# Weak values and the past of a quantum particle

Jonte R. Hance<EMAIL_ADDRESS>Quantum Engineering Technology
Laboratories, Department of Electrical and Electronic Engineering, University
of Bristol, Woodland Road, Bristol, BS8 1US, UK  John Rarity  Quantum
Engineering Technology Laboratories, Department of Electrical and Electronic
Engineering, University of Bristol, Woodland Road, Bristol, BS8 1US, UK  James
Ladyman<EMAIL_ADDRESS>Department of Philosophy, University of
Bristol, Cotham House, Bristol, BS6 6JL, UK

###### Abstract

We investigate four key issues with using a nonzero weak value of the spatial
projection operator to infer the past path of an individual quantum particle.
First, we note that weak measurements disturb a system, so any approach
relying on such a perturbation to determine the location of a quantum particle
describes the state of a disturbed system, not that of a hypothetical
undisturbed system. Secondly, even assuming no disturbance, there is no reason
to associate the non-zero weak value of an operator containing the spatial
projection operator with the classical idea of ‘particle presence’. Thirdly,
weak values are only measurable over ensembles, and so to infer properties of
individual particles from values of them is problematic. Finally, weak value
approaches to the path of a particle do not provide information beyond
standard quantum mechanics (and the classical modes supporting the
experiment). We know of no experiment with testable consequences that
demonstrates a connection between particle presence and weak values.

##  I Introduction

Weak values are obtained via weak measurement and pre- and post-selection, of
sub-ensembles from ensembles of particles [1]. They have been shown to be very
useful for metrology and state tomography tasks.[2, 3, 4]. Weak values have
also been used to infer the presence of individual quantum particles, where
standard quantum mechanics would not normally allow this. According to these
weak value approaches to particle presence, non-zero weak values show where a
quantum particle has been, even when this would normally be impossible due to
the pre- and post-selected states not being eigenstates of spatial projection
operators [5, 6]. Despite weak values only being defined over ensembles, in
these approaches, inferred particle presence is attributed to individual
particles.

One such approach is the Weak Trace Approach, according to which particles
leave traces along the paths they traverse, due to interaction (coupling) with
the environment, and such a trace is left even when this interaction is
insufficient to count as a projective measurement. A non-zero weak value of an
operator on a path (weak trace) is taken to show that an individual particle
travelled along that path [7] even in the limit of this interaction (coupling)
vanishing.

According to this approach there is a weak trace in a region when any operator
formed by the product of the spatial projection operator for that region with
an operator for a property of the particle (e.g. spin) has a non-zero weak
value. This can happen even if the weak value of the spatial projection
operator itself is zero. Weak traces are supposed to give us more information
about the particle than von Neumann measurement of the spatial projection
operator would allow. The Weak Trace Approach even allows discontinuous
particle trajectories, hinting at a mechanism by which seemingly disconnected
events can affect one another, such as in counterfactual communication [8].
Therefore, this approach has been controversial [9, 10, 11, 12, 13, 14, 15,
16, 17, 18].

Another approach connecting weak values and particle presence, implicit in the
quantum Cheshire cat [19], requires that the spatial projection operator at a
particular location has a non-zero weak value, for us to say the particle was
present at that location. In the quantum Cheshire cat, these approaches come
apart since the weak value of the spatial projection operator on a path can be
zero, but the weak value of the spin operator times the spatial projection
operator non-zero—in these cases, according to the Weak Trace Approach the
particle _was_ on the path with a zero weak value for the spatial projection
operator, contrary to [19].

In this paper, we show how one obtains the weak value of an operator. We then
give a case where the weak value of an operator containing the spatial
projection operator is unexpectedly non-zero (and so, by weak value
approaches, a particle has left an unexpected trace in its environment). After
this, we look specifically at the Weak Trace Approach. It is argued in [20]
that the Weak Trace Approach is justified because there is some non-vanishing
local interaction between the quantum particle and its environment along
whatever path it travels [20]. This implies the weak value of the spatial
projection operator along a path should always be non-zero whenever a particle
travels along that path, and so the particle was not present if the weak value
of the spatial projection operator is zero. However, there are cases where
according to the Weak Trace Approach a particle is present on a path, despite
the spatial projection operator for that path having weak value zero (e.g. in
the quantum Cheshire cat set-up).

We then investigate four key issues with weak value approaches.

First, even weak measurements disturb a system, so any approach relying on
such a perturbation to determine the location of a quantum particle will only
describe this disturbed system, not a hypothetical undisturbed state. We
highlight this using the case of a balanced interferometer tuned to have
destructive interference (i.e. no light exiting at its dark port) where such a
perturbation changes the nature of the system. The unperturbed state is the
vacuum, while the perturbed state has light present. While the measurement
effect can be made arbitrarily small, this is not the same as removing it
entirely. Further, we show attempts to respond to this line of argument by
saying there are no completely unperturbed systems, as weak interactions are
ubiquitous in nature (as in [20]), raises questions about why weak value
approaches only associate presence with the part of this interaction which is
of the same order of magnitude as the coupling between system and environment.

Secondly, even assuming no disturbance, there is no reason to associate the
non-zero weak value of an operator containing the spatial projection operator
with the classical idea of ‘particle presence’. Indeed, in some situations,
just taking the weak value gives features inconsistent with classical ideas
associated with a particle being present (e.g., giving discontinuous particle
trajectories, or particles not being in coarse-grainings of their location).

Thirdly, weak values are only measurable over ensembles, and so to infer
properties of individual particles from values of them is problematic at best.

Finally, weak value approaches to the path of a particle do not give us any
new physics beyond standard quantum theory [21], to explain the causes of
counterintuitive quantum effects (or even the paradoxes the approach itself
creates). They assume a connection between particle presence and weak values
without contributing testable new physics.

##  II Weak Values

We first present the derivation of a weak value of an operator, and how it is
argued this relates to the trace left by a particle on its environment.

The weak value $O_{w}$ of an operator $\hat{O}$ is defined as [1]

$O_{w}=\langle\hat{O}\rangle_{w}=\frac{\left\langle\psi_{f}\right|\hat{O}\left|\psi_{i}\right\rangle}{\langle\psi_{f}|\psi_{i}\rangle}$
(1)

As this weak value increases as $\langle\psi_{f}|\psi_{i}\rangle$ goes to
zero, weak value protocols are used in metrology to amplify signals from
delicate results so they can be observed experimentally [2, 3, 4]. This is at
the expense of postselection reducing success probability. Weak values however
also lead to a range of paradoxes [22].

In practice, the weak value is measured experimentally by taking the
postselected result of a weak measurement. To obtain this weak value, we first
couple our initial system $\left|\psi_{i}\right\rangle$ to our initial pointer
state, $\left|\phi\right\rangle$, by weakly measuring them with the probe
Hamiltonian $\hat{H}=\lambda\hat{O}\otimes\hat{P_{d}}/T$ for small coupling
constant $\lambda$ and state-probe interaction time $T$. This produces the
state

$\begin{split}\left|\psi_{w}\right\rangle&=e^{-\frac{i\hat{H}T}{\hbar}}\left|\psi_{i}\right\rangle\otimes\left|\phi\right\rangle\\\
&=e^{-\frac{i\lambda}{\hbar}\hat{O}\otimes\hat{P_{d}}}\left|\psi_{i}\right\rangle\otimes\left|\phi\right\rangle\end{split}$
(2)

where $\hat{P}_{d}$ is the momentum of that pointer. We then strongly measure
this weak-measured state $\left|\psi_{w}\right\rangle$ with the operator

$\hat{F}_{1}=\left|\psi_{f}\right\rangle\hskip-2.84526pt\left\langle\psi_{f}\right|\otimes\hat{I}_{d}$
(3)

Assuming $\hat{P}_{d}$ has Gaussian distribution around 0 with low variance
(so $\hat{X}_{d}$, the position of the pointer, has Gaussian distribution with
high variance), we can say

$\begin{split}&e^{-\frac{i\lambda}{\hbar}\hat{O}\otimes\hat{P_{d}}}=\sum_{k=0}^{\infty}\big{(}-\frac{i\lambda}{\hbar}\hat{O}\otimes\hat{P_{d}}\big{)}^{k}/k!\\\
&\;=\mathds{1}-\frac{i\lambda}{\hbar}\hat{O}\otimes\hat{P_{d}}+\mathcal{O}(\lambda^{2})\approx\mathds{1}-\frac{i\lambda}{\hbar}\hat{O}\otimes\hat{P_{d}}\end{split}$
(4)

so this strong measurement gives the result

$\begin{split}&\left|\psi_{f}\right\rangle\hskip-2.84526pt\left\langle\psi_{f}\right|e^{-\frac{i\lambda}{\hbar}\hat{O}\otimes\hat{P_{d}}}\left|\psi_{i}\right\rangle\otimes\left|\phi\right\rangle\\\
&\approx\left|\psi_{f}\right\rangle\hskip-2.84526pt\left\langle\psi_{f}\right|(\mathds{1}-\frac{i\lambda}{\hbar}\hat{O}\otimes\hat{P_{d}})\left|\psi_{i}\right\rangle\otimes\left|\phi\right\rangle\\\
&=\left|\psi_{f}\right\rangle\otimes\langle\psi_{f}|\psi_{i}\rangle(\mathds{1}-\frac{i\lambda}{\hbar}O_{w}\hat{P}_{d})\left|\phi\right\rangle\\\
&\approx\left|\psi_{f}\right\rangle\otimes\langle\psi_{f}|\psi_{i}\rangle
e^{-\frac{i\lambda}{\hbar}O_{w}\hat{P}_{d}}\left|\phi\right\rangle\end{split}$
(5)

This entails the position of our initial pointer state
$\left|\phi\right\rangle$, acting as our “readout needle”, shifts, with
measurement of the pointer position giving a read-out value $(\langle
x\rangle-a)$. For an initial Gaussian pointer state, this shift causes the
average position to move from $\langle x\rangle$ to $(\langle
x\rangle-\overline{a})$ due to the application of the effective operator
$\mathds{1}-i\lambda O_{w}\hat{P}_{d}/\hbar$. $a$ is distributed over a wide
range of values for many repeats, but the distribution of “$a$”s will be a
Gaussian centred on (the real part of) $\lambda O_{w}$. Peculiarly, this weak
value $O_{w}$ can be very far from any of the eigenvalues of $\hat{O}$, or
even imaginary [1, 23, 24]. This is odd, given this weak value appears in
exactly the place in the equation that an eigenvalue of $\hat{O}$ would for a
Von Neumann measurement (where the variance of $\hat{X}_{d}$ on the measured
state is vanishingly small). This led to the belief that the weak value
$O_{w}$ represented some fundamental value of the operator $\hat{O}$ between
measurements.

Due to their interrelatedness, we can graphically show where the spatial
projection operator has a non-zero weak value through the Two-State Vector
Formalism (TSVF). The TSVF considers both the backwards and forwards-evolving
quantum states, rather than just the forward as in standard quantum mechanics
[25]. (A similar intuition led to the development of the Double Inferential-
state Vector Formalism [26].) For some operator $\hat{O}$, the forwards
travelling initial state $\left|\psi_{i}\right\rangle$, and the backwards
travelling final state $\left\langle\psi_{f}\right|$, the TSVF gives out a
conditional probability amplitude
$\left\langle\psi_{f}\right|\hat{O}\left|\psi_{i}\right\rangle$, where
$\left\langle\psi_{f}\right|\left|\psi_{i}\right\rangle$ is referred to as the
Two-State Vector. This conditional probability amplitude is of the same form
as the numerator of the weak value, and so we can use the TSVF to graphically
plot where the weak value of an operator containing the spatial projection
operator is nonzero. This involves plotting the forward-evolving state
(possible paths the particle could have travelled via from its original
position) and the backwards-evolving state (possible locations the particle
could have come from to reach its final position). According to Weak value
approaches, a quantum particle is present wherever these states visibly
overlap (as shown in Fig. 1).

Figure 1: Nested-interferometer model, used to discuss cases where the Two-
State Vector Formalism contradicts “common-sense” continuous-path approaches
[7].

The nested-interferometer model, which we show in Fig. 1, was pointed out as a
case where the Two-State Vector Formalism contradicted “common-sense”
continuous-path approaches [7]. The inner interferometer is balanced such that
a photon entering from arm $D$ exits to detector $D3$. Consequently, the outer
interferometer is unbalanced, so there is an equal probability of a photon
introduced from the source ending in $D1$ or $D2$. When the photon ends at
$D2$, common sense would tell you it must have travelled via path $A$. This is
as, had it travelled via $D$ into the inner interferometer, it could not have
exited onto path $E$, or reached $D2$. However, according to the Weak Trace
Approach, while the photon never travelled paths $D$ or $E$, it travelled
along paths $B$ and $C$ as well as along path $A$.

##  III The Weak Trace Approach

Consider a case where the weak value of either the spatial projection
operator, or an operator formed by multiplying the spatial projection operator
with the operator for some property of a particle, along a certain path was
non-zero for a given $\left|\psi_{i}\right\rangle$ and
$\left\langle\psi_{f}\right|$. According to the Weak Trace Approach this
indicates a particle was on that path and left a weak trace along it [7]. This
is on two grounds. First, if it were a strong measurement, an operator
containing the spatial projection operator having non-zero eigenvalues would
be sufficient for a quantum particle to be present, so correspondingly a non-
zero weak value of this operator is a weak trace of particle presence.
Secondly, if a particle was present in a region, it should have non-zero
interaction with the local environment (as reflected by the subtle decoherence
its state would undergo), and so leave a “trace” along its local path. While
this interaction is weak, which it must be so that the environmental coupling
does not collapse position superpositions in the way a projective measurement
would, according to the Weak Trace Approach it is still in principle
detectable (i.e., the interaction would be of the same order of magnitude as
$\lambda$), and so the interaction would be equivalent in scale to that
required to give a non-zero weak value of an operator containing the spatial
projection operator along that path.

However, there are cases where the Weak Trace Approach entails that a particle
was present on a path, despite the spatial projection operator for that path
having weak value zero. In these cases, the Weak Trace Approach takes it that
a particle is present along a path even when the weak value of the spatial
projection operator along that path is zero, so long as the weak value of some
other operator multiplied by the spatial projection operator is nonzero (e.g.
the weak value of the spin along the right-hand path in the quantum Cheshire
cat set-up [19]) [7]. If the justification for the Weak Trace Approach is the
non-vanishing local interaction, as in [20], then, if a quantum particle
travels along a path, either: the weak value of the spatial projection
operator on that path should be non-zero; or we should consider the higher-
order terms in $\lambda$, as they would indicate additional information about
the path on which the particle travelled; or we should say a non-zero weak
value for the spatial projection operator is sufficient, but not necessary, to
indicate the presence of a quantum particle, as this non-vanishing local
interaction can exist along a path without there necessarily being a non-zero
weak value for this operator. Hence, the Weak Trace Approach implicitly
requires this third option, which raises the question of how it can be used to
infer that particles were never present in some specific regions (see the
nested interferometer, below). However, it could be that unlike the specific
case of the weak value of the spatial projection operator, there existing at
least one operator formed of the product of the spatial projection operator
for a location and some other operator, with a non-zero weak value, is both a
necessary and a sufficient condition for particle presence at that location.
Further, it could be that in the nested interferometer case, this condition
cannot be met at certain points, so the particle was not there according to
the Weak Trace Approach.

While such an argument resolves any formal contradiction in associating a non-
zero weak value for the product of some operator with the spatial projection
operator with presence, even when the weak value of the spatial projection
operator is zero, this still seems peculiar. A key part of the motivation
behind weak value approaches is some assumption that the weak value of an
operator tells us something about the property associated with that operator,
similarly to an eigenvalue obtained by measuring that operator. By this
assumption, while measuring the weak value of a spatial projection operator
infers some information about whether the particle was at the location
corresponding to that operator, measuring the weak value of some product of
the spatial projection operator with some other operator corresponding to a
property of the particle only infers some information about that property at
that location. It requires the separate assumption that a property of a
particle cannot be disembodied from that particle to then use the nonzero weak
value of such a product operator to then infer about the particle’s location.

##  IV Issues with the Weakness Assumption

The process of weak measurement and strong postselection given above leads to
a small shift in the position $X_{d}$ of our pointer system
$\left|\phi(x)\right\rangle$, with an average value

$\overline{a}=\lambda\text{Re}(O_{w})<<\Delta X_{d}$ (6)

Performing the measurement on a pre-and-postselected ensemble of $N$ particles
allows measurement of the shift to precision $\Delta X_{d}/\sqrt{N}$.
Therefore, according to the Weak Values Approach, as long as $N>(\Delta
X_{d})^{2}\mathcal{O}(\lambda^{-2})$, the presence of the particle is
revealed. This is still however a coupling—so long as $\lambda\neq 0$, there
is measurement. This is as it must be, because Busch’s Theorem says we cannot
gain information about the state of the system without disturbing it [27].

This is in tension with with the Weakness Assumption inherent to weak
measurement—that if we measure weakly enough, we can effectively see how the
system behaves when it is not measured. As argued above, weak interaction is
still interaction and so disturbs the system (albeit negligibly so for some
purposes). As long as there is a non-zero coupling, as in all weak value
experiments, the ‘negligibly’ small disturbance is still a disturbance, and
there are differences in observable effects between small coupling and no
coupling (see [28] for an example of such a difference).

According to popular interpretations of weak values they exist in the absence
of measurement and perturbation [29, 30, 21]. However, this is clearly not the
case [31, 32, 33, 34, 35, 36, 37, 38, 28]. To defend against this point [20]
explicitly abandons the Weakness Assumption, saying, “in a hypothetical world
with vanishing interaction of the photon with the environment, $[$the Weak
Trace definition$]$ is not applicable, but in the real world there is always
some non-vanishing local interaction. Unquestionably, there is an unavoidable
interaction of the photon with the mirrors and beam splitters of the
interferometer”.

This statement, while formally correct and sufficient to avoid the issues
associated with the Weakness Assumption, raises questions about why we should
only consider the part of the weak trace which is of $\mathcal{O}(\lambda)$,
especially when considered in the context of the approach taken practically in
obtaining spatial weak values. If the purpose of weak value approaches is to
identify the non-vanishing interaction a quantum particle has with its
environment, to trace its path, then a definition of this trace which neglects
some of these non-vanishing local interactions is only telling us part of the
story. Weak value approaches only consider the first-order trace, as per the
approximation given in Eq. 4, due to the supposedly vanishing nature of the
terms of second-order or higher in $\lambda$. Despite this, [20] specifically
says that these environmental terms provide non-vanishing local interaction. A
typical photon interaction with a mirror for instance transfers of order
$10^{-33}$ of its energy per reflection (for a 1500nm-wavelength photon and a
1 gram mirror), a suitably weak interaction that is mostly safely ignored.
Hence even the higher order terms are many orders of magnitude larger than the
strongest perturbation from the environment in optics experiments, and so the
invoking in [20] of the weak interaction naturally left by the quantum object
on its surroundings, while still having the weak trace ignore higher-order
terms, seems more an attempt to avoid the issues with the Weakness Assumption
than a physically well-motivated argument for believing the weak trace tells
us everything about the location of a quantum particle.

It could be argued that this is more an objection to the experimental methods
used to detect spatial weak values than the application of the concept
itself—however, it is consideration of these experimental methods that is
taken to motivate neglecting the higher-order terms (due to their comparative
undetectability)—so therefore, if we are instead looking more theoretically at
the non-vanishing local interactions, we should associate presence with all
terms, rather than just the first order terms in $\lambda$.

The need to consider higher-order coefficients of $\lambda$ has also been
discussed in [39, 40, 41, 42].

##  V Do weak values reveal particle presence?

In this Section, we argue that, even assuming no disturbance, there is no
reason to associate the non-zero weak value of the spatial projection operator
with the classical idea of particle presence. To make this argument, we first
give a key assumption for this section—that presence is a property typically
ascribed to classical particles, and so any attempt to form a definition of
presence for quantum particles should correspond to our intuitions about
classical presence, unless we have a good reason for it to deviate from this.

The classical conception of a particle presence—being present at a certain
place at a certain time—can be characterised as follows:

  1. i).

Every particle is located in space at all times.

  2. ii).

Particles cannot be on more than one path simultaneously.

  3. iii).

Particle trajectories are continuous (or at least as continuous as space is)
so particles cannot get from one place to another without passing through the
space in between.

  4. iv).

Particles interact with other objects/fields local to their location.

  5. v).

If a particle is on a path at a given time, and that path is within some
region, then the particle is also located in that region at that time.

  6. vi).

If a particle’s property is at a location, the particle must be at that
location too.

In standard quantum mechanics, when a particle is in an eigenstate of a
position operator, then it is attributed a position. However, quantum
particles can be in superpositions of position eigenstates. Whether they then
have positions at all, or even have two positions at once, is contentious and
interpretation-dependent. To require that there is always some location at
every time where conditions (i) and (ii) are satisfied is to advocate a hidden
variable approach to quantum mechanics, where the hidden variable is the
particle’s location. However, despite quantum tunnelling and other phenomena,
a reformulation of condition (iii) still applies in so far as the probability
current of quantum particles evolving according to the Schrödinger equation
obeys a continuity equation. (iv-vi) are also compatible with standard quantum
mechanics.

Particles that are not in an eigenstate of one of the spatial projection
operators demarcating coarse-grained locations of interest (e.g. specific
paths in a nested-interferometer experiment) can nonetheless have nonzero weak
values for any of those spatial projection operators (or any composite
operators formed of any of those spatial projection operators multiplied by
some other operator). Perhaps a condition for a particle being present at a
location in a quantum context is that it has a non-zero weak value for an
operator containing the spatial projection operator at that location. Whenever
a particle satisfies condition (ii), it also satisfies this condition, as its
forwards and backwards-travelling states always overlap. However, this can
only be a necessary condition for particle presence in the sense of (i) to
(vi) above, rather than a sufficient condition, as there are cases where:

  1. 1).

A particle has a non-zero weak value of an operator containing the spatial
projection operator at a location, but has no continuous path to/from this
location.

  2. 2).

A particle has a non-zero weak value of an operator containing the spatial
projection operator at a location, but a weak value of zero for an operator
containing the spatial projection operator at a coarse-graining of that
location (e.g. having a non-zero weak value of an operator containing the
spatial projection operator on path $B$, and a non-zero weak value of an
operator containing the spatial projection operator on path $C$, but a weak
value of zero for an operator containing the spatial projection operator for
the space composed of paths $B$ and $C$).

  3. 3).

A particle has a weak value of zero for the spatial projection operator along
a certain path, yet has a non-zero weak value for some other operator (e.g.
the spin operator in a given direction) multiplied by the spatial projection
operator along that path (e.g. in the quantum Cheshire cat set-up).

Using the two-state vector formalism and weak value tools, we can
quantitatively analyse the nested interferometer set-up to show Point (2). We
first define the forwards-travelling initial vector and backwards-travelling
final vector by the paths via which they evolve:

$\begin{split}\left|\psi_{i}\right\rangle=\frac{\sqrt{2}\left|A\right\rangle+\left|B\right\rangle+\left|C\right\rangle}{2}\\\
\left\langle\psi_{f}\right|=\frac{\sqrt{2}\left\langle A\right|+\left\langle
B\right|-\left\langle C\right|}{2}\end{split}$ (7)

Using these, and defining the spatial projection operator for arm $A$ as
$\hat{P}_{A}=\left|A\right\rangle\hskip-2.84526pt\left\langle A\right|$
(similarly for $B$ and $C$), we get the same weak values as in [7]:

$\begin{split}\langle\hat{P}_{A}\rangle_{w}=&\frac{\left\langle\psi_{f}\right|\left|A\right\rangle\hskip-2.84526pt\left\langle
A\right|\left|\psi_{i}\right\rangle}{\langle\psi_{f}|\psi_{i}\rangle}=1\\\
\langle\hat{P}_{B}\rangle_{w}=&\frac{\left\langle\psi_{f}\right|\left|B\right\rangle\hskip-2.84526pt\left\langle
B\right|\left|\psi_{i}\right\rangle}{\langle\psi_{f}|\psi_{i}\rangle}=\frac{1}{2}\\\
\langle\hat{P}_{C}\rangle_{w}=&\frac{\left\langle\psi_{f}\right|\left|C\right\rangle\hskip-2.84526pt\left\langle
C\right|\left|\psi_{i}\right\rangle}{\langle\psi_{f}|\psi_{i}\rangle}=-\frac{1}{2}\end{split}$
(8)

However, if we want to see if the particle was present in the inner
interferometer as a whole (as in either arm $B$ or $C$), we define
$\hat{P}_{BC}=\left|B\right\rangle\hskip-2.84526pt\left\langle
B\right|+\left|C\right\rangle\hskip-2.84526pt\left\langle C\right|$, as we are
allowed to do since projectors in standard quantum mechanics are linear. We
find

$\begin{split}\langle\hat{P}_{BC}\rangle_{w}&=\frac{\left\langle\psi_{f}\right|(\left|B\right\rangle\hskip-2.84526pt\left\langle
B\right|+\left|C\right\rangle\hskip-2.84526pt\left\langle
C\right|)\left|\psi_{i}\right\rangle}{\langle\psi_{f}|\psi_{i}\rangle}\\\
&=\frac{1}{2}-\frac{1}{2}=0\end{split}$ (9)

If we assumed a non-zero weak trace implies particle presence, this would mean
the photon was never in the inner interferometer (made up of arms $B$ or $C$)
overall. ([21, 43] explicitly say that weak values obey the sum rule, and so
allows us to say $\langle\hat{P}_{BC}\rangle_{w}$ must equal
$\langle\hat{P}_{B}\rangle_{w}+\langle\hat{P}_{C}\rangle_{w}$.) Taken
separately the summands are taken to mean that the particle was in arm B, and
was in arm C, respectively, yet the sum being $0$ is taken to mean that the
particle is not in their union (i.e., not in the inner interferometer at all);
this seems incoherent. (Aharonov et al consider a similar scenario in their
three-box experiment, and also discuss this idea of negative weak value in the
equivalent of arm C cancelling the positive weak value in the equivalent of
arm B [29]. This usefulness of considering the sign of the weak value for
resolving paradoxes is also discussed in [42].) This illustrates the
importance of the sign of the weak values, which the Weak Trace Approach, and
the graphical form of the TSVF (as in Fig.1), neglect.

For weak value experiments aiming to distinguish which-path information, we
can link this to the Visibility-Distinguishability Inequality [44, 45],

$\mathcal{D}^{2}+\mathcal{V}^{2}\leq 1$ (10)

where $\mathcal{D}$ is the distinguishability of which path the light
travelled, and $\mathcal{V}$ is the fringe-visibility at the output of the
interferometer. Therefore, doing anything which would increase the
distinguishability between the two paths (e.g. placing different tags on $B$
and $C$) will affect the interference pattern at the $BCE$ beamsplitter. Given
perfect interference is required to ensure all light that enters the inner
interferometer from $D$ exits into $D3$, anything causing distinguishability
between paths $B$ and $C$ will allow light to leak through onto $E$, and cause
a trace to show in $D2$. Therefore, it will never be showing what would happen
in an unperturbed system.

Given Eq. 10, $\langle\hat{P}_{BC}\rangle_{w}$ is a far better measure of
whether light reaching $D2$ was ever in the undisturbed inner interferometer.
This is as measuring $\langle\hat{P}_{BC}\rangle_{w}$ does not cause
distinguishability between paths $B$ and $C$ in the inner interferometer, and
so does not affect the interference pattern required to output all inner
interferometer light into $D3$. $\langle\hat{P}_{BC}\rangle_{w}$ being 0
provides support for a ‘common-sense’ path (i.e. light only travelling via $A$
to reach $D2$) in an unperturbed system.

An argument against looking at the weak value of the projector over multiple
paths (e.g. $\hat{P}_{BC}$) specific to the Weak Trace Approach is that the
weak trace is only defined as the effect of the particle on the environment
local to it—and combining the two weak values for these two paths is not a
local operation, so does not indicate anything about the weak trace. However,
this still goes against classical ideas of presence, given it shows that the
presence in each of the two arms must have opposite sign, so they can cancel
when added, and so implies that being realist about weak values implying
presence requires being realist in some way about negative presence. This
negative presence is yet another way the kind of presence that is attributed
by weak value approaches differs from classical presence.

Point (3) seems to support the Weak Trace Approach to particle presence,
whereby a non-zero weak value of any operator corresponding to a property of
the particle, multiplied by the spatial projection operator along a path,
implies the particle was present along that path. However, as we saw in
Section III, taking this to imply presence even when the weak value of the
spatial projection operator alone is zero, contradicts the argument in [20]
that the weak trace comes from the necessarily nonzero interaction between a
particle and its environment, given if this were the case, we would expect a
particle to necessarily have a nonzero weak value for the spatial projection
operator for any path along which it travels. This is a dilemma for weak value
approaches—they either contradict the rationale for weak values indicating
particle presence, or imply that a particle’s properties are not necessarily
in the same location as that particle.

From these three points, we see that the “nonzero weak value for an operator
containing the spatial projection operator” condition can be satisfied while
the conditions for a particle being present at a location are not satisfied.
Therefore, it is not a sufficient condition for particle presence in the
standard sense of the term.

##  VI Weak values are only defined over ensembles

It is important to note that the weak value approach to particle presence is
used to attribute presence to individual particles. This is despite the
experimental results of measurements of weak values necessarily being produced
by ensembles, because weak values are obtained by post-selection. It is non-
obvious that we can go from facts about ensembles to facts about individual
elements of that ensemble. As an example, we see that one can have a weak
value of the velocity of electrons in a given set-up that is greater than the
speed of light in a vacuum [46, 47, 48]. However, this does not necessarily
mean than any of the individual electrons in the ensemble used to obtain this
weak value travelled at superluminal speeds—as otherwise this would be a
violation of special relativity. Instead, similar to how group velocities of
wavepackets can be superluminal despite the phase velocity (or velocity of the
components) being below $c$, this superluminal weak value for speed is a fact
only about the ensemble, rather than any of its constituents. Similarly, there
is no reason to infer facts about the presence of individual quantum particles
from weak values of operators containing the spatial projection operator.

Note, recent work has shown anomalous weak values can be observed using
single-particle detection [49]—however, this requires the pre- and
postselected states to be protected, which effectively involves re-
initialising the preselected state on the particle after coupling. This
process is could be described as still considering an ensemble of results,
just embodied on the same particle at different times, rather than different
particles at the same time.

##  VII Why adopt weak value approaches to particle presence?

Figure 2: Two-State Vector Formalism analysis of Salih et al’s polarisation-
based single-outer-cycle protocol for counterfactual communication. Bob
communicates with Alice by turning off/on his switchable mirrors to determine
whether the photon goes to $D1$ or $D0$ respectively. We specify the
polarisation, given it determines direction of travel through the polarising
beamsplitters (PBSs). The forwards- and backwards-travelling states do not
overlap anywhere on the inner interferometer chain when there is a detection
at $D0$, meaning, by weak value approaches, the particle detected at $D0$ was
never at Bob. This means Bob’s ability to communicate with Alice is not
explained by weak value approaches any more than it is by standard quantum
mechanics.

Weak value approaches are intended to provide more (interpretational)
information about the underlying state of the system than standard quantum
mechanics. Vaidman [7] says that an issue with Wheeler’s “common-sense”
approach to particle trajectories [50], is that it is entirely operational:
not telling us anything about the underlying mechanisms at work, just the
final result. Similarly, [7] says “the von Neumann description of the particle
alone is not sufficient to explain the weak trace”—implying weak value
approaches provide something more than the standard von Neumann approach does.
We however turn this criticism back on weak value approaches – they do not
tell us anything about the underlying system either, beyond standard quantum
mechanics.

A key motivation for weak value approaches is to explain how the results of
interference are affected by changes to disconnected regions. They are
intended to show that phenomena such as Wheeler’s Delayed Choice, or Salih et
al’s Counterfactual Communication protocol [8, 51] and related effects [52,
53, 54, 55, 56] are not as “spooky” as they appear [57, 58, 59, 60]. Salih et
al however have shown that in their communication protocol, the weak value of
the spatial projection operator (or any compound operator including the
spatial projection operator) at Bob is always zero when Alice receives the
quantum particle (see Fig.2) [61]. Aharonov and Vaidman have also given an
altered protocol for counterfactual communication where the weak value of the
spatial projection operator (or any compound operator including the spatial
projection operator) at Bob is zero [62, 63]. Each of these results show that
weak value approaches do not explain this phenomenon.

While suggesting a time-symmetry to quantum processes through the TSVF, weak
value approaches do not imply any new physics beyond standard quantum theory
[21], to explain the causes of counterintuitive quantum effects when there is
a nonzero weak value of the spatial projection operator. These approaches
simply assume the particle was present wherever the weak value of an operator
containing the spatial projection operator is nonzero. Hence they posit
particle presence but contribute nothing testable to our physics. This label
confuses matters by oversimplifying a complex concept: what it means for a
specific particle to be sequentially “present” at a two specific places in
quantum field theory. It also causes a number of paradoxes by itself, such as
the discontinuous photon trajectories in the nested interferometer set-up
above.

##  VIII Conclusion

We have investigated four issues with both the Weak Trace Approach, and other
approaches where a nonzero weak value of the spatial projection operator is
implicitly assumed to indicate the presence of an individual quantum particle.
First, we have shown even weak measurements disturb a system, so any approach
relying on such a perturbation to determine the location of a quantum particle
will only describe this disturbed system, not a hypothetical undisturbed
state. Secondly, we have shown even assuming no disturbance, there is no
reason to associate the non-zero weak value of an operator containing the
spatial projection operator with the classical idea of ‘particle presence’.
Thirdly, we have discussed that weak values are only measurable over
ensembles, and so to infer properties of individual particles from values of
them is problematic at best. Finally, we have shown that weak value approaches
to the path of a particle do not contribute any new physics—the assumption of
a connection between particle presence and weak values does not give us
anything testable. This is not to say that weak values are not useful in
general—as discussed, they are very useful in metrology [2, 3, 4]—but that
these approaches specifically are not useful for helping identify the past
path of quantum particles.

Acknowledgements - We thank Hatim Salih and Sophie Inman for useful
discussions, and an anonymous reviewer who gave detailed and useful feedback
on multiple earlier versions of this paper. This work was supported by the
University of York’s EPSRC DTP grant EP/R513386/1, and the Quantum
Communications Hub funded by EPSRC grants EP/M013472/1 and EP/T001011/1.

## References

  * [1] Yakir Aharonov, David Z. Albert, and Lev Vaidman.  How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100.  Phys. Rev. Lett., 60:1351–1354, Apr 1988.  doi:10.1103/PhysRevLett.60.1351.
  * [2] P Ben Dixon, David J Starling, Andrew N Jordan, and John C Howell.  Ultrasensitive beam deflection measurement via interferometric weak value amplification.  Physical review letters, 102(17):173601, 2009.  doi:10.1103/PhysRevLett.102.173601.
  * [3] Andrew N Jordan, Julián Martínez-Rincón, and John C Howell.  Technical advantages for weak-value amplification: when less is more.  Physical Review X, 4(1):011031, 2014.  doi:10.1103/PhysRevX.4.011031.
  * [4] Jérémie Harris, Robert W Boyd, and Jeff S Lundeen.  Weak value amplification can outperform conventional measurement in the presence of detector saturation.  Physical review letters, 118(7):070802, 2017.  doi:10.1103/PhysRevLett.118.070802.
  * [5] Ariel Danan, Demitry Farfurnik, Shimshon Bar-Ad, and Lev Vaidman.  Asking photons where they have been.  Physical review letters, 111(24):240402, 2013.  doi:10.1103/PhysRevLett.111.240402.
  * [6] Lev Vaidman.  Tracing the past of a quantum particle.  Phys. Rev. A, 89:024102, Feb 2014.  doi:10.1103/PhysRevA.89.024102.
  * [7] Lev Vaidman.  Past of a quantum particle.  Phys. Rev. A, 87:052104, May 2013.  doi:10.1103/PhysRevA.87.052104.
  * [8] Hatim Salih, Zheng-Hong Li, Mohammad Al-Amri, and Muhammad Suhail Zubairy.  Protocol for direct counterfactual quantum communication.  Phys. Rev. Lett., 110:170502, Apr 2013.  doi:10.1103/PhysRevLett.110.170502.
  * [9] Zheng-Hong Li, Mohammad Al-Amri, and Muhammad Suhail Zubairy.  Comment on “past of a quantum particle”.  Phys. Rev. A, 88:046102, Oct 2013.  doi:10.1103/PhysRevA.88.046102.
  * [10] Lev Vaidman.  Reply to “comment on ‘past of a quantum particle’ ”.  Phys. Rev. A, 88:046103, Oct 2013.  doi:10.1103/PhysRevA.88.046103.
  * [11] Berthold-Georg Englert, Kelvin Horia, Jibo Dai, Yink Loong Len, and Hui Khoon Ng.  Past of a quantum particle revisited.  Phys. Rev. A, 96:022126, Aug 2017.  doi:10.1103/PhysRevA.96.022126.
  * [12] Robert B. Griffiths.  Particle path through a nested mach-zehnder interferometer.  Phys. Rev. A, 94:032115, Sep 2016.  doi:10.1103/PhysRevA.94.032115.
  * [13] Lev Vaidman.  Comment on “particle path through a nested mach-zehnder interferometer”.  Phys. Rev. A, 95:066101, Jun 2017.  doi:10.1103/PhysRevA.95.066101.
  * [14] Robert B. Griffiths.  Reply to “comment on ‘particle path through a nested mach-zehnder interferometer’ ”.  Phys. Rev. A, 95:066102, Jun 2017.  doi:10.1103/PhysRevA.95.066102.
  * [15] Hatim Salih.  Comment on “particle path through a nested mach-zehnder interferometer”.  Phys. Rev. A, 97:026101, Feb 2018.  doi:10.1103/PhysRevA.97.026101.
  * [16] Robert B. Griffiths.  Reply to “comment on ‘particle path through a nested mach-zehnder interferometer’ ”.  Phys. Rev. A, 97:026102, Feb 2018.  doi:10.1103/PhysRevA.97.026102.
  * [17] FA Hashmi, Fu Li, Shi-Yao Zhu, and M Suhail Zubairy.  Two-state vector formalism and quantum interference.  Journal of Physics A: Mathematical and Theoretical, 49(34):345302, 2016.  doi:10.1088/1751-8113/49/34/345302.
  * [18] L Vaidman.  Comment on ‘two-state vector formalism and quantum interference’.  Journal of Physics A: Mathematical and Theoretical, 51(6):068002, jan 2018.  doi:10.1088/1751-8121/aa8d24.
  * [19] Yakir Aharonov, Sandu Popescu, Daniel Rohrlich, and Paul Skrzypczyk.  Quantum cheshire cats.  New Journal of Physics, 15(11):113015, 2013.  doi:10.1088/1367-2630/15/11/113015.
  * [20] Uri Peleg and Lev Vaidman.  Comment on “past of a quantum particle revisited”.  Physical Review A, 99(2):026103, 2019.  doi:10.1103/PhysRevA.99.026103.
  * [21] Lev Vaidman.  Weak-measurement elements of reality.  Foundations of Physics, 26(7):895–906, 1996.  doi:10.1007/BF02148832.
  * [22] Yakir Aharonov and Daniel Rohrlich.  Quantum paradoxes: quantum theory for the perplexed.  John Wiley & Sons, 2005.  doi:10.1002/9783527619115.
  * [23] Boaz Tamir and Eliahu Cohen.  Introduction to weak measurements and weak values.  Quanta, 2(1):7–17, 2013.  doi:10.12743/quanta.v2i1.14.
  * [24] Justin Dressel, Mehul Malik, Filippo M Miatto, Andrew N Jordan, and Robert W Boyd.  Colloquium: Understanding quantum weak values: Basics and applications.  Reviews of Modern Physics, 86(1):307, 2014.  doi:10.1103/RevModPhys.86.307.
  * [25] Yakir Aharonov, Peter G Bergmann, and Joel L Lebowitz.  Time symmetry in the quantum process of measurement.  Physical Review, 134(6B):B1410, 1964.  doi:10.1103/PhysRev.134.B1410.
  * [26] Satosi Watanabe.  Symmetry of physical laws. part iii. prediction and retrodiction.  Rev. Mod. Phys., 27:179–186, Apr 1955.  doi:10.1103/RevModPhys.27.179.
  * [27] Paul Busch.  “no information without disturbance”: Quantum limitations of measurement.  In Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle, pages 229–256. Springer, 2009.
  * [28] Jonte R. Hance, James Ladyman, and John Rarity.  Is the dynamical quantum cheshire cat detectable?  arXiv preprint arXiv:2204.03374, 2022.  doi:10.48550/ARXIV.2204.03374.
  * [29] Yakir Aharonov, Eliahu Cohen, Ariel Landau, and Avshalom C. Elitzur.  The case of the disappearing (and re-appearing) particle.  Scientific reports, 7(1):531, 2017.  doi:10.1038/s41598-017-00274-w.
  * [30] Yakir Aharonov, Eliahu Cohen, Mordecai Waegell, and Avshalom C Elitzur.  The weak reality that makes quantum phenomena more natural: novel insights and experiments.  Entropy, 20(11):854, 2018.  doi:10.3390/e20110854.
  * [31] Bengt EY Svensson.  What is a quantum-mechanical “weak value” the value of?  Foundations of Physics, 43(10):1193–1205, 2013.  doi:10.1007/s10701-013-9740-6.
  * [32] Marcin Wieśniak.  Criticism of “asking photons where they have been”.  arXiv preprint arXiv:1407.1739, 2014.  URL: https://arxiv.org/abs/1407.1739.
  * [33] Hatim Salih.  Commentary: “asking photons where they have been” - without telling them what to say.  Frontiers in Physics, 3:47, 2015.  doi:10.3389/fphy.2015.00047.
  * [34] Bengt EY Svensson.  Non-representative quantum mechanical weak values.  Foundations of Physics, 45(12):1645–1656, 2015.  doi:10.1007/s10701-015-9951-0.
  * [35] Alon Ben-Israel and Lev Vaidman.  Comment on “non-representative quantum mechanical weak values”.  Foundations of Physics, 47(4):467–470, 2017.  doi:10.1007/s10701-017-0071-x.
  * [36] R. E. Kastner.  Demystifying weak measurements.  Foundations of Physics, 47(5):697–707, May 2017.  doi:10.1007/s10701-017-0085-4.
  * [37] BEY Svensson.  Response to comment on ‘non-representative quantum mechanical weak values’ by ben-israel and vaidman.  Foundations of Physics, 47(9):1258–1260, 2017.  doi:10.1007/s10701-017-0108-1.
  * [38] Asger C. Ipsen.  Anomalous weak values are caused by disturbance.  Foundations of Physics, 52(1):20, Jan 2022.  doi:10.1007/s10701-021-00534-w.
  * [39] Dilip Paneru and Eliahu Cohen.  Past of a particle in an entangled state.  International Journal of Quantum Information, 15(08):1740019, 2017\.  doi:10.1142/S0219749917400196.
  * [40] Yakir Aharonov, Eliahu Cohen, and Jeff Tollaksen.  Completely top–down hierarchical structure in quantum mechanics.  Proceedings of the National Academy of Sciences, 115(46):11730–11735, 2018.  doi:10.1073/pnas.1807554115.
  * [41] Danko Georgiev and Eliahu Cohen.  Probing finite coarse-grained virtual feynman histories with sequential weak values.  Physical Review A, 97(5):052102, 2018.  doi:10.1103/PhysRevA.97.052102.
  * [42] Mordecai Waegell, Eliahu Cohen, Avshalom Elitzur, Jeff Tollaksen, and Yakir Aharonov.  Quantum reality with negative-mass particles.  arXiv preprint arXiv:2201.09510, 2022.
  * [43] Alon Ben-Israel, Lukas Knips, Jan Dziewior, Jasmin Meinecke, Ariel Danan, Harald Weinfurter, and Lev Vaidman.  An improved experiment to determine the ‘past of a particle’ in the nested mach–zehnder interferometer.  Chinese Physics Letters, 34(2):020301, 2017.  doi:10.1088/0256-307X/34/2/020301.
  * [44] Gregg Jaeger, Abner Shimony, and Lev Vaidman.  Two interferometric complementarities.  Physical Review A, 51(1):54, 1995.  doi:10.1103/PhysRevA.51.54.
  * [45] Berthold-Georg Englert.  Fringe visibility and which-way information: An inequality.  Physical review letters, 77(11):2154, 1996.  doi:10.1103/PhysRevLett.77.2154.
  * [46] Daniel Rohrlich and Yakir Aharonov.  Cherenkov radiation of superluminal particles.  Phys. Rev. A, 66:042102, Oct 2002.  doi:10.1103/PhysRevA.66.042102.
  * [47] D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann.  Fast light, slow light, and phase singularities: A connection to generalized weak values.  Phys. Rev. Lett., 92:043601, Jan 2004.  doi:10.1103/PhysRevLett.92.043601.
  * [48] S. E. Ahnert and M. C. Payne.  Weak measurement of the arrival times of single photons and pairs of entangled photons.  Phys. Rev. A, 69:042103, Apr 2004.  doi:10.1103/PhysRevA.69.042103.
  * [49] Enrico Rebufello, Fabrizio Piacentini, Alessio Avella, Muriel A. de Souza, Marco Gramegna, Jan Dziewior, Eliahu Cohen, Lev Vaidman, Ivo Pietro Degiovanni, and Marco Genovese.  Anomalous weak values via a single photon detection.  Light: Science & Applications, 10(1):106, May 2021.  doi:10.1038/s41377-021-00539-0.
  * [50] John Archibald Wheeler.  The “past” and the “delayed-choice” double-slit experiment.  In Mathematical foundations of quantum theory, pages 9–48. Elsevier, 1978.  doi:10.1016/B978-0-12-473250-6.50006-6.
  * [51] Jonte R Hance, James Ladyman, and John Rarity.  How quantum is quantum counterfactual communication?  Foundations of Physics, 51(1):12, Feb 2021.  doi:10.1007/s10701-021-00412-5.
  * [52] Hatim Salih.  Protocol for counterfactually transporting an unknown qubit.  Frontiers in Physics, 3:94, 2016.  doi:10.3389/fphy.2015.00094.
  * [53] Hatim Salih.  From a quantum paradox to counterportation.  arXiv preprint arXiv:1807.06586, 2018.  URL: https://arxiv.org/abs/1807.06586.
  * [54] Jonte R Hance and John Rarity.  Counterfactual ghost imaging.  npj Quantum Information, 7:88, 2021.  doi:10.1038/s41534-021-00411-4.
  * [55] Hatim Salih, Jonte R Hance, Will McCutcheon, Terry Rudolph, and John Rarity.  Exchange-free computation on an unknown qubit at a distance.  New Journal of Physics, 23(1):013004, jan 2021.  doi:10.1088/1367-2630/abd3c4.
  * [56] Hatim Salih, Jonte R. Hance, Will McCutcheon, Terry Rudolph, and John Rarity.  Deterministic teleportation and universal computation without particle exchange.  arXiv preprint arXiv:2009.05564, 2020.  URL: https://arxiv.org/abs/2009.05564.
  * [57] Lev Vaidman.  Comment on “protocol for direct counterfactual quantum communication”.  Phys. Rev. Lett., 112:208901, May 2014.  doi:10.1103/PhysRevLett.112.208901.
  * [58] Lev Vaidman.  Counterfactuality of ‘counterfactual’ communication.  Journal of Physics A: Mathematical and Theoretical, 48(46):465303, Oct 2015.  doi:10.1088/1751-8113/48/46/465303.
  * [59] Lev Vaidman.  “counterfactual” quantum protocols.  International Journal of Quantum Information, 14(04):1640012, 2016\.  doi:10.1142/S0219749916400128.
  * [60] Lev Vaidman.  Analysis of counterfactual communication protocols.  Physical Review A, 99(5):052127, 2019.  doi:10.1103/PhysRevA.99.052127.
  * [61] Hatim Salih, Will McCutcheon, Jonte R. Hance, and John Rarity.  The laws of physics do not prohibit counterfactual communication.  npj Quantum Information, 8(1):60, May 2022.  doi:10.1038/s41534-022-00564-w.
  * [62] Yakir Aharonov and Lev Vaidman.  Modification of counterfactual communication protocols that eliminates weak particle traces.  Phys. Rev. A, 99:010103(R), Jan 2019.  doi:10.1103/PhysRevA.99.010103.
  * [63] Alon Wander, Eliahu Cohen, and Lev Vaidman.  Three approaches for analyzing the counterfactuality of counterfactual protocols.  Phys. Rev. A, 104:012610, Jul 2021.  doi:10.1103/PhysRevA.104.012610.