A strict experimental test of macroscopic realism in a superconducting flux qubit

1.2 MB·

by  George C. Knee

ARTICLE
Received26Feb 2016 | Accepted 15Sep 2016| Published 4 Nov2016 DOI:10.1038/ncomms13253 OPEN
A strict experimental test of macroscopic realism in
a superconducting flux qubit
6
1
0
2 George C. Knee1,*,w, Kosuke Kakuyanagi1,*, Mao-Chuang Yeh2,*, Yuichiro Matsuzaki1, Hiraku Toida1,

v Hiroshi Yamaguchi1, Shiro Saito1, Anthony J. Leggett2 & William J. Munro1
o
N

7
1
Macroscopic realism is the name for a class of modifications to quantum theory that allow

]
macroscopic objects to be described in a measurement-independent manner, while largely
h
p preserving a fully quantum mechanical description of the microscopic world. Objective
-
t collapse theories are examples which aim to solve the quantum measurement problem
n
through modified dynamical laws. Whether such theories describe nature, however, is not
a
u known. Here we describe and implement an experimental protocol capable of constraining
q
theories of this class, that is more noise tolerant and conceptually transparent than the
[

original Leggett–Garg test. We implement the protocol in a superconducting flux qubit,
3
v and rule out (by B84 s.d.) those theories which would deny coherent superpositions
8 of170nAcurrentsoveraB10nstimescale.Further,weaddressthe‘clumsinessloophole’by
2
determining classical disturbance with control experiments. Our results constitute strong
7
3 evidence for the superposition of states of nontrivial macroscopic distinctness.
0
.
1
0
6
1
:
v
i
X
r
a
1NTTBasicResearchLaboratories,NTTCorporation,3-1Morinosato-Wakamiya,Atsugi,Kanagawa243-0198,Japan.2DepartmentofPhysics,Universityof
IllinoisatUrbana-Champaign,Urbana,Illinois61801,USA.*Theseauthorscontributedequallytothiswork.wPresentaddress:DepartmentofPhysics,
UniversityofWarwick,GibbetHillRoad,CoventryCV47AL,UK.CorrespondenceandrequestsformaterialsshouldbeaddressedtoG.C.K.
(email:[email protected])ortoW.J.M.(email:[email protected]).
NATURECOMMUNICATIONS|7:13253|DOI:10.1038/ncomms13253|www.nature.com/naturecommunications 1
ARTICLE
NATURECOMMUNICATIONS|DOI:10.1038/ncomms13253
I
n their original paper, Leggett and Garg (LG) asked whether
the flux trapped in a superconducting ring was really ‘there’
when nobody looks1. The systems LG had in mind are Original Leggett-Garg
micrometre scale loops of superconducting material interrupted a LG
with one or more nonlinear elements known as Josephson S2
S
junctions.Suchcircuitsdefinetwopossiblestatesofmagneticflux 1
threading the loop, and modern variants2 are among the most G 〈Q2Q3〉
S
2
macroscopic candidates for a quantum bit (or qubit), the basic S
1
constituent of various proffered quantum-enhanced technologies 1 〈Q1Q3〉
such as the quantum computer. When a measurement is made, S
2
thequbitisfoundinoneofthetwopossiblestates|gior|eiwitha S
1
probability that oscillates in time. Observation of such so-called 2 〈Q1Q2〉
‘Rabi oscillations’ is consistent with a textbook quantum S2
mechanical prediction (which generally ascribes a nonzero 3 S1 t3
t
complex amplitude to each of the states), but not necessarily 2
t
inconsistent with a classical ‘value-definite’ description (which 1
pmroesmcreinbte)s3.tThhatedtheecasyyesntevmeloipseinofethxaecRtlaybioonsecislltaattieonastiasngyivegnivebny Control experimentS 〈Q3〉G dg
an empirical parameter T . Huge efforts have been invested in b 2 – or
eoxf-ttehned-ianrgt tvhailsucehaorfac8t5ermissti2(cre‘cfo.h4e)r,enwciethtimaev’iteowthteoccurrorsesnitngstatthee- G ⏐g〉 S1((cid:2) = 0 or ⏐(cid:3)g) 〉 or⏐e〉 S2 〈Q3〉2 de
qquuaannttuumm ceormropr-uctoartiroenct5i.oTnhethgreusihdionldgsquaensdtioennaobfliLnGg’slaarpgper-oscaaclhe 2 ⏐g〉 S1((cid:2) = 0 o⏐r (cid:3)g)〉 or⏐e〉 t2 t3
extendsbeyondtheirprototypicalsystem:isthereafundamental t1 Main experiment
mechanism preventing macroscopic superpositions from 〈Q 〉
persisting, or is the problem merely about resources? LG’s c S2 3G
name for the former position is ‘macroscopic realism’, G ⏐g〉 S1((cid:2) = (cid:3)/2) (cid:4) – d(cid:4)
oasr G‘mhaircarrodrie,alRisimm’inif,orWsehboerrt:anodbjePcetiavrele c(oGllRapWseP)mthoedoerlsy6,s7ucohr S2 〈Q3〉2
Penrose’s gravitationally induced collapse theory8 are specific ⏐g〉 S1((cid:2) = (cid:3)/2) (cid:4) t3
2 t2
examples which might make the quantum-classical divide well t
1
defined.
Motivatedbytheneedforastricttestwhichcouldruleoutthis
worldview, LG considered Q , Q , Q as the value taken by a Figure1|Asimplifiedtestofmacrorealism.(a)Leggett–Garg(LG)derive
1 2 3
macroscopicobservableQmeasuredatthreeconsecutivetimest , theirinequality-constraintonmacrorealismbyconsideringameasurement
1
t , t , respectively. LG made the assumption of ‘macrorealism ofabivalentobservableQatthreeconsecutivetimes(onanensemble‘G’
2 3
per se’ (MRPS): that these variables can each be assigned a value oftwo-levelsystems).Theinequalityistestedbygatheringtwo-time
±1 at all times. Then the constraint1 correlatorsfromseparateexperiments,eachwithameasurementomitted
atoneoftheinstants(ensemble‘(cid:2)1’andsoon).Thelowerpaneshowsour
Q Q þQ Q þQ Q (cid:2) (cid:3)1 ð1Þ
1 2 1 3 2 3 full,simplifiedexperimentalprotocol.Inourexperimenttheshuffling
will hold. An elementary consequence is operationsarepulsesinducedwithresonantmicrowaveradiationthat
causeapseudo-spinrotationbyavariableangley,creatingcoherent
hQ Q i þhQ Q i þhQ Q i (cid:2) (cid:3)1 ð2Þ
1 2 G 1 3 G 2 3 G superpositionsof|giand|ei.Theexpectationvalueofafinalmeasurement
where hyi denotes the average over a ‘grand’ ensemble (att )maybeinfluencedbythepresenceorabsenceofanearlieroperation
G 3
(or experimental arrangement) where all three observables O(att ).(b)Controlexperimentsdeterminetheworstcasedisturbance
2
(Q , Q , Q ) are measured. LG conjoined a premise they termed whenclassicalstatesareprepared:|eiispreparedfromthethermal
1 2 3
‘non-invasive measurability’ (NIM) to reach equilibriumstate|giwithay¼poperation.(c)Themainexperimentis
identical,onlyamaximallycoherentsuperpositionispreparedwitha
LGI: hQ Q i þhQ Q i þhQ Q i (cid:2) (cid:3)1; ð3Þ
1 2 (cid:2)3 1 3 (cid:2)2 2 3 (cid:2)1 y¼p/2operation.Thisgivesrisetoameasurementdisturbancenot
the Leggett-Garg inequality (LGI). Here, h...i (for i¼1, 2, 3) explainablemerelybyappealingtotheclumsinessrevealedinthecontrol
(cid:2)i
denotes an average over a ensemble identical to the grand experiments.
ensemble, with the exception that the observable Q is not
i
measured. If NIM is taken to mean that a suitably careful
measurement has no effect on the statistics of measurement LGI or variants thereof have been experimentally tested
outcomes before or after it, it is effectively the premise (and violated) in a wide variety of microscopic experimental
h...i ¼h...i ¼h...i ¼h...i : that the three ensembles in systems, sometimes with one or more of a variety of additional
(cid:2)3 (cid:2)2 (cid:2)1 G
which experiments are actually performed (see Fig. 1a) are assumptions. A review of these experiments may be found in
equivalent to the grand ensemble. Here, we include the ref. 10, but see also more recentexperimental testson acaesium
impossibility of backwards causation (sometimes called atom11, delocalized photoexcitations12 and a two-transmon
Induction9) in NIM. When ‘shuffling’ operations S and S system13. The demanding nature of LG tests has so far
1 2
(which induce coherent oscillations between the two states of influenced the slow progress of experiments toward larger
interest)intervene respectively between t andt ,andbetween t objects, meaning that experiments performed to date at best
1 2 2
and t , LGI can be violated by a quantum mechanical system. place only loose bounds on the critical macroscopicity.
3
If the system is sufficiently large (super-critically macroscopic), Here, we show that LG’s approach can be significantly
ontheotherhand,macrorealismpredictsthatnosuchviolationis streamlined, resulting in a conceptually cleaner and experimen-
possible. tallysimplifiedprotocol.Wegoontoapplyournewprotocoltoa
2 NATURECOMMUNICATIONS|7:13253|DOI:10.1038/ncomms13253|www.nature.com/naturecommunications
ARTICLE
NATURECOMMUNICATIONS|DOI:10.1038/ncomms13253
a Chip External B field Time
Flux
JBA qubit
O Readout
JBA output line JBA input line
S S
1 2
Qubit control line
b c
×10 Without With
0.12 measurement measurement Difference
Without measurement pulse S, S
1 2 pulse pulse
0.10 With measurement pulse
〉 0.08 ×10 0, (cid:3)/2 ±30..0055×31807–4 ±03..01247×21508–4 dg = ±–40..3082×61601–24
Q3 0.06 ×10 ×10
〈 0.04 ×10 (cid:3), (cid:3)/2 ±04..02525×21304–4 ±04..03791×11108–4 de = ±06..00195×81804–4
×10
00..0002 (cid:3)/2, (cid:3)/2 ±04..12233×51902–4 ±04..04333×81004–4 d(cid:4) = ±–60..1028×91708–84
g e (cid:4)
d
MR impossible MR possible
d(cid:4) dg de
–0.10 –0.08 –0.06 –0.04 –0.02 0.02
Figure2|Experimentalsetupandresults.(a)Oursuperconductingfluxqubit(yellow)isamicrometrescaledevice,featuresthreeJosephsonjunctions
(crosses),iscontrolledwithmicrowavepulses(orange)andisreadoutthroughmagneticcouplingtoaJosephsonBifurcationAmplifier(JBA,green)
drivenbyamodulatedvoltage(blue).Pulsesnottoscale.(b)TheexpectationvalueofthefinalmeasurementwithandwithoutthemeasurementpulseO,
forthethreepreparationsg,e,andr(representedbyaS pulseangleof0,pandp/2respectively,seeFig.1).Theerrorbarsrepresent10s.d.
1
(c)Thedifferencebetweentheexpectationvaluesisourmeasureofthedisturbanceoftheoperationatt .Themeasureismorenegativeforaquantum
2
preparation(r)thanforeitherclassicalpreparation(gore).(d)d violatesthemacrorealist(MR)boundwithahighdegreeofstatisticalsignificance.
r
superconducting flux qubit, thereby pushing the envelope of than an inequality18. The latter two points imply that any
macroscopicity. The experimental results place constraints onall non-zero measurement visibility V:¼(maxhQi (cid:3)minhQi )/
obs obs
possible macroscopic-realist theories, and should spur progress (maxhQi (cid:3)minhQi ) (relating observed statistics to ideal
ideal ideal
towards tests at higher levels of macroscopicity. ones) is sufficient to find a violation in principle, whereas
pffiffiffiffiffiffiffiffi
previously V4 2=3 was required.
Results
A simpler test. MRPS and NIM allow one to reach a simpler
Measurement invasiveness. The ever-present possibility of a
constraint
clumsy measurement procedure at t , giving rise to a violation,
2
hQ i ¼hQ i ; or ð4Þ rather than any inherent quantum effect, is as important a
3 G 3 (cid:2)2
loophole here as ever. The issue has so far only been addressed
with a priori arguments—those from the use of null result mea-
NDC: d:¼hQ i (cid:3)hQ i ¼0 ð5Þ
3 G 3 (cid:2)2 surements11,19, weak measurements20–22 or the use of an
withtheensemblesasdefinedpreviously.Wecallthisequalitythe additional ‘stationarity’ assumption23. By contrast, Leggett24
‘non-disturbance condition’ (NDC). This condition has been and later Wilde and Mizel25 have proposed that the problem
derived by others and has been termed a quantum witness14, canbeattackedexperimentally.Thisispreciselytheapproachwe
‘non disturbing measurement’15,16 or ‘no signalling in time’ adopt here: The classical disturbance of a measurement (which
condition17. It follows from the same assumptions as LGI we define shortly) may be determined in a control experiment,
(Supplementary Note 1 and Supplementary Table 1) and rather than assumed zero.
demands a zero effect of the choice of measurement at t on Building on these ideas, here we lay out a precise and
2
the statistics of a measurement at the later time t . Here we operational notion of macrorealism that may be tested in the
3
suggest,however,thattherequirementsforthemeasurementatt laboratory. Using conditional probabilities, define the dis-
2
areveryminimal—itwillbeclearshortlythatitneednotevenbe turbanceparameterd :¼[P(Q ¼ þ1|s,O)(cid:3)P(Q ¼ (cid:3)1|s,O)]
s 3 3
a measurement at all, but some generalized operation O about (cid:3)[P(Q ¼ þ1|s)(cid:3)P(Q ¼ (cid:3)1|s)] as a measure of how much
3 3
whichweneednotassumeanything.AllpertinentpropertiesofO disturbance is introduced to Q by applying O at t (compared
3 2
are to be obtained through experiment. with doing nothing) when the preparation of the system
Inspection of the NDC exposes a number of advantages over immediately before t is described by s. In a pair of control
2
LGI. First, that there is no need to measure at all at t . Second, experiments, determine d and d, where g and e are the states
1 g e
that only one-point averages, rather than two-point correlations thatthemeasurementrevealsreliably(thatis,with100%chance).
are required. Third, that the condition is an equality rather These are measures of classical disturbance: see Fig. 1b.
NATURECOMMUNICATIONS|7:13253|DOI:10.1038/ncomms13253|www.nature.com/naturecommunications 3
ARTICLE
NATURECOMMUNICATIONS|DOI:10.1038/ncomms13253
0.4 as an instance of ‘super-activation’26. Our definition of
0.3 0, (cid:2) macrorealism (7) is a noise-tolerant version of Maroney and
Timpson’s ‘operational eigenstate mixture macrorealism’16.
0.2
If the quantum disturbance d is significantly greater in
r
Q3 0.1 magnitudethanthegreatestclassicaldisturbance,thisimpliesthat
theshufflingoperationpreparedsomethingotherthanastatistical
0.0
mixture of g and e. On the quantum view, this would be a
–0.1
Without measurement pulse coherentsuperpositionofthepreparations.Onahidden-variable
–0.2 With measurement pulse view, where preparation of a pure quantum state is actually a
0.2
stochastic selection from a set of underlying states of reality {l},
(cid:2), (cid:2) rmightaccessanewsetof{l}notselectableviaeithereorg;oir
0.1 i
indeed represent a distribution over the same {l} that is further
i
from equilibrium with respect to O (ref. 16). It is worth noting
0.0
Q3 that the leading theories of macrorealism do not rely on such
–0.1 hiddenstates,andso(alongwithawholeclassoffuturetheories
subscribing to (7)) can indeed be ruled out by our approach.
–0.2
0.4 Experimental results. Now, let us test the protocol experimen-
0.3 2(cid:2), (cid:2) tally using a superconducting flux qubit, where O is a measure-
mentwhoseresultisnotinspected:SchildandEmary27callthisa
0.2
‘blind measurement’ but here we refer to it as a ‘measurement
Q3 0.1 pulse’ due to the way it is implemented in our system. We find
0.0 de4dg, and a violation of (7) dr(cid:3)dg¼ (cid:3)0.063177 which is
B84s.d. awayfrom zero (shown in Fig. 2). Despite ouruse ofa
–0.1
relatively low fidelity qubit, we are able to reach very low
–0.2 uncertainties by performing 7(cid:5)106 trials per data point (see
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Fig. 3). A more pristine flux qubit with increased visibility and
Microwave amplitude (√mW) longer coherence times could display an even stronger violation
ofthemacrorealistview.Ourstricttestofmacrorealismprovides
Figure3|Controlpulses.Tomeasured,d ,d,thecontrolcircuitryofour
g r e evidence for a superposition of magnetic moments equivalent to
systemwasarrangedinthreeseparateexperimentssothatthepseudo-spin
several hundred thousand static electron spins pointing in
rotationanglesofS andS were,respectively(0,y),(y,y),(2y,y)fora
1 2 opposite directions simultaneously. For further discussion on
rangeofanglesy.Wethenselectedthedatacorrespondingtoy¼p/2, measures of macroscopicity, see Supplementary Note 3.
(c.f.Fig.1b,c)whichcorrespondstoacertainmicrowavepulsepower
(shownwithadottedverticalline).Errorbarscorrespondtoa95% Discussion
confidenceinterval.Themicrowaveamplitudeshownisatthesignal NotethatthevisibilityofourmeasurementVE0.28isfarbelow
generator(itisattenuatedbeforereachingthefluxqubit).Forquantum that required to find a violation of the LGI—this showcases the
mechanicalcurves,seeSupplementaryNote2andSupplementaryFig.1. advantage of our scheme over standard tests of macrorealism.
By eschewing LG’s inequality, but upholding their methodology
Once the control experiments are completed, the main (as we have done here), improved bounds on macrorealist
experiment may begin to determine dr. r describes the net theories may be obtained more easily. This is in contrast with
preparation when g is followed by a shuffling operation S1 proposalsthattendtoincreasethecomplexitybothofthelogical
(see Fig. 1c). According to quantum mechanics, the preparation argument and of the experimental set-up; note that the
r is described by a density operator r¼jaj2r þjbj2r þC. requirements of a recent laboratory test of the LGI22 extend to
g e
Here, r ¼|g, eihg, e| are the density operators associated with high visibility, partial-strength, non-demolition measurements
g,e
preparations g, e respectively, a and b are complex numbers of two-time correlators via entanglement with a coherent
satisfying |a|2þ|b|2¼1, and C¼ðabwjgihejþh:c:Þ are off- ancilla—requirements that are not necessary in our approach.
diagonal ‘coherence’ terms. In this language the predictions of a Furthermore, our reasoning does not use any quantum
macrorealisttheory(forsuper-criticallymacroscopicsystems)are mechanical assumptions, which, if relied on, can otherwise
equivalent to those which follow from putting C¼0. vitiate the refutation of macrorealism. With the experimental
UndertheassumptionthatS merelypreparesweightedconvex protocol duly simplified, the challenge now is to perform strict
1
combinations(statisticalmixtures)ofthepreparationsassociated tests of macrorealism at much higher macroscopicities—a feat
with g and e, and does not affect the operation of the which should be possible as long as the classical disturbance of
measurement at t , it could be thought natural that the one’s measurement is not too high.
2
disturbance d should not be higher than the similarly weighted
r Methods
linear combination of each individual disturbance:
Qubitdesignandfabrication. Ourthree-junctionfluxqubit,fabricatedusing
d ¼jaj2d þjbj2d ; ð6Þ angledevaporation,isplacedatthecentreofatransmissionlineJosephson
r g e BifurcationAmplifier(JBA)resonatorwhereitismagneticallycoupled(seeFig.2).
TheHamiltonianis
or even the much weaker
E D
min(cid:3)dg;de(cid:4)(cid:4)dr (cid:4)max(cid:3)dg;de(cid:4) ð7Þ H¼2ðjhihhj(cid:3)jgihgjÞþ 2ðjhihgjþjgihhjÞ ð8Þ
whereDisthetunnellingenergyandEisthebiasenergy,andjhiandjgiare
to cover the possibility that the shuffling operation simply statesofdefinitesupercurrent.Theenergyeigenstatesare
deterministicallypreparestheworststate(thatis,theonewiththe
jgi¼sinxjhiþcosxjgi ð9Þ
highest susceptibility to disturbance by O). The fact that
(theoretically at least) d ¼d ¼0 but d a0 could be thought of jei¼cosxjhi(cid:3)sinxjgi ð10Þ
g e r
4 NATURECOMMUNICATIONS|7:13253|DOI:10.1038/ncomms13253|www.nature.com/naturecommunications
ARTICLE
NATURECOMMUNICATIONS|DOI:10.1038/ncomms13253
where2x¼arctan(D/E).Inourexperiment,D/h¼2.75GHzandE/hE2.90GHz 16.Maroney,O.J.E.&Timpson,C.G.Quantum-versusmacro-realism:what
(hisPlanck’sconstant),henceapproximately doestheLeggett–Garginequalityactuallytest?Preprintathttp://arxiv.org/abs/
jgi¼0:37jhiþ0:93jgi ð11Þ 1412.6139v1(2014).
17.Kofler,J.&Brukner,C.Conditionformacroscopicrealismbeyondthe
jei¼0:93jhi(cid:3)0:37jgi: ð12Þ Leggett–Garginequalities.Phys.Rev.A87,052115(2013).
18.Clemente,L.&Kofler,J.Nofinetheoremformacrorealism:limitationsofthe
Ourfluxqubithasaresonancefrequencyof4.0GHzwithapersistentcurrentof
I ¼170nA.ItischosentohaveanaveragecoherencetimeTE10ns(ordersof Leggett–Garginequality.Phys.Rev.Lett.116,150401(2016).
mpagnitudebelowthebestrecordedtime4)toshowtheadvantag2esofourapproach. 19.Knee,G.C.etal.ViolationofaLeggett–Garginequalitywithidealnon-invasive
measurements.Nat.Commun.3,606(2012).
20.Palacios-Laloy,A.etal.ExperimentalviolationofaBell’sinequalityintime
Preparation,controlandreadout. Initializationofourfluxqubit(operating withweakmeasurement.Nat.Phys.6,442–447(2010).
below10mK)isachievedbythermalrelaxation.Microwavelinesprovidea 21.Goggin,M.E.etal.ViolationoftheLeggett–Garginequalitywithweak
mechanismforapplyingresonantqubitcontrolpulsesoffixedduration(2ns), measurementsofphotons.Proc.NatlAcad.Sci.108,1256–1261(2011).
whichrotatethequbitstatebyanangleproportionaltotheamplitudeofthe 22.Groen,J.P.etal.Partial-measurementbackactionandnonclassicalweakvalues
appliedfield.ThesecondS2controlpulseisapplied18nsaftertheS1pulse. inasuperconductingcircuit.Phys.Rev.Lett.111,090506(2013).
Theeffectofthefixed-durationpulsesiscontrolledbymodulatingthepower 23.Huelga,S.F.,Marshall,T.W.&Santos,E.Proposedtestforrealisttheories
(seeFig.3).Tocalibratemicrowavepowerwiththeintendedrotationangleywe usingRydbergatomscoupledtoahigh-Qresonator.Phys.Rev.A52,
measureda(2y,0)sequence,andfittedasinusoidtothedata.Thedataagree R2497–R2500(1995).
qualitativelywithasimplequantummechanicalmodel(seeSupplementaryFig.1). 24.Leggett,A.J.Experimentalapproachestothequantummeasurementparadox.
Optionally,weapplyameasurementpulseO(atafrequencyof6.5GHzwith Found.Phys.18,939–952(1988).
totallength12nswith4nsrisingtime)totheJBA,turningitonatt2(betweenS1 25.Wilde,M.&Mizel,A.AddressingtheclumsinessloopholeinaLeggett–Garg
andS2).Thisoperationcanbethoughtofasameasurementofthefluxqubit,or testofmacrorealism.Found.Phys.42,256–265(2011).
(sincewedonotinspecttheresult)equivalently,asacompletelydephasing
26.Palazuelos,C.Superactivationofquantumnonlocality.Phys.Rev.Lett.109,
operationEðrÞ¼jaj2jgihgjþjbj2jeihej.Thequbitstateisfinallymeasuredatt,
3 190401(2012).
againthroughcouplingtotheJBAinthestandardmanner28–30.
27.Schild,G.&Emary,C.Maximumviolationsofthequantum-witnessequality.
Phys.Rev.A92,032101(2015).
Statistics. Variancesinmeasuredquantitieswerepropagatedaccordingto 28.Nakano,H.,Saito,S.,Semba,K.&Takayanagi,H.Quantumtimeevolutionina
qffiffiffiffiffiffiffiffi(cid:3)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(cid:4)ffiffi qubitreadoutprocesswithaJosephsonbifurcationamplifier.Phys.Rev.Lett.
elementaryrulesVar(dr(cid:3)dg)¼Var(dr)þVar(dg),giving|dr(cid:3)dg|/ Var dr(cid:3)dg 102,257003(2009).
E84.Weconcludethatourviolationofmacrorealismisofextremelyhigh 29.Siddiqi,I.etal.Rf-drivenJosephsonbifurcationamplifierforquantum
significance(SeeFig.2). measurement.Phys.Rev.Lett.93,207002(2004).
30.Lupascu,A.etal.Quantumnon-demolitionmeasurementofasuperconducting
Dataavailability. Allrelevantdataareavailablefromtheauthors. two-levelsystem.Nat.Phys.3,119–125(2007).
References Acknowledgements
1. Leggett,A.J.&Garg,A.Quantummechanicsversusmacroscopicrealism: WethankErikGaugerfordiscussions.WeacknowledgesupportfromtheMEXTGrant-
isthefluxtherewhennobodylooks?Phys.Rev.Lett.54,857–860(1985). in-AidforScientificResearchonInnovativeAreas‘Scienceofhybridquantumsystems’,
2. Chiorescu,I.,Nakamura,Y.,Harmans,C.J.P.M.&Mooij,J.E.Coherent grantnumbers15H05870&15H05867.
quantumdynamicsofasuperconductingfluxqubit.Science299,1869–1871
(2003). Author contributions
3. Omelyanchouk,A.N.,Shevchenko,S.N.,Zagoskin,A.M.,Il’ichev,E.&Nori,
G.C.K.andM-C.Y.conceivedanddesignedtheexperiment.G.C.K.andA.J.L.wrote
F.Pseudo-Rabioscillationsinsuperconductingfluxqubitsintheclassical
thepaper.K.K.,H.T.,H.Y.andS.S.performedtheexperiment.G.C.K.,M-C.Y.,Y.M.
regime.Phys.Rev.B78,054512(2008).
andW.J.M.modelledandrefinedtheprotocol.Allauthorscontributedtorefiningthe
4. Yan,F.etal.Thefluxqubitrevisitedtoenhancecoherenceandreproducibility.
manuscript.
Nat.Commun.(inthepress).
5. Barends,R.etal.Superconductingquantumcircuitsatthesurfacecode
thresholdforfaulttolerance.Nature508,500–503(2014). Additional information
6. Ghirardi,G.C.,Rimini,A.&Weber,T.Unifieddynamicsformicroscopicand SupplementaryInformationaccompaniesthispaperathttp://www.nature.com/
macroscopicsystems.Phys.Rev.D34,470–491(1986). naturecommunications
7. Pearle,P.Combiningstochasticdynamicalstate-vectorreductionwith
spontaneouslocalization.Phys.Rev.A39,2277–2289(1989). Competingfinancialinterests:Theauthorsdeclarenocompetingfinancial
8. Penrose,R.Ongravity’sroleinquantumstatereduction.Gen.Relativ.Gravit. interests.
28,581–600(1996).
Reprintsandpermissioninformationisavailableonlineathttp://npg.nature.com/
9. Leggett,A.J.Realismandthephysicalworld.Rep.Prog.Phys71,022001
reprintsandpermissions/
(2008).
10.Emary,C.,Lambert,N.&Nori,F.Leggett–Garginequalities.Rep.Prog.Phys. Howtocitethisarticle:Knee,G.C.etal.Astrictexperimentaltestofmacroscopic
77,016001(2014). realisminasuperconductingfluxqubit.Nat.Commun.7,13253doi:10.1038/
11.Robens,C.,Alt,W.,Meschede,D.,Emary,C.&Alberti,A.Idealnegative ncomms13253(2016).
measurementsinquantumwalksdisprovetheoriesbasedonclassical
trajectories.Phys.Rev.X5,011003(2015). Publisher’snote:SpringerNatureremainsneutralwithregardtojurisdictionalclaimsin
12.Zhou,Z.-Q.,Huelga,S.F.,Li,C.-F.&Guo,G.-C.Experimentaldetectionof publishedmapsandinstitutionalaffiliations.
quantumcoherentevolutionthroughtheviolationofLeggett–Garg-type
inequalities.Phys.Rev.Lett.115,113002(2015). This work is licensed under a Creative Commons Attribution 4.0
13.White,T.C.etal.Preservingentanglementduringweakmeasurement InternationalLicense.Theimagesorotherthirdpartymaterialinthis
demonstratedwithaviolationoftheBell–Leggett–Garginequality. articleareincludedinthearticle’sCreativeCommonslicense,unlessindicatedotherwise
npjQuantumInf.2,15022(2016). inthecreditline;ifthematerialisnotincludedundertheCreativeCommonslicense,
14.Li,C.-M.,Lambert,N.,Chen,Y.-N.,Chen,G.-Y.&Nori,F.Witnessing userswillneedtoobtainpermissionfromthelicenseholdertoreproducethematerial.
quantumcoherence:fromsolid-statetobiologicalsystems.Sci.Rep2,885 Toviewacopyofthislicense,visithttp://creativecommons.org/licenses/by/4.0/
(2012).
15.George,R.E.etal.Openingupthreequantumboxescausesclassically
undetectablewavefunctioncollapse.Proc.NatlAcad.Sci.110,3777–3781(2013). rTheAuthor(s)2016
NATURECOMMUNICATIONS|7:13253|DOI:10.1038/ncomms13253|www.nature.com/naturecommunications 5
1.0
0,θ
0.5
〉
0.0 Q3
〈
-0.5
11..00
θ,θ
0.5
〉
0.0 Q3
〈
-0.5
11..00
2θ,θ
0.5
〉
0.0 Q3
〈
-0.5
-1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
θ
Supplementary Figure 1: Theoretical predictions shown in equation (12), prodced by a simple
quantum-mechanical model. Red (black) curves correspond to the presence (absence) of the mea-
surement pulse O, intervening between two rotations by angles θ ,θ . Grey, dashed curves relate
1 2
to the case of no measurement pulse and an infinite T time. Compare with Figure 3 of the main
2
text.
6
Q Q inequality implication
1 2
1 -1 LG2(cid:48) d ≤ 0
1 -1 LG1(cid:48) d ≥ 0
1 1 LG3(cid:48) d ≥ 0
1 1 LG4(cid:48) d ≤ 0
-1 1 LG2(cid:48) d ≤ 0
-1 1 LG1(cid:48) d ≥ 0
-1 -1 LG3(cid:48) d ≤ 0
-1 -1 LG4(cid:48) d ≥ 0
Supplementary Table 1: Logical implications: By fixing the values of Q and Q and by using
1 2
multipleLeggett-Garginequalities,onecanderivethenon-disturbanceconditiond = 0.
7
Supplementary Note 1: Three types of measurement invasiveness, and the logical relation of
NDCtoLGI
InthemaintextweargueddirectlyfromNIMtotheNDC.Itispossiblehowever,toarguefrom
LGI : (cid:104)Q Q (cid:105) +(cid:104)Q Q (cid:105) +(cid:104)Q Q (cid:105) ≥ −1. (1)
1 2 ¯3 1 3 ¯2 2 3 ¯1
totheNDC.ItisclearthataslightmodificationfromLGIto
LG1(cid:48) : (cid:104)Q Q (cid:105) +(cid:104)Q Q (cid:105) +(cid:104)Q Q (cid:105) ≥ −1; (2)
1 2 G 1 3 ¯2 2 3 G
also follows via NIM, and captures all of the essential features of the LGI. The advantage is that
now only two ensembles need be discussed. One should in principle check for the disturbance of
anymeasurementwhichappearsinsomeensemblesand notinothersduringthecourseofthetest
ofmacrorealism. WiththefullmachineryofLGIinplace,thiswouldinvolvetesting
dI := (cid:104)Q Q (cid:105) −(cid:104)Q Q (cid:105) (3)
c 2 3 G 2 3 ¯1
dII := (cid:104)Q Q (cid:105) −(cid:104)Q Q (cid:105) (4)
c 1 3 G 1 3 ¯2
dIII := (cid:104)Q Q (cid:105) −(cid:104)Q Q (cid:105) (5)
c 1 2 G 1 2 ¯3
incontrolexperiments,andthenlookingforsatisfactionorviolationof
(cid:104)Q Q (cid:105) +(cid:104)Q Q (cid:105) +(cid:104)Q Q (cid:105) ≥ −1−dI −dII −dIII. (6)
1 2 ¯3 1 3 ¯2 2 3 ¯1 c c c
The assumption that dIII = 0 is often motivated by appeals to the unlikelihood of retrocausality
c
(calledInductionbyLeggett): dI = 0however,hasbeenoftenmistakenlyassumedintheliterature
c
despite its negation being quite plausible in a general theory. This is perhaps because it is true
8
according to quantum mechanics: any influence propagating from t to t is completely screened
1 3
off by a projective measurement at t . For a fuller discussion of these issues, see Ref.1. Our
2
simplified protocol however, negates the need to test for dI or dIII since we never use ensembles ¯1
c c
or ¯3. In reducing the number of distinct ensembles from four (G, ¯3,¯2,¯1) to two (G, ¯2), we need
onlyconsiderthepossibleeffectsofameasurement(ornot)att .
2
Theissueofexperimentallydeterminingthedisturbanceofmeasurementswasdiscussedina
proposalfromWildeandMizel2. Theycallameasurement(cid:15)-adroitifthetotalchangeinjointprob-
ability for outcomes of measurements made before and after the measurement is upper-bounded
by (cid:15), when the joint probability is conditioned on either i) performing the measurement or ii) not
performing the measurement. It is in this spirit that we pursue the experimental determination of
measurement disturbance – previously only a priori arguments for d = 0 (or its equivalent) have
c
beenemployed.
To see the logical connection between LGI and NDC, consider that under MRPS there are
only four possible values for Q and Q . For each combination, a pair of inequalities together
1 2
imply the non disturbance condition. Take the Q = 1 = −Q case as an example. The LG1(cid:48)
1 2
condition(2)becomesd ≥ 0andapartnerinequality
LG2(cid:48) : (cid:104)Q Q (cid:105) −(cid:104)Q Q (cid:105) −(cid:104)Q Q (cid:105) ≥ −1, (7)
1 2 G 1 3 ¯2 2 3 G
obtained by changing the sign of two correlators, becomes d ≤ 0. These two inequalities together
9
implyd = 0. Informalnotation(∧forlogicalconjunction,→forimplication,¬fornegation)
(Q = 1)∧(Q2 = −1)∧(LG1(cid:48)) → d ≥ 0 (8)
1
(Q = 1)∧(Q2 = −1)∧(LG2(cid:48)) → d ≤ 0 (9)
1
(d ≥ 0)∧(d ≤ 0) → d = 0 . (10)
Similar arguments can be made for the other three combinations, which make use of another pair
of inequalities LG3(cid:48) : −(cid:104)Q Q (cid:105) +(cid:104)Q Q (cid:105) −(cid:104)Q Q (cid:105) ≥ −1 and LG4(cid:48) : −(cid:104)Q Q (cid:105) −
1 2 G 1 3 ¯2 2 3 G 1 2 G
(cid:104)Q Q (cid:105) +(cid:104)Q Q (cid:105) ≥ −1.
1 3 ¯2 2 3 G
TheeightlogicalimplicationsrequiredtosecureallpossibilitiesareshowninSupplementary
Table 1. Given a violation of NDC, one can always choose a fixed assignment of ±1 values to Q
1
andQ suchthatatleastonesimplifiedLGinequalityisdulyviolated:
2
¬NDC → (¬LG1(cid:48))∨(¬LG2(cid:48))∨(¬LG3(cid:48))∨(¬LG4(cid:48)). (11)
Furthermore, given a fixed value assignment of Q and Q , the sign of the violation of the NDC
1 2
willinformastowhichoneofapairofLGinequalitiesisviolated.
SupplementaryNote2: IdealquantummechanicalpredictionsforFigure3ofthemaintext
Althoughourrefutationofmacroscopicrealismdoesnotrequirethatquantummechanicalpredic-
tions are reproduced, for completeness we show that the results in Figure 3 of the main text are
captured by a simple quantum mechanical model. Our qubit begins in the ground state |g(cid:105), and
then is subject to a pseudo-spin rotation through angle θ . Next, it either experiences environ-
1
10