Protective measurements of the wave function of a single system

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by  Lev Vaidman

Protective measurements of the wave function of a single system
L. Vaidman
Raymond and Beverly Sackler School of Physics and Astronomy
Tel-Aviv University, Tel-Aviv 69978, Israel
Myviewonthemeaningofthequantumwavefunctionanditsconnectiontoprotectivemeasure-
ments is described. The wave function and only the wave function is the ontology of the quantum
theory. Protective measurements support this view although they do not provide a decisive proof.
Abriefreviewofthediscoveryandthecriticismofprotectivemeasurementsispresented. Protective
measurements with postselection are discussed.
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INTRODUCTION procedure to observe it. It is not a decisive argument
1
0 because there are some limitations on the measurability
2 InthefirstgraduatecourseofquantummechanicsIre- of the wave function of a single particle. The quantum
state has to be “protected”, so it is possible to argue
b member askingthe question: “Canwe considerthe wave
e function as a description of a single quantum system?” that what is measured is not the wave function, but the
F “protection procedure”. Still, I can argue that it is bet-
I got no answer. Twelve years later, in South Carolina,
7 after I completed my Ph.D. studies at Tel Aviv Univer- ter to say that we measure the quantum state and not
2 sityunderthesupervisionofYakirAharonovinwhichwe its “protection”becausethere aremanydifferentprotec-
tions for the same wave function, all of which give rise
developed the theory of weak measurements [1], I asked
] to the same measurement results. Moreover, a protec-
h Aharonov: Canweuseweakmeasurementtoobservethe
p wave function of a single particle? tion procedure frequently protects many different wave
- functions, but the protective measurement finds the one
t At that time I alreadybecame a strongbeliever in the
n which is present.
many-worldsinterpretation (MWI) of quantum mechan-
a
u ics[2]andhadnodoubtthatasinglesystemisdescribed
q bythewavefunction. YakirAharonovneversharedwith
WHY I THINK THAT THE QUANTUM WAVE
[ me the belief in the MWI. When we realised that using
FUNCTION DESCRIBES A SINGLE QUANTUM
2 what is called now protective measurements, we can, un- SYSTEM (AND EVERYTHING ELSE)
v dercertainconditions,observethewavefunctionofasin-
6 gle quantum system, he was really excited by the result.
I want to believe that Science is capable of explaining
9
At1992IwasinvitedtoaconferenceontheFoundations
everything. We should not explain every detail since we
6
of Quantum Mechanics in Japan where I presented this
6 cannotstoreallrequiredinformation. Ithinkitisenough
. result: “TheSchr¨odingerwaveisobservableafterall!”[3]. to have a theory which can, given an unlimited storage
1
Then I wenthome to Tel Aviv where I finished writing a
0 and computational power, explain everything. The the-
letter which received mixed reviews in PRL, while Jeeva
4 oryshouldagreewithexperiment. Morespecifically,any
1 Anandan, working on the topic with Aharonov in South experiment, simple enough to allow theoretically to pre-
v: Carolina, wrote a paper accepted in PRA [4]. After ac- dict its outcome, should be in agreement with the the-
ceptance of the PRA paper it was hard to fight the ref-
i ory. Classical physics is not such a theory since many
X erees in PRL, but PLA accepted it immediately [5].
experimentslikeparticleinterference,atomstability,etc.
r I do not think that protective measurements provide contradict classical physics. Quantum mechanics is such
a
a decisive answer to my question in the graduate school. a theory. There is no experiment today contradicting its
I came to believe that a single quantum particle is com- prediction. The ontology of quantum theory is |Ψi and
pletely described by its Schr¨odinger wave before under- this is why I believe it is real.
standing protective measurements. And after the pub- Accepting that there is no ontology beyond the wave
lication of our work on protective measurement, many function, we admit that every time we perform a prepa-
physicists stil view it as an open question. This mani- ration procedure of a particular quantum state, we end
fest in the enormous interest to the Pusey, Barrett, and up exactly with the same situation. The failure of Bell
Rudolph (PBR) paper [6] entitled “On the reality of the inequalities, the PBR theorem and related results [8, 9]
quantum state” which puts strong constrains on the en- supportthis view. Then,ameasurementonanensemble
semble interpretation. Also, one of the “most read” Na- of identically prepared systems can be viewed as a mea-
turepapersis“Directmeasurementofthequantumwave- surement of the wave function of each system. So, even
function” [7]. without protective measurements one can accept the re-
The method of protective measurements provides a ality of the wave function of a single quantum system.
plausibility argument: it is more natural to attribute What might prevent us from considering quantum
the wave function to a single particle when there is a wavefunction as realis the collapseofthe wavefunction
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at quantum measurement. Today there is no a satisfac- of everything, the theory of a quantum wave function.
tory physical explanation of the collapse and it is not
plausible that such an explanation will be achieved due
to nonlocality and randomness of the collapse. WHAT IS AND WHAT IS NOT MEASURABLE
The approach according to which the wave function USING PROTECTIVE MEASUREMENT
is not something real, but represents a subjective infor-
mation, explains the collapse at quantum measurement Ifwearegivenasinglesysteminanunknownquantum
perfectly: itisjustaprocessofupdatingtheinformation state, there is no way to find out what is this state. It
the observer has. This approach seems to be highly at- would contradict the no-cloning theorem: if we can find
tractive. The problem is that it has not been successful outwhatisthestate,wecanpreparemanyothersystems
until now. No one provided an answer to the question: in this state. However,if we are given a single system in
“Information about what?” No good candidate for the a “protected” quantum state, we can find out its state.
underlyingontologyhasbeenproposed. Bellinequalities In general, a single system might not have a descrip-
related to quantum measurements performed on entan- tion as a pure state. Formally, we can consider a situa-
gled particles suggest that such an ontology, if we insist tion when our system is entangled with another system,
on locality, does not exist. Bell himself, however never andthepurestateofthecompositesystemsisprotected.
gave up looking for it. He even introduced the concept: Then we can observe the density matrix of our system
localbeables [10]. But in spite of much effort, no attrac- which provides its complete description. However, if the
tive theory of beables has been constructed. systemsareseparate,thenthelocalityofinteractionsdis-
One may say that a successful theory of nonlocal be- allows an efficient protection.
ables is provided by the de Broglie-Bohm theory, how- We canspecify the wavefunction ofthe systemby the
ever, it does not make the wave function epistemic. expectation values of a set of variables. Thus, measur-
Bohmians frequently say that “positions” represent pri- ing these expectation values is equivalent to the mea-
maryontology,butthewavefunctionisstillanontology. surement of the wave function. However, to argue that
And in some setups (surrealistic trajectories [11], weak protective measurements help us viewing the wave func-
measurements [12], protective measurements [13]) it is tion of a single system as a “real” entity, we need weak
the wave function which provides an explanation of the measurements of projections on small regions of space
observed results. to provide directly the spatial picture of the wave func-
Quantum theory not only corresponds extremely well tion. Also, weak measurements of local currents [16] al-
toourobservations,itisalsoaveryeleganttheoryexcept lowspecifyingthe phaseofthewavefunction. Ofcourse,
for the collapse of the wave function. At 1998 I heard onlythe relativephase canbe found, the overallphaseis
Gottfried saying at a conference in Erice [14] that: “The unobservable.
reduction postulate is an ugly scar on what would be a Ifthe wavefunction has a supportinseparateregions,
beautiful theory if it could be removed”. I firmly believe the situation is less clear. Local currents do not allow to
itcanberemoved. Thereisnoexperimentalevidencefor reconstructtherelativephasebetweentheseregions. An-
thecollapse,onlyourprejudicethattherearenomultiple other problem is the protection of such a state. Clearly,
copies of every one of us. Removal of the collapse leads alocalclassicalpotentialcannotleadto suchprotection:
totheMWI.AccordingtotheMWIeverythingisawave. energy cannot depend on the relative phase. Although
The Universe is a highly entangled wave. It has natural some nonlocal states can be measured using local inter-
decomposition into branches corresponding to different actions with help of measuring devices with entangled
worldsinwhichmacroscopicobjectsaredescribedbywell parts [17], many other states cannot be measured in a
localizedwavefunctions. “Inside”abrancheveryphoton nondemolition way [18].
emitted from a single photon source is described by its Anonlocalstateofequalsuperpositionofspatiallysep-
wave function. The same wave function which everyone aratedwavepackets 1 (|Ai+|Bi)canbeprotectedusing
√2
will associate with photons emitted by a laser replacing nonlocal measurements. If it is a photon state it can be
the single photon source. “swaped”toaBellstateoftwospins[19]. Bellstatescan
What are the alternatives? Only a minority remained bemeasuredinanondemolitionwayusingmeasurements
withahope to completethe ontologyofquantumtheory ofmodularsumsofspins(σA+σB)mod2,(σA+σB)mod2
z z x x
with“hiddenvariables”. Aconsistentoptionistoaccept [17]. Such measurements require measuring devices with
thatquantumtheoryisnotaboutwhattheNatureis,but entangled parts. Measurement of a fermion wave func-
about what we can say aboutit, as Bohr preached. This tion requires also an antiparticle with known phase [20].
approach developed into a popular trend of Qbism [15], Even if the protection of a particular wave function is
a kind ofmetaphysicalnihilism: “Inthe quantumworld, possible,itmightnotbeenoughforperformingaprotec-
maximalinformationisnotcompleteandcannotbecom- tive measurement of this wave fucntion. Nonlocal weak
pleted.” Thisnihilismseemsunnecessary,becauseweare measurementsaredifficult andcurrentproposals[21,22]
givenanunprecedentedlysuccessfuldeterministictheory are not efficient. The superposition of wave packets of
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unequal weight, α|Ai+β|Bi, 0 < |α| < |β| cannot be localized at zero, shifts to 1 or remains at 0. When the
measured in a nondemolition way [18], so in this case uncertaintyofthepointerissmall,∆≪1,weknowwith
there is no protection procedure. certainty the outcome of each trial. In the correspond-
Themainmotivationforprotectivemeasurementswas ing weak measurement, the pointer remains at 0 if the
a slogan: “what is observable is real”. I, however, not particle is not there, and it is shifted by a small value ǫ,
fully support it. I do look for an ontology that can ex- ǫ ≪ ∆, if it is present there. The analysis of weak mea-
plainallwhatweobserve,butIdonotwanttoconstrain surements showsthat, up to a verygoodapproximation,
“observation” to instantaneous nondemolition measure- aftertheinteractionwithonesystem,themeterisshifted
ments. Iamreadytoviewtherelativephasebetweentwo by ǫ|ψ|2. If we test now that the system remained in its
spatially separated parts of the wave function as “real”, originalstate,theprobabilityofthefailureisoftheorder
evenifitcanbeobservedonlylater,whenthewavepack- of ǫ2 . Now, we repeat this procedure with an ensemble
∆2
ets overlap. of N = 1 systems using the same measuring device. At
ǫ
theend,thepointerwillshowthedesiredvalue|ψ|2 with
almost unchanged uncertainty ∆. The probability that
THE METHODS OF PROTECTIVE all the systems will be found in its original state after
MEASUREMENTS AGNADINTHE INFORMATION the interaction is (1− ∆ǫ22)1ǫ. If we make ǫ ≪ ∆2, the
probability of even one failure goes to zero. It means
that we can use the same system every time. This is the
There are two methods for protection of states. First, Zeno-type protective measurements.
when it is a nondegenerate eigenstate of some Hamilto-
Instead of the test by the projection on the state of
nian. Second, based on the quantum Zeno effect, that a
interest, we can make measurements for which the state
frequentmeasurementsofavariableforwhichthestateis
wewantto measureis oneofthe eigenstates. If weknow
a nondegenerate eigenstate are performed. The strength
that we start with this state, we only need to perform
of the protection is characterized by the energy gap to
measurementinteractionswithoutactuallylookingatthe
a nearby eigenstate and the frequency of measurements
result. Theprocedureensuresthatwewillhaveourstate
in the Zeno type protection. If the strength of the pro-
for the duration of the procedure.
tection is known, the weakness parameter of weak mea-
This procedure is different from a measurement on an
surements ofthe wavefunction canbe calculated. Then,
ensemble not only because we use just one system. Al-
the whole wave function can be found. If the protection
thoughwehavemultiplecouplingswedonotgetmultiple
strengthis notknown,wecanchoosea weaknessparam-
outcomes and we do not calculate statistical average as
eter, make the weak measurement of the wave function
in the standard ensemble measurements.
and then repeat it. If the result is the same, we know
thatthemeasurementissuccessful. Ifnot,weshouldask The Zeno-type protection is more of a theoretical
foranothersample. Evenifwewillneedseveralsystems, construction. Performing frequent verification measure-
still, it is not a measurement on a large ensemble. ments might be a very difficult if not impossible task.
Intheprocessofprotectivemeasurementwegainsome Another type of protection is to have a Hamiltonian
information: thewavefunctionisspecifiedbymanymore with nondegenerate eigenstates. Experimentally, such
parametersthanthe strength ofthe protection. There is a protection is much simpler. We have a system with
also some gain of information beyond the information a nondegenerate ground state. Then we just have to
the party which arranges the protection must have. For waitlongenoughtime(whichcanbereducedusingsome
protection it is enough to know that the state is one of cooling procedure) and the wave function of the ground
thesetoforthogonalstates. Aftertheprotectionandthe state is there and protected. It does not require much
weakmeasurementwewillknowwhichstate ofthe setis priorknowledge. Weak measurementcoupling after long
given. enoughtimewillprovidetheinformationaboutthewave
Probably, the Zeno-protection method is easier to un- function. Adiabatic switching of the weak coupling on
derstand. First, consider a measurement of the wave and off will allow to make the procedure faster.
function on an ensemble. To know the absolute value In recent years, there is a great progress with cooling
ofthe wavefunctionin aparticularpoint, the projection ions and atoms in traps. Science magazines frequently
onasmallregionofspacearoundthis pointismeasured, show pictures of what looks like a wave function of a
and the value is the ratio of the number of times the trapped particle. But, as far as I understand, these pic-
particle was found there to the number of trials. On tures look like a “wave cloud” because of the width of
every measurement,one of the eigenvalues of the projec- the photons which are scattered by the ions. As far as I
tionoperator,0or1,is obtained. Thestatisticalaverage know, no protective measurements have been performed
provides the absolute value of the wave function there. yet. Recent “wave function microscopy” [23] which used
The projectionmeasurementcanbe modeledby avon aphotoionizationimagingwasalsoanensemblemeasure-
Neumannprocedureinwhichateachtrialapointer,well ment. AlthoughIhavenodoubtsaboutwhatwillbe the
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outcomes of protective measurements, I think it is of in- ticle should not change the outcomes of our protective
teresttomakeanefforttoperformthem,especiallysince measurementswhichconstituateapictureoftheforward
many are reluctant to associate a wave function with a evolvingwavefunction. Weakcouplingduringalongpe-
singlesystem. Ihopethatsomeprotectivemeasurements riod of time of the projection on every place x shows,
will be performed in the near future, maybe along the at the end, the value of |ψ(x)|2. For all outcomes of
lines of Nussinov’s proposal [24]. the postselectionmeasurementwe getthe same(approx-
The discussion above is about measurement of a sta- imately correct) result of the density of the wave there.
tionary wave function. Protective measurements require (The postselection which is very far from the center will
some period of time to observe the wave function which correspondtosomedistortedpicturesincetheweakcou-
is supposed to be constant during this period. Can it pling of protective measurements creates some small en-
shed light on the question of the reality of a nonstation- tanglement.) However, if we will test whether during all
ary wave function? If evolution is slow, unitary, and the the interaction time the coupling was to the same value,
protection is strong, then we can observe the wavefunc- we will discover that this is not so [26]. To make this
tion (with some limited precision) “in real time”. If we test we should perform these protective measurements
start with the preselected state at the beginning of pro- on a large pre- and postselected ensemble with all sys-
tection procedure, it will ensure that the wave function temsoriginallyinthegroundstate. Weshouldconsidera
will change in time appropriately and the weak coupling subensembleofparticlespostselectedatlocalizedstateat
of the measurement of the wave will not disturb it sig- positionx. Inthisexperiment,wewilllookatthepointer
nificantly. A nonunitary evolution which includes “col- not at the end, but at various intermediate times. This
lapses” of the wave function cannot be observed in this will allow to find the position of the pointer as a func-
way. The weak coupling will not be a problem, but we tion of time. We will see that the pointer does not move
cannotensurebyanyprotectionprocedurethatthewave in the same way during the time of the measurement.
functionwillcollapsetotherightstate. Thisisonemore Sometimes it moves fast and sometimes it almost stops.
reasonnottobelievethatthereisacollapseofaquantum Itdependsonthebackwardevolvingwavefunctionφ(x),
state in Nature. which changes with time. The pointer “feels” the weak
value of the projection operator (P ) = φ∗(x)ψ(x) and
x w φψ
it changes with time due to the time dependhen|cei of the
PROTECTIVE MEASUREMENT AND backwardevolving wave function.
POSTSELECTION
ThisanalysissuggeststhattheHamiltonianprotective
measurement does not really measure the density of the
Protective measurements support the approach ac-
wave function, but it measures a time average of a par-
cording to which the reality of a quantum system is its
ticle density which actually changes with time. To avoid
wave function. Aharonov and myself argued in many
this difficulty we can adda postselection of the state |ψi
publications that the complete descriptionof a quantum
attheendoftheprotectivemeasurement. Weknowthat
system is a two-state vector consisting of forward and
itwillsucceedwithcertaintyandthentheweakmeasure-
backward evolving wave functions [25]. How can these
mentpointerwillmovewiththesamevelocityduringthe
apparently contradicting approaches peacefully coexist?
whole process. Indeed, with the postselection, the weak
In the Zeno-type protective measurements, the back-
measurements show weak value of the pre- and posts-
ward evolving wave function is identical to the forward
elected system. When the backward wave function is
evolvingwavefunction. Evenifwepostselectsomeother
identical to the forwardwave function, the weak value is
state, the last verification measurement “collapses” the
equal to the expectation value, and it shows the density
backward evolving state to be identical to the forward
of the forward evolving wave function.
evolving one. It cannot be any other eigenstate of the
protection measurementsince the measurement specifies IntheframeworkoftheMWI,thereis,however,asat-
both forward and backward evolving states. isfactory explanationof the situation even without post-
For the Hamiltonian type protection the situation is selection of the original state |ψi. Let us consider again
different. Wemaystart,say,withagroundstateofahar- thecaseofthepostselectionmeasurementofposition. In-
monic oscillator and after performing a protective mea- deed,ineveryoneoutofthedifferentworldswithpostse-
surement of the wave function, to measure the particle lectionindifferentpositions,theprotectivemeasurement
position. Whatever outcome we will get, the backward measurestimeaverageoftheparticledensityasitisgiven
evolvingwavefunctionwillbeverydifferentfromthefor- by the weak value of the projection on a particular lo-
ward evolving wave function. It will not be a stationary cation. However,the time averagesare subjective to the
state, but an oscillating wave function. observersinworldswithdifferentpostselectionpositions.
Protective measurements cause almost no entangle- These arenotobjective realities. The objectiverealityis
ment between the system and the measuring device, so in the universe which incorporates worlds with all possi-
the postselection according to the position of the par- ble outcomes of the postselection measurement. In this
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world, the backward evolving wave function is identical (or backward) evolving wave function.
to the forward evolving wave function [27]. To test this I have to note that the protective measurement of the
we just can look at the preselected ensemble only, with- two-state vector reality is a highly theoretical concept.
outpostselection. Inthiscasethepointerwillmovewith Itsimplementationinlaboratoryrequireshighlyimprob-
constant small velocity ending in the final value |ψ(x)|2. able result of the postselection measurement.
In the world with a postselection, subjective reality
is best describedby the two-statevector. The protective
CRITIQUE OF PROTECTIVE MEASUREMENTS
measurementdescribedaboveshowedjustforwardevolv-
ing wave, we observe |ψ(x)|2 and can also observe the
local currentdefined by ψ(x). This is because only pres- There have been many papers criticising our work on
electedforwardevolvingwavewasprotected. Adifferent protective measurements. According to Paraoanu[30] it
postselected state cannot be orthogonal (such postselec- has been established that protective measurements pro-
tionisimpossible)andthusitcannotbeprotectedbythe gram does not work: “Clearly, to think about the wave
same Hamiltonian or by the Zeno measurements. The function as real, we would have to be able to measure it
backward evolving state φ(x) changes with time. The on a single quantum system. The question of whether
effective coupling to the pointer is through weak value this is possible was first raised in the context of the so-
which oscillates, but after averaging, it roughly repro- called“protective” (weakly disturbing) measurements in
ducestheexpectationvaluecorrespondingtotheforward the early 1990s, where it was answered in the negative
evolving wave. If, instead of the protection Hamiltonian [2].” HereliesonAlterandYamamoto’spaper[31]which
of the preselected state we arrange protection by other analyzesthecasewhenthequantumwavefunctionisnot
Hamiltonian (or frequent measurements) which protects protected and, not surprisingly, shows that in this case
the postselected backwardevolving state, then our weak the wave function cannot be found [32]. Another refer-
coupling of the measurement procedure will yield the enceofParaoanuis“ImpossibilityofMeasuringtheWave
backward evolving state φ(x). Forward evolving state FunctionofaSingle QuantumSystem”byD’Arianoand
will change in time and will lead to oscillating weak val- Yuen [33]. All what they claimed is that an unknown
ues averagingto the expectationvalues correspondingto and unprotected quantum state cannot be found (due
φ(x). to the no-cloning theorem). Regarding protective mea-
surementsthey actuallysaythatitworks,giventhatthe
At first, it seems that it is impossible to observe the
protection Hamiltonian is known. They add that known
two-state vector on a single system because the same
protection Hamiltonian means that its eigenstates are
Hamiltonian cannot protect both the forward evolving
known and that we can find which of the eigenstates
and a the nonorthogonal backward evolving wave func-
is given without breaking the no-cloning theorem and
tions. Usually, all Hamitonians are Hermitian so they
without the need for a weak coupling of protective mea-
protect the same set of forward and backward evolving
surements. They did not distinguish between the party
wave functions. However, if we pre- and postselect the
which can find the state and the party that protects the
whole system including the protection device, then for
stateanddonotconsiderthecasethatallwhatisknown
weakcouplingtheeffectiveHamiltonianwillbetheweak
is the strength of the protection of the state. Another
value of the Hamitonian. Weak value of Hermition op-
work Paraoanu cites is a paper with a provocative title
erator might be a complex number. Then, the effective
byUffink: “Howtoprotecttheinterpretationofthewave
Hamiltonians responsible for evolutions of the forward
function against protective measurements?”[34]. How-
and the backward evolving states are different. Thus,
ever, Gao recently criticised this paper [35] and Uffink
we can, in this way, arrange protection of both forward
retracted the main part of his objection [36].
and backward evolving wave functions even if they are
There are also authors who praised protective mea-
different. This is the protection of the two-state vector
surements. Unruh [37] viewed protective measurements
[28].
asademonstrationoftherealityofcertainoperatorsand
Thepossibilitytoprotectthetwo-statevectordoesnot not of the wave function, but he admitted that “protec-
necessarily mean that we can directly observe the two- tive measurements has broadened our understanding of
state vector. The weak coupling to the projection on a the quantum measurement process.” Ghose and Home
particular location x will yield its weak value (Px)w = [38] in “An analysis of the Aharonov-Anandan-Vaidman
φ∗(x)ψ(x). Since weak measurements do not disturb each model”wrote:“theAAVschemeservestocounteractthe
φψ
othhe|rsiignificantly,wecanmeasureinparallelweakvalues orthodox belief that quantum mechanics does not say
ofothervariables. Protectionofthetwo-statevectorwill anythingempiricallymeaningfulaboutanindividualsys-
allow then to view the full “weak measurement reality” tem.” Dikson [39] consideredprotective measurement as
[29] specified by the two-state vector. This will allow to “a good reply for the realist” against empiricist.
reconstructthe two-state vector, but not in a direct way It seems that many of the criticisms follow from mis-
as it was in the protective measurement of the forward understandingtriggeredbythe somewhatmisleadingex-
6
amplesinourPRApaper[4]. Oneoftheexampleswasa ments, Phys. Rev.A 41, 11 (1990).
particle in a superposition of two wave packets which is [2] L. Vaidman, Many-Worlds interpretation of quantum
difficult to observe using protective measurements. The mechanics, Stan. Enc. Phil., E. N. Zalta (ed.) (2002),
http://plato.stanford.edu/entries/qm-manyworlds/.
example was technically correct, since there was some
[3] Y. Aharonov and L. Vaidman The Schrdinger wave is
tunneling wave connecting the wave packets such that
abservable after all! in Quantum Control and Measure-
they were only “almost” separate. The Stern-Gerlach
ment, H. Ezawa and Y. Murayama (eds.) 99 (Elsevier
experiment hadits owndifficulties, in particular,related Publ., Tokyo,1993).
to divergenceless nature of the magnetic field. [4] Y. Aharonov, J. Anandan, and L. Vaidman, Meaning of
Probably the most harsh criticism was a comment by the wavefunction, Phys.Rev.A 47, 4616 (1993).
[5] Y. Aharonov and L. Vaidman, Measurement of the
Rovelli [40]: “We argue that the experiment does not
Schrodingerwaveofasingleparticle,Phys.Lett.A178,
provide a way for measuring noncommuting observables
38 (1993).
withoutacollapse,doesnotbearontheissueofthe”real-
[6] M. F. Pusey,J. Barrett, and T. Rudolph,On thereality
ityofthewavefunction,”anddoesnotaddanyparticular of thequantumstate, NaturePhys. 8, 476 (2012).
insight into our understanding (or nonunderstanding) of [7] J.S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and
quantummechanics.” I hope,however,thatafter reading C. Bamber, Direct measurement of the quantum wave-
my paper the following quotation of Rovelli’s comment function, Nature474, 188 (2011).
[8] R. Colbeck and R. Renner, Is a System’s Wave Func-
makeshismisunderstandingevident: “Tomaketheprob-
tioninOne-to-OneCorrespondencewithItsElementsof
lem particularly evident, consider the following experi-
Reality? Phys.Rev. Lett.108, 150402 (2012).
mental arrangement, which is entirely equivalent to the
[9] L. Hardy, Are quantum states real? Int. J. Mod. Phys.
Aharonov-Anandan-Vaidmanexperimentasfarasthein- B, 27, 1345012 (2013).
terpretation of quantum mechanics is concerned. First, [10] J.S. Bell, Beables for quantum field theory, in J.S. Bell,
measure the polarization of a quantum particle. Sec- Speakable and unspeakable in quantum mechanics, pp.
ond, write the outcome of this measurement on a piece 173180 (Cambridge: Cambridge University Press 1987).
[11] B.G.Englert,M.O.Scully,G.Su¨ssmann,andH.Walther,
of paper. At this stage assume that we do not know the
Surrealistic Bohm trajectories, Z. Naturforsch. A 47,
polarization and we do not know what is written on the
1175 (1992).
piece of paper. Then make the following protected mea-
[12] Y. Aharonov and L. Vaidman, About position measure-
surement: Read what is written on the piece of paper.” mentswhichdonotshowtheBohmian particleposition,
After our clarification [41], Dass and Qureshi in a pa- in Bohmian Mechanics and Quantum Theory: An Ap-
praisal, J.T. Cushing, A. Fine and S. Goldstein (eds.),
per [42] “Critique of protective measurements” “looked
pp. 141-154 (Kluwer, Dordrecht,1996).
atearliercriticisms ofthe idea,andconcludedthat most
[13] Y.Aharonov,M.O.Scully,andB.G.Englert,Protective
of them are not relevant to the original proposal.” They
measurementsandBohmtrajectories,Phys.Lett.A263,
argued, however, that “one can never perform a protec- 137 (1999).
tive measurement on a single quantum system with ab- [14] K.Gottfried,Doesquantummechanicsdescribethe“col-
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