradeoff
relation (Eq. (2)). The red curve is the bound on the tradeoff relation that
is implied by the assumption that there exists a noncontextual ontological
model of the experimental set-up (Eq. (1)).

It is interesting to note that Greenberger and Yasin Greenberger and Yasin
(1988) explicitly expressed surprise at how large the fringe visibility can be
for a given degree of path distinguishability in the quantum tradeoff relation
of Eq. (2). They did not, however, articulate their reasons for expecting
otherwise. While it is true that the simplest form of such a tradeoff would be
linear (so that an increase in path distinguishability implies a proportional
decrease in fringe visibility), this is not a compelling reason to expect
linearity. Our result, however, provides such a reason: if it were possible to
explain the experimental set-up classically, in the sense of reproducing its
predictions within a generalized-noncontextual ontological model, then the
tradeoff would be linear.

##  II The wave-particle duality relation

The fringe visibility $\mathcal{V}$ in a generic interference pattern is
defined as the normalized difference between the maximum and minimum
intensities of the pattern:

$\mathcal{V}=\frac{I_{\mathrm{max}}-I_{\mathrm{min}}}{I_{\mathrm{max}}+I_{\mathrm{min}}}.$
(3)

For the type of interference that is of interest in quantum theory, the
pattern is built up from detections of individual particles that exhibit
statistical variation in their location (rather than, for instance, describing
a continuous physical quantity, such as the height of a water wave). The
intensity at a given location tracks the number of detections in that location
and is consequently proportional to the probability of finding an individual
particle there, so that we can replace intensities with probabilities in the
expression for $\mathcal{V}$,

$\mathcal{V}=\frac{\mathbb{P}_{\mathrm{max}}-\mathbb{P}_{\mathrm{min}}}{\mathbb{P}_{\mathrm{max}}+\mathbb{P}_{\mathrm{min}}}.$
(4)

The expression further simplifies when we specialize to the case of the
measurement depicted in Fig. 2 (b). In such an interference pattern, there are
only two spatial locations of interest, namely, the two output ports of the
second beamsplitter. Consequently, the two probabilities of detection sum to
unity, trivializing the denominator.

Meanwhile, the path distinguishability $\mathcal{P}$ is defined as the
absolute value of the difference in the probability of finding the particle on
the left and the probability of finding it on the right for a which-way
measurement, such as the one depicted in Fig. 2 (a), i.e.,

$\displaystyle\mathcal{P}$ $\displaystyle=|\mathbb{P}(L)-\mathbb{P}(R)|.$ (5)

Quantum theory predicts that there is a tradeoff between the fringe visibility
and the path distinguishability which holds for all states, and hence for all
preparations of the type depicted in Fig. 2 (c), namely,
$\mathcal{V}^{2}+\mathcal{P}^{2}\leq 1$ (Eq. (2)). This relation was first
derived by Wootters and Zurek Wootters and Zurek (1979), who were considering
the double-slit experiment, and it was later rederived in the context of the
Mach-Zehnder interferometer by Greenberger and Yasin Greenberger and Yasin
(1988). 111Note that this relation is distinct from the one described by
Englert Englert (1996) (Eq. (10) therein) which concerns the tradeoff between
the fringe visibility and path distinguishability when one performs unsharp
versions of the which-phase and which-way measurements simultaneously, rather
than sharp versions of these as counterfactual alternatives.

The extreme cases in the tradeoff relation of Eq. (2) are
$(\mathcal{V},\mathcal{P})=(1,0)$ and $(\mathcal{V},\mathcal{P})=(0,1)$. The
point $(\mathcal{V},\mathcal{P})=(1,0)$ can be achieved by the preparation
procedure where the beamsplitter has reflectivity 1/2 (a 50-50 beamsplitter)
and the relative phase-shift between the two paths is 0 or $\pi$, because in
this case the which-phase measurement is perfectly predictable while the
which-way measurement is completely unpredictable. The point
$(\mathcal{V},\mathcal{P})=(0,1)$ can be achieved by the preparation procedure
where the beamsplitter has reflectivity 0 (i.e., no beamsplitter) or
reflectivity 1 (a perfect mirror), because in this case it is the which-way
measurement that is perfectly predictable and the which-phase measurement that
is completely unpredictable. If one wishes to make either of these quantities
nonzero, the other must become strictly less than unity. The particular manner
in which this occurs is what the wave-particle duality relation stipulates.

##  III Why the wave-particle duality relation is an uncertainty relation

As noted above, it is sufficient for our purposes to conceptualize the system
that passes between the preparation stage and the measurement stage as a dual-
rail qubit, that is, as a system described by a two-dimensional Hilbert space
where the basis $\\{|L\rangle,|R\rangle\\}$ denotes the two eigenstates of a
which-way measurement, corresponding to the two “rails” or spatial modes of
the interferometer on which the particle might be found; here, $L$ denotes the
left mode and $R$ denotes the right mode.222This is a first-quantized
description of the experimental set-up. In Ref. Catani  _et al._ (2021), it is
argued that in order to properly analyze questions about locality, the first-
quantized description should be understood to concern coarse-grainings of the
excitational degrees of freedom of field modes, rather than the motional
degrees of freedom of particles. Because we do not concern ourselves with
issues of locality here, we do not emphasize the distinction.

The which-way measurement depicted in Fig. 2 (a) corresponds to measuring the
dual-rail qubit in the $\\{|L\rangle,|R\rangle\\}$ basis. The Hermitian
operator associated with such a measurement when one associates the
$|L\rangle$ ($|R\rangle$) outcome with the eigenvalue $+1$ ($-1$) is given by
the operator $\hat{Z}:=|L\rangle\langle L|-|R\rangle\langle R|$, that is, the
Pauli-$Z$ operator relative to the $\\{|L\rangle,|R\rangle\\}$ basis. We term
this the which-way observable. Letting $\mathbb{P}_{Z}$ denote the probability
distribution over the outcome of this measurement (a random variable which we
denote by $Z$), the path distinguishability $\mathcal{P}$ defined in Eq. (5)
can be expressed as

$\displaystyle\mathcal{P}$
$\displaystyle=|\mathbb{P}_{Z}(+1)-\mathbb{P}_{Z}(-1)|$
$\displaystyle=|\langle Z\rangle|.$ (6)

Here, $\langle Z\rangle={\rm tr}(\hat{Z}\rho)$, where $\rho$ denotes the
quantum state of the dual-rail qubit associated to the preparation stage.

For a measurement with outcome $M$ in the set $\\{+1,-1\\}$, the absolute
value of its expectation value, $|\langle M\rangle|$, is a measure of the
predictability of the measurement. Its maximum value $|\langle M\rangle|=1$
occurs for a distribution that is weighted entirely on the outcome +1 or
entirely on the outcome $-1$, and its minimum value $|\langle M\rangle|=0$
occurs for a distribution that is uniform over the two outcomes.

Eq. (III) therefore shows that the path distinguishability $\mathcal{P}$ is
simply a measure of the predictability of the which-way measurement, and it is
consequently appropriate to describe $\mathcal{P}$ as the which-way
predictability.

We now consider the fringe visibility. Note that in its definition, Eq. (4),
the maximum and minimum probabilities are obtained by ranging over all the
values of the phase shift on one arm. These extrema are achieved for a pair of
phases, $\phi_{\rm max}$ and $\phi_{\rm max}+\pi$. For convenience, we
henceforth consider an interferometer wherein the path-lengths are chosen such
that it is possible to take $\phi_{\rm max}=0$ and that for this trivial phase
shift, it is the detector at the left output port of the second beamsplitter
that fires.

For such an interferometer, the beamsplitter transformation in the measurement
depicted in Fig. 2 (b) is represented by the Hadamard unitary (relative to the
basis $\\{|L\rangle,|R\rangle\\}$) and therefore the effective measurement on
the dual-rail qubit is of the basis $\\{|+\rangle,|-\rangle\\}$, where the two
states $|\pm\rangle:=\tfrac{1}{\sqrt{2}}(|L\rangle\pm|R\rangle)$ correspond to
the relative phase between the left and right modes being $0$ or $\pi$
respectively. It is for this reason that it is apt to call the measurement
depicted in Fig. 2 (b) a which-phase measurement. We adopt the convention that
the outcome $+1$ ($-1$) is associated to the firing of the detector at the
left (right) output port of the second beamsplitter, and hence to the case
where the relative phase is 0 ($\pi$). In this case, the Hermitian operator
associated to this measurement, the which-phase observable, is
$\hat{X}=|+\rangle\langle+|-|-\rangle\langle-|$, that is, the Pauli-$X$
operator relative to the $\\{|L\rangle,|R\rangle\\}$ basis. Letting
$\mathbb{P}_{X}$ denote the probability distribution over the outcome of this
measurement (a random variable which we denote by $X$), the fringe visibility
$\mathcal{V}$ defined in Eq. (4) can be expressed as

$\displaystyle\mathcal{V}$
$\displaystyle=\max\\{\mathbb{P}_{X}(+1),\mathbb{P}_{X}(-1)\\}-\min\\{\mathbb{P}_{X}(+1),\mathbb{P}_{X}(-1)\\}$
$\displaystyle=|\mathbb{P}_{X}(+1)-\mathbb{P}_{X}(-1)|$
$\displaystyle=|\langle{X}\rangle|.$ (7)

Eq. (7) shows that the fringe visibility $\mathcal{V}$ is simply the absolute
value of the expectation value of the $\pm 1$-valued variable $X$ that
describes the outcome of the which-phase measurement, and so is a measure of
the predictability of this measurement. It is consequently appropriate to
describe $\mathcal{V}$ as the which-phase predictability.

Given the identification of $\mathcal{V}$ and $\mathcal{P}$ as
predictabilities of the outcomes of the which-phase and which-way measurements
respectively, the wave-particle duality relation of Eq. (2) can be understood
as a tradeoff of predictabilities, or equivalently, as a tradeoff of
unpredictabilities or uncertainties. An uncertainty relation is a tradeoff of
predictabilities that holds for all states. Because the tradeoff between
which-path and which-way predictability is of this type, it describes an
uncertainty relation.

In fact, the wave-particle duality relation is simply a special case of an
uncertainty relation for an arbitrary qubit. One can quantify the
predictability of measurements of Pauli-$Z$ and Pauli-$X$ observables by
$|\langle Z\rangle|$ and $|\langle X\rangle|$ respectively. It follows from
the fact that the quantum state space of a qubit is the Bloch sphere that the
tradeoff relation that holds between these predictabilities is

$\langle Z\rangle^{2}+\langle X\rangle^{2}\leq 1.$ (8)

This can be understood as an uncertainty relation for a qubit. (See Appendix D
of Ref. Catani  _et al._ (2022) for an account of the history of
conceptualizing Eq. (8) as an uncertainty relation.) Given Eqs. (III) and (7),
we see that the quantum wave-particle duality relation of Eq. (2) is simply an
instance of this uncertainty relation, specialized to the case of the which-
way and which-phase observables of a dual-rail qubit.

It should be noted that Greenberger and Yasin Greenberger and Yasin (1988)
explicitly denied that their wave-particle duality relation could be
interpreted as an uncertainty relation. Our impression is that this arose from
having an overly narrow conception of what constitutes an uncertainty
relation. Many other authors, however, have recognized that wave-particle
duality relations wherein the which-way and which-phase measurements are
counterfactual alternatives (the type considered here) are instances of
uncertainty relations Jaeger  _et al._ (1995); Björk  _et al._ (1999); Luis
(2001); Liu  _et al._ (2012); Coles  _et al._ (2014, 2017).

##  IV Operational theories, ontological models and generalized
noncontextuality

We now briefly introduce the relevant preliminaries for discussing
noncontextuality for prepare-measure scenarios. For a given system, an
operational theory stipulates the possible preparations and measurements, as
well as the probability $\mathbb{P}(y|M,P)$ of obtaining each outcome $y$ of
measurement $M$ when performed on preparation $P$. When characterized in the
framework of GPTs Hardy (2001); Barrett (2007); Chiribella  _et al._ (2010),
one represents each preparation $P$ and each effect $[y|M]$ by real-valued
vectors $\vec{s}_{P}$ and $\vec{e}_{y|M}$, respectively, where the
probabilities are given by $\mathbb{P}(y|M,P)=\vec{s}_{P}\cdot\vec{e}_{y|M}$.

An ontological model of an operational theory provides an explanation of the
operational predictions of the latter in terms of processes (stochastic maps)
on an underlying classical state space Harrigan and Spekkens (2010). An
ontological model associates a set $\Lambda$, known as the ontic state space,
with each given system. (For our purposes this set can be taken to be finite
without loss of generality.) An element $\lambda\in\Lambda$ is known as an
ontic state, and encodes all physical properties of the system. The
ontological model represents each preparation $P$ in the operational theory as
a probability distribution $\mu(\lambda|P)$ over ontic states. Each effect
$[y|M]$ is represented by a conditional probability distribution
$\xi(y|M,\lambda)$ describing the probability of obtaining outcome $y$ given
that measurement $M$ was implemented on a system with ontic state $\lambda$.
We can associate $\mu(\lambda|P)$ and $\xi(y|M,\lambda)$ with vectors, denoted
(respectively) by $\vec{\mu}_{P}$ and $\vec{\xi}_{y|M}$. An ontological model
of an operational theory must reproduce the predictions of that theory as
follows,

$\displaystyle\mathbb{P}(y|M,P)$
$\displaystyle=\sum_{\lambda\in\Lambda}\xi(y|M,\lambda)\mu(\lambda|P)=\vec{\xi}_{y|M}\cdot\vec{\mu}_{P}.$
(9)

Such a model satisfies the principle of generalized
noncontextuality333Henceforth, we shall use the term “noncontextuality” as a
shorthand for “generalized noncontextuality”. if operationally equivalent
laboratory procedures are represented as identical processes in the
ontological model Spekkens (2005). In the case of preparation procedures
(which is all we require in this work), this can be formalized as follows. Two
preparation procedures $P$ and $P^{\prime}$ are operationally equivalent if
they lead to the same statistics for all possible measurements, so that
$\forall M:\mathbb{P}(y|M,P)=\mathbb{P}(y|M,P^{\prime})$. We denote this
equivalence relation by $P\simeq P^{\prime}$. Then, an ontological model is
noncontextual if any two such preparations are represented by the same
probability distribution over ontic states:

$P\simeq P^{\prime}\implies\vec{\mu}_{P}=\vec{\mu}_{P^{\prime}}.$ (10)

The representation of operational procedures as real-valued GPT vectors
(introduced above) throws away all information about those procedures except
for their operational equivalence class. As a consequence, a noncontextual
ontological representation of an operational theory is one wherein all
procedures associated to the same GPT state vector are represented by the same
probability distribution over ontic states. For example, if two different
mixtures of GPT states are equal, the corresponding mixtures of their
representations as probability distributions must also be equal:

$\displaystyle\sum_{i}w_{i}\vec{s}_{i}=\sum_{j}w^{\prime}_{j}\vec{s}^{\;\prime}_{j}\;\;\implies\;\;\sum_{i}w_{i}\vec{\mu}_{i}=\sum_{j}w^{\prime}_{j}\vec{\mu}^{\;\prime}_{j},$
(11)

where $\\{w_{i}\\}_{i}$ and $\\{w^{\prime}_{j}\\}_{j}$ are probability
distributions.

##  V Witnessing contextuality via the wave-particle duality relation

###  V.1 A noncontextuality no-go theorem based on wave-particle duality

Our main result is an application of a result from Ref. Catani  _et al._
(2022) concerning how to witness contextuality via uncertainty relations. We
therefore begin by summarizing this prior result.

Consider an arbitrary operational theory and two binary-outcome measurements
therein, denoted $M$ and $M^{\prime}$. We will also use $M$ and $M^{\prime}$
to denote the outcomes of these measurements, which are assumed to take values
in the set $\\{+1,-1\\}$.

We will consider the tradeoff between the $M$-predictability and the
$M^{\prime}$-predictability for an operational state that satisfies a certain
condition, termed $A_{1}^{2}$-orbit realizability (which is defined relative
to $M$ and $M^{\prime}$). It consists of two subconditions. The first is that
the operational state has equal-predictability counterparts relative to $M$
and $M^{\prime}$: if the operational state is $\vec{s}_{1}$, then one can find
three other operational states, $\vec{s}_{2}$, $\vec{s}_{3}$, and
$\vec{s}_{4}$, such that the quadruple of states give equal predictabilities
for the $M$ and $M^{\prime}$ measurements, but vary over all possible signs of
the expectation values for the pair:

$\displaystyle\langle M\rangle_{\vec{s}_{1}}=\langle
M\rangle_{\vec{s}_{2}}=-\langle M\rangle_{\vec{s}_{3}}=-\langle
M\rangle_{\vec{s}_{4}},$ $\displaystyle\langle
M^{\prime}\rangle_{\vec{s}_{1}}=-\langle
M^{\prime}\rangle_{\vec{s}_{2}}=-\langle
M^{\prime}\rangle_{\vec{s}_{3}}=\langle M^{\prime}\rangle_{\vec{s}_{4}}.$ (12)

Notice that all four of these states assign the same values to $|\langle
M\rangle|$ and $|\langle M^{\prime}\rangle|$ and therefore any tradeoff
between $|\langle M\rangle|$ and $|\langle M^{\prime}\rangle|$ applies to all
of them. The second subcondition is that this quadruple of states satisfies
the operational equivalence relation

$\frac{1}{2}\vec{s}_{1}+\frac{1}{2}\vec{s}_{3}=\frac{1}{2}\vec{s}_{2}+\frac{1}{2}\vec{s}_{4}.$
(13)

(Note that some operational equivalence relation among the states is always
required in order for the principle of noncontextuality to imply constraints
on the ontological model.)

In summary, a state satisfies the $A_{1}^{2}$-orbit-realizability condition
relative to $M$ and $M^{\prime}$ if it is part of a quadruple of states
satisfying Eqs. (V.1) and (13). The condition is referred to as
“$A_{1}^{2}$-orbit-realizability” because any such quadruple of states is an
orbit of $A_{1}^{2}$, the Coxeter group describing the abstract version of the
group of reflections across two planes.

An operational theory is said to have $A_{1}^{2}$-symmetry relative to $M$ and
$M^{\prime}$ if all operational states in the theory satisfy the
$A_{1}^{2}$-orbit-realizability condition relative to $M$ and $M^{\prime}$.

For example, consider qubit quantum theory. The only pairs of measurements
relative to which all states satisfy the $A_{1}^{2}$-orbit realizability
condition are pairs of measurements associated to complementary bases (i.e.,
orthogonal directions in the Bloch sphere). This fact can be inferred from the
geometry of the Bloch sphere, as we now explain (see Fig. 4). Consider two
observables, $M$ and $M^{\prime}$, whose +1 outcomes are associated to
directions that subtend an angle strictly greater than $90^{\circ}$ in the
Bloch sphere. Now consider the Bloch vector $\vec{s}_{1}$ that describes a
pure state that assigns positive expectation values to both $M$ and
$M^{\prime}$. The state $\vec{s}_{1}$ is an example of one that does not
satisfy the $A_{1}^{2}$-orbit-realizability condition relative to $M$ and
$M^{\prime}$. This is because, to satisfy the condition, there must be an
operational state $\vec{s}_{2}$ that satisfies $\langle
M\rangle_{\vec{s}_{2}}=\langle M\rangle_{\vec{s}_{1}}$ and $\langle
M^{\prime}\rangle_{\vec{s}_{2}}=-\langle M^{\prime}\rangle_{\vec{s}_{1}}$, but
this places $\vec{s}_{2}$ outside the Bloch sphere, so that it does not
describe a valid quantum state. If $M$ and $M^{\prime}$ describe a pair of
observables whose +1 outcomes are associated to directions that subtend an
angle strictly less than $90^{\circ}$ in the Bloch sphere, then any pure state
that assigns a positive expectation value to $M$ and a negative expectation
value to $M^{\prime}$ provides an example of one that does not satisfy the
$A_{1}^{2}$-orbit-realizability condition, by similar logic. In this way, we
reach the conclusion that $M$ and $M^{\prime}$ must correspond to orthogonal
directions in the Bloch sphere. For any pair of orthogonal directions
(corresponding to a pair of complementary measurements), it is easy to see
that every state in the Bloch sphere satisfies the $A_{1}^{2}$-orbit-
realizability condition, as illustrated in Fig. 4(b).

Figure 4: (a) Example of a pair of quantum measurements, $M$ and $M^{\prime}$,
relative to which qubit quantum theory fails to have $A_{1}^{2}$-symmetry. (b)
By contrast, for a pair of complementary measurements, such as Pauli $Z$ and
Pauli $X$, qubit quantum theory has $A_{1}^{2}$-symmetry.

The which-way and which-phase measurements in quantum theory are an instance
of a pair of complementary measurements, and so every state of a dual-rail
qubit satisfies $A_{1}^{2}$-orbit-realizability relative to the which-way and
which-phase measurements. In other words, the dual-rail qubit has the
$A_{1}^{2}$-symmetry property.

The significance of having the $A_{1}^{2}$-symmetry property is that it is
under this condition that noncontextuality implies a bound on the
predictability tradeoff for the pair of measurements in question.
Specifically, the operational equivalence of Eq. (13), together with the
specific instance of noncontextuality described in Eq. (11), implies
nontrivial constraints on the ontological representations of the states. These
constraints have consequences for the possible measurement statistics, which
are derived in Ref. Catani  _et al._ (2022), and which we recall here:

######  Proposition V.1.

Consider an operational theory having $A_{1}^{2}$-symmetry relative to the
pair of measurements $M$ and $M^{\prime}$. If the operational theory admits of
a noncontextual ontological model, then the tradeoff of predictabilities for
this pair of measurements must satisfy the following bound for all states:

$|\langle M\rangle|+|\langle M^{\prime}\rangle|\leq 1.$ (14)

If we take $M$ and $M^{\prime}$ to be the which-way and which-phase
measurements respectively, so that $|\langle M\rangle|=|\langle
Z\rangle|=\mathcal{P}$ (where we have used Eq. (III)), and $|\langle
M^{\prime}\rangle|=|\langle X\rangle|=\mathcal{V}$ (where we have used Eq.
(7)), then we get the following corollary:

######  Corollary V.1.1.

Consider an operational theory having $A_{1}^{2}$-symmetry relative to which-
way and which-phase measurements. If the operational theory admits of a
noncontextual ontological model, then the wave-particle duality relation in
the operational theory must satisfy the bound

$\mathcal{V}+\mathcal{P}\leq 1.$ (15)

As we noted above, a dual-rail qubit has the $A_{1}^{2}$-symmetry property
relative to the which-way and which-phase measurements. Consequently, the fact
that it can saturate the wave-particle duality relation of Eq. (2), namely,
$\mathcal{V}^{2}+\mathcal{P}^{2}\leq 1$, and thereby violate the bound of Eq.
(15), implies that the functional form of its wave-particle duality relation
witnesses contextuality. This is illustrated in Fig. 3.

Imagine now some operational theory that describes an alternative to quantum
theory, i.e., a foil to quantum theory Spekkens (2016). Suppose that it allows
for a which-way measurement and a which-phase measurement, and that it has
$A_{1}^{2}$-symmetry relative to these measurements. Such theories also have
tradeoff relations between the predictabilities of the which-way and which-
phase measurements, and these can be distinct from the quantum one. For
instance, one can construct four examples of such tradeoff relations from the
four foil theories presented in Ref. Catani  _et al._ (2022) if one takes the
pair of complementary measurements described in each of these to be which-way
and which-phase measurements. Under this mapping, the various forms of
uncertainty relation for complementary measurements that are achieved for
these different foil theories (described in Eqs. (3)-(6) of Ref. Catani  _et
al._ (2022)) yield corresponding forms of the wave-particle duality relation.

###  V.2 How to implement an experimental test of contextuality based on wave-
particle duality

It is also possible to implement an experimental test of contextuality via the
phenomenology of wave-particle duality relations. Here again, we leverage a
result from Ref. Catani  _et al._ (2022).

Any experimental test of contextuality should not presume the correctness of
quantum theory, but should instead allow that some other operational theory
describes the experiment. It follows that one cannot assume that the
operational theory governing the experiment satisfies the $A_{1}^{2}$-symmetry
property. Instead, one simply needs to experimentally identify two
measurements $M$ and $M^{\prime}$ and an experimentally realizable operational
state that satisfies the $A_{1}^{2}$-orbit-realizability condition relative to
$M$ and $M^{\prime}$ and then test the noncontextual bound on these.

The result can be summarized as follows.

######  Proposition V.2.

Consider an experimentally realizable state that satisfies the
$A_{1}^{2}$-orbit-realizability condition relative to measurements $M$ and
$M^{\prime}$. If the measured values of the $M$-predictability and the
$M^{\prime}$-predictability for this state violate the inequality of Eq. (14),
the experiment does not admit of a noncontextual ontological model.

The specialization of this result to the case where $M$ and $M^{\prime}$ are
the which-way and which-phase measurements, can be summarized as follows:

######  Corollary V.2.1.

Consider an experimentally realizable state that satisfies the
$A_{1}^{2}$-orbit-realizability condition relative to the which-way and which-
phase measurements. If the measured values of the path distinguishability
$\mathcal{P}$ and the fringe visibility $\mathcal{V}$ for this state violate
the inequality $\mathcal{V}+\mathcal{P}\leq 1$ (Eq. (15)), the experiment does
not admit of a noncontextual ontological model.

For concreteness, we now describe a quantum set-up that, according to the
theoretical ideal, allows for the preparation of a state that saturates the
quantum wave-particle duality relation while also satisfying the
$A_{1}^{2}$-orbit-realizability condition.

Consider the preparation procedure depicted in Fig. 2 (c), but where the phase
shift $\phi$ is limited to be 0 or $\pi$, and the reflectivity is limited to a
pair of values $r$ and $1-r$ which are bounded away from 0 and 1. In quantum
theory, these procedures are represented by the following quadruple of pure
states of the dual-rail qubit
$\\{\rho_{i}=|\psi_{i}\rangle\langle\psi_{i}|\\}_{i=1,\dots,4}$, where

$\displaystyle|\psi_{1}\rangle$
$\displaystyle=\sqrt{r}|R\rangle+\sqrt{1-r}|L\rangle,$
$\displaystyle|\psi_{2}\rangle$
$\displaystyle=\sqrt{r}|R\rangle-\sqrt{1-r}|L\rangle,$
$\displaystyle|\psi_{3}\rangle$
$\displaystyle=\sqrt{1-r}|R\rangle-\sqrt{r}|L\rangle,$
$\displaystyle|\psi_{4}\rangle$
$\displaystyle=\sqrt{1-r}|R\rangle+\sqrt{r}|L\rangle.$ (16)

One easily verifies that these form an $A_{1}^{2}$-orbit relative to the
which-way and which-phase observables.

A simple interferometric implementation of these four states is depicted in
Fig. 5. Rather than requiring two distinct beamsplitters, one with
reflectivity $r$ and the other with reflectivity $1-r$, one uses the
reflectivity-$r$ beamsplitter to simulate the reflectivity-$(1-r)$
beamsplitter by following it up with a swap operation between the modes.

Figure 5: Preparation procedures associated to each of the four pure quantum
states in Eq. (V.2).

In such an experiment, one is aiming to implement the theoretical ideal of
each of the four preparation procedures described in Fig. 5 as well as the
theoretical ideal of the which-way measurement and the which-phase measurement
depicted in Figs. 2 (a) and b) respectively. However, the idealization of
purity and sharpness is never realized in practice. Rather, one realizes noisy
and inaccurate versions of the pure states of Eq. (V.2) and of the sharp
measurements $\\{|L\rangle,|R\rangle\\}$ and $\\{|+\rangle,|-\rangle\\}$. It
follows that one must characterize precisely which states and effects are
realized in the experiment. This can be achieved using the tomography scheme
of Refs. Mazurek  _et al._ (2021); Grabowecky  _et al._ (2022).

The latter scheme does not presume the correctness of quantum theory, but
rather fits the data to states and effects in the framework of generalized
probabilistic theories. This aspect of the scheme—that it is theory-
agnostic—enables one to adjudicate between quantum theory and alternative
operational theories, such as those that do admit of a noncontextual model or
those that manifest greater inequality violations than quantum theory permits.
In this way, one obtains real-valued vector representations of the two
realized measurements, as well as real-valued vector representations
$\vec{s}_{1},\vec{s}_{2},\vec{s}_{3},\vec{s}_{4}$ of the realized preparation
procedures.444Without presuming the correctness of quantum theory, one cannot
know in advance what sets of preparations and measurements are tomographically
complete, so the plausibility of the hypothesis of completeness depends on how
much work has been done in attempting to falsify it. The fact that the
hypothesis could in principle be falsified in the future by some exotic
laboratory procedure is the most significant loophole in experimental tests of
noncontextuality. See the introduction of Ref. Mazurek  _et al._ (2021);
Grabowecky  _et al._ (2022) or Refs. Pusey  _et al._ (2019); Schmid and Selby
for more discussion of this point.

The four states that are actually realized in the experiment, by virtue of
being noisy and inaccurate versions of the theoretical ideals, will generally
only approximately satisfy the $A_{1}^{2}$-orbit-realizability condition
relative to the realized measurements (Eqs. (V.1) and (13)). However, one can
always define a quadruple of secondary states lying in the convex hull of
those that were realized, such that these do satisfy the condition. Every
state that is a convex mixture of the realized states must be included in the
operational theory, since the state space is presumed to be convexly closed.
As a result, these secondary states are known to be included in the set that
are experimentally realizable. Hence, even though the states that were
actually realized in the experiment may not satisfy the $A_{1}^{2}$-orbit-
realizability condition exactly, one can always define secondary states that
do, and if these violate the noncontextual bound, one has witnessed the fact
that the experimental data cannot be explained by a noncontextual model. (See
Ref. Mazurek  _et al._ (2016a) for a discussion of the technique of secondary
procedures.)

Evidently, choosing the reflectivity $r=3/4$ for the beamsplitter in the
preparation stage provides the best opportunity for witnessing contextuality,
as this is the value that, in the ideal version of the experiment, leads to
the largest violation of the noncontextual bound.

##  VI Discussion

As seen in Fig. 3, the quantum tradeoff curve intersects the noncontextual
bound at $(\mathcal{V},\mathcal{P})=(1,0)$ and
$(\mathcal{V},\mathcal{P})=(0,1)$. As noted earlier, the
$(\mathcal{V},\mathcal{P})=(1,0)$ point corresponds to the preparation of a
which-phase eigenstate and the $(\mathcal{V},\mathcal{P})=(0,1)$ point
corresponds to the preparation of a which-way eigenstate. These are the only
cases that are typically referenced by researchers who claim that interference
phenomena capture the essence of quantum theory. In other words, they are the
only cases that appear in the TRAP phenomenology. In Ref. Catani  _et al._
(2021), it was shown that these cases admit of a noncontextual ontological
model, thereby undermining the conventional claim. The present article
provides an alternative way to see that this is the case, and hence provides
an alternative way of undermining the conventional claim.

The TRAP phenomenology of interference is fully captured by preparations of
the dual-rail qubit that only make use of a 50-50 beamsplitter and relative
phase shifts that only take values 0 or $\pi$ (together with the which-way and
which-phase measurements). For any departure from this class of preparation
procedures, the question arises of whether one can provide a classical
explanation of it. Of course, to vindicate the claim that a particular
experimental set-up resists classical explanation, one must prove a no-go
theorem, a methodological point that was emphasized in Sec. V.B.4 of Ref.
Catani  _et al._ (2021).

We noted in the introduction that it is actually straightforward to prove a
noncontextuality no-go theorem using a dual-rail qubit. It suffices to
translate into the concrete framework of a dual-rail qubit any of the existing
proofs based a single qubit. There are many to choose from.555Those that
leverage only operational equivalences among the preparations include the
first proof of the failure of generalized noncontextuality, based on states
forming a star of David in the Bloch sphere Spekkens (2005), the proofs based
on the probability of success in 2-bit or 3-bit parity-oblivious multiplexing
Spekkens  _et al._ (2009), the proof based on the probability of error-free
state discrimination of a pair of nonorthogonal states Schmid and Spekkens
(2018), the proof based on the probability of unambiguous discrimination of a
pair of nonorthogonal states Flatt  _et al._ (2022), and the proof based on
the probability of success in probabilistic cloning of a pair of nonorthogonal
states Lostaglio and Senno (2020). In addition, there are those that also
leverage operational equivalences among the measurements. These include the
first proof based an assumption of noncontextuality for unsharp measurements,
given in Sec. V of Ref. Spekkens (2005), which was also used in Ref. Mazurek
_et al._ (2016b), and the proof based on three unsharp measurements that are
jointly measurable pairwise but not triplewise Kunjwal and Ghosh (2014). There
are even proofs that leverage operational equivalences among transformations.
These include the original proof of the failure of transformation
noncontextuality Spekkens (2005), the proof arising from different
decompositions of the completely depolarizing channel in terms of Clifford
unitaries Lillystone  _et al._ (2019), and proofs based on the transformative
aspect of measurements, which concern pre- and post-selected scenarios Pusey
(2014); Pusey and Leifer (2015); Kunjwal  _et al._ (2019).  Furthermore,
because many of these only make use of states and effects in a single plane of
the Bloch sphere666Of the ones listed in footnote 5, the only ones that are
not of this type, i.e., the only ones that extend out of a single plane of the
Bloch sphere, are the 3-bit scheme of Ref. Spekkens  _et al._ (2009), and the
proofs in Refs. Kunjwal and Ghosh (2014) and Lillystone  _et al._ (2019).  it
follows that simple modifications of the experimental set-up appearing in the
TRAP phenomenology are sufficient to obtain a no-go result.

For instance, if, in the preparation stage, one keeps the reflectivity of the
beamsplitter at $\frac{1}{2}$ (as is the case for the TRAP phenomenology) but
one allows relative phase shifts in the full region $[0,2\pi]$ rather than
simply 0 or $\pi$, then this is sufficient to prepare any state in the
$\hat{x}{-}\hat{y}$ plane of the Bloch sphere (relative to the association of
Pauli-$Z$ and Pauli-$X$ with the bases $\\{|L\rangle,|R\rangle\\}$ and
$\\{(1/\sqrt{2})(|L\rangle+|R\rangle),(1/\sqrt{2})(|L\rangle-|R\rangle)\\}$).
If one has more than a single measurement in this plane, for instance, one
allows measurements of the bases
$\\{(1/\sqrt{2})(|L\rangle+|R\rangle),(1/\sqrt{2})(|L\rangle-|R\rangle)\\}$
and
$\\{(1/\sqrt{2})(|L\rangle+i|R\rangle),(1/\sqrt{2})(|L\rangle-i|R\rangle)\\}$,
that is, both the Pauli-$X$ and Pauli-$Y$ observables, then one can derive a
no-go result. However, such a result cannot obviously be interpreted in terms
of wave-particle duality. Indeed, it is not clear whether it would have any
significance in terms of aspects of the phenomenology of interference that
were of prior interest to researchers.

But now consider the case where, in the preparation stage, the relative phase
shift is still constrained to be 0 or $\pi$ (as is the case for the TRAP
phenomenology), but the reflectivity of the beamsplitter is allowed to be
anything in the range $[0,1]$. This is also sufficient to prepare any state in
the plane of the Bloch sphere spanned by the eigenstates of the which-way and
which-phase observables (i.e., the $\hat{x}{-}\hat{z}$ plane). Together with
the which-way and which-phase measurements, this is sufficient to derive a no-
go result. The no-go result described in this article is an instance of this
case. Because it refers to the quantum tradeoff between the fringe visibility
and the path distinguishability, which has been previously thought to be
significant in the context of interferometry, it satisfies the desideratum we
articulated in the introduction.

Recent work by Wagner et al. Wagner  _et al._ (2022a) also considered what
aspects of the phenomenology of interference might resist explanation in terms
of a noncontextual ontological model. In particular, they translate the qubit
proofs of contextuality of Refs. Galvão and Brod (2020); Wagner  _et al._
(2022b) into the language of a dual-rail qubit. This no-go result does not (as
far as we can tell) directly relate to phenomena that have been previously
thought to be significant in the context of interferometry. Nonetheless,
Wagner et al. also demonstrate a no-go result based on the probability of
succeeding in the task of quantum interrogation, which was of prior interest
in the interferometric setting. Quantum interrogation is a generalization of
the bomb-testing task considered by Elitzur and Vaidman Elitzur and Vaidman
(1993) wherein one seeks to maximize the probability of detecting the presence
of a detector without causing it to fire (or of detecting a bomb without
exploding it). The aspect of the task that Wagner et al. show can witness
contextuality is the dependence of the efficiency (a function of the
probability of success in the task) on a parameter describing the
confusability of the states (related to the nonorthogonality of the states in
the quantum case). Since the probability of success in this task has been of
prior interest in the context of interferometry, this result provides the type
of answer to the question of “what is nonclassical about the phenomenology of
interference?” that we are looking for. It is a complementary answer to the
one we have given here.

Given that there has already been an experiment showing that the quantum wave-
particle duality relation considered here (wherein the which-way and which-
phase measurements are counterfactual alternatives) can be saturated to good
approximation Liu  _et al._ (2012), the question naturally arises whether this
experiment already demonstrated the nonclassical aspects of the phenomenology
of interference highlighted in this article. The answer is: not quite.
Although it did confirm that there exist states that can approximately
saturate the ideal quantum tradeoff, it did not verify that these states
satisfy the $A_{1}^{2}$-orbit realizability condition. Only if the latter
condition is satisfied is one justified in inferring from the assumption of
noncontextuality that the tradeoff is bounded by a linear curve. Furthermore,
in order to allow for the possibility of a failure of operational quantum
theory in the experiment (for instance, to allow for the possibility of
violations of the noncontextual bound greater than what quantum theory
allows), it is important to characterize the preparations and measurements
using the technique of theory-agnostic tomography Mazurek  _et al._ (2021);
Grabowecky  _et al._ (2022). It is relative to these characterizations that
one can identify a quadruple of preparations that satisfy the
$A_{1}^{2}$-orbit-realizability condition, and then test whether these yield
values of fringe visibility $\mathcal{V}$ and path distinguishability
$\mathcal{P}$ that violate the noncontextuality inequality
$\mathcal{V}+\mathcal{P}\leq 1$. In this way, one can implement a direct
experimental test of contextuality via the wave-particle duality relation.
This relation, therefore, is an aspect of the phenomenology of interference
that can witness nonclassicality.

## Acknowledgments

This research was supported by Perimeter Institute for Theoretical Physics and
by funding from QuantERA/2/2020, an ERA-Net cofund in Quantum Technologies,
under eDICT project. 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. RWS was also supported by the
Natural Sciences and Engineering Research Council of Canada (Grant No.
RGPIN-2017-04383). ML was supported in part by the Fetzer Franklin Fund of the
John E. Fetzer Memorial Trust and by grant number FQXi-RFP-IPW-1905 from the
Foundational Questions Institute and Fetzer Franklin Fund, a donor advised
fund of Silicon Alley Community Foundation. LC acknowledges funding from the
Einstein Research Unit ‘Perspectives of a Quantum Digital Transformation’. DS
acknowledge support by the Foundation for Polish Science (IRAP project, ICTQT,
contract no.2018/MAB/5, co-financed by EU within Smart Growth Operational
Programme). GS is thankful to the group QUANTUM from Bari for the invaluable
scientific and human support they provided during the early stage of this
project.

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