# Lying particles

Lev Vaidman  Raymond and Beverly Sackler School of Physics and Astronomy, Tel-
Aviv University, Tel-Aviv 69978, Israel

###### Abstract

The common feature of several experiments, performed and proposed, in which
particles provide misleading evidence about where they have been, is
identified and discussed. It is argued that the experimental results provide a
consistent picture when interference amplification effects are taken into
account.

##  1 Introduction

Perhaps the most significant difference between classical and quantum physics
is that quantum particles do not have well-defined trajectories (unless we
accept an arguably fringe Bohmian interpretation, [1]). Standard quantum
mechanics does not have a clear concept of the location of a particle. There
is no answer to the question: Through which slit did the particle pass in a
two-slit interference experiment? Still, there is a growing interest in
discussing which-path questions for particles passing through interferometers,
notably following Wheeler’s controversial proposal, [2].

Wheeler suggested assigning a well-defined path to the particle in the
interference experiment of the wave packets when only one wave packet had a
continuous trajectory from the source to the detector. I proposed an
alternative approach, [3], in which the particle was present in the locations
where it left a significant trace in the environment. According to my
definition, a pre and postselected particle can be in several places
simultaneously, and, moreover, the particle might have disconnected regions of
presence. My proposal created a significant controversy, [4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18].

To demonstrate the usefulness of my definition, I led an experiment, [19], in
which the photons themselves told us where they had been inside an
interferometer, which only increased the controversy, [20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. Although the experiment was supposed
to demonstrate the local trace left by photons in different locations, because
of the difficulty of performing conditional counting, the measurement pointer
was the photon itself: the photon’s transversal degree affected by the local
interaction with the environment. The justification of this modification in
the experiment of [19] was the fact that the record in this degree of freedom
of the photon was not distorted during the time the photon wave packet was
moving toward the detector.

I also argued that there was another way to interpret this experiment as a
demonstration of the presence of the particle in various interferometer paths.
All interactions are local. If the pre and postselected photons bring
information about a disturbance which was solely introduced in one location,
then the photon tells us that it was there. This sounds like a very strong
argument, but it turned out to be a subtle issue. This is what I analyze in
this report. The analysis demonstrates the core of the resolution of the
debates [36] and [37], [38] and [39], [40] and [41], [42] and [43].

##  2 The general setup

We consider a particle entering an interferometer with multiple paths in a
particular input state that reaches a particular output port. The
interferometer has mirrors and beam splitters that cause a well-defined
unitary evolution of the degree of freedom of the path of the particle. In
addition, the particle has another degree of freedom with a specified state
$|\Phi\rangle$ at the input port which is observed at the output port. For my
analysis, I assume that if the interferometer is undisturbed, the state of
this degree of freedom remains unchanged everywhere inside the interferometer.
The presence of the pre and postselected particle in a particular path of the
interferometer is then characterized by the effect of the local disturbance in
this path on the state of the particle at the output port. Any disturbance of
the non-path degree of freedom in a particular path can always be expressed as

$|\Phi\rangle\rightarrow\mathcal{N}(|\Phi\rangle+\epsilon|\Phi_{\perp}\rangle),~{}~{}~{}~{}\epsilon\ll
1,$ (1)

where $\langle\Phi|\Phi_{\perp}\rangle=0$. The non-path degree of freedom of
the state in the output port is:

$|\Phi\rangle_{out}=\mathcal{N}(|\Phi\rangle+\alpha\epsilon|\Phi_{\perp}\rangle).$
(2)

The parameter $\alpha$ characterizes the modification of the effect of the
interaction and thus the amount of presence of the particle in the respective
path.

Figure 1:  A particle tells us about its presence. a) Reference signal from
the path certainly taken by the particle. b)-d) Obtaining the signal of
presence of the particle (modification of the state reaching the detector) in
different paths of pre and postselected particle when a beam splitter is
added. The ellipse with $\epsilon$ indicates the location where the
disturbance is introduced to test for the presence of the particle.

As expected, the one-path “interfereometer” has a measure of presence in the
path $\alpha=1$, Fig. 1a. If we add a beamsplitter, Fig. 1b-d, but consider
the particle which reaches the detector, then the particle has presence

$\alpha_{\rm IN}=\alpha_{\rm OUT}=1,~{}~{}\alpha_{\rm REF}=0.$ (3)

If we consider a balanced Mach-Zehnder interferometer (MZI) Fig. 2a, tuned to
constructive interference, then we are not surprised to obtain

$\alpha_{\rm IN}=\alpha_{\rm OUT}=1,~{}~{}\alpha_{A}=\alpha_{B}=\frac{1}{2}.$
(4)

If MZI has beam splitters reflecting $90\%$, which is also tuned to
constructive interference, Fig. 2b, we obtain

$\alpha_{\rm IN}=\alpha_{\rm
OUT}=1,~{}~{}\alpha_{A}=0.9,~{}~{}\alpha_{B}=0.1.$ (5)

If we tune this interferometer to the minimum intensity at the detector, Fig.
2c, then the experiment shows marks of the presence in the arms of the
interferometer which are less intuitive

$\alpha_{\rm IN}=\alpha_{\rm
OUT}=1,~{}~{}\alpha_{A}=\frac{9}{8},~{}~{}\alpha_{B}=-\frac{1}{8}.$ (6)

However, I do not consider these results to be incorrect. They provide the
same characterizations of the presence of a pre and postselected particle at
particular locations as given by the weak value of the projection on these
locations. They describe the universal property of modifications of all weak
couplings at these locations, [44]. The particles do not lie about their
presence (in the above sense), which can be larger than 1 or negative, or even
corresponding to a complex number. The presence can be anomalously large; the
only constraint is that at any moment of time one particle should have a total
presence in all paths of an interferometer equal to 1.

Figure 2:  A Mach-Zehnder interferometer. a) Balanced MZI tuned to
constructive interference. [A device inserted into arm $B$ which introduces
relative phase $\pi$ in the non-path degree of freedom makes the particles lie
about its presence.] b) Unbalanced MZI tuned to constructive interference. c)
Unbalanced MZI tuned to maximally destructive interference.

Particles also do not lie when they tell us about a surprising picture of
presence in nested MZI, [3, 19], see Fig. 3a.

$\alpha_{\rm IN}=\alpha_{\rm
OUT}=1,~{}~{}\alpha_{C}=1,~{}~{}\alpha_{E}=\alpha_{F}=0,~{}~{}\alpha_{A}=1,~{}~{}\alpha_{B}=-1.$
(7)

I consider these results to be a correct demonstration of the presence of the
particle in the interferometer, since it faithfully represents the trace left
on the paths by the particle. Every mirror in the path gets a kick equal to
$\alpha\sqrt{2}p$ where $p$ is the momentum of the particle. (Note that in
experiment [19] a special arrangement with disturbances with different
frequencies was implemented. This allowed the measurement of the presence in
all paths simultaneously, but instead of $\alpha$, only the absolute value of
the signal $|\alpha|$ was observed.)

Figure 3:  Nested Mach-Zehnder interferometer. a) The interferometer with a
particle present in a disconnected region, cf. Fig. 2b. of [19]. b) The same
interferometer with a device introduced in path E which affects the non-path
degree of freedom of the particle making it to lie about where it has been.
The traces left by the particle on the parts of the interferometer are
essentially the same in the two cases.

##  3 Interferometers with particles lying about where they have been

Let us now consider cases where particles lie about their presence in a
particular path of the interferometer. Our setup is the same as above, but we
add in one of the paths a device which affects the non-path degree of freedom
in the following way. It adds the relative phase $\pi$ to the component
$|\Phi_{\perp}\rangle$ which is created by the disturbance (1):

$a|\Phi\rangle+b|\Phi_{\perp}\rangle\rightarrow
a|\Phi\rangle-b|\Phi_{\perp}\rangle.$ (8)

This corresponds to a modification of the experiment of [19] proposed by [38].
They inserted a Dove prism inside an interferometer. The non-path degree of
freedom of the particle was its transversal motion of the photon. The
disturbance was a transversal shift. The shift of a well-aligned Gaussian
profile created an orthogonal component which was an odd function in the
transverse direction. The action of the Dove prism on such a component is to
flip its sign, i.e., to add phase $\pi$. The other proposals ([36, 40, 42])
have an equivalent mathematical structure.

Introducing the relative phase to the component that is not present changes
nothing, so placing the device makes no effect if the interferometer is not
disturbed, but it causes the particles to lie (in some cases) about where they
have been by spoiling the mathematical equivalence between the local trace on
the environment and the observed trace on the non-path degree of freedom of
the particle. The records created locally on the particle are distorted during
the passage of the particle to the output port. I will show two examples of
how it happens.

Let us add this device at the beginning of the path $B$ of the balanced MZI
described in Fig. 2a. Then, our method of characterization of the presence of
the particle by observing the effect of introducing the disturbance (1) in
different paths on the non-path degree of freedom at the output port will
lead, instead of (4) to

$\alpha_{\rm IN}=0,~{}~{}\alpha_{\rm
OUT}=1,~{}~{}\alpha_{A}=\frac{1}{2},~{}~{}\alpha_{B}=-\frac{1}{2}.$ (9)

Clearly, this description does not make sense. We know that the particle
entered the interferometer. The method itself tells us that it left the
interferometer, but the method also tells us that the particle was not present
at the input port. The malfunction of the method is transparent: the
orthogonal component created at the input port splits into a superposition in
the arms $A$ and $B$ and the phase device in the arm $B$ causes destructive
interference in the output port. The absence of the signal from the input port
does not ensure that the particle was not there.

The second example is a nested interferometer of [19], which has already been
the subject of great controversy. Fig. 3b shows explicitly the phase $\pi$
device added in the arm $B$. Then, our method of characterization of presence
by observing the effect leads instead of (7) to

$\alpha_{\rm IN}=3,\alpha_{\rm
OUT}=1,~{}~{}\alpha_{C}=1,~{}~{}\alpha_{E}=2,~{}~{}\alpha_{F}=0,~{}~{}\alpha_{A}=1,~{}~{}\alpha_{B}=-1.$
(10)

These results are neither present a sensible classical picture, nor describe
faithfully a modification of the weak couplings of the particle with the
environment which are, in fact, not affected by the phase shifting device and
remain to be described correctly by (7).

##  4 Discussion

In Fig. 3b. I show the experiment with a nested interferometer in which we get
a signal from the path $E$, but I claim that it was not there. How can the
particle bring a signal from the place it was not present? It contradicts the
idea of causality and locality of all interactions.

The answer to this question is not simple. In fact, there is some kind of
particle presence in $E$, also in the interferometer described in Fig. 3a
without $\pi$ shifter, which I named “secondary” presence, [45]. The main
property of the secondary presence is that although there is no trace of the
first order (in $\epsilon$) in $E$, placing a block in $E$ will affect the
first order traces elsewhere (in $A$ and $B$). Also, when the interferometer
is not ideal (and there are no ideal interferometers) there will be a higher
order trace in $E$.

The question to be answered is how the phase shifter causes particles to lie
about their presence showing a strong signal in $E$ that looks like a signal
of ordinary presence. This is due to the anomalous sensitivity to the presence
in $E$ when the phase device is inserted. If we add a nondemolition
measurement of the presence in $E$ and condition our observation of the signal
on finding the particle in $E$, our signal instead of $\alpha_{E}=2$ will be
$\alpha_{E~{}[{\rm conditioned~{}on~{}presence~{}in~{}E}]}\rightarrow\infty$.
This follows from the definition of $\alpha$ in (2), since the state at the
output port is $|\Phi_{\perp}\rangle$.

The locality of interaction tells us that if the particle brings a signal from
a particular location, the particle must have some presence there, but the
“amount of presence” is not characterized directly by this signal. It is
characterized by the ratio between the obtained signal and the reference
signal that would be obtained if the particle were localized there. Since in
the nested interferometer with the phase shifter the signal is finite, but the
localized particle in this experiment leads to an unbounded signal, the ratio
is zero. The method tells us that the particle was not present there. See the
analysis of a realistic experiment in [42, 43].

I suggest adopting the definition according to which the particle passing
through an interferometer was in a particular path if a local disturbance in
this path (if it was performed) can be observed in the output state of the
particle. Every path which fulfills this criterion is the path where, by this
definition, the particle was. In many cases, then, the particle is present in
more than one path simultaneously. This definition contradicts the fact that a
single particle is never found in two places simultaneously. The setup I
proposed cannot demonstrate this contradiction, because the proposal
introduces disturbances in the paths one by one and not simultaneously
together. Let us see what happens if we introduce disturbances in two places
simultaneously.

We consider a particle passing MZI described in Fig. 2a and try to observe its
presence in arms $A$ and $B$. To obtain evidence that it is present
simultaneously in two places we need that the particle will have two non-path
degrees of freedom $|\Phi_{A}\rangle$ and $|\Phi_{B}\rangle$ affected by the
disturbances in the two paths. Then, the non-path degree of freedom state of
the particle will be

$|\Phi_{A}\rangle|\Phi_{B}\rangle\rightarrow\mathcal{N}(|\Phi_{A}\rangle|\Phi_{B}\rangle+\frac{\epsilon}{2}|\Phi_{A\perp}\rangle|\Phi_{B}\rangle+\frac{\epsilon}{2}|\Phi_{A}\rangle|\Phi_{B\perp}\rangle).$
(11)

There is a first order signal both from path $A$ and from path $B$, so at this
stage, by definition, the particle was in two places. However, we will never
be able to find one particular particle in two places simultaneously.
Performing analysis of this state on an ensemble of particles starting at the
source and ending up at the output port we will observe these first order
traces, so we will be able to use the definition. This, however, does not
provide direct information about every particle in the ensemble. A particular
measurement which occasionally (with probability larger than 0) finds state
$|\Phi_{A\perp}\rangle$ tells us that the particle was in path $A$ and finding
$|\Phi_{B\perp}\rangle$ tells us that the particle was in path $B$. The
corresponding degrees of freedom in state (11) are entangled and the term
$|\Phi_{A\perp}\rangle|\Phi_{B\perp}\rangle$ is not present.

We do not have to adopt the many-world interpretation [46] to appreciate the
result of this paper, but it seems to me that the MWI provides the most
satisfactory resolution of the apparent paradox of claiming that the particle
in some sense is in two places simultaneously, in spite of the fact that we
are not able to find the particle simultaneously in two places. In a world the
particle passes through the interferometer, it has “memory” of order
$\epsilon$ of being both in $A$ and in $B$. However, at the moment when a
measurement verifying the presence of $|\Phi_{A\perp}\rangle$ is performed,
the world splits into two, one in which we know that it was in $A$ and one in
which we have no decisive information. Due to the entanglement of degrees of
freedom, in the world we know about the presence in the path $A$ the component
of order $\epsilon$ signifying the presence in path $B$ is erased. This is why
we never find definite evidence of simultaneous presence in the two arms of
the interferometer.

## Acknowledgments

I thank Jan Dziewior and Gregory Reznik for very helpful discussions. This
work has been supported in part by the National Science Foundation Grant No.
1915015 and the U.S.-Israel Binational Science Foundation Grant No. 735/18 and
by the Israel Science Foundation Grant No. 2064/19.

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