## Abstract

Wave–particle duality, superposition and entanglement are among the most counterintuitive features of quantum theory. Their clash with our classical expectations motivated hidden-variable (HV) theories. With the emergence of quantum technologies, we can test experimentally the predictions of quantum theory versus HV theories and put strong restrictions on their key assumptions. Here, we study an entanglement-assisted version of the quantum delayed-choice experiment and show that the extension of HV to the controlling devices only exacerbates the contradiction. We compare HV theories that satisfy the conditions of objectivity (a property of photons being either particles or waves, but not both), determinism and local independence of hidden variables with quantum mechanics. Any two of the above conditions are compatible with it. The conflict becomes manifest when all three conditions are imposed and persists for any non-zero value of entanglement. We propose an experiment to test our conclusions.

## Introduction

Quantum mechanics is proverbially counterintuitive^{1,2}. For many years,
thought experiments were used to dissect its puzzling properties, while hidden-variable
(HV) models strived to explain or even to remove them^{1,2,3,4}. The
development of quantum technologies^{5,6} enabled us not only to perform
several former gedanken experiments^{1,2}, but also to devise new ones^{7,8,9,10,11}. One can gain new insights into quantum foundations by
introducing quantum controlling devices^{10,11,12} into well-known
experiments. This has led, for example, to a reinterpretation^{11,12,13,14} of Bohr’s complementarity principle^{15}.

Wave–particle duality is best illustrated by the classic Wheeler delayed-choice
experiment (WDC)^{16,17,18}, Fig. 1a,b. A photon
enters a Mach–Zehnder interferometer (MZI) and its trajectory is coherently
split by the beamsplitter BS_{1} into an upper and a lower path. The upper path
contains a variable phase shift *ϕ*. A random number generator controls
the insertion (*b*=1) or removal (*b*=0) of a second
beamsplitter BS_{2}. If BS_{2} is present, the interferometer is closed
and we observe an interference pattern depending on the phase shift *ϕ*.
If BS_{2} is absent, the MZI is open and the detectors measure a constant
probability distribution independent of *ϕ*. Thus, depending on the
experimental setup, the photon behaves in two completely different ways. In the case of
the closed MZI, the interference pattern suggests that the photon travelled along both
paths simultaneously and interfered with itself at the second beamsplitter
BS_{2}, hence showing a wave-like behaviour. However, if the interferometer
is open, since always only one of the two detectors fires, one is led to the conclusion
that the photon travelled only one path, hence displaying a particle-like behaviour.

The complementarity of the interferometer setups required to observe particle or wave
behaviour obscures the simultaneous presence of both properties, allowing the
(objective) view that, at any moment of time, a photon can be either a particle or a
wave. The WDC experiment uncovers the difficulty inherent in this view by randomly
choosing whether or not to insert the second beamsplitter (BS_{2}) after the
photon enters the interferometer (Fig. 1a). This delayed choice
prevents a possible causal link between the experimental setup and the
photon’s behaviour: the photon should not know beforehand if it has to behave
like a particle or like a wave.

The delayed-choice experiment with a quantum control (Fig. 1c)
highlights the complexity of space–time ordering of events, once parts of the
experimental setup become quantum systems^{11}. The quantum-controlled
delayed-choice experiment has been recently implemented in several different
systems^{19,20,21,22,23}. To ensure the quantum behaviour of the
controlling device, one can either test the Bell inequality^{23} or use an
entangled ancilla^{19}.

The theoretical analysis of the quantum WDC involved so far a single binary HV
*λ* describing the classical concepts of wave/particle. Here, we
introduce a full HV description for both the photon A and the ancilla. We analyse the
relationships between the concepts of determinism, wave–particle objectivity
and local independence of HV in the entanglement-controlled delayed-choice experiment.
We show that, when combined, these assumptions lead to predictions that are different
from those of quantum mechanics, even if any two of them are compatible with it. We
propose and discuss an experiment to test our conclusions.

## Results

### Notation

We use the conventions as in refs 3, 12; *q*(*a*, *b*,…) are the
quantum-mechanical probability distributions and *p*(*a*, *b*, .
. . , Λ) the predictions of HV theories with a HV Λ. We
consider either a single HV Λ, which fully determines behaviour of
the system, or refine it as Λ_{1}, Λ_{2}
pertaining to different parts of the system. For simplicity, we assume
Λ is discrete; the analysis can be easily generalized to the
continuous case.

### Quantum system

The system we analyse consists of three qubits: a photon A and an entangled pair
BC (Fig. 1d). We denote the measurement outcomes for the
photon A as *a*=0, 1 and for the two ancilla qubits as *b*
and *c*; the corresponding detectors are D_{A}, D_{B} and
D_{C}. The system is prepared in the initial state ; for , BC is a maximally
entangled EPR pair.

Photon A enters a MZI in which the second beamsplitter is quantum-controlled by
qubit B. The third qubit C undergoes a *σ*_{y}
rotation followed by a measurement in the
computational basis. The state before the measurements is

The counting statistics that result from the particle-like state and the wave-like state are discussed below (equations (3 and 4) and Methods).

### Constraints on HV theories

Our strategy is to show that *q*(*a*, *b*, *c*) cannot result
from a probability distribution *p*(*a*, *b*, *c*,
Λ) of a HV theory satisfying the requirements of
wave–particle objectivity, local independence and determinism. Any
viable HV theory should satisfy the adequacy condition: namely, it should
reproduce the quantum statistics by summing over all HVs Λ:

We encapsulate the additional classical expectations into three assumptions (see Box 1 for the formal definitions of the concepts we consider in this section).

For a given photon, we require the property of being a particle or a wave to be
objective (intrinsic), that is, to be unchanged during its lifetime. This
condition selects from the set of adequate HV theories those models that have
meaningful notions of particle and wave^{11}. For each photon, the
HV Λ should determine unambiguously if the photon is a particle or a
wave, thus allowing the partition of the set of HVs into two disjoint subsets, =_{p}∪_{w}, where the
subscript indicates the property, particle or wave.

The particle (wave) properties are abstractions of the particle (wave) counting statistics in open (closed) MZI, respectively. The behaviour of a particle (wave) in a closed (open) MZI is not constrained; this allows for significant freedom in constructing HV theories. Experimentally, the wave or particle behaviour depends only on the photon and the settings of the MZI:

for all values of *a*, *b*, *c* and Λ.

By replacing the single-qubit ancilla with an entangled pair, one can take
advantage of both the quantum control and the space-like separation between
events. The rationale behind the third qubit C is that it allows us to choose
the rotation angle *α* after both qubits A (the photon) and B
(the quantum control) are detected. This is not possible in the standard quantum
WDC^{11}, Fig. 1c, where the quantum
control B has to be prepared (by setting the angle *α*) before it
interacts with A. As discussed in Methods, there is a unique assignment of
probabilities that satisfies all the requirements of adequacy,
wave–particle objectivity and determinism. Adopting this assignment,
we reach the same level of incongruity as in ref. 11, since the probability *p*(*λ*) of photon
A being a particle or a wave is determined by the entanglement between B and
C,

This incongruity becomes an impossibility when the photon A and the entangled
pair BC are prepared independently. In this case, their HVs are generated
independently; that is, a single HV Λ not only has the structure
Λ=(Λ_{1},Λ_{2}),
where the subscripts 1 and 2 refer to the photon A and the pair BC,
respectively, but the prior probability distribution of HV has a product form.
To realize this condition experimentally, we rely on the absence of the
superluminal communication and a space-like separation of the two events.

Unlike the typical Bell-inequality scenarios, we have a single measurement setup
which involves two independent HV distributions. Moreover, by performing the
rotation *R*_{y}(*α*) and the detection
D_{C} sufficiently fast, such that the information about A and
Λ_{1} cannot reach the detector D_{C}, the
detection outcome is determined only by Λ_{2}. Since being a
wave (particle) is assumed to be an objective property of A,
*λ*=*λ*(Λ_{1})
is a binary function of the HV Λ_{1} only.

### Contradiction

We show in Methods that for *η*≠0, 1 (these two cases
correspond to an always closed or opened MZI), the requirements of adequacy,
wave–particle objectivity, determinism and local independence are
satisfied only if

This proves our main theoretical result: determinism, local independence and
wave–particle objectivity are not compatible with quantum mechanics
for any *α*≠±*π*/4,
±3*π*/4. We will later discuss how exactly a HV
theory that satisfies the three classical assumptions is inadequate.

### Proposed experiment

In Fig. 2, we show the proposed experimental setup for the entanglement-controlled delayed-choice experiment. Two pump pulses (blue) are incident on two nonlinear crystals and generate via spontaneous parametric down-conversion two pairs of entangled photons (red). One of the photons is the trigger and the other three are the photons A, B, C, with BC being the entangled pair.

Photons A and B are held in the lab (with appropriate delay lines) and together
they implement the controlled MZI. The central element is the quantum switch,
which is the controlled-Hadamard gate
*C*(*H*)=(*W*⨂*I*)*C*(*Z*)(*W*⨂*I*),
where . The photonic controlled-*Z* gate
*C*(*Z*) is implemented with a partially polarizing beamsplitter
and is done probabilistically via post-selection^{24,25}. Optical
wave plates perform single-qubit rotations (gates *H*, *ϕ*
and *W*) on photon A. Photon C is sent through a channel at a distant
location, then measured in a rotated basis. Two independent lasers generate the
two photon pairs (Fig. 2 (refs 26, 27)); in this case, we can use
equation (6) to describe independent probability
distributions for Λ_{1} and Λ_{2}.

## Discussion

In this section, we consider how exactly a HV theory, which satisfies the three
classical assumptions, fails the adequacy test. The interference pattern measured by
the detector D_{A0} is
*I*_{A}(*ϕ*)=Tr(*ρ*_{A}|0〉〈0|),
with
*ρ*_{A}=Tr_{BC}|*ψ*〉〈*ψ*|,
the reduced density matrix of photon A. The data can be postselected according to
the outcome *c* resulting in *I*_{A|c}. The visibility of
the interference pattern (Methods) is
*V*=(*I*_{max}−*I*_{min})/(*I*_{max}+*I*_{min}),
where the min/max values are calculated with respect to *ϕ*. The
postselected visibility for *c*=0 is (Fig.
3)

The full (non-postselected) visibility is
*V*_{A}=1−*η* and gives
information about the initial entanglement of the BC pair. On the other hand, if one
assumes that the HV are distributed according to equation (6)
and satisfy the wave–particle objectivity and determinism, the visibility
is independent of *c*,

in contrast with the quantum-mechanical prediction (Fig. 3). Details of this calculation are in Methods.

This incompatibility between the basic tenets of HV theories and quantum mechanics
has two remarkable features. First, the contradiction is revealed for any,
arbitrarily small, amount of entanglement. This test is in sharp distinction with
Bell-type experiments insofar as our result is free from inequalities.
Wave–particle objectivity, revealed only statistically, is more intuitive
and technically milder than the assumption of sharp values of quantum incompatible
observables. Second, in our setup, any two of the classical ideas together are
compatible with the quantum-mechanical predictions. This fact, and the way we
arrived at the contradiction, invite questions concerning the internal consistency
of classical concepts^{28}.

## Methods

### Quantum-mechanical analysis

The initial state of photons A, B and C is

The ancilla qubits B and C are maximally entangled for . The final state before measurement is given by equation (1). From it, we calculate the quantum statistics
*q*(*a*, *b*, *c*), where each of *a*, *b*,
and *c* take the values {0, 1}. The probability distribution for
*c*=0 is

where the four entries correspond to the values (*a*,
*b*)=(00, 01, 10, 11). For *c*=1, we obtain

This in turn yields

For the probability distributions for *b* and
*c* are equal. If , B and C are no longer
maximally entangled and the symmetry between them is broken: a rotation
*α* on C no longer corresponds to a rotation
*α* on B. The conditional probabilities are

and from Bayes’ rule
*q*(*b*|*c*)=*q*(*c*|*b*)*q*(*b*)/*q*(*c*).

### Solution to the three constraints

We now show that it is possible to construct a HV model that is adequate,
objective and deterministic. The unknown parameters at our disposal are 16
probabilities *p*(*a*, *b*, *c*, *λ*). These
probabilities are derived from the underlying distribution
*p*(Λ) summed over appropriate domains. At this stage, we do not
enquire about the connection with the HV Λ. The probabilities
*p*(*a*, *b*, *c*, *λ*) satisfy seven
adequacy constraints, equations (13) and (14), plus the normalization constraint. The adequacy conditions can
be written as

In addition, equation (7) and the standard rules for the conditional probabilities, such as

imply the existence of four additional constrains,

The resulting linear system has a four-parameter family of solutions. However, a
straightforward calculation shows that for all these solutions
*p*_{4}(*a*, *b*, *c*, *λ*), the
resulting statistics in an open/closed MZI is independent of
*λ*,

that is, the statistics of D_{A} is determined solely by the state of the
interferometer.

We can avoid the reintroduction of wave–particle duality using a special solution

which imposes the *b*–*λ* correlation (compare
ref. 11). As a result,

and since the probabilities are positive,

the eight above probabilities are zero individually. The system appears overconstrained, but it still has a unique solution

In particular,

### Deriving the contradiction

In addition to the partition of according to the values
of *λ*=p, w, we will use the decomposition of the set
of HV according to the outcomes of D_{C}. The two branches
*c*=0, 1 correspond to the partition

where for Λ∈_{c}, the
outcome of D_{C} is *c*. The assumption of local independence
implies a Cartesian product structure

of the set of HV, where the subsets depend on the experimental setup. When the superscripts 1 and 2 on are redundant, we may omit them.

Now, we show that under the assumptions of adequacy and the three classical
assumptions of the wave–particle objectivity, determinism and local
independence, it is impossible to derive the solution *p*(*a*,
*b*, *c*, *λ*) with any arrangement of the
probabilities *p*(Λ). The probability of the outcome *c*
satisfies

To simplify the calculations, we enumerate the variables
Λ_{1,2} by the indices *i*, *j*, respectively.
The domain corresponds, according to the
hypothesis, to the index set *J*_{c} of
Λ_{2}, and the domains and
to the index sets *I*_{p}
and *I*_{w} of Λ_{1}, respectively. In
particular,

for some . The prior distribution of HV and the
domains of summation can depend on the parameters *η*,
*ϕ* and *α*.

The putative behaviour of a wave (*λ*=w) in an open
(*b*=0) interferometer and of a particle
(*λ*=p) in a closed (*b*=1) one
is characterized by two unknown distributions *x*_{ij},
*i*∈*I*_{w} and *y*_{ij},
*i*∈*I*_{p}, respectively

allowing for a possible dependence on a value of Λ_{2}. The
remaining two sets of variables are the probability distributions for *b*
conditioned on the values of HVs Λ:

The requirement of adequacy means that the proposed HV theory reproduces the
quantum statistics given above. For compactness, we refer to the probability of
having the HV values , , as *p*_{ij}, using the same convention as for
*x*_{ij}, *y*_{ij},
*z*_{ij} and *υ*_{ij}.
For *c*=0, we have

with analogous expressions for *c*=1. Adding and subtracting equations (39) and (41) we obtain,
respectively

Adding equations (40) and (42) yields

which on substitution back into equation (40) results in

Four additional equations (giving a total of seven independent equations) are
obtained for *j*∈ *J*_{1} with cos^{2}
*α*↔sin^{2}
*α*.

From equation (26), it follows that
*υ*_{ij}=0, *i* ∈
*I*_{w} and *z*_{ij}=1, *i*
∈ *I*_{p}. Hence, for *c*=0, only two
equations are not automatically satisfied,

The corresponding equations for *c*=1 are

which are in agreement with *q*(*c*), equation
(18).

Now we use the product structure of the probability distribution, equation (6),

Using equations (28) and (30), we find that

Adding the pairs of equations in (47) and (48) and summing over the index
*i*, we express the adequacy condition ,

but on the other hand, for *η*≠0, 1 summing over *i* in
each of these four equations separately and using equation
(50) we get

These equations can be satisfied for any *η* only if

resulting in the contradiction (for arbitrary *α*)
cos 2*α*=0.

### Experimental signature

The interference pattern measured by the detector D_{A} is
*I*_{A}(*ϕ*)=Tr(*ρ*_{A}|0〉〈0|),
with
*ρ*_{A}=Tr_{BC}|*ψ*〉〈*ψ*|
the reduced density matrix of photon A. The data can be postselected according
to the outcome *c* resulting in *I*_{A|c}. The
intensity (signal) measured by detector D_{A} for *c*=0
(and no post-selection on *b*) is:

giving the visibility

A similar calculation gives the visibility for *c*=1

The full intensity measured by detector D_{A} (without postselecting on
*c*) is and the corresponding
visibility

Thus the visibility of detector D_{A} gives information about the
entanglement of the BC pair.

We now calculate the visibilities predicted by a non-trivial HV theory that is assumed to satisfy the three classical assumptions. Using equation (26), we rewrite the counting statistics as

For the product probability distribution above, we get

for *j*=0, 1 separately, where . As
a result,

giving

for the visibilities in HV theories.

## Additional information

**How to cite this article:** Ionicioiu, R. *et al.* Is
wave–particle objectivity compatible with determinism and locality?
*Nat. Commun.* 5:4997 doi: 10.1038/ncomms5997 (2014).

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## Acknowledgements

D.R.T. thanks Perimeter Institute for support and hospitality. We thank Lucas Céleri, Jim Cresser, Berge Englert, Peter Knight, Stojan Rebić, Valerio Scarani, Vlatko Vedral and Man-Hong Yung for discussions and critical comments and Alla Terno for help with visualization. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. R.I. acknowledges support from the Institute for Quantum Computing, University of Waterloo, Canada, where this work started.

## Author information

## Affiliations

### Department of Theoretical Physics, National Institute of Physics and Nuclear Engineering, 077125 Bucharest-Măgurele, Romania

- Radu Ionicioiu

### Research Center for Spatial Information - CEOSpaceTech, University Politehnica of Bucharest, 313 Splaiul Independentei, 061071 Bucharest, Romania

- Radu Ionicioiu

### Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

- Thomas Jennewein
- & Robert B. Mann

### Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

- Thomas Jennewein
- & Robert B. Mann

### Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y6

- Robert B. Mann

### Department of Physics and Astronomy, Macquarie University, Sydney, New South Wales 2109, Australia

- Daniel R. Terno

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### Contributions

T.J. and R.I. conceived the entanglement-controlled protocol. D.R.T. performed the hidden-variable analysis. R.I. and T.J. produced the experimental design. R.B.M. and D.R.T. analysed the experimental signatures. All authors contributed to the writing of the manuscript. D.R.T. coordinated the project.

### Competing interests

The authors declare no competing financial interests.

## Corresponding author

Correspondence to Daniel R. Terno.

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