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arXiv:1703.02959v3 [quant-ph] 26 Jan 2018
The Quantum Cheshire Cat effect: Theoretical basis and
observational implications
Q. Duprey,1S. Kanjilal,2U. Sinha,3D. Home,2and A.Matzkin1
1Laboratoire de Physique Th´eorique et Mod´elisation (CNRS Unit´e 8089),
Universit´e de Cergy-Pontoise, 95302 Cergy-Pontoise cedex, France
2Center for Astroparticle Physics and Space Science (CAPSS),
Bose Institute, Kolkata 700 091, India
3Raman Research Institute, Sadashivanagar, Bangalore, India
Abstract
The Quantum Cheshire Cat (QCC) is an effect introduced recently within the Weak Measure-
ments framework. The main feature of the QCC effect is that a property of a quantum particle
appears to be spatially separated from its position. The status of this effect has however remained
unclear, as claims of experimental observation of the QCC have been disputed by strong criticism
of the experimental as well as the theoretical aspects of the effect. In this paper we clarify in what
precise sense the QCC can be regarded as an unambiguous consequence of the standard quantum
mechanical formalism applied to describe quantum pointers weakly coupled to a system. In light
of this clarification, the raised criticisms of the QCC effect are rebutted. We further point out
that the limitations of the experiments performed to date imply that a loophole-free experimental
demonstration of the QCC has not yet been achieved.
1
I. INTRODUCTION
Weak measurements were introduced [1] in 1988 as a theoretical scheme for minimally
perturbing non-destructive quantum measurements. It is indeed well-known that a stan-
dard quantum measurement irremediably destroys the premeasurement state of a system by
projecting it to an eigenstate of the measured observable, say B. It is therefore impossible,
according to this “standard view”, to know anything about a given system property, repre-
sented by an observable A, as the system evolves from a prepared initial state towards the
final eigenstate obtained after measuring B.
The weak measurement scheme introduced by Aharonov, Albert and Vaidman [1] aims to
measure Awithout appreciably modifying the system evolution relative to the case without
the measurement. This is achieved by means of a weak coupling between the system observ-
able Aand a dynamical variable of an external degree of freedom (that we will designate
here by the term “quantum pointer”). This weak coupling entangles the quantum pointer
with the system, until the final projective measurement of Bcorrelates the obtained system
eigenstate with the quantum state of the weak pointer. The resulting state of the weak
pointer has picked up a shift (relative to its initial pre-coupled state) proportional to the
real part of a quantity known as the weak value of A.
While the meaning of weak values has been debated since their inception [2–5], several
experimental implementations of the weak measurement protocol have been carried out:
weak values have thus been measured for different observables in many quantum systems
[6–13]. Concurrently, theoretical schemes based on weak measurements have been proposed
with practical [14–16] or foundational [17–21] aims. Among the latter, Aharonov et al [22]
introduced an interferometric based scheme baptized the Quantum Cheshire Cat (QCC).
A QCC situation takes place when at some location (say, region I) the weak value of an
observable representing a system property (eg, polarization) vanishes while the weak value
of the system’s spatial projector is non-zero. At some other location (say region II) the
opposite takes place (the weak values of the spatial projector and system property are
zero and non-zero respectively). Echoing the features of Lewis Carroll’s eponym character,
Aharonov et al. loosely described this Cheshire cat situation in Ref. [22] as seeing the grin
2
(the polarization) without the cat (the photon in region I)1; they further wrote in [22] that
the QCC scheme implies “physical properties can be disembodied from the objects they belong
to”. A tentative experimental realisation of the QCC effect employing neutrons published
soon after [24] concluded “that the system behaves as if the neutrons go through one beam
path, while their magnetic moment travels along the other”. Very recently, an experiment
with single photons similar to the neutron one was reported [25].
The Quantum Cheshire cat phenomenon reported in Refs [22, 24] has raised a string of
criticisms, in particular in the published works [5, 26–29]. The common trend in many of
these criticisms is to view the QCC effect as a false paradox, an illusion that would have an
arguably simpler interpretation. Unfortunately, rather than making their point using unam-
biguous conceptual and technical terms, most of the works criticizing the Quantum Cheshire
Cat effect did not analyze the weak measurements framework (generally even avoiding to
mention weak values). Interpretational claims (that ultimately depend on the interpreta-
tion of the standard quantum formalism, not on the specificities of weak values) were not
always distinguished from practical pointer readouts, and the theoretical QCC effect was
not carefully discriminated from the tentative experimental implementations.
Hence, instead of demystifying the Quantum Cheshire Cat, the criticism that has been
made may have brought additional confusion. To be fair, it must be pointed out that some
of the well-known works dealing with weak measurements, such as the QCC paper [22] put
the stress on delivering a simple take-home message without providing the detailed formal
arguments that would explain and justify the main message.
In this work, our aim will be to carefully scrutinize the Quantum Cheshire Cat effect
in order to lift the confusion on what this effect is really about. We will discriminate the
theoretical, ideal QCC from the shortcomings that can inevitably appear in any experimental
realization. We will disentangle the interpretational aspects from the technical terms in
which the effect is couched. We will in particular argue through the analysis of the meaning
of vanishing weak values that spatial separation between a particle and its properties can
be consistently defined, provided one is willing to relax ascribing properties to a quantum
system solely through the eigenstate-eigenvalue link.
1As we will see below, the property (the grin) is not the polarization itself but a specific polarization
component. A generalization that would apply to any polarization component was suggested in Ref. [23].
3
In order to do so we will first recall the aspects of weak measurements that will be
relevant to describe the Quantum Cheshire Cat (Sec. 2). We will then define, in Sec. 3,
the QCC effect from the weak values obtained when precise coupling conditions are met,
detailing the weak measurement process that was omitted in the original QCC paper [22]
or in subsequent QCC related works [30]. Indeed, a proper account of this detailed process
appears to be one of the two crucial ingredients necessary in order to dispel the confusion
that has emerged around the QCC effect. Sec. 4 will be devoted to the description of the
tentative experimental implementation of the QCC with neutrons and with single photons.
We will in particular highlight several essential differences with the ideal theoretical scheme.
The published criticism of the QCC effect will be examined in Sec. 5 and compared to the
precisely defined QCC introduced in Sec. 3. In the Discussion section (Sec. 6) we will detail
the technical and conceptual aspects of the arguments given in the criticisms; we will also
explain under which assumptions it is legitimate to interpret the effect as “disembodiment”.
This will be seen to depend on the general interpretation of the quantum formalism that is
favoured. We will finally give our conclusions in Sec. 7; anticipating on our assessments, we
will conclude that: (i) the Quantum Cheshire Cat effect is a well defined quantum feature
derived from the standard quantum formalism that can be interpreted as a spatial separation
of a particle from one of its properties if some assumptions regarding property ascription
are made; (ii) the QCC effect as predicted theoretically has not yet been experimentally
observed; (iii) most of the works rebuking the QCC effect produced a substantial criticism
of the experimental attempts to observe the Quantum Cheshire Cat, but a criticism of the
ideal QCC can only be undertaken within a proper conceptual framework able to account for
the relation between weakly coupled pointers and the properties of the measured quantum
system. This is in our view necessary in order to analyze the issue of spatial separation of
a quantum particle from one of its properties in a pre and postselected situation.
II. WEAK MEASUREMENTS AND WEAK VALUES
A. Protocol: Preselection, Unitary coupling, Postselection
The basic idea underlying the weak measurement (WM) approach is to give an answer
to the question:“what is the value of a property of a quantum system at some intermediate
4
time while the system evolves from an initial state |ψ(ti)ito a final state |χ(tf)i?”, where
|χ(tf)iis the result of a standard projective measurement made at time tf. As we are only
interested here in applying WM to derive the Quantum Cheshire Cat effect, we will restrict
our discussion to the property corresponding to a bivalued observable A, with eigenstates
and eigenvalues denoted by A|aki=ak|aki, k = 1,2.
Suppose that initially (at t=ti) the system is prepared (preselected) into the state |ψ(ti)i.
Let |ϕ(ti)idesignate the initial state of the quantum pointer. The total initial quantum state
is the product state
|Ψ(ti)i=|ψ(ti)i|ϕ(ti)i.(1)
We assume the pointer is local (its wavefunction has compact support in configuration
space), and that the system and the pointer will interact during a brief time interval τ
centered around t=tw(physically corresponding to the time during which the system and
the quantum pointer interact). The interaction between the system and the quantum pointer
is given by the Hamiltonian
Hint =g(t)AP. (2)
Ais the system observable that couples to the momentum Pof the pointer. g(t) is a
smooth function non-vanishing only in the interval tw−τ/2< t < tw+τ /2 and such that
g≡Rtw+τ/2
tw−τ/2g(t)dt appears as the effective coupling constant. Recall that the coupling (2) is
the usual interaction employed to account for projective measurements of A(von Neumann’s
impulsive model): in that case g(t) is sharply peaked and each |akiis correlated with an
orthogonal state of the strongly coupled pointer. In a projective measurement the collpase
of the pointer projects the system state to a random eigenstate |ak0i. Here instead gwill be
small, the weakly coupled quantum pointer does not collapse, and the system will undergo at
postselection a genuine projective measurement that will consequently project the quantum
pointer to a specific final state, as we now detail.
Let U(tw, ti) be the unitary operator describing the system evolution between tiand tw
(we disregard the self-evolution of the pointer state). After the interaction (t > tw+τ /2)
5
the initial uncoupled state (1) has become entangled 2:
|Ψ(t)i=U(t, tw)e−igAP U(tw, ti)|ψ(ti)i|ϕ(ti)i(3)
=U(t, tw)e−igAP |ψ(tw)i|ϕ(ti)i(4)
=U(t, tw)X
k=1,2
e−igakPhak|ψ(tw)i|aki|ϕ(ti)i.(5)
At time tfthe system undergoes a standard projective measurement: a given observable
ˆ
Bis measured and the system ends up in one of its eigenstates |bki. We filter the results
of this projective measurement by keeping only a chosen outcome, say bf. The system is
thus postselected in the corresponding state |bfi. We will denote the postselected state by
|χf(tf)i ≡ |bfi; in most of the paper we will be dealing with a single postselected state and
drop the second label, writing |χ(tf)ior |χfiinstead. After postselection, the state of the
pointer, given by Eq. (5), becomes
|ϕ(tf)i=X
k=1,2
[hχ(tw)|akihak|ψ(tw)i]e−igakP|ϕ(ti)i,(6)
where we have used hχ(tw)|=hχ(tf)|U(tf, tw). ϕ(x, tf) is then given by a superposition of
shifted initial states
ϕ(x, tf) = X
k=1,2
[hχ(tw)|akihak|ψ(tw)i]ϕ(x−gak, ti).(7)
This expression is similar to the usual von Neumann projective measurement: the first
step of a projective measurement (the premeasurement, in which each eigenstate |akiof the
measured observable is correlated with a given state ϕ(x−gak) of the pointer) is identical
here, but in a projective measurement the second step is a projection to an eigenstate
akf
of A.
Let us now assume the coupling gis sufficiently small so that e−igakP≈1−igakPholds
2In Eq. (3) the interaction appears to take place precisely at tw; this “midpoint rule” holds provided τis
small relative to the system evolution timescale (see Appdx A in the Supp. Mat. of Ref. [17]).
6
for each k3. Eq. (6) becomes
|ϕ(tf)i=hχ(tw)|ψ(tw)i1−igP hχ(tw)|A|ψ(tw)i
hχ(tw)|ψ(tw)i|ϕ(ti)i(8)
=hχ(tw)|ψ(tw)iexp (−igAwP)|ϕ(ti)i(9)
where
Aw=hχ(tw)|A|ψ(tw)i
hχ(tw)|ψ(tw)i(10)
is the weak value of the observable Agiven pre and postselected states |ψiand |χire-
spectively (we will sometimes employ instead the full notation Aw
hχ|,|ψito specify pre and
postselection). For a localized pointer state, expanding to first order the terms ϕ(x−gak, ti)
in Eq. (7) leads to Eq. (9): when Awis real, the overall shift ϕ(x−gAw, ti) is readily
seen to result from the interference due to the superposition of the slightly shifted terms
ϕ(x−gak, ti), as shown early on in Ref. [31]. Note that the shift gAwis very small so it
needs to be evaluated from the statistics over a large number of trials.
Summing up, we see that a weak measurement contains 4 steps: preselection, weak
coupling through the Hamiltonian (2), postselection and readout of the quantum pointer.
B. Weak values
1. Observable average
Awis in general a complex quantity. Following Eq. (9) the real part of the weak value
Awappears as the shift brought to the initial pointer state |ϕ(ti)iby the interaction with
the system via the coupling Hamitonian (2). The weak values are generally different from
the eigenvalues, but obey a similar relation with regard to the computation of expectation
values. Indeed, the expectation value of Ain state |ψ(tw)i, given in terms of eigenvalues by
hψ(tw)|A|ψ(tw)i=X
f|haf|ψ(tw)i|2af,(11)
3The exact condition for the asymptotic expression to hold implies that the higher order terms of order
mobey kgmPmhχ(tw)|Am|ψ(tw)ik ≪ kgP hχ(tw)|A|ψ(tw)ik <1 for Eq. (8) and kg P Awk ≪ 1 for Eq.
(9). These conditions take precise forms for specific pointer wavefunctions, in particular when ϕ(x, ti) is
Gaussian, see eg Ref. [31].
7
can also be written in terms of weak values as
hψ(tw)|A|ψ(tw)i=hψ(tw)|U†(tf, tw)U(tf, tw)A|ψ(tw)i(12)
=hψ(tw)|U†(tf, tw)X
f|bfihbf|U(tf, tw)A|ψ(tw)i(13)
=X
f|hχf(tf)|ψ(tf)i|2hχf(tw)|A|ψ(tw)i
hχf(tw)|ψ(tw)i(14)
=X
f|hχf(tf)|ψ(tf)i|2Aw
hχf|,|ψi(15)
where we have used hχf(tf)|=hbf|and hχf(tw)|=hχf(tf)|U(tf, tw) denotes the postselected
state hbf|evolved backward in time up to t=tw. Eq. (15) is expressed in terms of the
probabilities of obtaining a postselected state |bfi(instead of the probability of obtaining
an eigenstate), with the rationale that the probabilities of obtaining the postselected states
are not modified by the weak coupling.
2. Vanishing weak values
Let us start by first looking at the case of null eigenvalues. In a standard projective
measurement, a vanishing eigenvalue implies that the state of the measurement pointer is
left, untouched, remaining in the initial state: the coupling has no effect on the pointer. But
the system state does change, as it is projected to the eigenstate associated with the null
eigenvalue for the measured observable. For example imagine a beam of atoms incoming
on a beamsplitter, after which the quantum state of each atom can be described by the
superposition |ui+|liof atoms traveling along the upper or lower paths; if a measurement
of the projector onto the upper path Πu≡ |ui hu|yields 0, then the quantum state has
collapsed to |liand indeed one is certain to find the atom on the lower path. If the atom
has some integer spin, then measuring the spin projection along some direction can yield a
null eigenvalue. The atom spin state is projected to the corresponding eigenstate (as can be
verified by making subsequent measurements) corresponding to no spin component along
that direction.
For weak measurements, the main difference is that the system’s state is not projected af-
ter the weak coupling. Instead, the overall state evolves according to e−igAP |ψ(tw)i|ϕ(ti)i ≈
|ψ(tw)i|ϕ(ti)i−iA (g|ψ(tw)i)P|ϕ(ti)i.The system state appears as essentially unperturbed
8
except for the small fraction g|ψ(tw)ithat couples to the quantum pointer. A(g|ψ(tw)i)
is precisely the post-interaction (or premeasured) part of the system state, corresponding
to the slight change in the system state produced by the weak coupling to the quantum
pointer. The weak value appears as the imprint of this coupling left on the pointer, con-
ditioned on the final projective measurement. A vanishing weak value correlates successful
postselection with the quantum pointer having been left unchanged despite the interaction
with the system. The reason is that the quantity hχ(tw)|A|ψ(tw)i, which is the numerator
in the definition (10) of Awvanishes. We will adopt here Feynman’s terminology [32] and
designate this quantity by the term “transition element” 4.
Hence when a weak value vanishes the coupling has no effect (though the pointer is left
unchanged) as in the case of vanishing eigenvalues, but the implication is not relative to
eigenvectors but to the transition elements between A(g|ψ(tw)i) and the postselected state
|χ(tw)i. Put differently, if the postselected state is obtained, then whenever Aw= 0 the
property represented by Acannot be detected by the weakly coupled quantum pointer. For
example when the weak value of a projector (Πu)w
hχ|,|ψi= 0, this implies (i) no effective
action of the coupling on the quantum pointer (that is left unchanged) with respect to
the postselected state |χfiand (ii) that the state |χficannot be reached from the initial
state |ψ(ti)iby the part of the system wavefunction that has interacted with the pointer in
the region where Πuwas weakly measured 5. If instead some spin observable Sγis weakly
measured in some region and hSγiw
hf|,|ii= 0 this implies again that (i) there is no effective
action on the quantum pointer of the coupling between Sγand the pointer variable, and
(ii) the final spin state |ficannot be obtained by premeasuring Sγon the initial spin state
|ii,i.e. the fraction of the spin state that is perturbed by the interaction with the pointer
and thereby transforms as Sγ|iidoes not reach |fi. We will discuss in Sec. VI how these
statements can be further interpreted. We will for the moment note that the argument
encapsulated by (ii) can be logically restated as: since the final state is reached when
postselection is successful, a null weak value (i.e., a vanishing transition element) implies
that for this postselected system state, the property represented by the weakly measured
observable cannot have been detected by the quantum pointer. Hence in some loose sense
4In the literature, this term is also called “transition matrix element” or “transition amplitude”
5Note that this reasonings hold for asymptotically weak couplings, those that do not affect the postselection
probabilities |hχf(tf)|ψ(tf)i|2; otherwise terms beyond the linear expansion (8) would contribute.
9
FIG. 1: The ideal Quantum Cheshire Cat setup based on a 2-path interferometer. A quantum
pointer is placed on each path. Each pointer interacts locally with the particle, coupling the
particle observable Aj= Πjor Aj= (σx)j(where j=I, I I) to the pointer dynamical variable
PIor PII resp. (gis the coupling strength). The particle then travels along the interferometer
essentially unperturbed until it gets detected (SF is a spin flipper, required for postselection of the
spin state). Successful postselection (detection in the upper port) is correlated with pointers in final
states exp −igAw
jPj|ϕj(ti)iwith the weak values Πw
I= 1,(σx)w
I= 0 and Πw
II = 0,(σx)w
II = 1 for
pointers Iand II respectively.
(to be refined below), that property “was not there”, ie in the region where the system and
weak pointer interacted.
III. THE QUANTUM CHESHIRE CAT EFFECT
The Quantum Cheshire Cat effect, as introduced in Ref. [22] is based on the Mach-
Zehnder interferometer shown in Fig. 1. Rather than elaborating on the original argument
[22], we will instead give a proper account of the QCC in terms of weak measurements, by
filling the gaps left in the theoretical account given in [22]. We will mostly remain here at a
technical level (interpretations will be discussed later in Sec. VI).
Assume that a quantum particle with spin 1/2 enters the interferometer shown in Fig. 1 in
10
state |ψii|siiwhere |ψiiis a localized wavepacket and |siithe spin state in which the system
was prepared. Let us label by Iand II the upper and lower arms of the interferometer, and
assume a quantum pointer in state |ϕI(ti)isits on arm I. Hence the total system-pointer
initial state is
|Ψ(ti)i=|ψii|sii|ϕI(ti)i.(16)
The system will then enter the interferometer, evolving to
U(t, ti)|ψii|sii=1
√2(|ψIi+|ψII i)|sii(17)
and then locally interact along arm Ithrough the interaction Hamitonian (2) given here
by g(t)AIPIwhere index Iemphasizes the coupling takes place only on branch Iand PIis
the pointer’s linear momentum. Ais a system observable that will be taken to be either the
spatial projector or some spin component. After the interaction the state vector is given by
Eq. (4) that takes here the form
|Ψ(t)i=U(t, tw)e−igAIPI|ψIi|sii+|ψII i |sii|ϕI(ti)i/√2.(18)
Assume finally that the system is postselected after exiting the interferometer (|ψIiand
|ψII ithus overlap) in the state
|χfi= (|ψIi|sii+|ψII i |−sii)/√2.(19)
To be definite we will choose the initial state to be prepared in state |sii=|+ziso |−sii=
|−zi.
Let us now examine the final state of the quantum pointer, that will depend on the
observables to which it was weakly coupled. Then, after postselection, the pointer state is
given by Eq. (9):
|ϕI(tf)i=1
√2hχf|U(tf, tw) (|ψIi+|ψII i)|siiexp (−igAw
IPI)|ϕI(ti)i(20)
=1
2exp (−igAw
IPI)|ϕI(ti)i.(21)
Hence the final state of the quantum pointer that was weakly coupled to AIdepends on the
weak value
Aw
I=hχf|U(tf, tw)AIU(tw, ti)|ψii|sii
hχf|U(tf, ti)|ψii|sii(22)
= [hψI|h+z|]AI[|ψIi|+zi].(23)
11
For AI= ΠI(projector to the spatial wavefunction of arm I, ΠI≡ |IihI|) and AI= (σx)I
(the spin component along the xaxis, where Irecalls the coupling takes place along path
I), we obtain by using Eq. (23):
Πw
I= 1 (σx)w
I= 0.(24)
Similarly, a quantum pointer coupled to AII can be placed along path II. Its final quan-
tum state |ϕII (tf)iis obtained after postselection (to the same state |χfi) exactly as above,
yielding
|ϕII (tf)i=1
2exp (−igAw
II PI I )|ϕI I (ti)i(25)
with
Aw
II = [hψI I |h−z|]AII [|ψII i|+zi].(26)
This gives us for AII = ΠI I and AI I = (σx)I I respectively
Πw
II = 0 (σx)w
II = 1.(27)
The conjunction of Eqs. (24) and (27) defines the quantum Cheshire Cat effect. Indeed,
the quantum pointer coupled to the system on path Ionly detects the spatial wavefunction,
but the interaction with the spin component σxhas no effect. Conversely along arm II, a
quantum pointer picks up a shift due to the coupling with the system’s spin component σx
but the coupling to the spatial wavefunction along path II has no effect on the pointer.
Following our discussion on null weak values in Sec. II B 2, the fact that the particle
cannot be seen on path II means that the slight change brought to the spatial wavefunction
of the system by the interaction with the weakly coupled pointer on path II cannot be
postselected in the state |χfi(the transition element hχf|ΠII |ψiivanishes). Similarly the
spin component σxis not detected on arm Ibecause the postselected spin state |+zicannot
be reached by the part σx|+ziof the spin state that has been modified by coupling (the
transition element h+z|σx|+zivanishes). Note that in principle different weak measure-
ments can be made jointly, on both arms, or subsequently, on the same arm (since for an
asymptotically weak interaction, to lowest order, only the unperturbed part of the system
state is taken into account when the system interacts with a subsequent weak pointer).
12
IV. EXPERIMENTAL IMPLEMENTATIONS OF THE QUANTUM CHESHIRE
CAT
A. Experiment in a neutron interferometer
A few months after the publication of the Quantum Cheshire cat paper by Aharonov
et al [22], an experimental realization of the QCC effect was implemented [24] with single
neutrons in a Mach-Zehnder-like triple-Laue interferometer. Due to strong experimental
constraints, the scheme employed in the experiment was significantly different from the
ideal QCC scheme described in Sec. III. In particular, it was not experimentally feasible to
couple quantum pointers to the neutron inside the interferometer; instead, the weak values
were inferred from the intensities of the detected neutron signal after postselection. While
the weak values determined experimentally were in excellent agreement with Eqs. (24) and
(27) defining the QCC effect, the fact that weak measurements were not made (the weak
values were instead inferred from intensities measured after making transformations) has
implications concerning the validity of the observation of the QCC effect, as we now detail
below.
Rather than summarizing the experiment as described in Ref. [24], we will highlight the
differences with the theoretical QCC proposal. For simplicity we will employ the same pre
and postselected states employed in the theoretical proposal [22] and in Sec. III (in the
experiment pre and postselected states were inverted relative to the theoretical proposal,
but this has no consequence for our present discussion).
The crucial difference between the QCC theoretical proposal and the neutron experiment
is the lack of quantum pointers interacting with the neutron (or the lack of additional de-
grees of freedom of the neutron that would act as such pointers). Hence there is no such
thing as the state vector |ϕI(ti)iin equation (16) or in any of the equations below equation
(16). This is of course a radical departure from the weak measurement formalism introduced
in Sec. II. Instead of relying on weak measurements, the neutron QCC experiment is based
on introducing external potentials along the arms. These interactions modify the postse-
lected intensity (ie, the neutrons detected in the postselected state) relative to the intensity
obtained without the interaction in that arm.
The spatial projector weak values are obtained by inserting an absorber on arms Ior
13
II. The absorber is not a quantum pointer: it is modeled by an external decay potential
Vj=e−iMjwhere jstands for either arm Ior II ;Mis the absorption coefficient. For M
small, some simple manipulations lead to [24]
Iabs
j=|hχf|U(t, ti)|ψii|+zi|21−2MjΠw
j=I01−2MjΠw
j(28)
where I0=|hχf|U(t, ti)|ψii|+zi|2,obtained from Eqs. (17) and (19), defines the detected
reference intensity after postselection. Hence the experimental observation of Iabs
jwhen the
absorber is placed on arm jallows to extract the weak value Πw
j. But no weak measurement
has been made: instead of a weak interaction, we have imposed a strong interaction that
happens with a small probability; instead of a pointer whose state would reflect the effect
of the coupled observable on the quantum state of the pointer, we are inferring the presence
of neutrons along path Iby the fact that the relative number of postselected neutrons
decreases. Hence a postselected neutron does not carry any signature of the interaction
(precisely because the neutrons that have interacted with the absorber on path Ihave been
absorbed and have thus vanished). This does not entail that one cannot conclude from
Eq. (28) that the neutrons can only reach postselection by going through path I: indeed
when an absorber is placed along path Ithis is reflected by the fact that Iabs
I/I0<1
while an absorber along path II has no effect on the intensity, Iabs
2/I0= 1.However this
conclusion does not rely on weak measurements nor on weak values but only on the relative
intensities, consistent with the fact that no coupling to a quantum pointer has taken place:
no observable has been weakly measured. Instead, the conclusion Πw
II = 0 is made from the
lack of backaction of the strong interaction with the absorber on the postselected neutron
intensity (see Fig. 2).
For the spin component weak value the same objection can be made, now with more
serious consequences. The external potential employed is the one for a magnetic moment in a
magnetic field Bjoriented along the xaxis, Uj=−γ(σx)jBj/2 where γis the gyromagnetic
ratio and the index jmeans that Bjis non-zero only in a region along arm j(and thus
affects only the magnetic moment γ(σx)jon arm j). Since RdtUj=α(σx)j/2,where αis
the precession angle induced by the magnetic field, we have for small α
e−iRdtUj/~U(tw, ti)|ψii|sii=1 + iα
2(σx)j−α2
8Πj(|ψIi|sii+|ψII i |sii)/√2.(29)
Note that the second order term is needed because postselection leads here to intensities,
14
not to pointer shifts [compare with Eq. (18)]. Postselection indeed yields
Imag
j=|hχf|U(t, ti)|ψii|+zi|21 + α2
4
(σx)w
j
2
−α2
4Πw
j(30)
where I0=|hχf|U(t, ti)|ψii|+zi|2is again the reference intensity and Imag
jthe detected
intensity when a magnetic field is applied on arm j.
In the neutron experiment, Eq. (30) is employed to compute
(σx)w
j
from the observed
relative intensities Imag
j/I0. The term Πw
jwas taken from the value inferred experimentally
from Eq. (28), though it would also have been consistent to use δI,j instead. Indeed (σx)2
j=
1j(the identity along arm j) but upon postselection 1j(|ψIi |sii+|ψI I i|sii) vanishes for
j=II. The upshot is that while the theoretical prediction (σx)w
I= 0 can be recovered
experimentally from Imag
Iby fitting Eq. (30), the magnetic field along arm Inevertheless
has an effect on the spin σxthrough the last term −α2/4 of Eq. (30), a point that was made
in Ref. [29]. In this particular experiment this effect can be claimed to be systematic (in
the sense that it doesn’t depend on the field orientation – ie, which spin component couples
with the field), but it is there nonetheless: the appearance of higher order terms is generic
when inferring weak values from intensity measurements, rather than making genuine weak
measurements.
B. Experiment with single photons
A QCC experiment with photons was carried out very recently [25]. The authors make
clear from the start that their setup is based on the one implemented in the neutron exper-
iment, where the photon polarization takes the place of the neutron spin (the polarization
state along the interferometer arms is |Hior |Viinstead of the spin states |+ziand |−zi,
and the circular polarisation weak value replaces (σx)w
j). Indeed, leaving aside the specifici-
ties of the photon experiment (use of a Sagnac interferometer, coincidence detection with
the heralded photon), the scheme employed in Ref. [25] in order to display the QCC effect is
identical to the neutron experiment: the presence weak value Πw
jis inferred from the photon
counts with/without an absorber placed along the arms, and the polarization weak value
is inferred from photon counts with/without a rotation of the polarization induced by half
wave plates. This similarity can be seen directly by comparing Eqs. (3) and (6) of Ref. [25]
with Eqs. (28) and (30) given above for the neutron experiment respectively.
15
FIG. 2: Schematic representation of the differences between the ideal Quantum Cheshire Cat setup
proposed theoretically and the experimental implementation with neutrons [24]. These differences
also apply to a similar experiment performed with photons in a Sagnac-like interferometer [25] (see
text).
Hence, our remarks made above for the neutron experiment also hold for this photon
experiment: no weak measurements were made, and inferring weak values from measured
intensities does not give an unambiguous demonstration of the Cheshire cat effect. It is
noteworthy that the authors of Ref. [25] point out in their Conclusion that their observed
measurement statistics can be understood without recourse to weak values.
V. CRITICISM OF THE QCC IN THE LITERATURE
A. General remarks
The Quantum Cheshire Cat scheme has been criticized by several groups [5, 26–29]. At
the basis of the criticism there is the underlying idea that claiming separation of a property
from a particle is preposterous. However the criticism was not always carefully formulated.
Most of the works did not clearly discriminate the original theoretical proposal from the
neutron experimental implementation (which as we have shown above are very different).
They were also not rigorous when making assertions concerning the disembodiment, often
16
using terms in plain English (which can be at best an interpretation of the QCC effect) rather
then scrutinizing the technical definition of the QCC effect in terms of weak measurements;
actually most of these works with the exception of Ref. [5] did not base their arguments on
the characteristics of the weak measurements framework, that was generally ignored.
As a result, the criticism did not deliver clarification. We study below the main arguments
that were given in the criticisms and assess their relevance. The first class of arguments
involves the detected intensities, a second type of arguments invokes interference, while
other arguments attempt to discard the possibility of disembodiment.
B. Intensities
The work by Stuckey et al [29] criticizes the QCC neutron experiment only, on the basis
that the detected intensities when the spin weak value is (σx)w
I= 0 still shows the effect
of the magnetic field (they do accept that the absorber’s effect on the intensities is a proof
that the particle did not take path II). Their agument starts from the identity
e−iασx/2= cos α
2I−isin α
2σx.(31)
Stuckey et al point out that the magnetic field has an effect on the intensities through the
cos α
2term: to lowest order this term is quadratic, exactly as the term containing the weak
value contribution. So they conclude that this contradicts the statement according to which
the spin component σxdid not travel through arm I. This argument is technically correct
as far as intensities are concerned; it is equivalent to the observation we made below Eq.
(30). However this argument does not involve weak measurements (although we know from
our discussion above that the QCC is defined from weak measurements that leave the post-
selected intensities of the system undisturbed), nor does Ref. [29] propose an alternative
framework for a non-destructive measurement that would ascribe a non-zero value to the
spin component on arm I, despite the fact that (σx)w
I= 0. Still it is valuable to remark that
Stuckey et al [29] leave open the possibility of observing a genuine QCC.
In a very different work Atherton et al [28] performed a classical optics experiment based
on a Mach-Zehnder type interferometer. They start with polarized beams and compare the
intensities obtained on one of the output ports with or without an absorber or a polarization
rotating plate inserted on either arms Ior II (a polarizer ensures postselection of the
17
detected beam). This setup indeed mimicks the neutron experiment described in Sec. IV A.
The observed electric field intensities display the same behavior as the neutron counts. On
this basis, Atherton et al conclude that there is nothing quantum about the QCC effect, that
in their view is an “illusion”. This classical physics experiment has the merit of highlighting
the fact that the observed intensities in the neutron experiment do not need to be interpreted
with the weak measurements formalism, since no quantum weak values can be obtained
with classical beams. However attempting to draw conclusions on the value of quantum
observables from a classical optics experiment is an impossible task: the classical beams
travel through both arms, so what could the Cheshire cat effect mean in this context?
Instead a single neutron obeys the standard quantum rule of projective measurements and
cannot be found simultaneously on both arms. Atherton et. al. do not define precisely the
analogue of the quantum observable σxand there is no mention of anything that can play the
role of weak values in their work, although they are necessary to precisely define the QCC
effect. Therefore the findings of Ref. [28] are only relevant to the specific implementation
of the neutron experiment summarized in Sec. IV A, not to the QCC effect itself.
C. Interference
In an interesting paper, Correa et al [26] assert that the QCC effect arises from “simple
quantum interference”. They mean by that statement that a quantum pointer remaining
unchanged (as this happens for vanishing weak values) results from the interference of almost
perfectly overlapping final pointer states. This remark is uncontroversial – this is of course
the way weak measurements work in general [see Eq. (6)], as shown early in a 1989 work
[31]. Significantly, Correa et al. explicitly introduce the quantum pointer states (our states
|ϕ(ti)iof Secs. II and III) but treat the pointer-system interactions in an ad-hoc way,
by stating in words how these pointer states are transformed, rather than introducing an
interaction Hamiltonian such as our equation (2). Hence weak values do not explicitly appear
in their treatment, but the motion of the weakly coupled pointer states are recovered by the
superposition of the quantum state after postselection, see our Eq. (7) above.
However in our view the Quantum Cheshire Cat effect does not depend on the underlying
mathematical formulation of the pointers motion, but on giving a physical meaning to these
motions, as we now discuss for the most salient physical property claimed to characterize
18
the QCC effect, disembodiment.
D. Disembodiment
The main objective of the criticisms seems to undermine the claim made in the original
proposal by Aharonov et al. that Eqs. (24)-(27) could imply disembodiment of the particle
from one of its properties. Correa et al [26] write that their interference argument allows
them to interpret the QCC phenomenon without appealing to disembodiment, but they do
not produce a full reasoning that would support this claim. Indeed, the interference ac-
count of the quantum pointer dynamics does not endorse nor disprove the “disembodiment”
claim: the superposition of state vectors is a Hilbert space feature, that at least according
to standard quantum mechanics is only a mathematical description aimed to compute prob-
abilities. The authors of Ref. [26] do not elaborate on whether their argument implies going
beyond this standard view, for example by endowing the state vectors with some ontological
features that would then propagate along both arms. Instead, whether the motion of the
quantum pointers weakly coupled to the particle’s position or spin component is indicative
of disembodiment depends on a framework ascribing properties to a quantum system in the
absence of a projective measurement. The weak measurements formalism constitutes such
a framework, and discarding the possibility of disembodiment implies either replacing the
WM formalism with some alternative proposition, or refuse that quantum properties can be
defined beyond projective measurements. This point will be further discussed in Sec. VI.
Michielsen et al [27] also objected on disembodiment by running numerical experiments.
Strictly speaking their work only applies to the neutron experiment. They simulate the ob-
served interferences with/without absorbers/spin rotators on arms Iand II. The simulation
is based on a discrete event learning model, in which particles act as messengers in such a
way that the particle counts in the outgoing ports of a Mach-Zehnder interferometer quickly
converge towards the quantum probabilities. Such a model was previously employed to re-
produce intensities in neutron interferometric experiments [33], so Ref [27] is an extension of
that previous work so as to include the absorber/rotator interactions. In any case Michielsen
et al. do not consider quantum pointers and weak measurements in their model, and while
it would be interesting to investigate if the Quantum Cheshire Cat can be properly formu-
lated within the discrete event learning model by including the coupled quantum pointers
19
explicitly, their results are not relevant to the QCC effect as we have properly defined it.
VI. DISCUSSION
A. General remarks
The main property characterizing the Quantum Cheshire Cat seen here is that for a fixed
initial and final state of the system (photon, neutron), the state of a quantum pointer weakly
coupled to the system’s spatial wavefunction is modified for a pointer placed along path I
but not for a pointer placed along path II. If the quantum pointer is coupled to the spin
component σxinstead, the opposite behavior is obtained. Since neither the neutron nor
the single photon experiments have recourse to weak pointers, criticizing the experiments
done so far as not having realized the QCC is legitimate. However we have seen that some
authors of the criticism did not clearly discriminate the experimental implementations from
the ideal QCC. For instance Atherton et al [28] and Michielsen et al [27] criticize the neu-
tron experiment, but from there cast suspicion on the ideal theoretical scheme as being an
“illusion”.
While doing so is strictly speaking inconsistent, the common element underlying the
criticism of the Quantum Chehsire Cat formulated in Refs [26–29] is to refute the idea of
a spatial separation between the particle and one of its properties. Unfortunately, rather
than starting from the technical definition of the QCC and from there prove that this
definition cannot imply a spatial separation, Refs [26–29] do not specify the assumptions they
make concerning the possibility of ascribing properties to a quantum system in the absence
of a projective measurement. The weak measurements formalism proposes a framework
accounting for such properties, and from there a technical definition for spatial separation
is obtained in terms of weak values. This is arguably different than taking the term “spatial
separation” in a literal sense, that would disregard the well-known conceptual difficulties of
the standard quantum formalism in giving an unambiguous account of the physical state
and properties of a system. These difficulties are ultimately due to the fact that contrary
to classical mechanics or classical optics, the relationship between the theoretical terms of
quantum mechanics and physical reality are unknown, and most often denied. Hence relying
on pointers to assess the value of the property of a quantum system is crucial; this in turn
20
hinges on employing an explicit conceptual and interpretative framework.
B. Technical and conceptual aspects
The Quantum Cheshire Cat is technically defined by Eqs. (24) and (27) in the context
of weak measurements with postselection that do not affect the coherence of the system. So
the first question is whether one can have Eqs. (24) and (27) (in the context of weak mea-
surements) while still being able to assess the particle can be found on path II , or that spin
component σxcan be found along path I(conditioned on successful postselection). In order
to answer this question, the authors of Refs. [26–29] do not take into account the fact that
some type of measurement or interaction (presumably different from the weak measurement
protocol) needs to be proposed. Otherwise it is impossible to assert anything about the sys-
tem properties. In particular, relying on the state vector, as done in [26, 29], is insufficient:
every quantum physicist agrees that the total state vector is in a superposition state along
both paths, and that postselection will imply a certain correlation due to interference. But
the quantum axiomatics remain silent on the meaning of the state vector, that may be taken
as a simple computational tool (this is the standard view conveyed in textbooks), or as an
element of a more elaborate ontology, but clearly not in a literal manner as representing a
classical field that would propagate simultaneously along both arms of the interferometer.
C. Interpretations
This leads us to the second question: does the technical definition of the QCC recalled
in the preceding paragraph necessarily imply some form of spatial separation, or disembod-
iment? The answer depends on how the quantum pointer’s motions are interpreted. Indeed,
the pointers measure weak values, and null weak values are null transition elements. And
transition elements are well-defined quantities in standard quantum mechanics.
Now having pointers measuring transition elements does not fit with the eigenstate-
eigenvalue link (by which the value of the property represented by the observable is associ-
ated with a quantum state of the system). This would be impossible given the aim of the
WM scheme, as recalled at the beginning of Sec. II: a given weak value is not associated
with a given state of the system, but with the transition of the (time-evolved) preselected
21
state to the postselected state induced by the weak coupling between system and pointer
observables. As we have seen in Sec. II B 2, vanishing transition amplitudes imply that the
final postselected state cannot be reached by the part of the system state that has been
perturbed by the weak interaction with the quantum pointer.
Hence Πw
II = 0 implies that the transformation generated on the preselected state by
premeasuring ΠII does not reach the postselected state. If one upholds an interpretation in
which a property relies on projection to an eigenstate, then measuring a vanishing transition
element has no bearing on a statement concerning property ascription. This is the criticism
developed by Sokolovski [5]. In the terminology of Ref. [5], the transition amplitudes belong
to “virtual paths”, that do not describe the real path of a particle. Indeed, according to this
view, only a strong projective measurement can tell us if the particle is or not in arm II.
A projective measurement creates a “real path” that precludes the possibility of measuring
any other property on the arm in which the particle is not found (otherwise the uncertainty
principle would be violated), so that by that account a real path cannot accommodate the
idea of spatial separation. If we do not perform a projective measurement, but measure
instead a null weak value, then we know that the transition element hχf|ΠII |ψiivanishes,
but this only characterizes the reaction of the system to a small perturbation and should
not be taken as a measurement of the position of the particle.
While this point of view is consistent, it is also possible to go further in interpreting
transition elements as characterizing system properties. The null weak value operationally
means that the pointer state corresponding to the particles detected in the postselected state
is unaffected by the presence of weak interactions. Accordingly Πw
II = 0 implies that the
particle’s presence cannot be detected by a quantum pointer weakly interacting with the
spatial wavefunction on path I I and be detected in the chosen postselection state |χfi[Eq.
(19)]: such a correlation is forbidden by standard quantum mechanics, as the transition
element hχf|ΠII |ψiivanishes. On this basis it is possible to uphold that the final state
cannot have been reached by taking path II (otherwise the weakly coupled pointer along
arm II would have been displaced upon postselection). In a crude sense, we can say that
the particle has not been in arm II .
The same reasoning can be made concerning the spin component. (σx)w
I= 0 because
the transition element h+z|(σx)I|+zivanishes. A vanishing transition element means that
the transformed part of the spin state resulting from coupling σxweakly to a quantum
22
pointer on path Iis orthogonal to the postselected state (hence the postselected cannot be
reached by the fraction of the system state coupled to the pointer). Therefore σxcannot
be found on path Iin this sense: the premeasurement of σxon path Islightly changes
the spin state, but for the stipulated postselection, this slight change has no effect, leaving
the state of the quantum pointer undisturbed. Note that this is very different from the
projective measurement process yielding a null eigenvalue, given that a vanishing eigenvalue
is associated with a particular eigenstate (see Sec. II B 2).
It should be stressed that relying on transition elements to assess the value of properties
in pre/post-selected systems does not necessarily call for paradoxes. This can be understood
as the confirmation that the effect of the superposition principle (or sum over paths) can
be observed by a local weak coupling of a system observable with a quantum pointer. In
general, the transition element on either path will be not vanishing, yielding an observable
effect on the pointers placed on both arms of the interferometer. However for specific choices
of pre/postselected states, a given system observable may generate a transition to the final
state only along a given arm, while such a transition cannot take place along the other.
VII. CONCLUSION
In this paper, we have bridged the gap, both at a theoretical and at a conceptual level,
between the original Quantum Cheshire Cat proposal [22], and the various experimental
realizations and criticisms that have been formulated. In doing so, we have clarified the
meaning of the QCC effect and dispelled the considerable degree of confusion that was seen
to arise from the raised criticisms of the Quantum Cheshire Cat proposal.
We have argued that “disembodiment” can be said to hold if it is defined in terms of
transition elements for the system observables. As we have seen, a quantum pointer after
postselection detects the system observable to which it is weakly coupled only when the
relevant transition element for that observable does not vanish. By a suitable choice of
pre and post-selected states, the spatial wavefunction can only be detected by a quantum
pointer placed on one of the paths (and not the other) while the spin component is only
seen on the other path.
We conclude by summarizing our main results:
•The Quantum Cheshire Cat effect is a well defined quantum feature derived from
23
the standard quantum formalism for pre- and post-selected states of a system; the
interpretation of the effect in terms of spatial separation of a particle from one of its
properties hinges on the issue of the relation between property ascription and weakly
coupled pointers;
•The QCC effect as predicted theoretically has not yet been experimentally observed,
as the experimental realizations done so far have not been able to properly implement
the weak measurement protocol;
•Most of the works criticizing the QCC effect did not introduce a proper framework in
order to analyze the issue of spatial separation of a quantum particle from one of its
properties in a pre and postselected situation, so their criticism is incomplete.
Acknowledgements Partial support from the Templeton Foundation (Project 57758)
is gratefully acknowledged. AM thanks the Institute for Quantum Studies (Chapman Uni-
versity) for hospitality at the time this work was completed.
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