Quantum weak values and the 'which way?' question

Abstract

Uncertainty principle forbids one to determine which of the two paths a quantum system has travelled, unless interference between the alternatives had been destroyed by a measuring device, e.g., by a pointer. One can try to weaken the coupling between the device and the system, in order to avoid the veto. We demonstrate, however, that a weak pointer is at the same time an inaccurate one, and the information about the path taken by the system in each individual trial is inevitably lost. We show also that a similar problem occurs if a classical system is monitored by an inaccurate quantum meter. In both cases one can still determine some characteristic of the corresponding statistical ensemble, a relation between path probabilities in the classical case, and a relation between the probability amplitudes if a quantum system is involved.

Full text

Quantum weak values and the “which way?” question
A. Urangaa,b, E. Akhmatskayaa,c, and D. Sokolovskib,c,d
aBasque Center for Applied Mathematics (BCAM),
Alameda de Mazarredo 14, 48009, Bilbao, Spain
bDepartmento de Qu´ımica-F´ısica, Universidad del Pa´ıs Vasco, UPV/EHU, Leioa, Spain
cIKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, 48009, Bilbao, Spain and
dEHU Quantum Center, Universidad del Pa´ıs Vasco, UPV/EHU, 48940 Leioa, Spain
(Dated: July 16, 2024)
Uncertainty principle forbids one to determine which of the two paths a quantum system has
travelled, unless interference between the alternatives had been destroyed by a measuring device,
e.g., by a pointer. One can try to weaken the coupling between the device and the system, in
order to avoid the veto. We demonstrate, however, that a weak pointer is at the same time an
inaccurate one, and the information about the path taken by the system in each individual trial
is inevitably lost. We show also that a similar problem occurs if a classical system is monitored
by an inaccurate quantum meter. In both cases one can still determine some characteristic of the
corresponding statistical ensemble, a relation between path probabilities in the classical case, and a
relation between the probability amplitudes if a quantum system is involved.
I. INTRODUCTION
There is a well known difficulty with determining the
path taken by a quantum system capable of reaching a
known final state via several alternative routes. Accord-
ing to the Uncertainty Principle [1] such determination is
possible only if an additional measuring device destroys
interference between the alternatives. However, the de-
vice inevitably perturbs the system’s motion, and alters
the likelihood of its arrival at the desired final state. The
knowledge of the system’s past must, therefore, be incom-
patible with maintaining the probability of a successful
post-selection intact.
A suitable measuring device can be a pointer [2], de-
signed to move only if the system travels the chosen
path, so that finding it displaced at the end of experi-
ment could constitute a proof of the system’s past. A
somewhat naive way around the Uncertainty Principle
may be the use a pointer coupled to the system only
weakly, thus leaving interference between the paths al-
most intact. Perhaps the small change of the pointer’s
final state could provide “which path?” (“which way?”)
information previously deemed to be unavailable?
The idea is not new, and was applied, for example,
to an optical realisation of a three-path problem [3]-[4].
The conclusion of that the photons can be found in a
part of the setup they can neither enter nor leave, and
must therefore have discontinuous trajectories, was sub-
sequently criticised by a number of authors [5]-[7] for
both technical and more fundamental reasons. A sim-
ilar treatment of a four-path “quantum Cheshire cat”
[8]-[10] model suggests a possibility of separating a sys-
tem from its property, to wit, electrons detached from
their charges, and an atom’s internal energy “disembod-
ied” from the atom itself. (For further discussion of the
model the reader is referred to [11]). The case for a quan-
tum particle (or, at least, of some of its “properties”) be-
ing in several interfering pathways at the same time was
recently made in [12].
Here our more modest aim is to analyse, in some detail,
the validity of the approach in the case of the simplest
“double slit” (two-path) problem.
The rest of the paper is organised as follows. Section II
briefly describes the well known quantum double-slit ex-
periment. A classical analogue of the problem is studied
in Sections III-VI. A simple two-way quantum problem
is analysed in Sects. VII-XI. Section XII contains our
conclusions.
II. QUANTUM “WHICH WAY?” PROBLEM
One of the unanswered questions in quantum theory,
indeed its “only mystery” [13], concerns the behaviour of
a quantum particle in a double-slit experiment shown in
Fig.1. The orthodox [1] view is as follows. With only two
observable events, preparation and final detection, it is
impossible to claim that the particle has gone via one of
the slits (paths) and not the other. This is because the
rate of detection by the detector in Fig.1may increase if
one of the paths is blocked [1].
Neither is it possible to claim that both paths were
travelled at the same time, since an additional inspec-
tion never finds only fraction of a photon in one of the
paths [13]. However, such an inspection destroys the in-
terference between the paths, and alters the probability
of detection. The problem is summarised in the Uncer-
tainty Principle [13]: “It is impossible to design any ap-
paratus whatsoever to determine through which hole the
particle passes that will not at the same time disturb the
particle enough to destroy the interference.”
The approach of [3] may suggest a way around this ver-
dict. If two von Neumann pointers [2], set up to measure
projectors on the paths (e.g., on the states |ψ1⟩and |ψ2⟩
in Fig.1) are coupled to the particle only weakly, interfer-
ence between the paths can be preserved. If, in addition,
both pointers are found to “have moved”, albeit on aver-
age, the “weak traces” [3] left by the particle will reveal
arXiv:2407.10360v1 [quant-ph] 14 Jul 2024

2
BS1
BS2
Detector Time
FIG. 1. An optical realisation of a “double-slit” experiment.
At time t1, a photon’s wave packet |I⟩is split into two at the
first beam splitter (BS1). Its parts, |ψ1⟩and |ψ2⟩, travel both
optic fibres, t2, and are recombined into |F⟩after the second
BS2, t3. The two observed events are initial preparation, and
final detection of the photon. Where was the photon between
these two events?
its presence in both paths at the same time. The idea
appears to contradict the Uncertainty Principle and, for
this reason, deserves our attention. We start the investi-
gation by looking first at inaccurate pointers designed to
monitor a classical stochastic system in Fig.2.
III. CONSECUTIVE MEASUREMENTS OF A
CLASSICAL SYSTEM
Our simple classical model is as follows. (We ask for
reader’s patience. The quantum case will be discussed
shortly). A system (one can think of a little ball rolling
down a network of connected tubes shown in Fig.2) is
introduced into one of the two inputs at t=t1, with a
probability wi,i= 0,1. It then passes through states j
and k, where j, k = 0,1 at the times t2, and t3, respec-
tively. The experiment is finished when the system is
collected in a state l,l= 0,1 at t=t4. From each state
i, the system is directed to one of the states jwith a prob-
ability p(j←i), similarly from jto k, and finally from k
to l. There are altogether eight paths {l←k←j←i},
each travelled with a probability
P(l←k←j←i) = p(l←k)p(k←j)p(j←i),(1)
p(k←j) = δjk ,
where δjk is the Kroneker delta. (The choice of this
design will become clear shortly.)
1
0
0
0
0
1
1
10
0
Pre-
selection
Post-
selection Time
FIG. 2. A classical two-state system can reach a final state
by taking one of the the eight paths with a probability given
by Eq.(1). The system can be monitored with the help of
pointers (red arrows), which move if the system is detected
[cf. Eq.(2)]. Also shown (in dark blue) is the two-way problem
of Sect. IV.
We make the following assumptions.
1. Alice, the experimenter, knows the paths probabil-
ities in Eq.(1), but not the input values wi.
2. She cannot observe the system directly, and relies
on the readings of pointers with positions fn,n=
1,2...5, installed at different locations, as shown in
Fig.2. If the system passes through a location, the
corresponding pointer is displaced by a unit length,
fn→fn+ 1, otherwise it is left intact.
3. The pointers are, in general, inaccurate, since their
initial positions are distributed around zero with
probabilities Gn(fn) (see Fig.2). Their final posi-
tions are, however, determined precisely. We will
consider the distributions Gnto be Gaussians of
widths ∆fn,
Gn(fn) = s1
π∆f2
n
exp −f2
n
∆f2
n,(2)
ZGn(fn)dfn= 1, Gn(fn)−−−−−→
∆fn→0δ(fn).
The experiment ends just after t=t4, when Alice’s ob-
served outcomes are the five numbers fn,n= 1,...,5.
These are distributed with a probability density

3
ρ(f1, f2, f3, f4, f5) = X
i,j,k,l=0,1
wiP(l←k←j←i)G1(f1−i)G2(f2−j)G3(f3−k)G4(f4−l)G5(f5+j−1).(3)
Equation (3) is not particularly useful, since wiare un-
known. However, by making the first pointer accurate,
∆f1→0, G1(f1)→δ(f1−i) where δ(x) is the Dirac
delta, she is able to pre-select those cases where, say,
f1= 0, and collect only the corresponding statistics.
Now the (properly normalised) distribution of the re-
maining four readings does not depend on wi,
ρ0(f2, f3, f4, f5) = X
j,k,l=0,1
P(l←k←j←0) ×(4)
G2(f2−j)G3(f3−k)G4(f4−l)G5(f5+j−1),
and Alice has a complete description of the pre-selected
ensemble.
Alice can also post-select the system by selecting, e.g.,
the cases where it ends in a state 1 at t=t4. With
G4(f4)→δ(f4−1), the remaining random variables f2,
f3, and f5are distributed according to [cf. Fig.2and
Eq.(1)]
ρ1←0(f2, f3, f5) = (5)
Pj=0,1PjG2(f2−j)G3(f3−j)G5(f5+j−1)
P0+P1
,
where we introduced a shorthand
Pj≡P(1 ←j←j←0), j = 0,1.(6)
Equation (4) suggests a simple, yet useful, general crite-
rion.
•Alice can determine the system’s past location only
when she obtains a pointer’s reading whose likeli-
hood depends only on the probabilities of the sys-
tem’s paths passing through that location.
For example, at least three accurate readings (f1,f4and
one of f2,f3, or f5,) are needed if Alice is to know which
of the eight paths shown in Fig.2the system has travelled
during each trial. With Gn(fn) = δ(fn) for n= 1,2,4, a
trial can yield. e.g., values f1= 0, f2= 1, and f4= 1.
The likelihood of these outcomes is given by the proba-
bility P(1 ←1←1←0) in Eq.(1), and Alice can be
certain that the route {1←1←1←0}has indeed been
travelled.
IV. A CLASSICAL “TWO-WAY PROBLEM”
Consider next a pre- and post-selected ensemble with
two routes connecting the states 0 at t=t1and 1 at
t=t4(shown in Fig.2in dark blue). As a function of
the second pointer’s accuracy, ∆f2, the distribution of its
readings (5) changes from a bimodal, when the pointer
is accurate,
ρ1←0(f2)≡Zρ1←0(f2, f3, f5)df3df5
=P0G2(f2) + P1G2(f2−1)
P0+P1
−−−−−→
∆f2→0
P0δ(f2) + P1δ(f2−1)
P0+P1
(7)
to the original broad Gaussian for an inaccurate pointer,
ρ1←0(f2)−−−−−→
∆f2→∞
G2(f2−z),(8)
displaced as a whole by
z=P1
P0+P1
.(9)
Equation (8) reflects a known property of Gaussians, to
our knowledge first explored in [14], and discussed in de-
tail in Appendix A. The transformation of two peaks (7)
into a single maximum (8) is best described by the catas-
trophe theory [15]. For example, for P0=P1, a pitchfork
bifurcation converts two maxima and a minimum into a
single maximum for ∆f2=√2 (see Appendix B).
With a sufficiently accurate pointer, ∆f2≪1, a read-
ing always lies close to 0 or 1, and in every trial Alice
knows the path followed by the system.
With a highly inaccurate pointer, ∆f2≫1, not a sin-
gle reading f2can be attributed to one path in preference
to the other, and the route by which the system arrived
at its final state is never known (see Appendix C). In-
deed, for P0=P1, even the most probable outcome,
f2= 1/2 is equally likely to occur if the system takes
path {1←0←0←0}, or {1←1←1←0},
ρf2=1
2=1
2G21
2+G2−1
2,(10)
and the “which way?” information is clearly lost.
Still, something can be learnt about a pre- and post-
selected classical ensemble, even without knowing the
path taken by the system. Having performed many trials,
Alice can evaluate an average reading,
⟨f2⟩ ≡ Zf2ρ1←0(f2)df2=z. (11)
The quantity zin Eqs.(9) and (11) is the relative (i.e.,
renormalised to a unit sum) probability of travelling the
path {1←1←1←0}, and is independent of ∆f2.
Thus, by using an inaccurate pointer, Alice can still es-
timate certain parameters of her statistical ensemble.

4
V. TWO INACCURATE CLASSICAL
POINTERS AND A WRONG CONCLUSION
A word of caution should be added against an at-
tempt to recover the “which way?” information with
the help of Eq.(11). For two equally inaccurate pointers,
∆f2= ∆f5≡∆f≫1, [cf. Fig.2] the distribution of the
readings tends to a single Gaussian shown in Fig.3a (see
also Appendix A),
ρ1←0(f2, f5) = (12)
P0G2(f2)G5(f5−1) + P1G2(f2−1)G5(f5)
P0+P1
−−−−−→
∆f→∞
G2(f2−z2)G5(f5−z5),
where
z2=P1
P0+P1
, z5=P0
P0+P1
= 1 −z2.(13)
It may seem that (the reader can already see where we
are going with this),
(i) Each pointer in Eq.(12) “moves” only when the sys-
tem is in its path.
(ii) Eq.(12) suggests that both pointers have moved (al-
beit on average).
(iii) Hence, the system must be travelling both paths at
the same time.
To check if this is the case, Alice can add an accurate
pointer (f3) acting at t=t3(see Fig.2). If parts of the
system were in both places at t2, the same must be true
at t3, since Alice made sure that no pathway connects the
points j= 0 and k= 1. The accurate pointer should,
therefore, always find only a part of the system. Needless
to say, this is not what happens. With an additional
accurate pointer in place, three dimensional distribution
(5) becomes bimodal, again (see Fig.3b)
ρ1←0(f2, f3, f5) = (P0+P1)−1×(14)
P0G2(f2)G5(f5−1)δ(f3)+
P1G2(f2−1)G5(f5)δ(f3−1).
An inspection of statistics collected separately for f3= 0
or f3= 1 shows that at t=t2only one of the two
pointers moves in any given trial. Contour plots of the
densities in Eqs.(12) and (14) are shown in Figs.3a and b,
respectively. The fallacy (i)-(iii), evident in our classical
example, will become less obvious in the quantum case
we will study after a brief digression.
VI. CLASSICAL “HIDDEN VARIABLES”
Before considering the quantum case, it may be in-
structive to add a fourth assumption to the list of
Sect.III.
b)
a)
FIG. 3. a) Distribution of the readings ρ1←0(f2, f5) of two
inaccurate pointers with P0=P1and ∆f2= ∆f5= 10, mon-
itoring a classical system at t=t2[cf. Eq.(12)]; b) The dis-
tribution ρ1←0(f2, f3, f5) in Eq.(14) with an accurate pointer,
∆f3→0, added at t=t3. Note that integration of the distri-
bution in (b) over df3recovers the distribution shown in (a).
4. In Alice’s world all pointers have the property that
an accurate detection inevitably perturbs the sys-
tem’s evolution.
For example, whenever the pointer f3moves, the prob-
abilities p(l←1) in Eq.(1) are reset to p′(l←1,∆f3).
Thus, P1changes to P′
1(∆f3), while P0remains the same.
The change may be the greater the smaller is ∆f3, and
P′
1(∆f3→ ∞) = P1. Now the system, accurately ob-
served in the state 1 at t3arrives in state 1 at t=t4,
say, less frequently than it would with no pointer (f3) in
place, P0+P′
1(∆f3→0) < P0+P1. So, where was the
unobserved system at t=t3?
Empirically, the question has no answer. To ensure the
arrival rate is unchanged by observation, Alice can only
use an inaccurate pointer, ∆f3≫1, which yields no
“which way?” information. Performing many trials, she
can, however, measure both the probability of arriving in
1 at t=t4,W1(t4) = P0+P′
1(∆f3≫1) ≈P0+P1and
the value of z=P′
1(∆f3≫1)/[P0+P′
1(∆f3≫1)] ≈

5
P1/[P0+P1] [cf. Eqs.(8) and (9)]. She can then evaluate
unperturbed path probabilities,
P1≈zW1(t4), P0≈(1 −z)W1(t4).(15)
Having observed that P0and P1are both positive, and do
not exceed unity, Alice may reason about what happens
to unobserved system in the following manner. Available
empirical data is consistent with the system always fol-
lowing one of the two paths with probabilities in Eq.(15).
However, with the available instruments, it is not possi-
ble to verify this conclusion experimentally.
This is as close as we can get to the quantum case
using a classical toy model. We consider the quantum
case next.
VII. CONSECUTIVE MEASUREMENTS OF A
QUBIT
A quantum analogue of the classical model just dis-
cussed is shown in Fig.4. Experiment, in which Alice
monitors the evolution of a two-level quantum system
(qubit) with a Hamiltonian ˆ
Hsby means of five von Neu-
mann pointers, begins at t=t1and ends at t=t4. With
no transitions between the states |b0(1)⟩and |c1(0)⟩there
are altogether eight virtual (Feynman) paths which con-
nect the initial and final states. Just before t1the qubit
may be thought to be in some state |Ψin⟩, and the eight
path amplitudes are given by (i, j, k, l = 0,1)[cf. Fig.4]
As(Fl←cj←bj←Ii) = (16)
a(Fl←cj)a(cj←bj)a(bj←Ii),
where a(bj←Ii) = ⟨bj|ˆ
Us(t2−t1)|Ii⟩etc., and ˆ
Us(t)≡
exp(−iˆ
Hst) is the qubit’s own evolution operator.
We note the following.
1. Alice the experimenter knows the path amplitudes
in Eq.(16), but not the system’s input state |Ψin⟩.
(If she did, the experiment would begin earlier, at
the time |Ψin⟩was first determined.)
2. Alice cannot look at the system directly, and has
access only to von Neumann pointers [2] with posi-
tions fn, and momenta λn,n= 1, ...5, (see Fig.4).
The pointers are briefly coupled to the system at
t=tn, (t5≡t2), via
ˆ
Hint
n=−i∂fnˆπnδ(t−tn),ˆπn=|Xn⟩⟨Xn|,(17)
|Xn⟩=|I1⟩,|b1⟩,|c1⟩,|F1⟩,|b0⟩, n = 1, ...5.
and have no own dynamics.
3. The pointers, initially in states |Gn⟩, are inaccu-
rate, with initial positions distributed around zero
0
Pre-
selection
Post-
selection Time
0
FIG. 4. A two-level quantum system can reach final states
|Fl⟩,l= 0,1, via eight virtual paths whose probability am-
plitudes are given by Eq.(16). The system is monitored by
means of von Neumann pointers (red arrows), set to measure
projectors ˆπn[cf. Eq.(17)]. Also shown (in maroon) is the
“double-slit problem” of Sect.VIII.
with probability amplitudes Gn(fn)≡ ⟨fn|Gn⟩.
We will consider Gaussian pointers,
Gn(fn) = 2
π∆f2
n1/4
exp −f2
n
∆f2
n,(18)
ZG2
n(fn)dfn= 1, G2
n(fn)−−−−−→
∆fn→0δ(fn).
4. A pointer perturbs the qubit’s evolution, except
in the limit ∆fn→ ∞. Indeed, replacing fn
with f′
n=fn/∆fnchanges ˆ
Hint in Eq.(17) to
ˆ
Hint′=ˆ
Hint/∆fn, and a highly inaccurate pointer
decouples from the qubit [11]. Vice versa, a weakly
coupled pointer is, necessarily, an inaccurate one.
As in the classical case, to be able to make statistical
predictions, Alice needs to make the first measurement
accurate, ∆f1→0, G2
1(f1)→δ(f1), and pre-select, e.g.,
only those cases where f1= 0, preparing thereby the
system in the state |I0⟩. The rest of the readings are
distributed according to (we use ˜ρto distinguish from

6
the classical distributions of Sects. III-V)
˜ρ0(f2, f3, f4, f5) = X
l=0,1|G4(f4−l)|2×(19)
X
j=0,1
G3(f3−j)G2(f2−j)×
G5(f5+j−1)As(Fl←cj←bj←I0)
2
.
As in the classical case, Alice can also post-select the
qubit, e.g., in a state |F1⟩, by choosing ∆f4→0,
G2
4(f4)→δ(f4), and collecting the statistics only if
f1= 0 and f4= 1. The distribution of the remaining
three readings is given by
˜ρ1←0(f2, f3, f5) = W1(t4)−1×(20)
X
j=0,1
G3(f3−j)G2(f2−j)G5(f5+j−1)As
j
2
,
where we introduced a shorthand
As
j≡ As(F1←cj←bj←I0), j = 0,1.(21)
The normalisation factor W1(t4)≡Rρ1←0(f2, f3, f5)
df2df3df5is the probability of reaching the final state |F1⟩
with all three pointers in place, which depends on the
pointers’ accuracies,
W1(t4) = |As
0|2+|As
1|2+ 2J2J3J5Re [As
0
∗As
1],(22)
Jn≡ZGn(fn)Gn(fn−1)dfn= exp −1
2∆f2
n.
The general rule of the previous Section can be extended
to the quantum case as follows.
•Alice may ascertain the qubit’s condition, repre-
sented by a state in its Hilbert space, only when
she obtains a pointer’s reading whose probability
depends only on the system’s path amplitudes for
the paths passing through the state in question.
As in the classical case, three accurate measurements al-
low one to determine the path followed by the qubit.
For example, with ∆f1,∆f2,∆f4→0, outcomes f1= 0,
f2= 1, f4= 1, whose probability is
P(1,1,0) ≡Zϵ
−ϵ
df2Z1+ϵ
1−ϵ
df4Z∞
−∞
df3df5(23)
טρ0(f2, f3, f4, f5)−−−−−−−−→
∆f2,∆f4→0|As
1|2,
indicates that qubit has followed the path {F1←c1←
b1←I0}(see Fig.4).
VIII. A QUANTUM “DOUBLE-SLIT”
PROBLEM
The simple model shown in Fig.4has the essential fea-
tures of the setup shown in Fig.1, and is simple to anal-
yse. Two paths connect the initial and final states, |I0⟩
at t=t1and |F1⟩at t=t4; pointers f2and f5monitor
the presence of the qubit in each path at t=t2, and the
pointer f3can be used for additional control. For sim-
plicity, Alice can decouple two pointers from the qubit
by sending
∆f3,∆f5→ ∞, J3, J5→1.(24)
As a function of the remaining pointer’s accuracy, ∆f2,
the distribution of its readings (20) changes from bi-
modal,
˜ρ1←0(f2)≡Z˜ρ1←0(f2, f3, f5)df3df5(25)
−−−−−→
∆f2→0|As
0|2δ(f2) + |As
1|2δ(f2−1)
|As
0|2+|As
1|2,
to a single broad Gaussian,
˜ρ1←0(f2) = |G2(f2)As
0+G2(f2−1)As
1|2
|As
0|2+|As
1|2+ 2Re [As∗
0As
1](26)
−−−−−→
∆f2→∞
G2
2(f2−˜z),
displaced as a whole by
˜z= Re As
1
As
0+As
1,(27)
where we have used Eq.(D4) of Appendix D. The trans-
formation between the two forms is similar to transfor-
mation of the classical probability from (7) to (8) (see
Appendix D).
As in the classical case, with an accurate pointer,
∆f2≪1, a reading is always either 0 or 1, and in every
trial Alice knows the path followed by the qubit.
For a highly inaccurate pointer, ∆f2≫1, there is not
a single reading f2which can be attributed to one path in
preference to the other (cf. Appendix C), so Alice never
knows how the qubit arrived at its final state. Indeed,
even the probability of the most likely reading, f2= ˜z,
contains contributions from each path,
˜ρ1←0(f2= ˜z) = |As
0G2(˜z) + As
1G2(˜z−1)|2
|As
0+As
1|2.(28)
However, Alice may gain information about a pre- and
post-selected ensemble even without knowing the path
chosen by the qubit. Having performed many trials (it
will take more trials the larger is ∆f2), she can evaluate
the average reading, i.e. first moment,
⟨f2⟩1←0≡Zf2˜ρ1←0(f2)df2= (29)
|As
1|2+J2Re[As
0
∗As
1]
|As
0|2+|As
1|2+ 2J2Re[As
0
∗As
1]−−−−−→
∆f2→∞
˜z.
There is no contradiction with the Uncertainty Principle,
which permits knowing the amplitudes As
i, [and, there-
fore, their particular combination (27)]. What the Prin-
ciple forbids is using this knowledge to answer, among
other things, the “which way?” question. We illustrate
this by the next example.

7
IX. TWO INACCURATE QUANTUM
POINTERS, AND ANOTHER CONCLUSION
NOT TO MAKE
As in the classical case, Alice can employ at t=t2
two highly inaccurate pointers, ∆f2= ∆f5≡∆f≫
1, which measure projectors on the states |b0⟩and |b1⟩,
respectively. Now, by Eq.(D7), the distribution of the
readings is Gaussian,
˜ρ1←0(f2, f5) = (30)
|As
0G2(f2)G5(f5−1) + As
1G2(f2−1)G5(f5)|2
|As
0|2+|As
1|2+ 2J2J5Re [As∗
0As
1]
−−−−−→
∆f→∞
G2
2(f2−˜z2)G2
5(f5−˜z5),
where
˜z2= Re As
1
As
0+As
1,˜z5= Re As
0
As
0+As
1= 1 −˜z2.
(31)
And, as in the classical case, we encourage the reader
to avoid the following reasoning (see section VIII).
(i) A pointer “moves” (“weak trace” [3] is produced)
[cf. Eq.(26)] only when the qubit is in the state
upon which the projection is made.
(ii) Eq.(30) suggests that both pointers have moved (al-
beit on average).
(iii) Hence, there is experimental evidence of the qubit’s
presence in both states at t=t2and, therefore, in
both paths connecting |I0⟩with |F1⟩.
As in the classical case, we find the fault with using the
position of the maximum of the distribution (30). As
was shown in the previous Section, an inaccurate quan-
tum pointer looses the “which way?” information. The
information cannot, therefore, be recovered by employ-
ing two, or more, such pointers to predict the presence
of the qubit in a given state.
In [13] it was pointed out that assuming that in a
double slit experiment the particle passes through both
slits at the same time may lead to a wrong prediction.
Namely, only a part of an electron, or photon, would need
to be detected at the exit of a slit, and this is not what
happens in practice. Next we briefly review the argument
of [13] in the present context.
X. A “WRONG PREDICTION”
If not convinced by the argument of the previous Sec-
tion, Alice can follow the advice of [13], and attempt to
study qubit’s evolution in more detail. In particular, she
can add an accurate pointer acting at t=t3(see Fig.4),
in order to detect only a part of the qubit travelling along
the path {F1←c1←b1←I0}. If the distribution (30)
b)
a)
c)
FIG. 5. a) Distribution of the readings ˜ρ1←0(f2, f5) of two
inaccurate quantum pointers with As
0=−As
1/2 and ∆f2=
∆f5= 10, monitoring a qubit at t=t2[cf. Eq.(30)]; b)
The distribution ˜ρ1←0(f2, f3, f5) in Eq.(32) with an accurate
pointer, ∆f3= 0, added at t=t3. c) The result of integrating
the distribution in (b) over df3. Note that in (c) one does not
recover the distribution shown in (a).
is a proof of the qubit being present in both paths at t2
in any meaningful sense, this must be the only logical
expectation. Since there is no path connecting |b0⟩with
|c1⟩(see Fig.4), two parts of the qubit cannot recombine
in |c1⟩at t3> t2.
However, at t=t3Alice finds either a complete qubit,

8
or no qubit at all. As, ∆f3→0, the distribution (20)
becomes bimodal in a three dimensional space (f2,f3,f5)
˜ρ1←0(f2, f3, f5)−−−−−→
∆f3→0|As
0|2G2
2(f2)δ(f3)G2
5(f5−1)
|As
0|2+|As
1|2
(32)
+|As
1|2G2
2(f2−1)δ(f3−1)G2
5(f5)
|As
0|2+|As
1|2.
Possible values of f3are 0 and 1, and only one of the
pointers acting at t=t2is seen to “move” in any given
trial. Contour plots of the densities in Eqs.(30) and (32)
are shown in Figs.5a and b, respectively. Note that inte-
grating the density in Fig.5b over df3does not reproduce
that in Figs.5a, but rather the density (12)[cf. Fig.3a] for
a classical system with P0= 1/5 and P1= 4/5, shown in
Figs.5c.
One can still argue that Alice does not compare like
with like, since the added accurate pointer perturbs
qubit’s evolution in a way that makes it choose the path
{F1←c1←b1←I0}. The difficulty with this ex-
planation is well known in the analysis of delayed choice
experiments [16]. Decision to couple the accurate pointer
may be taken by Alice after t2, and cannot be expected
to affect the manner in which the qubit passes through
the states |b0⟩and |b1⟩. This argument usually serves as
a warning against naive realistic picture for interpreting
quantum phenomena [16]. Conclusion in the case studied
here is even simpler. “Weak traces” are not faithful indi-
cators of the system’s presence at a given location, and
using them as such leads to avoidable contradictions.
XI. QUANTUM “HIDDEN VARIABLES”
In order to keep the rate of successful post-selections
intact, W1(t4) = |As
0+As
1|2, Alice may only use weakly
coupled and, therefore, inaccurate pointers. These, as
was shown above, yield no information as to whether the
qubit was in the state |b0⟩or |b1⟩at t=t2in any given
trial, so the question remains unanswerable in principle.
The same is true for classical inaccurate pointers in Sect.
VI, but there it was possible to deduce the probabilities,
P0and P1, with which the system travels each of the
two paths [cf. Eq.(15)]. In quantum case, an attempt
to find directly unobservable “hidden” path probabilities
governing statistical behaviour of an unobserved system
fails for a simple reason. With no a priori restrictions
on the signs of As
i, the measured ˜zin Eq.(27) can have
any real value (see, e.g., [17]). For a negative ˜z, the
“probability” ascribed to the path {F1←c1←b1←I0},
P1= ˜z W (t4)<0 (33)
will also have to be negative. Thus, P1cannot be re-
lated to a number of cases in which the system follows
the chosen path [18], and a realistic explanation of the
double-slit phenomenon fails, as expected.
XII. SUMMARY AND DISCUSSION
In summary, a weakly coupled pointer, employed to
monitor a quantum system, is, by necessity, an inaccurate
one. As such, it looses information about the path taken
by the system in any particular trial, yet one can learn
something about path probability amplitudes.
A helpful illustration is offered by a classical case,
where a stochastic two-way system is observed by means
of a pointer, designed to move only if the system takes
a particular path, leading to a chosen destination. The
pointer can be rendered inaccurate by making random its
initial position. For an accurate pointer, the final distri-
bution of reading consists of two non-overlapping parts,
and one always knows which path the system has trav-
elled. For a highly inaccurate pointer, the final distribu-
tion is broad, and not a single reading can be attributed
to one path in preference to the other.
Distribution of initial pointer’s positions can be cho-
sen to be a Gaussian centred at the origin. It is a curi-
ous property of broad Gaussians, that the final pointer’s
reading repeats the shape of the original distribution [cf.
Eq.(7)], shifted by a distance, equal to the probability of
travelling the chosen path, conditional on reaching the
desired destination [cf. Eq.(8)]. Transition from two
maxima to a single peak, achieved when the width of
the Gaussian reaches the critical value, is sudden, and
can be described as the cusp catastrophe [see Appendix
B]. Thus, although the “which way” information is lost
in every trial, one is still able to determine parameters
(path probabilities) of the relevant statistical ensemble,
e.g., by looking for the most probable final reading, or
by measuring the first moment of the distribution. For a
broad Gaussian these tasks would require a large number
of trials.
The same property of the Gaussians may be respon-
sible for a false impression that two inaccurate pointers
[cf. Eqs.(12) and Fig.3a] move simultaneously (albeit on
average), and that this indicates the presence of the sys-
tem in both paths at the same time. The fallacy is eas-
ily exposed by employing one more accurate pointer (see
Fig.3b), or simply by recalling that the system cannot be
split in two.
Although the quantum case is different, parallels with
the classical example can still be drawn. The accuracy
of a quantum pointer depends on the uncertainty of its
initial position, i.e., on the wave function (18). Weak-
ening the coupling between the pointer and the system
has the same effect as broadening the initial state. The
distribution of the readings of an accurate pointer con-
sists of two disjoint parts, and one always knows which
path has been taken, at the cost of altering the proba-
bility of a successful post-selection. The only way to the
probability intact is to reduce the coupling to (almost)
zero, but then there is not a single reading which can be
attributed to a particular path.
Owing to the already mentioned property of Gaussians
(see Appendix D), the most likely reading of a highly

9
inaccurate pointer is given by the real part of a quantum
“weak value” (27), the relative (i.e., normalised to a unit
sum) path amplitude. Unlike the classical “weak values”
in Eq.(9) which must lie between 0 and 1, their quantum
counterparts in Eq.(27) can have values anywhere in the
complex plane [17]. As in the classical case, employing a
weakly coupled quantum pointer allows one to determine
certain parameters (probability amplitudes rather than
probabilities) of the quantum ensemble [cf. Eq.(15) and
Eq.(33)].
Equally inadvisable is using the joint statistics of two
weak inaccurate quantum pointers [cf. Eq.(30)], as an ev-
idence of quantum system’s presence in both pathways at
the same time. Firstly, for the reasons similar to those
discussed in the classical case and, secondly, since this
would lead to a wrong prediction [13]. An additional ac-
curate pointer always detects either an entire qubit, or
no qubit at all, albeit at the price of destroying interfer-
ence between the paths. In the setup shown in Fig.4, the
parts of the qubit, presumably present in both paths,
have no means to recombine by the time the accurate
measurement is made, hence a contradiction. A similar
problem occurs with the interpretation of delayed choice
experiments [16], to which we refer the interested reader.
Our concluding remarks can be condensed to few sen-
tences. Unlike the probabilities, the probability ampli-
tudes, used to describe a quantum system, are always
available to a theorist. Weak measurements only de-
termine the values of probability amplitudes, or of their
combinations. Uncertainty principle forbids to determine
the path taken by a quantum system, unless interference
between the paths is destroyed [1]. Hence the weak val-
ues have little to contribute towards the resolution of the
quantum “which way?” conundrum.
[1] R. P. Feynman, R. Leighton and M. Sands, The Feynman
Lectures on Physics III (Dover Publications, Inc., New
York, 1989).
[2] J. Von Neumann, Mathematical Foundations of Quantum
Mechanics (Princeton University Press, Princeton, 1955),
pp. 183-217, Chap. VI.
[3] L. Vaidman, Past of a quantum particle, Phys. Rev. A
87, 052104 (2013).
[4] A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman,
Asking photons where they have been, Phys. Rev. Lett.
111, 240402 (2013).
[5] P. L. Saldanha, Interpreting a nested Mach-Zehnder in-
terferometer with classical optics, Phys. Rev. A 89,
033825 (2014).
[6] R. B. Griffiths, Particle path through a nested Mach-
Zehnder interferometer, Phys. Rev. A 94, 032115 (2016).
[7] D. Sokolovski, Asking photons where they have been in
plain language,Phys. Lett. A 381, 227 (2014).
[8] Y. Aharonov, S. Popescu, D. Rohrlich, and P.
Skrzypczyk, Quantum cheshire cats,New. J. Phys. 15,
113015 (2013).
[9] T. Denkmayr, H. Geppert, S. Sponar, H. Lemmel, A.
Matzkin, J. Tollaksen, and Y. Hasegawa, Observation of
a quantum Cheshire Cat in a matter-wave interferometer
experiment,Nat. Commun. 5, 4492 (2014).
[10] Q. Duprey, S. Kanjilal, U. Sinha, D. Home, and A.
Matzkin, The quantum cheshire cat effect: theoretical
basis and observational implications,Ann. Phys. 391, 1
(2018).
[11] D. Sokolovski, The meaning of ”anomalous weak values”
in quantum and classical theories,Phys. Lett. A 379,
1097 (2015).
[12] S. N. Sahoo, S. Chakraborti, S. Kanjilal, S. et al.,Un-
ambiguous joint detection of spatial ly separated properties
of a single photon in the two arms of an interferometer,
Commun. Phys. 6, 203 (2023).
[13] R. P. Feynman, The Character of Physical Law (M.I.T.
press, Cambridge, Mass, London, 1985).
[14] Y. Aharonov, D. Albert, and L. Vaidman, How the result
of a measurement of a spin component of the spin of a
spin-1/2 particle can turn out to be 100,Phys. Rev. Lett.
60, 1351 (1988).
[15] E. C. Zeeman, Catastrophe Theory: Selected Papers
(Addison-Wesley Educational Publishers Inc, 1977).
[16] X-s. Ma, J. Kofler, and A. Zeilinger, Delayed-choice
gedanken experiments and their realizations ,Rev. Mod.
Phys. 88, 015005 (2016).
[17] D. Sokolovski, D. Alonso Ramirez, and S. Brouard Mar-
tin, Speakable and unspeakable in quantum measure-
ments, Ann. Phys. (Berlin) 535, 2300261 (2023).
[18] R. P. Feynman, Simulating physics with computers,Int.
J. Theor. Phys. 21, 467 (1982).
[19] Handbook of Mathematical Functions, edited by M.
Abramowitz and I. A. Stegun (Harri Deutsch, Thun,
1984).
ACKNOWLEDGEMENTS
D.S. acknowledges financial support by
the Grant PID2021-126273NB-I00 funded by
MICINN/AEI/10.13039/501100011033 and by ”ERDF
A way of making Europe”, as well as by the Basque
Government Grant No. IT1470-22.
A.U. and E.A. acknowledge the financial support by
MICIU/AEI/10.13039/501100011033 and FEDER, UE
through BCAM Severo Ochoa accreditation CEX2021-
001142-S / MICIU/ AEI / 10.13039/501100011033;
“PLAN COMPLEMENTARIO MATERIALES AVAN-
ZADOS 2022-2025 “, PROYECTO No:1101288 and
grant PID2022-136585NB-C22; as well as by the Basque
Government through ELKARTEK program under
Grants KK-2023/00017, KK-2024/00006 and the BERC
2022-2025 program. This work was also supported
by the grant BCAM-IKUR, funded by the Basque
Government by the IKUR Strategy and by the European
Union NextGenerationEU/PRTR.

10
Appendix A: Some properties of Gaussian
distributions
Consider a function
F(f) = Aexp −(f−a)2
∆f2+Bexp −(f−b)2
∆f2(A1)
with arbitrary real A,B,a, and b. For ∆f≪ |b−a|,F
has two maxima at f=aand f=b, and a single min-
imum between them. We are interested in the opposite
limit, ∆f≫ |b−a|, where Fhas a single maximum at
z=aA +bB
A+B,(A2)
easily found by solving ∂fF(f) = 0 in the limit ∆f→ ∞.
Note that if Aand Bhave opposite signs, zcan lie outside
the interval [a, b]. In fact, F(f) can be approximated by
a single Gaussian
F(f)−−−−−→
∆f→∞
˜
F(f)=(A+B) exp −(f−z)2
∆f2,(A3)
to which it converges point-wise. Indeed, putting x=
f/∆f, and expanding the exponentials in Eqs.(A1) and
(A3) in Tailor series we find

F(x)−˜
F(x)|
F(x)≈(b−a)2(2x2−1)
∆f2(A4)
×
AB
(A+B)2
+o(∆f−2),
so that the relative error of the approximation (A3) can
be made small for any given f.
We note further that in the limit ∆f→ ∞, the first mo-
ments Fand ˜
Fagree [⟨fn⟩F≡RfnF(f)df / RF(f)df ,
n= 1,2...]
⟨f⟩F=⟨f⟩˜
F=z, (A5)
but the second moments, each of order of ∆f2, differ by
a finite quantity,
⟨f2⟩F− ⟨f2⟩˜
F=AB(a−b)2
A+B,(A6)
so Fand ˜
Fcan, at least in principle, be distinguished.
An approximation, similar to the one in Eq.(A3),
can also be obtained in two dimensions, by considering
F(
f) = Aexp "−(
f−a)2
∆f2#+Bexp "−(
f−
b)2
∆f2#,
(A7)
where y = (y1, y2) is a two dimensions vector, and (y)2≡
y2
1+y2
2. As ∆f→ ∞ we find
F(
f)−−−−−→
∆f→∞
(A+B) exp "−(
f−z)2
∆f2#(A8)
where
zi=aiA+biB
A+B, i = 1,2.(A9)
Appendix B: Connection with catastrophe theory
To study the transformation of two maxima and a min-
imum of the function F(f) in Eq.(A1) into a single max-
imum, we choose a special case a= 0 and b= 1. The
structure of the extrema of F(f) is determined by two
parameters, R=B/A and ∆f, and corresponds, there-
fore, to cusp singularity case of the Catastrophe Theory
[15]. In the symmetric case, R= 1, F(f) = F(f+ 1) and
there is always a single extremum at f= 1/2. The first
and second derivatives at f= 1/2 are given by
∂fF1
2,∆f= 0,(B1)
∂2
fF1
2,∆f=−4
∆f4∆f2−2,
and the three extrema coalesce at ∆f=√2, where
∂fF(1/2,∆f) = 0. This is a case of pitchfork bifurcation
[15], shown in Fig.6a. Other cases are shown in Figs.6b
and 6c. Note that with R < 0 the single maximum which
survives as ∆f→ ∞ lies outside the interval 0 ≤f≤1.
Appendix C: The likelihood of discovering which
way a classical system went
Consider a probability distribution,
ρ(f) = 1
2∆f√πexp −f2
∆f2+ exp −(f−c)2
∆f2,
(C1)
with c > 0, and search for a range of f, where the
first term can be neglected, compared to the second
one. For example, one may look for readings where
the ratio between the two terms does not exceed some
ϵ≪1. Such readings would occur for f > fϵ, where
fϵ= ∆f2|ln ϵ|/2c+c/2.The probability of finding a read-
ing of this kind is, therefore, P(f > fϵ) = R∞
fϵρ(f)df. Re-
placing exp h−f2
∆f2iby a larger term exp h−(f−c)2
∆f2i, we
have [erfc(x) is the complementary error function [19]]
P(f > fϵ)<2−1erfc fϵ−c
∆f−−−−−→
∆f→∞
(C2)
c
2√π∆f|ln ϵ|exp −∆f2|ln ϵ|2
4c2→0.

11
c)
b)
a)
FIG. 6. Maxima (solid) and minima (dashed) of F(f) vs. f
and ∆f(a.u.). a) Pitchfork bifurcation for R= 1, b) same
as (a) but for R= 4/5. c) same as (b) but for a negative
R=−1/2. Note that in this case one has an “anomalous”
value z=−1.
The probability of finding a value f, which can be at-
tributed to only one of the two terms in Eq.(C1), vanishes
rapidly for ∆f≫c.
Appendix D: More properties of Gaussian
distributions
Consider next a function
F(f) = 
Aexp −(f−a)2
∆f2+Bexp −(f−b)2
∆f2
2
(D1)
with complex valued Aand B, and real aand b
A=AR+iAI, B =BR+iBI.(D2)
Equation (D1) can be rewritten as
F(f) = ARexp −(f−a)2
∆f2+BRexp −(f−b)2
∆f22
+AIexp −(f−a)2
∆f2+BIexp −(f−b)2
∆f22
.
(D3)
Applying (A3) to each term in the curly brackets, and
then to the sum of the results, yields
F(f)−−−−−→
∆f→∞
˜
F(f) = |A+B|2exp −2(f−Re[z])2
∆f2,
(D4)
with zstill given by Eq.(A2), but with complex valued
Aand B,
z=a(AR+iAI) + b(BR+iBI)
(AR+BR) + i(AI+BI).(D5)
Extension to two dimensions can be done in a similar
manner as in Appendix A. For 
f= (f1, f2), a = (a1, a2)
and 
b= (b1, b2)
F(
f) = 
Aexp "−(
f−a)2
∆f2#+Bexp "−(
f−
b)2
∆f2#
2
.
(D6)
we find
F(
f)−−−−−→
∆f→∞ |A+B|2exp "−2(
f−Re[z])2
∆f2#,(D7)
where z is still given by Eq.(A9) with complex valued A
and B.