Experimental Violation of Two-Party Leggett-Garg Inequalities with Semi-weak Measurements

0.24 MB·English

by  J. Dressel

Experimental Violation of Two-Party Leggett-Garg Inequalities
with Semi-weak Measurements
J. Dressel, C. J. Broadbent, J. C. Howell, and A. N. Jordan
Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
(Dated: January 26, 2011)
Wegeneralize thederivation of Leggett-Garg inequalities to systematically treat a larger class of
experimental situations by allowing multi-particle correlations, invasive detection, and ambiguous
detector results. Furthermore, we show how many such inequalities may be tested simultaneously
withasinglesetup. Asaproofofprinciple,weviolateseveralsuchtwo-particleinequalitieswithdata
obtained from a polarization-entangled biphoton state and a semi-weak polarization measurement
based on Fresnel reflection. We also point out a non-trivial connection between specific two-party
1 Leggett-Garg inequality violations and convex sums of strange weak values.
1
0
PACSnumbers: 42.50.Xa,03.65.Ta,42.50.Dv
2

n
a To better understand and identify the apparent di-
J vision between macroscopic and microscopic behavior,

5 Leggett and Garg have distilled common implicit as-
2 sumptions about the macroscopic world into a set of ex-

plicit postulates that they dub macrorealism (MR) [1].
]
h From these postulates, they construct inequalities anal-
p ogous to Bell inequalities [2] but involving multiple cor-
-
t relations in time. Such Leggett-Garginequalities (LGIs)
n mustbesatisfiedbyanytheorycompatiblewithMR,but FIG.1. MRmeasurementschematic. Anobjectpairispicked
a from an ensemble ζ at time t . At t object 1 of the pair in-
may be violated by quantum mechanics. As such, LGI 0 1
u
teractswithanimperfectdetectorforthepropertyA ,which
q violations have received increasing interest as signatures 1
reports a generalized value α . At t both objects interact
[ ofdistinctlyquantumbehaviorinqubitimplementations 1 2
withunambiguousdetectorsforthepropertiesB1andB2that
1 [3–5], and have been recently confirmed experimentally report values b1 and b2. The two-party LG correlation C is
v in both solid-state [6] and optical systems [7]. constructed from themeasured results.
7 Inthis Letter,wedemonstrateatechnique forsystem-
1 interaction, or (b) ambiguous by reporting results that
atically deriving generalized LGIs that admit multiple
9 only correlate probabilistically with the object state due
parties, invasive detection, and/or ambiguous detector
4
to inherent detector inefficiencies or errors.
. results by considering a specific two-particle experimen-
1
talsetupwiththreemeasurements. Weproceedtoexper- For convenience we consider dichotomic properties in
0
imentally violateseveralsuchtwo-partyLGIs simultane- what follows, though the discussion can be easily ex-
1
1 ously with a single data set producedfrom a setup using tended. Unambiguousdetectoroutcomeswillbeassigned
: a semi-weak polarization measurement on an entangled the (arbitrary) values 1,1 corresponding to the two
v {− }
i biphoton state. The contextual values (CV) analysis of possiblestatesofthe propertybeingmeasured. Ambigu-
X
quantum measurement [8] suggests a direct comparison ous detectors will be calibrated to report the same en-
r between the classical and quantum treatments. Finally, sembleaverageas anunambiguousdetectorfor the same
a
we show that specific two-party LGIs are equivalent to property. To do so, their outcomes must be assigned
constraintsonconvexsumsofconditionedaverages(CA), generalized values α S from an expanded set S, with
∈
whicharethe generalizationsofthe quantumweakvalue minS 1 and maxS 1, to compensate for the im-
≤ − ≥
to an arbitrarymeasurement setup [8, 9]. The technique perfect state correlation of the outcomes. Such general-
may be easily extended to check data from a setup with izedvaluesaretheclassicalequivalentofquantumCV[8]
any number of measurements and parties. and may be determined by measuring pure ensembles of
Generalized LGIs.—AMR theory consistsofthreekey either 1.
±
postulates: (i) if an object has several distinguishable Wenowderiveaspecifictwo-partygeneralizedLGIfor
states available to it, then at any given time it is in only aparticularexperimentalsetup,keepinginmindthatthe
one of those states; (ii) one can in principle determine method may be extended to any setup. Consider a pair
which state it is in without disturbing that state or its of MR objects that interacts with a sequence of detec-
subsequent dynamics; and, (iii) its future state is deter- tors as shown in Fig. 1. At time t the pair is picked
0
mined causally by prior events [1]. Furthermore, we ac- fromaknownensembleζ. Attimet object1ofthepair
1
knowledge that physical detectors may be imperfect by interacts with an imperfect detector for the dichotomic
being(a)invasive byalteringtheobjectstateduringthe property A , which reports a generalized value α S .
1 1 1
∈
2
λ/4
Finally, at time t objects 1 and 2 interact with unam-
2 λ/2
biguous detectors for the dichotomic properties B and
1 SPDC pol.
B2,respectively,whichreportthevaluesb1,b2 ∈{−1,1}. λ/2 pol.-θ
For each object pair, we can keep all three results attn.
to construct the correlation product α b b , or we can mirror
1 1 2 λ/2 λ/4
ignore some results as non-selective measurements [10] coverslip
to construct the alternate quantities α , b , b , α b ,
1 1 2 1 1 pol.-θ
α b , or b b . Since the latter terms involve voluntary
1 2 1 2
loss of information after the measurement has been per-
FIG. 2. (color online) Experimental setup. A 488 nm laser
formed, we can compute them all from the same data
produces degenerate down-converted photon pairs. The po-
set. Exploitingthisfreedom,weconstructthecorrelation ◦
larization of the photon in the lower arm is rotated by 45
C = α1 +α1b1b2 b1b2 for each measured pair, which withahalf-waveplate,thenundergoessemi-weakpolarization
−
lies in the range, 1 2minS C 2maxS 1. measurement in the{h,v}basis using Fresnelreflection (A )
1 1 1
−| − |≤ ≤| − |
We repeat this procedure many times and that encodes the information in the resulting spatial modes,
average the results of C to obtain, C = and is finally projected into the {θ,θ⊥} basis with polariz-
h i ers set at angle θ (B ). The polarization of the photon in
P(α ζ)P(b ,b ζ,α )(α +α b b b b ), 1
Pα1,b1,b2 1| 1 2| 1 1 1 1 2− 1 2 theupperarm isprojected intothe{h,v}basiswith another
where P(α ζ) is the probability of detecting α given
1| 1 polarizer (B2). The half and quarter waveplates prior to the
the initial ensemble ζ, and P(b ,b ζ,α ) is the proba-
1 2 1 polarizers areused for tomography of theinputstate; during
|
bility of detecting b1 and b2 given the initial ensemble ζ datacollectiontheyareremovedfromthelowerarmandused
and the possibly invasive detection of α . to switch between h and v polarization in the upperarm.
1
Generally, we cannot separate the sums due to the
α -dependence of P(b ,b ζ,α ), so the best guaranteed Conditioned averages.—A single-object LGI, 3
bspo1eucniadlscaarsee,−i|f1t−he2dmeit1necSt21o||r≤fohr1CAi≤is|2umnaamxSbi1g−uo1u|s. Athsena hhAav1e+aAo1Bne1-−toB-o1nie≤co1r,rwesapsocnodnesnidceerewditihna[4n]aunpdpesrho−bwonunt≤do
1
minS1 = 1, maxS1 =1, and we find the LGI, to the averageof A1 conditionedon the positive value of
− B : A 1. Three other LGIs similarly correspondto
1 1
3 A +A B B B B 1. (1) h i≤
1 1 1 2 1 2 thebounds A 1,and 1 A 1,aschecked
− ≤h − i≤ 1 −1
h i≥− − ≤ h i≤
Alternatively, if the detector is noninvasive so that experimentally in [7].
P(b ,b ζ,α ) = P(b ,b ζ) then the sums do sep- We now extend these results to the two-object case
1 2 1 1 2
arate a|nd we can averag|e A first to find, C = using (1). First we define a marginal probability
1
PhuAesb11i,b2≤3P,1(b,11e,,ab12ch|ζa)tn(edhrAmw1eic(aa1gn+aitnba1krbee2c)oo−nvelybr1(bt1h2))r..eeThSpeionrsecsfeiobhrl−ee,i1vaan≤ly- oPdfe(fibmn1,eeba2as|ζuc,roAinn1gd)itb=io1nPaalnαdp1rPobb(2αab1gi|ilζvi)tePyn(obaf1n,myb2e|aζres,usαur1il)nt.goTαf1hAegn1ivweanes
viola{t−ion−of (1}) will imply that at leastone ofthe postu- the measurement of b1 and b2 as, P(α1 ζ,b1,b2) =
|
lates (i-iii) of MR does not hold, or that the detector for P(α1 ζ)P(b1,b2 ζ,α1)/P(b1,b2 ζ,A1). Therefore,theav-
| | |
A1 is both invasive and ambiguous. erage of A1 conditioned on the measurements of b1 and
WecanconstructmanysuchLGIsfromthesamedata. b2 is b1,b2hA1i=Pα1P(α1|ζ,b1,b2)α1.
For example, the three detectors in Fig. 1 allow the con- Usingthisdefinition,werewritetheupperboundof(1)
estarrulicetri,onwohficthhec2a3n−b1encoonmtrbiivnieadlcworirtehlatthioenttherremescloisetffied- aansdPinb1s,eb2rtPt(hbe1,pb2o|sζs,ibAl1e)v(abl1u,be2shAfo1ri(b11+anb1db2b)2−tob1fibn2)d≤th1e
cients 1,0,1 [11]. Ignoring an overall sign, we can CA constraint,
constru{c−t (323−}1 − 1)/2 = 1093 nonzero LGI correla- 1,1 A1 p++ −1,−1 A1 p− 1, (2)
tions bounded in a similar manner to (1). The subset h i h i ≤
of(322−1 1)/2=13single-objectLGIscanbe obtained where p± = P( 1, 1)/(P(1,1) + P( 1, 1)), and
− ± ± − −
by ignoring the B detector. Furthermore, if a fourth P(i,j) = P(i,j ζ,A ). The degeneracy of the product
2 1
|
detector for A were added before the detector for B , value b b resultsin anupper bound fora convex sum of
2 2 1 2
we could test (324−1 1)/2 = 7174453 such LGIs. One CAs,incontrasttothesingle-objectresultin[4]. Asuffi-
−
is formallyidentical to the CHSH-Bell inequality [2] (see cient conditionforviolating(2)isforbothCAstoexceed
also [12]), but tests MR and not Bell-locality. 1simultaneously. Conversely,ifallCAswereboundedby
Forcontrast,theoriginalapproachin[1]combinessep- 1, then it would be impossible to violate (2) or (1).
arate experiments for each correlation between ideal de- Quantum formulation.—Projective quantum measure-
tectors to form a single LGI. Our approachuses a single ments produce averages of eigenvalues analogous to the
experimental setup to determine all 2M 1 correlations results of an unambiguous detector, but non-projective
−
between M general detectors to form a large number of quantum measurements produce averages of contextual
LGIs. Hence we obtain an exponential improvement in values [8] which need not lie in the eigenvalue range and
experimental complexity for large M. are therefore analogous to the results of an ambiguous
3
|h|hh|>vv|>hvv>> <hh| <hv| <vh| <vv| --00..11550..ìæàìæàìæàaìæàìæàLìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæàìæà -----10864202ìæàòìæàòìæàòbìæàòìæàòLìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàòìæàò
0 20 40 60 80 Θ 0 20 40 60 80 Θ
FIG.3. (color online) Real(left) and imaginary (right) parts
ofthereconstructeddensitymatrixinthe{h,v}basis. Yellow FIG.4. (coloronline)Inalldataplots,solidlinesindicatethe-
and red represent positive and negative values, respectively. ory and points indicate experimental data. a) hσˆz(1)i (green,
flat), θhσˆz(1)i (red, decreasing), and θ⊥hσˆz(1)i (blue, increas-
detector. By measuring A1 weakly we can find quantum ing). b) θ,hhσˆz(1)i (red, bottom right), and θ⊥,vhσˆz(1)i (blue,
mechanical violations of (1) and (2). bottomleft),violatingnegativebounds,unlike θ⊥,hhσˆz(1)i(or-
eraStpoerciρfiˆcaalnlyd, mifewaseursetaArt wgiethneraal2ly-obsujeccht tdheantsitAyˆ op=- ange, increasing), and θ,vhσˆz(1)i (green, decreasing).
1 1
a Πˆ = α Eˆ (where a are the eigenval-
Pa1 1 a1 Pα1 1 α1 { 1} lation stage. The reflected light passes though a polar-
utheesCcoVrrecsoprroensdpionngdtiongthtoe pthroejPecOtiVonMs{{EΠˆˆαa11}=anMˆdα†{1αMˆ1}α1a}r)e, iWzaetiaolniganntahlyezceorvaenrsdlicpoaunpdlesthinetmoiarrtohrirtdofibbeepraarnadlleSlPwAiDth.
andthenmeasureB B projectivelysuchthatBˆ Bˆ = anincidenceangleof 40◦ relativetotheincomingbeam.
1 2 1 2
b b Πˆ Πˆ , we will find that the avera⊗ge cor- Finally,weoptimizethe fiberincouplingandbalancethe
Pb1,b2 1 2 b1 ⊗ b2
relation C = A +A B B B B has the form, collection efficiencies with attenuators so that the coin-
1 1 1 2 1 2
h i h − i
cidences between the upper arm and either of the lower
C = P(α ;b ,b ρˆ)(α +α b b b b ), (3) armsdifferbyonlyafewpercentwhenthemirroristaken
h i X 1 1 2| 1 1 1 2− 1 2
α1,b1,b2 in and out of the beam path.
The coverslip acts as a polarization-dependent beam-
where P(α1;b1,b2|ρˆ) = Trh(cid:16)Mˆα†1Πˆb1Mˆα1 ⊗Πˆb2(cid:17)ρˆi is splitter measuring A1 = σˆz. Averaging over the 3 nm
the probability of measuring outcome α of the general bandwidth and the thickness variation ( 150 0.6µm)
1 ∼ ±
measurementofA,followedbyajointprojectionofb b . produces an average Fresnel reflection similar to that of
1 2
The appearance of the CV instead of the eigenvalues of a single interface, with horizontal (h) polarization rela-
Aˆ in (3) combined with the non-separable probability tivetothetableexhibitingzeroreflectionnearBrewster’s
P(α ;b ,b ρˆ) allows violations of the LGI (1). angle and vertical (v) polarization exhibiting increasing
1 1 2
The left| side of (2) follows from (3), where reflection with incident angle.
P(b ,b ρˆ,A ) = P(α ;b ,b ρˆ) and A = Fora pure state of polarization ψ =αh +β v with
1α2P| (α1;b ,b Pρˆ)α/1P(b ,1b ρ1ˆ,A2|) is a qub1a,nb2thum1iCA α2+ β 2 =1, the resulting state|afiter pa|sising t|hirough
aPsαd1efi1ned i1n [18] 2th|at con1ver2g|es to1 a weak value [9] in |th|e cov|e|rslip is ψ′ = (γαh + η¯β v )r (γ¯αh +
| i | i | i | i − | i
the limit of minimal measurement disturbance. ηβ v )t , where j , j t,r , specify the transmit-
| i | i | i ∈ { }
Experimental setup.—To implement Fig. 1 we use the ted and reflected spatial modes of the coverslip, and the
polarization of an entangled biphoton with the setup reflection and transmission probabilities for h- and v-
shown in Fig. 2. A glass microscope coverslip mea- polarized light are Rh = γ2, Rv = η¯2, Th = γ¯2, and
sures a Stokes observable A1 semi-weakly as described Tv = η2, such that Ri +Ti = 1. Written this way, the
below,andpolarizersmeasureStokesobservablesB and coverslip reflection can be viewed as a generalization of
1
B projectively. We produce degenerate non-colinear the weak measurement in [15] and discussed in [8].
2
type-II down-conversion by pumping a 2 mm walkoff- From ψ′ , we find the measurement operators for
compensated BBO crystal [13] with a narrowband 488 the back-|acition of the coverslip outcomes to be Mˆ =
r
nmlaser. Thedown-convertedlightpassesthroughauto- γΠˆ +η¯Πˆ andMˆ =γ¯Πˆ +ηΠˆ ,whereΠˆ ,i h,v ,are
h v t h v i
∈{ }
matedpolarizationanalyzersand3nmbandpassfiltersat polarization projectors. The corresponding POVM ele-
976nmineacharmbeforebeingcoupledintomultimode ments are Eˆ = R Πˆ +R Πˆ and Eˆ = T Πˆ +T Πˆ ,
r h h v v t h h v v
fibers connected to single photon avalanche photodiodes with which we can expand the polarization Stokes op-
(SPAD). We detect coincidences using a 3 ns window. erator as σˆ = Πˆ Πˆ = α Eˆ +α Eˆ , as discussed
z h v r r t t
−
We perform state tomographywith maximum likelihood before (3), where α = (T +T )/(R R ) and α =
r h v h v t
−
estimation[14],whichgiesthestateshowninFig.3with (R +R )/(R R ) are the CV.
h v h v
− −
concurrence C = 0.794, and purity Tr ρˆ2 = 0.815, and We determine the values of R and R with cali-
(cid:2) (cid:3) h v
which resembles the pure state ψ =(hv +ivh )/√2. bration polarizers before the coverslip, yielding R =
h
| i | i | i
After the state tomography, we remove the half- and 0.0390 0.0007 and R = 0.175 0.001. The reflected
v
± ±
quarter-waveplatesfromthe lowerarmandinserteither armislargelyprojectedtov,whilethetransmittedarmis
amirrororacoverslipusingacomputer-controlledtrans- only weakly perturbed, making the total coverslip effect
4
0123456æàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæà --001...115055.. æàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæàæà
0 20 40 60 80 Θ 0 20 40 60 80 Θ
FIG.5. (coloronline)LGIcorrelationh−σz(1)−σz(1)σθ(1)σz(2)− FIG.6. (color online) LGI correlations hσz(1)σz(2)+σz(2)σθ(1)−
σθ(1)σz(2)i (red,squares)and thecorrespondingconvexsumof σz(1)σθ(1)i (red, circles), and h−σz(1)σz(2)+σz(2)σθ(1)+σz(1)σθ(1)i
theCAs θ,hh−σˆz(1)iand θ⊥,vh−σˆz(1)i(blue,circles), bothvio- (blue, squares) violating their upper bounds of 1 for nearly
latingtheirupperboundsof1inthesamedomainofθ. Com- theentire θ domain.
pare to Fig. 4 (b) and note that the LGI violation includes
theregion where thetwo CAs both exceed their bounds. pointsindicateexperimentaldataandincludePoissonian
error bars. The small discrepancies between theory and
a semi-weak measurement. The CV, α = 13.1 0.1 data can be explained by sensitivity to the state recon-
r
− ±
andα =1.57 0.01,arecorrespondinglyamplifiedfrom struction and additional equipment imperfections. The
t
±
the eigenvalues of σˆ . violations indicate either that MR is inconsistent with
z
Results.—To complete the state preparation, we place experimentorthatthesemi-weakmeasurementdeviceis
a half-wave plate before the coverslip in the lower arm both invasive and ambiguous in the MR sense.
and rotate the polarization by 45◦ to produce a state Conclusion.—We have illustrated the derivation of
similar to ψ′′ = (ha +ivd )/√2. We then measure generalized single-setup LGIs allowing for multiple par-
| i | i | i
(1) by choosing the observables A , B , and B to be ticles and measurements with more realistic detectors
1 1 2
the Stokes observables σˆ(1), σˆ(1) and σˆ(2), respectively, by considering a two-particle example, and have demon-
z θ z
stratedsimultaneousviolationsofseveralsuchtwo-party
where σˆ is the σˆ operator rotated to the θ,θ basis
θ z ⊥
{ } LGIs using the same data set from a biphoton polariza-
(e.g. σˆ0◦ = σˆz and σˆ45◦ = σˆx). By changing the single
tion experiment. Due to the single setup, any dataset
parameter, θ, we can explore a range of observables.
may be similarly examined for inherent LGI violations.
Fig.4showsthevariousaveragesofσˆ(1). Averagingall
z ThisworkwassupportedbytheNSFGrantNo. DMR-
results for orthogonal settings on σˆ(1) and σˆ(2) gives the
θ z 0844899, ARO Grant No. W911NF-09-1-0417, and a
expectation value σˆz(1) , which is properly constant and DARPA DSO Slow Light grant.
h i
near zero for all θ since the reduced density operator is
almostfullymixed. Averagingonlytheresultsfortheor-
[1] A. J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857
(2) (1)
thogonalsettingsofσˆz givesthesingleCAs θ σˆz and (1985); A. J. Leggett, J. Phys.: Condens. Matter 14,
h i
σˆ(1) , which are also well-behaved. Finally, averaging R415 (2002).
θo⊥nhlyzthei resultsforspecificsettingsgivesthedoubleCAs [2] J. S. Bell, Physics 1, 195 (1965); J. F. Clauser, et al.,
σˆ(1) , σˆ(1) , σˆ(1) , and σˆ(1) , which can Phys. Rev.Lett. 23, 880 (1969).
θ,vh z i θ⊥,hh z i θ,vh z i θ⊥,vh z i [3] R. Ruskov, A.N. Korotkov, and A. Mizel, Phys. Rev.
exceedtheeigenvaluerangeforsomerangeofθduetothe Lett. 96, 200404 (2006).
non-local correlations in the entangled biphoton state. [4] A.N. Jordan, A.N. Korotkov, and M. Bu¨ttiker, Phys.
Using the same set of data, Fig. 5 shows the upper Rev. Lett. 97, 026805 (2006); N. S. Williams and A. N.
(1) (1) (1) (2) (1) (2) Jordan, Phys. Rev.Lett.100, 26804 (2008).
boundoftheLGI−3≤h−σz −σz σθ σz −σθ σz i≤ [5] M. W. Wilde and A.Mizel, (2010), arXiv:1001.1777.
1 being violated in the same range of θ that the appro-
[6] A. Palacios-Laloy, et al.,Nature Physics 6, 442 (2010).
priateconvexsumof θ,hh−σˆz(1)iand θ⊥,vh−σˆz(1)iviolates [7] M. E. Goggin, et al.,(2010), arXiv:0907.1679.
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We can violate several more LGIs using the same set Lett. 104, 240401 (2010).
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hσz(1)σz(2)+σz(2)σθ(1)−σz(1)σθ(1)i,andh−σz(1)σz(2)+σz(2)σθ(1)+ [10] LKe.tKt.ra6u0s,,1S3t5a1te(s1,9e8ff8e)c.tsandoperations: fundamentalno-
σz(1)σθ(1)ithatbetweenthemviolateanupperboundover tions of quantum theory (Springer-Verlag, 1983).
nearly the whole range of θ, for illustration. [11] Allowingothercoefficients,assuggestedin[16],produces
even more possibilities.
AllsolidcurvesinFigures4,5,and6arequantumpre-
[12] S. Marcovitch and B. Reznik,(2010), arXiv:1005.3236.
dictions analogous to (3) using the measurement opera-
[13] P. G. Kwiat, et al., Phys.Rev.Lett. 75, 4337 (1995).
tors, CV, and the reconstructed initial state. They also [14] D. F. V. James, et al.,Phys. Rev.A 64, 052312 (2001).
include compensation for a few percent deviation in the [15] G. J Pryde,et al., Phys.Rev.Lett. 94, 220405 (2005).
thickness of the half-wave plate in the upper arm. The [16] M. Barbieri, Phys. Rev.A 80, 034102 (2009).