# Counterfactuality, back-action, and information gain in multi-path
interferometers

Jonte R. Hance  0000-0001-8587-7618<EMAIL_ADDRESS>School of
Computing, Newcastle University, 1 Science Square, Newcastle upon Tyne, NE4
5TG, UK  Quantum Engineering Technology Laboratories, Department of Electrical
and Electronic Engineering, University of Bristol, Woodland Road, Bristol, BS8
1US, UK  Tomonori Matsushita  0000-0001-7713-5374 Graduate School of Advanced
Science and Engineering, Hiroshima University, Kagamiyama 1-3-1, Higashi
Hiroshima 739-8530, Japan  Holger F. Hofmann  0000-0001-5649-9718
<EMAIL_ADDRESS>Graduate School of Advanced Science and Engineering,
Hiroshima University, Kagamiyama 1-3-1, Higashi Hiroshima 739-8530, Japan

###### Abstract

The presence of an absorber in one of the paths of an interferometer changes
the output statistics of that interferometer in a fundamental manner. Since
the individual quantum particles detected at any of the outputs of the
interferometer have not been absorbed, any non-trivial effect of the absorber
on the distribution of these particles over these paths is a counterfactual
effect. Here, we quantify counterfactual effects by evaluating the information
about the presence or absence of the absorber obtained from the output
statistics, distinguishing between classical and quantum counterfactual
effects. We identify the counterfactual gain which quantifies the advantage of
quantum counterfactual protocols over classical counterfactual protocols, and
show that this counterfactual gain can be separated into two terms: a semi-
classical term related to the amplitude blocked by the absorber, and a
Kirkwood-Dirac quasiprobability assigning a joint probability to the blocked
path and the output port. A negative Kirkwood-Dirac term between a path and an
output port indicates that inserting the absorber into that path will have a
focussing effect, increasing the probability of particles arriving at that
output port, resulting in a significant enhancement of the counterfactual
gain. We show that the magnitude of quantum counterfactual effects cannot be
explained by a simple removal of the absorbed particles, but originates
instead from a well-defined back-action effect caused by the presence of the
absorber in one path, on particles in other paths.

##  I Introduction

The term ‘counterfactual’ has been applied in very different ways in quantum
mechanics, and the link between these disparate uses is not immediately
obvious. On the one hand, the notion of ‘counterfactual definiteness’ has been
invoked as an implicit assumption of Bell inequalities and the similar class
of inequalities whose violation has been used to demonstrate quantum
contextuality—in this context, the ‘counterfactual definiteness’ refers to the
hypothetical assignment of outcomes to a specific measurement, when an
incompatible measurement has been carried out instead. The experimental
violation of the corresponding inequality can then be seen as an indication
that counterfactual definiteness does not hold when dealing with quantum
phenomena [1]. On the other hand, counterfactual computation [2, 3] allows us
to see what the result of a quantum computation would be without running the
computation, based on interference effects from the output of the computation.
This effect is based on interaction-free measurement [4, 5, 6, 7], where
supposedly information is gained about a quantum system through measurement
without any physical interaction with the system. Other ‘counterfactual’
effects have also been demonstrated based on interaction-free measurement:
counterfactual communication [8, 9, 10, 11] allows us to send messages (or
even quantum information [12, 13, 14, 15, 16]) optically across a channel,
without any light travelling across that channel when information is sent, and
counterfactual imaging [17, 18, 19] allows us to see what the optical
properties of a sample would be, without sending any light into that sample.
All counterfactual protocols based on interaction-free measurement make use of
interference effects to work, and the role of blocked paths, on the
interference effects observed in the output statistics of particles that did
not take these paths, are what originally led to these protocols being
referred to as ‘counterfactual’. However, the counterfactuality of these
protocols has been heavily debated [20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
11, 30]. Ref. [30] attempted to clarify the role of quantum effects in such
counterfactual protocols, comparing two of the protocols for counterfactual
quantum communication to classical methods of communicating using
counterfactuals, to identify whether there was anything special that set apart
the quantum protocols from closely-related classical methods. They concluded
that the quantum protocols combined (wave-like) interference effects with
(particle-like) discrete detection events, making use of the wave-particle
duality that is a unique feature of quantum mechanics. However, the use of
specific protocols made it difficult to identify the general role or meaning
of quantum interference in these protocols, and provided no quantitative
measure of the difference between quantum and classical counterfactuals.

In this paper, we go back to the principle underpinning all counterfactual
protocols inspired by interaction-free measurement—that there is a manifest
difference between the probability distributions we observe over the outcomes
of an interferometer, when comparing the case when one path within the
interferometer is blocked, with the case where it isn’t; and that this
difference is greater than the change to the distributions just caused by the
loss of particles in the blocked path alone. In Section II, we quantify the
counterfactual effects caused by inserting an absorber into one path of a
general interferometric setup by considering the statistical distance between
the output probabilities with or without the absorber. This statistical
distance represents the information that can be obtained regarding the
presence or absence of the absorber, where the information that can be gained
by simply removing the fraction of particles that is absorbed is upper bounded
by the absorption probability. Our results show that quantum interference
effects can increase this information significantly, allowing us to identify
the ‘counterfactual gain’ above the classical statistical distance which is
associated with these non-classical effects. A non-zero counterfactual gain is
only observed when there are quantum coherences between the blocked path and
the paths taken by the particles detected in the output of the interferometer.
Therefore, the counterfactual gain quantifies the effect of constructive and
destructive interferences between the counterfactual result and its
alternatives on the output distribution of photons that seemingly do not
interact with the absorber. This provides an intuitive indicator of the
difference between the quantum scenario and the classical notion of absorption
as a removal of particles without any effects on the remaining particles.

We separate this counterfactual gain into two terms: one term associated with
the amplitude of the blocked path in the output port, referred to as Elitzur-
Vaidman term due to its role in the original Elitzur-Vaidman bomb tester
protocol, and one term corresponding to a Kirkwood-Dirac quasiprobability [31,
32, 33, 34], which is absent in the Elitzur-Vaidman case. This Kirkwood-Dirac
term represents a generally non-positive joint probability of the blocked path
and the output port, corresponding directly to the reduction in each output
probability caused by the removal of particles in the blocked path. This
Kirkwood-Dirac term is subtracted from the counterfactual gain, reflecting the
fact that a mere removal of particles does not result in any counterfactual
gain. However, a negative Kirkwood-Dirac quasiprobability will increase the
counterfactual gain, and a non-zero counterfactual gain is observed whenever
at least one of the outcomes has a negative Kirkwood-Dirac term.

In Section III, we discuss the typical characteristics of counterfactual
effects observed in the output of a multi-path interferometer. We compare two
cases: one equivalent to the original Elitzur-Vaidman bomb tester, where the
counterfactual gain originates from one output port having probability zero of
the photon arriving in it in the case without an absorber, with a Kirkwood-
Dirac term of zero; and another where the initial output probabilities are all
equal, and a negative Kirkwood-Dirac term results in a counterfactual focusing
effect, where the presence of the absorber results in a significant increase
of the detection probability at one of the output ports. The focusing effect
described here shows that the presence of the absorber redirects the paths of
the photons that are not absorbed. The precise form of this redistribution is
analysed in Section IV, where we derive an explicit expression for the
measurement back-action of the non-detection of a particle in the path of the
absorber. In Section V, we illustrate these effects using Hofmann’s three-path
interferometer [35], showing how strong and counterintuitive this this
counterfactual gain can be in a scenario with a negative Kirkwood-Dirac term.
Finally, we summarise our argument in Section VI.

The counterfactual gain we demonstrate in this paper results from the loss of
coherence between states, which occurs when one of the states is lost by
observation—these quantum counterfactual effects cannot be explained simply by
removing the contributions to the output probabilities from the photons which
are lost to the absorber. Previous discussions of quantum counterfactuals only
highlighted this in specific situations, whereas we show this more generally:
that the underlying effect results from initial coherences which change when
one option is removed. The quantitative measure of counterfactual gain
demonstrates the difference between quantum contributions and classical
counterfactuals without qualitative arguments about the observation of events
that would not occur when the path is not blocked. Our theory of
counterfactual effects thus identifies the essential quantum effects behind a
much wider range of scenarios than those considered previously.

##  II Statistical analysis of counterfactual effects

In this Section, we show that quantum mechanical counterfactuals differ from
classical counterfactual logic due to the loss of coherence between the states
that have been excluded by the observation of a negative result, and the
states which are not excluded. We then develop a way to analyse the benefit
such coherence provides for counterfactual discrimination tasks—the
‘counterfactual gain’.

Let us start by considering what we would expect from a classical (particle-
like) attempt to discriminate whether or not an absorber has been inserted
into the arm of a single-particle interferometer. If the only effect of an
absorber inserted in a path $a$ of an interferometer was the absorption of any
photons which pass through that path, the observed probabilities at the output
ports would just reduce in proportion to how the absorbed photons would been
distributed across the outputs if there had not been a absorber. Due to the
loss of coherence between the path with the absorber and the other paths,
quantum mechanics predicts much greater changes to the output probabilities
than this classical constraint would allow. We should therefore quantify these
additional changes to the probability distribution in order to identify the
counterfactual gain between quantum and classical counterfactual effects.

Figure 1: A generic multi-path interferometer, in a situation where path $a$
can be either blocked or not blocked by an absorber. As discussed in the text,
the counterfactual scenario, where a single-photon goes through the
interferometer, is not absorbed by the blocker, and goes to one of the
detectors corresponding to outcomes $m$ (for $m$ between $m_{1}$ and $m_{n}$,
can and should have differing probability distributions for each of the
outcomes $m$ than the case where the absorber is not placed in path $a$.

Without any absorber, the output probabilities for the photon arriving at each
output port $m$ are given by $P(m)$. If all of the photons which pass through
path $a$ are absorbed, the probability that the photon is absorbed is $P(a)$.
The remaining photons, corresponding to a fraction of $1-P(a)$ of the total
number of photons sent sequentially through the interferometer, will arrive at
the output ports $m$ with probabilities of $P(m|X_{a})$, where $X_{a}$
represents the counterfactual condition that the photon did not pass through
path $a$.

Consider a scenario where we are trying to guess correctly whether an absorber
is present or absent. We are assuming this can be treated as being completely
unbiased—based on the information we have, it is equiprobable whether or not
the absorber was inserted. This assignment of equiprobability can be justified
in a Bayesian sense, as our prior being complete ignorance as to whether or
not the absorber was inserted. If our guess is based on a single trial, the
optimal strategy is to compare the probability $P(m)$ without the absorber to
the probability $P(m|X_{a})$ in the presence of an absorber. If the absorber
reduces the probability of the outcome ($P(m|X_{a})<P(m)$), the best guess
upon observing a photon arriving at output $m$ is that the absorber was
absent. While classically the absorber can only either reduce the probability
or keep it the same, to consider quantum scenarios we also need to take into
account the possibility that the presence of the absorber somehow increases
the probability of the outcome $m$ ($P(m|X_{a})>P(m)$), in which case the best
guess upon observing $m$ is that the absorber is present. These counter-
intuitive outcomes are essential for a counterfactual gain in our ability to
identify the presence of a detector: even though the photon was not absorbed,
the outcomes $m$ with $P(m|X_{a})>P(m)$ provide (non-classical) evidence for
the presence of the absorber. We can now quantify the total effect of this
counterfactual gain by finding the average error probability. Since our guess
will be wrong whenever the outcome $m$ was caused by the less likely
condition, this error probability is given by

$P_{\text{error}}=\frac{1}{2}\sum_{m}\text{Min}\left[P(m),P(m|X_{a})\right]$
(1)

In the case of a classical counterfactual situation ($P(m|X_{a})<P(m)$ for all
$m$), the presence of the absorber will not be detected unless the photon is
actually absorbed. This results in an average error probability of
$(1-P(a))/2$ for a classical counterfactual situation, where the absorption
simply removes some of the photons from the outputs $m$. For the evaluation of
the counterfactual gain, it is convenient to express the error probability
using the statistical distance,

$\Delta_{a}\equiv\frac{1}{2}P(a)+\frac{1}{2}\sum_{m}\left|P(m)-P(m|X_{a})\right|,$
(2)

so that the error probability can be given as

$P_{\text{error}}=\frac{1}{2}-\frac{1}{2}\Delta_{a}$ (3)

The statistical distance between the probabilities with and without the
absorber describes the suppression of errors in the guessing game described
above. In the classical counterfactual case, the statistical distance
$\Delta_{a}$ is equal to the probability of absorption given by $P(a)$.

The counterfactual gain, caused by the presence of quantum coherence between
the absorbed path $\left|a\right\rangle$ and the other paths in the
interferometer, is given by the difference between the statistical distance
$\Delta_{a}$ and the probability of absorption $P(a)$. As explained above, it
originates from outcomes $m$ with $P(m|X_{A})>P(m)$. Using Eq. (2), we find
that the counterfactual gain can be expressed as

$\Delta_{a}-P(a)=\sum_{P(m|X_{A})>P(m)}\left(P(m|X_{a})-P(m)\right).$ (4)

If the presence of an absorber in path $a$ increases the probability that
photons in other paths arrive at output $m$, obtaining that outcome is
evidence for the presence of the absorber and the increase in probability is
added to the statistical distance that describes the effects of the absorber
on the observable statistics. On closer inspection, the counterfactual gain
corresponds to the difference between the increase in the probability of
detecting the absorber gained by assigning absorber detection to the outcome
$m$, and an increase in the probability of false positives associated with the
probability $P(m)$ of obtaining the same result in the absence of the
absorber.

We can now apply the statistical analysis to a single particle interferometer
of arbitrary size (as we show in Fig. 1). The interference effects that
characterise the interferometer are given by the inner products of the Hilbert
space vectors representing the paths. For any input state $\hat{\rho}$, the
counterfactual output statistics are given by

$\begin{split}P(m|X_{a})&=\left\langle
m\right|\left(\mathds{1}-\left|a\right\rangle\hskip-2.84526pt\left\langle
a\right|\right)\hat{\rho}\left(\mathds{1}-\left|a\right\rangle\hskip-2.84526pt\left\langle
a\right|\right)\left|m\right\rangle\\\ &=P(m)-2\varrho(a,m)+\left|\langle
m|a\rangle\right|^{2}P(a),\end{split}$ (5)

where

$\varrho(a,m)=\mbox{Re}\left[\langle m|a\rangle\left\langle
a\right|\hat{\rho}\left|m\right\rangle\right]$ (6)

is the real part of the Kirkwood-Dirac quasiprobability describing the joint
quantum statistics of the two outcomes $a$ and $m$. The counterfactual gain is
then given by

$\begin{split}\Delta_{a}-&P(a)=\\\ &\sum_{P(m|X_{A})>P(m)}\left(\left|\langle
m|a\rangle\right|^{2}P(a)-2\varrho(a,m)\right).\end{split}$ (7)

The condition for inclusion in the sum can also be given in terms of the
positivity of the contribution,

$\left|\langle m|a\rangle\right|^{2}P(a)>2\varrho(a,m).$ (8)

The outcome $m$ contributes to the counterfactual gain whenever the Kirkwood-
Dirac term $\varrho(a,m)$ is below a probability product representing the
probability of a sequential non-absorbing observation of the photon, first at
$a$ and then at $m$.

In the original counterfactual scenario of Elitzur and Vaidman [6], the
counterfactual gain was obtained for an outcome with $P(m)=0$, representing
the additional condition that false positives should be avoided. In that case,
the corresponding Kirkwood-Dirac term is zero as well, so that the
counterfactual gain is represented entirely by the sequential probabilities
$|\langle m|a\rangle|^{2}P(a)$. In the following, we therefore refer to these
contributions to the counterfactual gain as the Elitzur-Vaidman terms.

Note that the Kirkwood-Dirac term is a quasiprobability of two non-commuting
observables, which can be obtained directly through weak measurement and
postselection [36], or indirectly through strong measurement [37]. As a
quasiprobability of $a$ and $m$, the roles of the $a$\- and $m$-terms in the
quasiprobability are equivalent and interchangeable. The Kirkwood-Dirac
contribution has the appearance of a joint probability for the reversible
sequence of events $(a,m)$, but it can take negative values. The Elitzur-
Vaidman term can instead be interpreted as the probability of a regular
sequence of events. It is not symmetric ($m$ definitely occurs after $a$), but
is necessarily positive. We can therefore associate the Kirkwood-Dirac term
with the non-classical correlations between $a$ and $m$ in the initial state,
while the Elitzur-Vaidman term is more directly related to the statistics of
sequential measurements.

Any outcome $m$ with a Kirkwood-Dirac term of zero contributes its whole
positive Elitzur-Vaidman term to the counterfactual gain. It is easy to
construct a scenario where this is the case, by ensuring the initial
probability for $m$ is zero, in which case the Kirkwood-Dirac term is zero as
well. This was the scenario given by Elitzur and Vaidman [6], which is usually
associated with counterfactual gain in quantum systems. We observe that the
interpretation of this process as interaction-free would be more consistent
with the subtraction of a joint probability $\rho(a,m)$ from each outcome
probability $P(m)$. To recover this classical situation in quantum mechanics,
$a$ needs to be an eigenstate of $\rho$, so that the Kirkwood-Dirac term and
the Elitzur-Vaidman term are equal. Eq. (2) then reduces to a contribution
from the Elitzur-Vaidman terms, which now represent a back-action free joint
probability of $a$ and $m$. In the absence of coherence between $a$ and the
other paths in the interferometer, this classical joint probability lowers the
outcomes $P(m|x_{a})$ with respect to the probabilities $P(m)$ observed in the
absence of the absorber, so there will be no counterfactual gain. If the
overlap $\langle a|m\rangle$ of $a$ and $m$ is nonzero, quantum coherences
between $a$ and states orthogonal to it are needed to achieve a probability of
$P(m)=0$. Even though it may appear to be ‘classical,’ a Kirkwood-Dirac term
with a value of zero is already an indicator of quantum coherence, which is
why the Elitzur-Vaidman bomb tester is a quantum effect despite involving no
negative Kirkwood-Dirac terms. However, all negative Kirkwood-Dirac terms
increase the counterfactual gain beyond what the Elitzur-Vaidman term
contribution alone would allow.

##  III Application to a multi-path interferometer with equally distributed
outcomes

The typical introductory example of the Elitzur-Vaidman bomb tester is based
on a balanced 50:50 Mach-Zehnder interferometer, arranged so one of the
detectors has a zero detection probability when the absorber is absent
($P(m_{1})=0$). As shown above, we can generalise this scenario to a multi-
path interferometer, characterised by the output probability $P(m=1)=0$ and a
specific probability $P(a)$ for the absorption of the photon if the absorber
(the ‘bomb’) is inserted into path $a$. For simplicity, we now assume that the
remaining $N-1$ output ports each have an equal probability of receiving the
photon. Likewise, the reduction in the number of photons that arrive at the
$N-1$ output ports when the absorber is inserted should be equally
distributed.

The essence of the Elitzur-Vaidman Bomb Tester scenario is that the presence
of the absorber can be detected even when no absorption happens. If absorption
is a classical process, it should be impossible to get information about
whether the absorber would have absorbed the photon if it had been in path
$a$, when the arrival at any output port $m$ shows that it had not been
absorbed in path $a$. It is interesting to ask what features of quantum
mechanics allow us to probe what could have happened if the present outcome
had not been observed. As Ref. [30] mentions, two classical limits of
optically probing the presence of an absorber using a similar setup are to
consider light as completely wavelike (which we can think about as being
equivalent to sending a coherent state from the source into the first beam
splitter), or to consider light as completely particle-like (which we can
think about as being equivalent to the photon being on a maximal mixture of
all the paths). The first case agrees with our intuition about what
interference effects we should see when the bomb doesn’t work (all light going
to output $m_{1}$), and when the bomb does work (light being divided equally
between the two outputs)—however, in this case, some light will always be
absorbed, meaning our inference that the absorber is present is obviously not
interaction-free, since light will always interact with the measured object.
In the second case, we retain the 50% probability of the bomb blowing up if it
works (given the input photon is put into an equal superposition of the two
paths), but detections are always distributed evenly between the two outputs,
meaning there is no way to distinguish these cases where the bomb would blow
up, but doesn’t, from cases where the bomb wouldn’t blow up. Comparing this
second (mixed state) case to the actual Elitzur-Vaidman bomb tester shows that
coherence between the two paths the photon could have travelled on is
necessary to allow us to distinguish cases where the bomb would have blown up
(but didn’t) from cases where it wouldn’t have blown up.

We can now define a specific version of the Elitzur-Vaidman-style scenario by
selecting a value of $P(a)$. Let us consider the case of $P(a)=1/3$. The
initial state can then be written as

$\left|\psi\right\rangle=\frac{1}{\sqrt{3}}\left|a\right\rangle+\sqrt{\frac{2}{3}}\left|b\right\rangle,$
(9)

where $\left|b\right\rangle$ represents an arbitrary superposition of states
orthogonal to $\left|a\right\rangle$. Eq. (9) can thus be used to describe
arbitrarily complicated multi-path interferometers, where only the state
$\left|a\right\rangle$ is associated with a specific physical path in the
interferometer. Next, we define the outcome $\left|m_{1}\right\rangle$ so that
we satisfy the condition $P(m_{1})=0$ in the absence of the absorber. The
output state $\left|m_{1}\right\rangle$ which provides the greatest
counterfactual gain for this input state $\left|\psi\right\rangle$ is

$\left|m_{1}\right\rangle_{EV}=\sqrt{\frac{2}{3}}\left|a\right\rangle-\frac{1}{\sqrt{3}}\left|b\right\rangle.$
(10)

We can define an arbitrary number of additional outputs, corresponding to the
total number of paths in the interferometer. However, the main result is
independent of the number of paths. The counterfactual gain of the Elitzur-
Vaidman scenario is given by a single Elitzur-Vaidman term,

$\left(\Delta_{a}-P(a)\right)_{EV}=|_{EV}\langle m_{1}|a\rangle|^{2}P(a).$
(11)

For the values given above, the counterfactual gain is $2/9$. In the Elitzur-
Vaidman scenario, this is equal to the probability $P(m_{1}|X_{a})=2/9$,
corresponding to the probability of detecting the presence of the absorber
without any absorption.

Now consider the additional counterfactual gain that can be obtained when the
condition $P(m_{1})=0$ is lifted. We can then identify two terms within Eq.
(7) which describe the actual effects of the absorber on the output
statistics, the Elitzur-Vaidman term $|\langle m|a\rangle|^{2}P(a)$ and the
Kirkwood-Dirac term $\varrho(a,m)$. Without coherence, the Kirkwood-Dirac term
is always equal to the Elitzur-Vaidman term (i.e., $|\langle
m|a\rangle|^{2}P(a)=\varrho(a,m)$). This value corresponds to the distribution
of the particles lost at $a$ in $m$. Also note that, if the Kirkwood-Dirac
term was a true joint (Kolmogorovian) probability, an interaction-free removal
of $a$ would result in the subtraction of the Kirkwood-Dirac term (i.e.,
$-\varrho(a,m)$) from the original probability $P(m)$, resulting in
$P(m|X_{a})<P(m)$ and the absence of any counterfactual gain. There would be
no Elitzur-Vaidman term, but the negative effect of the positive Kirkwood-
Dirac term would be halved.

Counterfactual gain is a consequence of the removal of destructive
interferences between $a$ and paths orthogonal to $a$ in the output $m$. For
$P(m)=0$, the amplitude from $a$ and the amplitude from orthogonal paths
cancel exactly. When the absorber is inserted, the amplitude from $a$ is
removed and the uncompensated magnitude of the amplitude from $a$ appears in
the output $m$. The detection probabilities associated with this amplitude are
represented by the Elitzur-Vaidman term. (Note that this is consistent with
Eq. (5).) Counterfactual gain with non-vanishing Kirkwood-Dirac terms
represent destructive interferences that do not cancel $P(m)$ completely,
because the amplitude from path $a$ and the amplitude from the other paths are
not equal in $m$. Negative Kirkwood-Dirac terms are obtained when the
amplitude from path $a$ is smaller than the amplitude from the other paths
with which it destructively interferes, so that the removal of the amplitude
from $a$ leaves a much larger amplitude behind. A negative Kirkwood-Dirac term
will therefore enhance the counterfactual gain by ‘uncovering’ an amplitude
that is much larger than the amplitude from $a$ described by the Elitzur-
Vaidman term.

The original Elitzur-Vaidman scenario starts with a heavily biased probability
distribution due to the condition that $P(m_{1})=0$ if the absorber is absent.
Without this condition, it is possible to start with an equal distribution of
output probabilities over all $N$ output ports. For a nine-path
interferometer, the probabilities of each outcome are $P(m)=1/9$ and the
outcome $m$ with a negative Kirkwood-Dirac term $\varrho(m,a)$ can be defined
as

$\left|m_{1}\right\rangle_{KD}=\frac{1}{\sqrt{3}}\left|a\right\rangle-\sqrt{\frac{2}{3}}\left|b\right\rangle.$
(12)

The Kirkwood-Dirac term for this outcome is $\varrho(m_{1},a)=-1/9$. Due to
its negativity, the Kirkwood-Dirac term increases the counterfactual gain by
two times its absolute value,

$\left(\Delta_{a}-P(a)\right)_{KD}=2|\varrho(m_{1},a)|+|_{KD}\langle
m_{1}|a\rangle|^{2}P(a).$ (13)

It should be noted that the Elitzur-Vaidman term depends on the relation
between path $a$ and the output $\left|m_{1}\right\rangle_{KD}$ given by Eq.
(12). Here, it has a value of $1/9$. The total counterfactual gain of this
scenario is equal to $1/3$, or 1.5 times the value of the Elitzur-Vaidman
scenario for $P(a)=1/3$.

Since the basic mechanism is destructive interference between the amplitude of
path $a$ and the amplitude of all the other paths, it is possible to derive
the optimal counterfactual gain for any given absorption probability $P(a)$ by
maximising the interference effects at the output ports. The details of the
derivation of the maximal counterfactual gain are given in Appendix A, but it
leads to the result

$\left(\Delta_{a}-P(a)\right)\leq\frac{1}{2}\left(\sqrt{(4-3P(a))P(a)}-P(a)\right).$
(14)

We can see by differentiating Eq. (14) that the counterfactual gain has an
overall maximal value of 1/3, and this value can only be obtained at
$P(a)=1/3$. The example given above is therefore the optimal scenario for
counterfactual gain. If we limit ourselves to the Elitzur-Vaidman scenario
($P(m_{1})_{EV}=0$), the optimal result is obtained with $|_{EV}\langle
m_{1}|a\rangle|^{2}=1-P(a)$, and the upper bound of counterfactual gain is

$\left(\Delta_{a}-P(a)\right)_{EV}\leq P(a)\left(1-P(a)\right).$ (15)

Perhaps not surprisingly, the Elitzur-Vaidman scenario achieves its maximal
counterfactual gain of 1/4 at $P(a)=1/2$, the fully symmetric and balanced
case described by a two-path interferometer.

The reason why the Kirkwood-Dirac term was not included in the original
Elitzur-Vaidman scenario is the requirement that the output $m_{1}$ only has
non-zero probability when the absorber is in place. $P(m_{1})_{EV}=0$ means
that there are no false positives. For the nine path case, the initial
distribution of outcomes is

$\begin{split}P(m_{1})_{EV}=0,\\\
P(m_{i})_{EV}=1/8,\,i\in\\{2,3,...,9\\}.\end{split}$ (16)

When the absorber is inserted, the probability of absorption is $P(a)=1/3$ and
the remaining outcome probabilities are

$\begin{split}P(m_{1}|X_{a})_{EV}=2/9,\\\
P(m_{i}|X_{a})_{EV}=1/18,\,i\in\\{2,3,...,9\\}.\end{split}$ (17)

Even though the total probability of finding the photon in outcomes $m_{2}$ to
$m_{9}$ is larger than the probability of finding it in outcome $m_{1}$, the
redistribution of photons maximises the probability in $m_{1}$.

If we allow false positives, the initial distribution can be perfectly
isotropic, with

$P(m_{i})_{KD}=1/9,\,i\in\\{1,2,...,9\\}.$ (18)

The insertion of the absorber now has the effect of focusing the photons in
the output onto $m_{1}$, even though there was no indication in the initial
state of the special role $m_{1}$ plays in the protocol. This special role is
entirely defined by $m_{1}$’s relation with the path $a$ into which the
absorber is placed. In the optimal case, the outcome probabilities in the
presence of the absorber are

$\begin{split}&P(m_{1}|X_{a})_{KD}=4/9,\\\
&P(m_{i}|X_{a})_{KD}=1/36,\,i\in\\{2,3,...,9\\}.\end{split}$ (19)

The presence of the absorber has increased the probability of the photon
arriving at output port $m_{1}$ by a factor of four. The Bayesian likelihood
that the absorber was present when an outcome of $m_{1}$ is obtained is
therefore 80 %. Incidentally, this is equal to the probability of a false
negative inferred from outcomes $m_{2}$ to $m_{9}$.

##  IV Back-action

The reason we can detect the presence of the absorber even when the particle
is not absorbed is that the presence of the absorber somehow increases the
probability that a particle arrives at some of the outcomes $m$. Since the
absorption process itself only removes particles, this increase in probability
means that the presence of the absorber causes a redistribution of particles
that are not absorbed, suggesting that absorption is not the only interaction
between the absorber and the particles. Interestingly, this possibility is not
considered in the original Elitzur-Vaidman scenario. Instead, the claim that
non-absorption is interaction-free suggests an intrinsic difference between
classical statistics and quantum statistics that might explain why the
interaction-free removal of particles in $a$ can somehow increase the number
of particles observed at $m$ without any redistribution of the remaining
particles. A natural representation of such a relation would be a
quasiprobability that can assign negative joint probabilities to paths and
outcomes. The Kirkwood-Dirac distribution is a suitable candidate; its
potential negativity could explain counterfactual increases in the output
probabilities without any back-action-related change to the initial
statistics. This makes it somewhat ironic that the Elitzur-Vaidman scenario is
the one where the Kirkwood-Dirac term is always necessarily zero. By itself,
the Elitzur-Vaidman term does not qualify as a quasiprobability, since it
fails to reproduce the outcome statistics $P(m)$ observed in the absence of
absorptions when a sum over an orthogonal set of paths $a$ is performed.
Moreover, its values are necessarily positive, and the addition of the term to
the modified outcome probability $P(m|X_{a})$ cannot be explained by the
removal of particles in $a$.

If the absorption process merely removed a fraction $P(a)$ of the input
photons, the original outcome statistics $P(m)$ could be recovered by adding
the contributions that originated from photons propagating through $a$. Since
the probability that a photon that starts out in path $a$ will continue to $m$
is given by $|\langle m|a\rangle|^{2}$, the total contribution of photons
passing through $a$ to the outcome $m$ should be given by the Elitzur-Vaidman
term. If we assume that there is no interaction between absorbers and the
photons that are not absorbed, it should be possible to recover the output
probability $P(m)$ by adding the outcomes for photons that passed through $a$
to the outcomes for photons that did not pass through $a$. It is possible to
verify experimentally that the Elitzur-Vaidman term describes photons that
passed through $a$ by blocking all of the other paths in an inversion of the
original counterfactual experiment. The addition of the outcome distributions
of the two experiments shows a well-defined deviation from the original
distribution,

$\begin{split}P(m|X_{a})+|\langle
m|a\rangle|^{2}P(a)=P(m)+\chi_{B}(m|a)\end{split}$ (20)

where the modification of the original probability distribution is given by

$\chi_{B}(m|a)=2\left(|\langle m|a\rangle|^{2}P(a)-\varrho(a,m)\right).$ (21)

This modification $\chi_{B}$ quantifies the unavoidable back-action effect
caused by distinguishing photons that pass through $a$ from photons that do
not pass through $a$. The only case where there is no back-action effect
($\chi_{B}=0$) is the case where

$\left|\langle m|a\rangle\right|^{2}P(a)=\varrho(a,m).$ (22)

Aside from the trivial case of $\langle m|a\rangle=0$, this only happens when

$\hat{\rho}\left|a\right\rangle=P(a)\left|a\right\rangle.$ (23)

As expected, the back-action is a consequence of quantum coherence between
$\left|a\right\rangle$ and orthogonal path states in the input. The presence
of the absorber necessarily removes this coherence, whether or not the photon
is actually absorbed, and this loss of coherence has an irreversible effect on
the output distribution of photons that are not absorbed. The argument that
the counterfactual outcomes conditioned by $X_{a}$ are not affected by any
back-action could only be maintained by arguing that the back-action in Eq.
(20) originates exclusively from photons passing through $a$. However, this
assumption is problematic because the data for this contribution can be taken
by inserting absorbers in all of the alternative paths. The outcome statistics
of photons that passed through $a$ are necessarily obtained when there is no
detector in $a$, and the back-action in this path must originate from the
absorbers in the other paths. The correct division of the back-action between
photons in $a$ and photons that did not pass through $a$ should be completely
symmetric, so that the two counterfactual scenarios are described
consistently. This can be done by considering the Kirkwood-Dirac term as a
valid representation of the undisturbed joint statistics of $a$ and $m$. When
absorbers are inserted into all paths except for $a$, the back-action converts
the non-positive Kirkwood-Dirac term of $a$ and $m$, which represents the
joint statistics of the initial quantum state, into the Elitzur-Vaidman term,
which describes the sequence of projective measurements that are actually
performed,

$|\langle m|a\rangle|^{2}P(a)=\varrho(a,m)+\frac{\chi_{B}(m|a)}{2}.$ (24)

This effect accounts for exactly half of the total back-action, indicating
that the back-action is always equally distributed between the photons that
pass through $a$ and the photons which do not. The back-action of the absorber
in path $a$ on photons that are not absorbed by it is therefore described by

$P(a|X_{a})=\left(P(m)-\varrho(a,m)\right)+\frac{\chi_{B}(m|a)}{2},$ (25)

correctly reproducing the original relation in Eq. (5) where the fsubtraction
of two times the Kirkwood-Dirac term is split into a term representing the
removal of photons in $a$ from the outcomes $m$, while the other half
represents part of the back-action effect $\chi_{B}/2$, which ensures that no
negative probabilities will be observed. It seems characteristic of quantum
counterfactuals that the quasiprobability subtracted to represent the absence
of the absorbed photons in the output also appears as part of the modification
of the probabilities caused by the unavoidable back-action effect that
provides the counterfactual gain.

Eq. (25) reduces to a particularly simple form in the Elitzur-Vaidman case,
where both $P(m)$ and $\varrho(a,m)$ are zero. In that case, the Elitzur-
Vaidman term describes the redistribution of photons by the unavoidable back-
action associated with the presence of the absorber in $a$. It may be worrying
that the absorber has an effect on particles in other paths, but the only
alternative would be to interpret the addition of the Elitzur-Vaidman term as
the removal of a negative joint probability that just happens to resemble the
(positive) probability of a sequential measurement of $a$ and $m$. As we
argued above, such an interpretation would result in different joint
statistics of $a$ and $m$ when the absorber is placed in a different path
$a^{\prime}$. Moreover, the loss of coherence between $\left|a\right\rangle$
and orthogonal components in the initial quantum state makes it difficult to
argue that the interaction between the absorber and the photons is limited to
the photons that it actually absorbs. Instead, the description of back-action
provided in the above analysis is fully consistent with alternative placements
of the absorber, as well as the effects of non-absorbing/non-demolition
measurements of the photon in path $a$.

The increase of the outcome probabilities that describe the counterfactual
gain in Eq. (4) originate from the sum of the back-action effect $\chi_{B}$
and a correlation effect described by the Kirkwood-Dirac quasiprobability. One
peculiar aspect revealed by the analysis above is that the counterfactual gain
can be calculated by ignoring all outcome probabilities that decrease when the
absorber is inserted. A back-action-free counterfactual gain would only be
possible if the joint probability that a photon passes through $a$ and then
arrives at $m$ were negative. Ironically, the Kirkwood-Dirac term seems to
describe such a situation, suggesting that part of the counterfactual gain
originates from non-classical correlations that do not depend on the magnitude
of the back-action. However, this negative Kirkwood-Dirac term contributes an
equal amount of counterfactual gain through the back-action, with an
additional back-action-related gain contributed by the Elitzur-Vaidman term.
There is no counterfactual gain without back-action, and statistical
contributions from negative Kirkwood-Dirac terms are always accompanied by at
least as much counterfactual gain from the back-action. In other words, back-
action is necessary for the observation of counterfactual gain, while a
negative Kirkwood-Dirac term is sufficient for the observation of such a gain.

To deepen our understanding of the role of statistics, it is worth looking at
the case where counterfactual gain is observed in the presence of positive
Kirkwood-Dirac terms. In this case, the Elitzur-Vaidman term must be at least
twice the size of the positive Kirkwood-Dirac term, so that the loss of
photons due to the correlation between $a$ and $m$ is more than compensated by
the redistribution of photons due to back-action. In classical statistics, a
joint probability would be limited by the marginal probabilities, so that one
might expect that if $\varrho(a,m)\leq P(m)$, there must be a minimal rate of
false positives whenever the Kirkwood-Dirac term is greater than zero.
Although quantum theory also requires a minimal rate of false positives for
non-zero Kirkwood-Dirac terms, the actual quantitative bound for non-zero
Kirkwood-Dirac terms is

$\left|\varrho(a,m)\right|\leq\sqrt{P(m)\;\;\left|\langle
m|a\rangle\right|^{2}P(a)}.$ (26)

This bound is also the geometric mean of the initial output probability $P(m)$
and the Elitzur-Vaidman term. In the absence of back-action, the Kirkwood-
Dirac term is equal to the Elitzur-Vaidman term, so the probability $P(m)$
must be equal to or larger than these two terms. As shown in Eq. (8),
counterfactual gain requires that the Kirkwood-Dirac term is less than half
the size of the Elitzur-Vaidman term, so that the back-action adds more
photons to the output than the statistical correlation removes. This condition
is satisfied whenever the probability $P(m)$ is smaller than one quarter of
the Elitzur-Vaidman term,

$P(m)<\frac{1}{4}|\langle m|a\rangle|^{2}P(a).$ (27)

This is a sufficient condition for the observation of counterfactual gain.
Whenever the rate of false positives drops below one-quarter of the Elitzur-
Vaidman term, the unavoidable back-action effects associated with the
possibility of absorption in path $a$ necessarily increase the probability of
the outcome $m$ more than the losses of photons at the absorber reduce it.

Counterfactual gain is primarily the result of an interaction between the
absorber and photons that were not absorbed. Quantum mechanics defines this
interaction by introducing a new kind of relation between the back-action and
non-classical correlations between the path $a$ and the outcome $m$. The
Elitzur-Vaidman term represents the purest form of back-action, since it
describes the situation where no photons are observed at $m$ when there is no
absorber present. Effectively, the Elitzur-Vaidman term quantifies the amount
of incoherent scattering that happens when the back-action effect removes the
coherence between $\left|a\right\rangle$ and the orthogonal state components.
Counterfactual gains much larger than the Elitzur-Vaidman term are possible
when the probability $P(m)$ is larger than the Elitzur-Vaidman term, so long
as the Kirkwood-Dirac term is negative. In principle, we can obtain a
counterfactual gain where the back-action contribution is only slightly higher
than the statistical contribution; this occurs when the probability $P(m)$ is
much larger than the Elitzur-Vaidman term. However, this limit is
characterised by a high rate of false positives, making it more difficult to
observe the quantum advantage achieved by the counterfactual gain.

##  V Statistics and back-action in Hofmann’s Three-Path Interferometer

It may help to illustrate the rather intricate relation between non-classical
statistics and back-action effects in an interferometer specifically designed
to illustrate the paradoxical aspects of quantum statistics associated with
negative Kirkwood-Dirac terms [35]. The layout of this interferometer, shown
in Fig. 2, breaks down the three-path interferences into sequential
interferences at five two-path beamsplitters, identifying a total of five
different measurement contexts with the different paths available to each
photon.

Figure 2: Hofmann’s three-path interferometer [35], where paths are
sequentially interfered at beamsplitters. The interferometer is aligned so
that paths equivalent to the original input paths are reconstructed at the
output, albeit with the position of paths $\left|1\right\rangle$ and
$\left|2\right\rangle$ swapped.

As we have seen above, negative Kirkwood-Dirac terms describe a specific kind
of destructive interference, where the amplitude associated with the blocked
path inside the interferometer is smaller than the contribution from the other
paths. In the three-path interferometer, such a situation can be constructed
systematically, by letting the amplitude of the path $\left|F\right\rangle$
which can be blocked by the absorber interfere destructively with the parallel
path $\left|P_{2}\right\rangle$ in the output port $\left|D_{2}\right\rangle$,
so that the probability of finding a photon in $\left|D_{2}\right\rangle$ is
zero in the absence of the absorber. In the final stage of the interferometer,
this empty port interferes with a third path ($\left|S_{2}\right\rangle$ in
Fig. 2), splitting into the outputs $\left|1\right\rangle$ and
$\left|3\right\rangle$, while the output $\left|2\right\rangle$ is described
by maximal constructive interference between $\left|F\right\rangle$ and
$\left|P_{2}\right\rangle$. If the reflectivities of the beam splitters are
laid out in the appropriate manner, the initial state can be written as an
equal superposition of the outputs,

$\left|N_{F}\right\rangle=\frac{1}{\sqrt{3}}(\left|1\right\rangle+\left|2\right\rangle+\left|3\right\rangle),$
(28)

while the blocked state $\left|F\right\rangle$ is given by

$\left|F\right\rangle=\frac{1}{\sqrt{3}}(\left|1\right\rangle+\left|2\right\rangle-\left|3\right\rangle).$
(29)

The Kirkwood-Dirac terms of the outputs with respect to the blocked path
$\left|F\right\rangle$ are

$\begin{split}\varrho(F,1)&=1/9,\\\ \varrho(F,2)&=1/9,\\\
\varrho(F,3)&=-1/9.\end{split}$ (30)

The Elitzur-Vaidman terms for the three outputs are all equal to $|\langle
m|F\rangle|^{2}P(F)=1/27$. It may also be worth noting that the empty path
$\left|D2\right\rangle$ has a Kirkwood-Dirac term of $\varrho(D_{2},3)=0$
given by the sum of $\varrho(F,1)$ and $\varrho(F,3)$ and an Elitzur-Vaidman
term of $|\langle D_{2}|F\rangle|^{2}P(F)=2/27$. If blocking path $F$ only
removed the Kirkwood-Dirac terms, the output pattern would be $(2/9,2/9,4/9)$,
with no change in $P(D_{2})$. However, the back-action effect adds a
redistribution of particles, so that the actual distribution of $P(m|X_{F})$
is $(4/27,4/27,16/27)$ (as shown in Fig. 3). Back-action reduces the lower
probabilities by an additional $2/27$, and adds $4/27$ to the counterfactual
gain obtained from output $\left|3\right\rangle$. The back-action also results
in a non-zero particle count in $D_{2}$ given by $P(D_{2}|X_{F})=2/27$.
However, the total counterfactual gain of output $\left|3\right\rangle$ is
equal to $7/27$, or $3.5$ times higher than the counterfactual gain of
$D_{2}$. This massive amplification of the blocking effect serves as a useful
example of the practical benefits of this work, which could have potential
uses in metrology or imaging of optically-sensitive samples.

Figure 3: A comparison of the detection probabilities on different paths, in
the standard three-path interferometer case, and the case where a blocker is
inserted on path $\left|F\right\rangle$. These probabilities are given for the
input state
$\left|N_{F}\right\rangle=(\left|1\right\rangle+\left|2\right\rangle+\left|3\right\rangle)/\sqrt{3}$.

Is it possible to distinguish the effects of non-classical statistics from the
additional back-action effects by considering the relation between measurement
contexts? One might object that the negative value of the Kirkwood-Dirac term
$\varrho(F,3)=-1/9$ could be replaced by a different quasiprobability able to
explain the counterfactual effects. As shown in the previous section, this is
only prevented by the observation that the Elitzur-Vaidman terms represent the
propagation from $F$ to the output ports when the other ports (e.g. $P_{2}$
and $S_{2}$) are blocked, and these terms would have to be described by the
same quasiprobability in order to explain the counterfactual effects. In the
present scenario the Elitzur-Vaidman terms for all outputs are $1/27$, so the
prediction for a removal of these particles would be a homogeneous reduction
of all output probabilities to $8/27$. Conversely, the distribution of
particles from $F$ in the outputs would have to be $(5/27,5/27,-7/27)$
according to the counterfactual effects observed when $F$ is blocked.
Consistency between the distributions observed when path $F$ is blocked and
when only $F$ is open requires that the back-action re-distributes the
particles from low probability outputs to high probability outputs, with the
change in probability given by $\chi_{B}=(-2/27,-2/27,4/27)$.

##  VI Discussion

This paper was motivated by the fact that the presence of an absorber in a
quantum interferometer can be detected not just by the missing particles it
absorbs, but also by its effects on the particles it didn’t absorb. This
effect is usually identified with the appearance of photons in originally
empty output ports, as in the Elitzur-Vaidman bomb tester. Here, we have shown
that a much stronger effect can be obtained when the initial output
probability is not zero. This additional counterfactual gain is represented by
a negative Kirkwood-Dirac term that describes the quasiprobability of the
particle propagating from $a$ to $m$ when path $a$ is not blocked. When path
$a$ is blocked, this quasiprobability is subtracted twice from the output
probability (resulting in an increase in this probability for anomalous
negative values of the Kirkwood-Dirac term) and an additional term is added
corresponding to the probability that a particle is first detected in $a$ and
then continues onward to $m$. This additional Elitzur-Vaidman term is
associated with the decoherence caused by blocking the path, and we have shown
above that the addition of the difference between the Elitzur-Vaidman term and
the Kirkwood-Dirac term describes the back-action effect that must occur
because the absorber at $a$ distinguishes between particles in $a$ and
particles not in $a$. This back-action is the reason why quantum mechanics
does not allow the removal of photons from one path without changing the
propagation of photons in the other paths.

Much confusion has been caused by the assumption that non-detection of a
particle requires no interaction with the particle that was not detected. Our
results show that this assumption is not consistent with the theoretical
description of the process. Quantum mechanics does not limit an absorber
solely to interacting with the particles that it absorbs, but necessarily
requires an additional interaction term that removes any coherences between
the path the absorber is on, and the other paths that the particle may
actually be on. It is interesting that this back-action is defined in terms of
fundamental statistics, rather than any particular dynamical process
associated with this decoherence. The Kirkwood-Dirac term is a consistent
expression of non-classical statistics as joint probabilities for the blocked
path and each output, and the Elitzur-Vaidman terms represents the incoherent
propagation of particles detected in $a$. The back-action therefore amounts to
a replacement of the coherence represented by the Kirkwood-Dirac term with the
lack of coherence represented by the Elitzur-Vaidman term. It is interesting
to note that the coherence is described in terms of the contribution of $a$ to
the output $m$, which means that this coherence term will be zero if we set
the condition that there are no particles in $m$ unless the absorber is
inserted (no false positives). This results in the inequality in Eq. (26) that
defines upper and lower bounds of the Kirkwood-Dirac term based on the
Elitzur-Vaidman term and the rate of false positives. Our results show that
the dynamics of quantum processes are fundamentally linked to non-classical
statistics, making it possible to implement quantum information processing by
defining the dynamics entirely in terms of the necessary back-action caused by
the obstruction of selected paths, or any other method that eliminates a
specific set of possibilities in a quantum system. The key to all
counterfactual operations is the well-defined characteristics of the back-
action dynamics necessarily induced by these obstructions in the remaining
paths.

Our work also has an interpretational component: we are keen to point out that
it is strange that in quantum counterfactual scenarios we intuit that there
should be no change of the physical properties in unabsorbed particles. It is
misleading to say that such a change is interaction-free, given we do not know
what the physics of this interaction are. The result we give above is not just
a technical result—it should enhance our understanding of the effects on
propagation caused by the absorption of any particle. Our work should clear up
the misunderstanding that the absorber should only have an effect when it
absorbs a photon, and so that the effects of this absorption can be quantified
as the results of this loss alone. Quantum counterfactual effects are a
typical signature of contextuality, given they are not a simple effect of
‘here is a value, here is how it changes due to a physical effect’, but
instead show that measuring in different context produces different results.
This is the same as the difference between a change in a force classically,
and a change in context quantum mechanically (note, Wagner et al previously
linked interaction-free measurement in a two-path Mach-Zehnder interferometer
to contextuality and quantum advantage [38], but didn’t identify the Kirkwood-
Dirac negativity-related behaviour describe). The absorber serves to insert an
eigencontext. A particle which doesn’t experience this eigencontext is
different from a particle which does. This is the counterfactual. Unlike
classical counterfactuals, the effects of quantum counterfactuals are more
than just the effects of the removed particles. The argument that ‘if the
particle didn’t travel via the absorber’s path, it cannot know whether or not
the absorber was there’ is not true when the idea of considering ‘which path’
the particle went on only comes into existence when apply this eigencontext.
Particle absorption cannot be seen as complete; it is not sufficient to define
the physics of the situation, and so we also need to consider whether the
photon was measured in the ‘absorbed or not’ basis. The causal connections
between the coherence at the start of the interferometer and the coherence at
end of the interferometer are destroyed if we measure in the ‘which path’
basis.

Using the ideas given above, we can argue that counterfactual effects are a
manifestation of contextuality that has been overlooked in the original
treatment of contextuality by Kochen and Specker [39], causing unnecessary
confusion about the nature of quantum contextuality. Specifically, Kochen and
Specker have placed too much emphasis on the possibility that compatible
measurements may exert an influence on each other, while downplaying the
possibility that contextuality originates from the well-defined relation
between the outcomes of incompatible measurement contexts [40, 41, 42, 43].
Counterfactual effects illustrate this relation between incompatible
measurements by introducing a minimal change of context represented by the
exclusion of a single possibility. This minimal change of context has a
profound impact on the output statistics because the output ports now
represent a fundamentally different measurement context with respect to the
input state. The undisturbed output measurement is not compatible with any
information about the excluded possibility, whether counterfactual or
affirmative. As we have shown above, counterfactuals describe a necessary and
well-defined relation between incompatible measurements, making it possible to
characterise the relation between different contexts in terms of the back-
action dynamics that relates them to each other. More specifically,
counterfactual effects establish a link between contextuality and quantum
interferences by identifying the modification of interference effects with the
change of measurement context.

In conclusion, quantum mechanics allows us to use counterfactual effects to
probe the non-classical relations between different measurement contexts. In
this paper, we have provided the theoretical framework for a complete
characterisation of counterfactual effects, describing both the statistical
relation between the blocked path and the output ports in the absence of the
observed, and the necessary back-action effect that modifies the original
statistics. It is important to understand that this back-action effect is very
different from the classical assumption that an absorption merely removes a
particle and does nothing else. The information gained by a non-detection of
the particle requires a form of dynamics that accounts for the necessary
decoherence effects described by the formalism, establishing a fundamental
relation between information and dynamics that distinguishes quantum mechanics
from classical physics.

Acknowledgements — JRH was supported by Hiroshima University’s Phoenix
Postdoctoral Fellowship for Research for part of the time this paper was being
written. TM was supported by JST, the establishment of university fellowships
towards the creation of science technology innovation, Grant Number
JPMJFS2129.

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##  Appendix A Derivation of Maximal Counterfactual Gain

The counterfactual gain that can be achieved depends on the fraction of
particles that will be absorbed, as expressed by the probability $P(a)$. For a
fixed value of $P(a)$, we can find the maximal counterfactual gain by
considering the square roots of the probability distributions $P(m)$ and
$P(m|X_{a})$ as vectors, where $P(a)$ defines the squared length of the vector
$(\sqrt{P(m|X_{a}})$ as

$\sum_{m}P(m|X_{a})=1-P(a).$ (31)

By normalisation, the length of the vector $(\sqrt{P(m)})$ is of course one.
We can then express the statistical distance using an inner product of the sum
and difference vectors,

$\begin{split}\Delta_{a}=\frac{1}{2}P(a)+\frac{1}{2}\sum_{m}\left|\sqrt{P(m|X_{a})}-\sqrt{P(m)}\right|\left(\sqrt{P(m|X_{a})}+\sqrt{P(m)}\right)\end{split}$
(32)

The lengths of the sum and difference vectors are given by

$\sum_{m}(\sqrt{P(m|X_{a})}\pm\sqrt{P(m)})^{2}=2-P(a)\pm
2\sum_{m}\sqrt{P(mP(m|X_{a})}.$ (33)

The maximal statistical distance is obtained when the sum and the difference
vectors are parallel to each other and the inner product is equal to a product
of the two lengths,

$\Delta_{a}\leq\frac{1}{2}P(a)+\frac{1}{2}\sqrt{\left(2-P(a)\right)^{2}-4\left(\sum_{m}\sqrt{P(m)P(m|X_{a})}\right)^{2}}.$
(34)

This upper bound is not yet complete, since it still depends on the inner
product of the two vectors $(\sqrt{P(m)})$ and $(\sqrt{P(m|X_{a})})$. Since
the inner product is subtracted, we need to find the lower bound of

$\sum_{m}\sqrt{P(m)P(m|X_{a})}=\sum_{m}\left|\langle
m|\psi\rangle\right|\left|\left\langle
m\right|\left(\hat{I}-\left|a\right\rangle\hskip-2.84526pt\left\langle
a\right|\right)\left|\psi\right\rangle\right|.$ (35)

Since the sum of the absolute value is larger than the absolute value of the
sum,

$\sum_{m}\left|\langle m|\psi\rangle\right|\left|\left\langle
m\right|\left(\hat{I}-\left|a\right\rangle\hskip-2.84526pt\left\langle
a\right|\right)\left|\psi\right\rangle\right|\geq\left|\sum_{m}\langle\psi|m\rangle\left\langle
m\right|\left(\hat{I}-\left|a\right\rangle\hskip-2.84526pt\left\langle
a\right|\right)\left|\psi\right\rangle\right|.$ (36)

The sum of the projectors $\left|m\right\rangle\hskip-2.84526pt\left\langle
m\right|$ for the outcomes $m$ is equal to the identity operator $\hat{I}$,

$\sum_{m}\left|m\right\rangle\hskip-2.84526pt\left\langle m\right|=\hat{I}$
(37)

so the lower bound is given by

$\sum_{m}\sqrt{P(m)P(m|X_{a})}\geq 1-P(a).$ (38)

By substituting this lower bound into Eq. (34) we obtain the bound given by
Eq. (14),

$\Delta_{a}\leq\frac{1}{2}P(a)+\frac{1}{2}\sqrt{P(a)(4-3P(a))}.$ (39)

The bound of the counterfactual gain given by Eq. (14) is obtained by
subtracting $P(a)$ on both sides. It is easy to confirm that the maximal
counterfactual gain overall is a gain of $1/3$ at $P(a)=1/3$.

For comparison, the maximal counterfactual gain obtained when the there are no
false positives ($P(m)=0$ for any outcome with $P(m|X_{a})>P(m)$) is obtained
when $|\langle m|a\rangle|^{2}=(1-P(a))$, so that

$\Delta_{a}-P(a)\leq P(a)(1-P(a)).$ (40)

As expected, the maximal gain of $1/4$ is achieved for $P(a)=1/2$. In addition
to this reduction of the maximal gain, the additional condition of no false
positives used in the Elitzur-Vaidman scenario makes it more difficult to
observe counterfactual gain at low absorption probability. If you are aiming
to maximise counterfactual gain, as defined by statistical distance, there
seems to be no reason to avoid the occurrence of false positives.