Various quantum nonlocality tests with a simple 2-photon entanglement source

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by  Enrico Pomarico

Various quantum nonlocality tests with a simple 2-photon entanglement source
Enrico Pomarico, Jean-Daniel Bancal, Bruno Sanguinetti, Anas Rochdi, and Nicolas Gisin
∗
Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland
Nonlocality is a fascinating and counterintuitive aspect of Nature, revealed by the violation of a
Bell inequality. The standard and easiest configuration in which Bell inequalities can be measured
hasbeen proposed byClauser-Horne-Shimony-Holt(CHSH).However,alternative nonlocality tests
can also be carried out. In particular, Bell inequalities requiring multiple measurement settings
can provide deeper fundamental insights about quantum nonlocality as well as offering advantages
in the presence of noise and detection inefficiency. In this article we show how these nonlocality
testscan beperformed usinga commercially available source of entangled photon pairs. Wereport
the violation of a series of these nonlocality tests (I3322, I4422 and chained inequalities). With the
1 violation of the chained inequality with 4 settings per side we put an upper limit at 0.49 on the
1 local content of thestates prepared by thesource (instead of 0.63 attainable with CHSH).Wealso
0 quantify the amount of true randomness that has been created during our experiment (assuming
2 fair sampling of thedetected events).

n
a
J I. INTRODUCTION any light-speed communication between them from the

2 measurement settings are set until the detection events
1 Nonlocalityisoneofthemostcounterintuitiveandfas- haveoccurred. Atthe momentbothloopholeshavebeen
cinating aspects of Nature revealed by quantum theory. closed, but not yet within the same experiment.
]
h Indeed, the fact that two separated systems appear to Nowadays,performingnonlocalitytestswithentangled
p work in a joint way, independently of the distance sep- photons is relatively easy. Indeed, the technology neces-
- arating them, does not have a counterpart in the classi- sary for generating entanglement is well known and en-
t
n cal world. In particular, this bizarre effect is predicted tanglementsourcescanbeextremelycompact,cheapand
a by quantum mechanics for two entangled systems and stable in time. Moreover,polarization entanglement can
u manifest itself in the correlations of the outcomes of the be analyzed in a practical and accurate way. Therefore,
q
measurements performed on the two systems. This kind the experimental study of entanglement is not confined
[
of”spooky actionsat a distance”was the argumentthat to a few privileged laboratories, but can be also carried

1 Einstein, Podolski and Rosen used to claim the incom- out as part of undergraduate laboratory courses.
v pleteness of the quantum mechanics [1]. They pointed TheCHSHinequalityrequiresarathersimpleconfigu-
3
outthatonecouldrestorelocalityinphysicsbyassuming ration: two binary measurement settings on each of two
1
3 the existenceoflocalvariablesdeterminingthe resultsof entangled particles. However, in recent years, new types
2 the measurements. However, in 1964 John Bell showed ofBellinequalities,involvingalargernumberofmeasure-
. that the correlations of the results of the measurement ment settings or outcomes with respect to CHSH, have
1
on entangled particles are stronger with respect to what beeninvestigatedfromatheoreticalpointofview. These
0
1 is expected by any physical theory of local hidden vari- inequalitiescangetconclusionsunattainablewithCHSH:
1 ables. By measuring them, a simple inequality (i.e. the fromafundamentalpointofview,theyallowforadeeper
: Bellinequality[2]),basedonthelocalityassumption,can understandingofthenonlocalcorrelations,whereasfrom
v
be violated. a practical one, they represent useful tools in the pres-
i
X Testingexperimentallynonlocalcorrelationspredicted enceofnoiseanddetectioninefficiency[5],asweshowin
r by quantum mechanics has become a concrete idea af- the next paragraph.
a
ter the Clauser-Horne-Shimony-Holt (CHSH) formula- In this paper,we show that, with a commercialsource
tion [3] of the original Bell inequality. Since the CHSH- of photon pairs entangled in polarization (QuTools [6]),
Bell tests performed by Aspect in 1981 [4], several ex- a series of nonlocality tests alternative to CHSH can
periments have confirmed consistent violations of the be made. In particular, we report the violation of Bell
CHSH inequality. However, the experimental imperfec- inequalities requiring multiple binary measurement set-
tions, which all these tests are susceptible to, open loop- tings. To our knowledge these inequalities (with the ex-
holesthatcanbeexploitedbyalocaltheorytoreproduce ception of I3322 [7]) have not been measured before.
the experimental data. Two relevant loopholes are the In section II we justify the interest in this kind of
detection andthe locality loophole. The former relies on inequalities from a fundamental and practical point of
the fact that particles are not always detected in both view. In section III we describe the partial tomogra-
channels of the experiment. The latter is related to the phy of the density matrix of the states prepared by the
necessity of separating the two sites enough to prevent sourceandtheoptimizationofthesettingsforenhancing
theviolationoftheBellinequalities. InsectionIVwere-
port the violation of inequalities inequivalent to CHSH,
in particular the I [8] and two I inequalities [9].
3322 4422
∗ [email protected] Then, we show the violations of chained inequalities [10]
2
by non-maximally entangled states, unlike CHSH.
Anothersetofinequalitiesthathaverecentlyattracted
attention is represented by the chained inequalities [10],
which are generalizations of CHSH with multiple set-
tings. In recent years, several theoretical models have
attempted to provide a better understanding of quan-
tum nonlocality. One of these is the Elitzur, Popescu,
and Rohrlich (EPR2) approach [11], according to which
theobserveddatacouldbeexplainedassumingthatonly
FIG. 1. Nonlocality tests where Alice and Bob measure the afractionofthephotonpairsproducedinanexperiment
N binaryoperators{A1,...,AN}and{B1,...,BN}respectively possesses nonlocal properties, while the remaining part
on the photon pairs produced by the entanglement source in gives rise to purely local correlations. In this scenario
themiddle.
chained inequalities are a useful tool for studying the lo-
cal content of the quantum correlations. For instance,
they have been used to prove that maximally entangled
quantumstatesinarbitrarydimensionshaveazerolocal
from 3 to 6 settings per side. In section VA we inter-
component [12] and to decrease the upper bound on the
pret the violation of the chained inequalities according
local content of nonmaximally entangled states [13].
to a specific nonlocality approachintroduced by Elitzur,
Popescu, and Rohrlich (EPR2) [11]. This allows us to Fromamorepracticalpointofview,inequalitiesbased
put an upper limit on the local content of the prepared onmultiple settings are alsointeresting. Recently, it has
statesthatisstrongerthantheoneattainablebyCHSH. been shown that in the presence of high dimensional en-
Finally, we show that the observed violations allow one tanglement, that is when the quantum systems sharing
to certify that true random numbers have been created entanglement have dimensions larger than two, these in-
duringtheexperiment. Throughoutthisworkweassume equalities can tolerate a detection efficiency of 61.8% for
fair sampling of the detected events, which allows us to closingthedetectionloophole[5]. Thisvalueislowerwith
avoid detection loophole issues. respect to the limit imposed by CHSH in experiments of
entangled qubits.
II. BELL INEQUALITIES WITH MULTIPLE
MEASUREMENT SETTINGS
III. OPTIMIZATION OF THE MEASUREMENT
A CHSH-Bell test requires the measurement of each SETTINGS FOR A SPECIFIC STATE
photon of an entangled pair in two different bases and
the estimation of the correlations in the four possible
Entangled states prepared in the laboratory (or by a
combinations of bases. Its practical implementation is
commercial source) are not perfect in terms of purity
conceptually easy and needs minimum experimental ef-
and degree of entanglement. In this case the measure-
fort with respect to other nonlocality tests. Moreover,
ment settings needed to observe the largest possible vi-
the CHSH test is particularly robust against the noise
olation of a given inequality do not necessarily coincide
presentinrealexperiments. However,otherBellinequal-
with those that are optimal for a maximally entangled
ities, requiringa largernumber ofmeasurementsettings,
state. In order to find these best measurement settings,
schematicallyrepresentedinFigure1,canleadtoconclu-
a knowledgeofthe state is required. Usually,a complete
sions about the quantum correlationsthat are nontrivial
reconstructionofthestateisnotpossibleorevennotnec-
and sometimes inaccessible to CHSH. In general,finding
essary. TheQuToolssourceset-up[6],thatweuseinour
all the Bell inequalities for a given setup is computa-
experiment,projectsthephotonsontolinearpolarization
tionally hard, therefore the researchis limited to a small
states (c.f. description of the experimental setup in the
numberofsettingsperside. Eveninthiscase,thesetests
next section), so we can perform a partial tomography
canprovideinterestinginsightsaboutquantumnonlocal-
of the state. Once we know the state on the equatorial
ity. Inthecaseofthreepossible2-outcomemeasurements
plane of the Bloch sphere corresponding to linear polar-
per side, only one inequality, called I , is inequivalent
3322 izations, we can optimize the measurement settings in
to CHSH [8]. Note that this inequality is relevant in the
this plane.
sense that it can be violated by specific mixed 2-qubit
states that do not violate CHSH. The I inequality
3322
hasalsobeenusedtoshowthatthreequbitscansharebi-
partite non-locality between more than two subsystems,
A. Partial tomography of a quantum state
a result that cannot be obtained with CHSH [8]. In the
case of four 2-outcome measurements per side, only a
partial list of inequalities I has been given [9]. Some The density matrix ρ of a two-qubit state can always
4422
oftheseinequalitiesaremaximallyviolated,surprisingly, be written in the basis composed by the identity and
1
3
the Pauli matrices {σ ,σ ,σ } as Note that this table has the same form as the one intro-
x y z
duced in [8], but here coefficients correspondto expecta-
1
ρ= ⊗ + a σ ⊗ + b ⊗σ tions values and not to probabilities.
4(cid:16)1 1 X i i 1 X j1 j Itisclearfromeq. (6)that,foragivenquantumstate,
i=x,y,z j=x,y,z
the term I is a function of the measurement angles on
+ X ci,jσi⊗σj(cid:17), (1) Alice’s ({α1,...,αN}) and Bob’s site ({β1,...,βN}), that
i,j=x,y,z is
where ai = hσi ⊗1iρ, bj = h1⊗σjiρ and ci,j = hσi ⊗ I =I(α1,...,αN,β1,...,βN). (8)
σ i are 15 real coefficients which completely define the
j ρ
state. A complete tomography [14] allows to determine This function is not linear in terms of its 2N variables,
the value of all these coefficients. but numericaloptimizations can be used in orderto find
In our case we only measure linear polarizations, so a the optimal measurement settings for Alice and Bob,
completeknowledgeofthestateisnotnecessaryinorder which will provide the largest possible value of I for the
to predict all possible measurement statistics. Indeed, state under consideration.
only the 8 coefficients a , b , c with i,j =x,z are use- For the nonlocality tests reported in this paper,
i j i,j
ful. These coefficients can be measured with our setup, we perform a partial tomography of the polarization
realizing a partial tomography of the generated state. entangled states that we prepare, as explained in the
previous section. This allows us to reconstruct the state
in the plane of the Bloch sphere corresponding to linear
B. Optimization of the settings polarizations. Then, we numerically find the optimal
measurement settings for this state, restricting them
to lie on the xz plane. These settings are used for the
The measurement of a qubit along a particular angle
experimental violation of the Bell inequalities taken in
θ in the xz plane of the Blochsphere can be represented
consideration.
by the measurement operator
O(θ)=cosθσ +sinθσ . (2)
z x
IV. NONOLOCALITY TESTS WITH MULTIPLE
If the two-qubit state (1) is shared between Alice and
SETTINGS
Bob, the following marginal values are expected if Alice
measures along angle α and Bob along β:
A. The entanglement source
E(α)=hA(α)⊗ i =cosαa +sinαa , (3)
ρ z x
1
E(β) =h ⊗B(β)i =cosβb +sinβb . (4) We use the commercial entanglement source sold by
ρ z x
1
the QuToolscompany[6],to whichwe broughtsomemi-
Moreover,the joint correlations are found to be nor modifications. This source generates photon pairs
entangled in polarization from Spontaneous Parametric
E(α,β)=hA(α)⊗B(β)i
ρ Down Conversion (SPDC) in a bulk BBO crystal. The
=cosαcosβcz,z+cosαsinβcz,x nonlinearcrystaliscutforaTypeIIphase-matchingand
+sinαcosβc +sinαsinβc . (5) is pumped by a continuous wave diode laser at 405nm.
x,z x,x
Photon pairs at 810nm are generated, filtered and col-
A Bell inequality I with N settings per side [15] can lected into single mode fibers. Linear polarizers at the
be generally defined by the following formula AliceandBobsitecanmeasurepolarizationsrespectively
at the angles α and β with respect to the vertical direc-
N N N
tion, as shown by the sketch in figure 2. They allow
I = n E(α )+ m E(β )+ l E(α ,β )≤I ,
X i i X j j X ij i j L to perform a partial tomography of the entangled state.
i=1 j=1 i,j=1 Notice that a full tomography could be done by just in-
(6)
serting quarter wave plates. Photons are then detected
where α and β are the angles of the measurements in
i i using a Silicon Avalanche PhotoDiodes (Si-APDs) with
Alice and Bob site respectively, n , m and l are coeffi-
i j ij efficiencies at the wavelength of the photons of 48% and
cients defining the inequality and I is the local bound.
L 55% respectively. The coincidences are measured by a
The inequality I can be represented schematically as a
Time-to-DigitalConverter(TDC)andacoincidencerate
table
of4.2kHzismeasuredfor15mWofinputpumppowerin
m m ... m theabsenceofthelinearpolarizers. Otherdetailsonthis
1 2 N
  source can be found in [18]. For a pair of polarization
n l l ... l
1 11 12 1N directionsmeasuredbythetwopolarizers,thenumberof
I = n l l ... l ≤I . (7)
 2 21 22 2N  L coincidences in a time interval of 20s is measured. The
 
 ... ... ... ... ...  error associated to the coincidence rate is estimated ac-
 
nN lN1 lN2 ... lNN  cordingtopoissonianstatistics. The inequalitiesthatwe
4
The error associated to a correlation term is obtained
simply by statistical propagation of the error in the co-
incidencesonwhichitdepends. Themarginalterms,the
last four in the table, have been measured twice using
twodifferentbasesonBob’ssite. Thedifferencebetween
these two values reflects in errors which are larger with
respect to the other expectation terms. Note that errors
on the measured expectation terms in table I need to be
consideredinordertodefinefromthepartialtomography
FIG. 2. Simple sketch of the polarization entanglement adensitymatrixwhichisdefinitepositive. Insomecases,
source. Linear polarizers at the Alice and Bob site are used in order to avoid the negativity of the density matrix
for the settings necessary for the measurement of the Bell attainable from a tomography, some techniques, such as
inequalities. themaximumlikelihoodestimation,needtobeemployed
[14]. It is evident that the singlet state prepared with
the commercial source has some imperfections. There
is a mixture component in the state since the expecta-
want to test require to measure a large number of corre-
tion values are slightly different to the theoretical ones.
lations. One correlation term is given by measuring one
The state is alsounbalancedinthe two orthogonalbases
photoninonebasisandtheotherphotoninanotherone.
{|Hi,|Vi} and {|+i,|−i}, which confirms that it is not
Measuring one photon in one basis means projecting it
maximally entangled.
into two orthogonalpolarizationdirections. So the mea-
The problem of optimization of the measurement set-
surement of one correlation term require four different
tings for the nonlocality tests is necessarily limited to
settings for the two polarizers.
the plane onwhich we have limited the tomography. We
measure 9 inequalities requiring in total 332 settings of
the linear polarizers in the Alice and Bob site. These
B. The characterization of the state
optimal settings have in some cases differences of some
degreeswithrespecttothestandardsettingsusedforob-
We perform a partial tomography of the state gener- tainingamaximumviolationwithaperfectsingletstate.
atedbythesourceontheplaneoftheBlochspherecorre-
spondingtothelinearpolarizations. Wedeterminethe 8
correlationvaluesindicatedinthesectionIIIA,demand-
ingtomeasureoneachsidetheoperatorsinpolarization:
C. CHSH inequality
σ ≡|HihH|−|VihV|,
z
σ ≡|+ih+|−|−ih−|,
x
First of all, in order to check the validity of our opti-
≡|HihH|+|VihV|≡|+ih+|+|−ih−|,
mizationmethod,wemeasuretheCHSHinequalityusing
1
the standardsettingsforthe singletstate,then the same
where |Hi and |Vi correspond to the states of horizon-
inequality with the settings optimized for the prepared
tal andverticalpolarizationsrespectively and|+(−)i=
state. According to the notation given in IIIB, we can
1 (|Hi ± |Vi). In the Table I a list of the measured
√2 represent the CHSH inequality by the following table
expectationterms for the partialtomographyofthe pre-
pared state is given.
0 0
 
Expectation term Exp Th ICHSH = 0 1 1 ≤2. (9)
 
hσz⊗σzi −0.9649±0.0012 -1 0 −1 1
hσx⊗σxi −0.9344±0.0017 -1
hσz⊗σxi 0.1053±0.0045 0 For the CHSH test with optimal settings we obtain a
hσx⊗σzi −0.0201±0.0048 0 value of 2.731±0.015, clearly enhanced with respect to
hσz⊗ i 0.065±0.034 0 2.691± 0.015, obtained by adopting the standard set-
1
hσx⊗ i 0.036±0.014 0 tings. These two values correspond to violations of the
1
h ⊗σzi −0.078±0.020 0 local bound of respectively 49 and 46 standard devia-
1
h ⊗σxi −0.015±0.019 0 tions. Thisconfirmsthevalidityofthesettings’optimiza-
1 tion. Actually,weexpectedfromthepartialtomography
TABLEI.Listoftheexpectationtermsforthepartialtomog- avalueforCHSHof2.683withoptimalsettingsand2.662
raphy of the prepared entangled state. The measured values with the standard ones. This little discrepancy could be
(Exp)can becompared tothetheoretical ones (Th) that are explained by small variations of the prepared state be-
expected from a perfect singlet state. tween the tomography and the final measurements.
5
D. Inequalities inequivalent to CHSH IL Iexp Itom Iexp−IL pnoise(%)
(σ units)
We then measure Bell inequalities inequivalent to ICHSH 2 2.731 ± 0.015 2.683 49 27
CHSH, in particular I3322 [8] and two different types of I3322 4 4.592 ± 0.024 4.769 25 13
I , called AS and AS [9, 19]. These inequalities are
4422 1 2 AS1 6 7.747 ± 0.026 7.750 67 23
facets of their corresponding Bell polytope [15]. They
AS2 10 12.85 ± 0.030 12.819 95 22
are thus optimal to detect nonlocality for some correla-
tionsinscenariosinvolvingthreeandfoursettings. Note TABLE II. Measurement of the CHSH inequality and of the
thatI3322 askstomeasurenotonlyjointcorrelations,but inequalitiesinequivalenttoCHSH.IListhelocalbound,Iexp
alsofourmarginalprobabilities. Onthecontrary,thetwo is the value of the Bell parameter obtained experimentally
I inequalities that we want to measure are the only with the optimized settings, Itom is the expected value from
4422
two of this kind to require correlation terms uniquely. the partial tomography, Iexp −IL is the difference between
This makes their measurement simpler from a concep- the obtained value and the local bound in terms of number
tual and experimental point of view. In the following, of standard deviations σ and pnoise(%) is the critical level of
whitenoisethatcanbeaddedtothesystemwhilestillkeeping
the coefficients of these inequalities are given
a violation.
1 1 0
 
1 −1 −1 −1 E. Chained inequalities
I = ≤4, (10)
3322  
1 −1 −1 1 
0 −1 1 0  Finally,wemeasurechainedinequalitieswithanumber
of settings per side from 2 to 6. The chained inequality
with N settings for Alice ({α ,α ,...,α }) and N for
1 2 N
Bob ({β ,β ,...,β }) can be written as a correlationin-
1 2 N
0 0 0 0 equality as
 
0 1 1 1 1
AS1 =0 1 1 1 −1≤6, (11) Ic(hNa)in =E(α1,β1)+E(β1,α2)+E(α2,β2)+...+
0 1 1 −2 0  +E(αN,βN)−E(βN,α1)≤2(N −1). (13)
 
0 1 −1 0 0 
Contrary to the previous inequalities, only the chained
inequality with N = 2, which is equivalent to CHSH, is
a facet of the Bell polytope.
0 0 0 0 Since the chained inequalities require 2N correlation
 
terms,thenumberofsettingsrequiredfortheirmeasure-
0 2 1 1 2
AS2 =0 1 1 2 −2≤10. (12) mseettnitngscsapleesrlsinideea.rlIynwthitehTNa.bWleeIItIhuaslilsimt oitfothuersmelveaessutroed6
 
0 1 2 −2 −1 inequalitiesisgiven. ForN =2wehavetheresultforthe
 
0 2 −2 −1 −1 CHSH inequality with optimal settings. All the inequal-
ities are violated in a way consistent with the expected
Note that this notation is different with respect to that results and with a large number of standard deviations
used in [9]. We call Iexp the experimental value of the of difference with respect to the local bound. However,
Bell parameter. For each of the three different inequali- itisinterestingtonotethatthelargerthenumberofset-
ties, we observe a violation of the local bound I (Table tings, the weaker the violation. Indeed, the fraction of
L
II). IntheTableIItheresultforCHSHisthatwithopti- noise that we can add to the system and still keep the
malsettings. TheviolationsforthetwoI inequalities violationsdecreasesforanincreasingnumberofsettings.
4422
are stronger than for I . We evaluate this aspect by
3322
considering the resistance to noise of the three different N IL Iexp Itom Iexp−IL pnoise (%)
inequalities. If some white noise were addedwith proba- (σ units)
bility p to our state ρ, we would obtain a state with
noise
2 2 2.731 ± 0.015 2.683 49 27
adensitymatrixρ =p +(1−p )ρ. Now,it
noise noise14 noise 3 4 4.907 ± 0.019 4.925 48 18
is easy to show that the violation wouldnot be observed
anymore if p > 1− IL . For I adding 13% of 4 6 7.018 ± 0.023 6.999 44 15
noise Iexp 3322
white noise to the state compromises the violation. For 5 8 8.969 ± 0.026 8.996 37 11
the other inequalities the tolerance to the noise is higher 6 10 10.91 ± 0.028 10.954 33 8
andthe CHSHinequalityconfirmstobethe mostrobust
to noise. TABLE III.Measurement of thechained inequalities with N
settings per side.
Inallthecasestheobtainedviolationsagreequitewell
with what we expected with the tomography.
6
V. APPLICATION OF THE CHAINED
INEQUALITIES
As recalled in the section II, the violation of the
chained inequalities can be linked with numerous prob-
lems. In the following we briefly mention what the vio-
lations we observed allow us to conclude with respect to
the EPR2 model of nonlocality and to true randomness.
A. EPR2 nonlocality
Chained inequalities can be used to put an upper
bound on the localcontent of the preparedstate accord-
ing to the EPR2 approach [11]. Indeed, as already ex- FIG.3. LargestlocalpartassociatedtoaWernerstatefordif-
plained in the section II, one can imagine that only part ferent values of V =1,0.98,0.96,0.94,0.92 (from the bottom
of the photon pairs produced by the source has nonlo- to thetop).
cal properties, while the other one behaves in a purely
local way. In such a situation, the measured value of
I(N) decomposes as a sum of the local bound I with
chain L measure is 0.491±0.012 with a chained inequality of 4
probability p and of some possibly larger value I with
L settings per side. This result means that at least half of
probability 1−p . The latter satisfies the no-signaling
L the photon pairs produced by the source are nonlocal, a
(NS) principle, i.e. the impossibility of faster-than-light
result that cannot be shown with the CHSH inequality.
communication. Therefore
To our knowledge, this is the first experiment that fixes
Iexp =p I(N)+(1−p )I(N). (14) an upper bound on the local part of the quantum cor-
L L L NS
relations according to the EPR2 approach [16]. Similar
Since the local and the no-signaling values of the I workinprogressisexpectedtoshowalowervalueofthis
quantity areatmostI(N) =2(N−2)andI(N) =2N re- quantity [17].
L NS
spectively,thelocalpartoftheproducedstateisbounded
by the maximum value pmax :
L
Iexp
p ≤pmax =N − . (15)
L L 2
To illustrate this bound, let us see how it applies
to noisy singlet states, i.e. Werner states ρ =
W
V|ψ ihψ |+(1−V) . For these states the maximum
− − 14
value ofI(N) is 2NV cos( π ). Thus, the maximum local
2N
probability pmax associated to a Werner state ρ de-
L W
pends on the visibility V of the state and it is given by
pmax = N(1−V cos( π )). The behaviour of this func-
L 2N
tion is shown in the figure 3 for different values of the
visibility V. For the pure singlet state (V = 1), pmax
L
tends to 0 for an infinite number of settings, confirming
the fact that the singlet state is fully nonlocal [12]. On
the contrary, when V is smaller than 1, pmax decreases
L FIG. 4. Largest local part as a function of the number of
until a certainlimit value. Therefore,the number of set- settingsperside. Theredlinejoinstheexpectedvaluesforour
tings which are useful for lowering the upper bound on state. Theminimummeasuredvalueofpmax is0.491±0.012
L
the local content of the state is limited and changes ac- with a chained inequality of 4 settings per side.
cording to the visibility V.
We observe a similar effect for the state produced by
our source. In figure 4 we represent the violations of the
measured chained inequalities in terms of local proba-
bilities, as calculated from (15). The red line joins the B. Randomness certified by the no-signaling
expected values for pmax obtained by the knowledge of principle
L
thestate. OurstateismorecomplexthanaWernerstate
but it can be approximated by a Werner state with vis- When measuring a singlet state, locally random out-
ibility V = 0.94. The minimum value of pmax that we comes can be observed. It has been recently shown that
L
7
violationofaBellinequalitycancertifythatthisrandom- estboundonthemarginalprobabilitythatwecancertify
ness truly emerges during the experiment, in the sense hereis P (A|X)=0.684±0.014,achievedforthe CHSH
∗
thatnoalgorithmcanpossiblypredictthemeasuredout- inequality. However, if we consider the bound imposed
comes [20]. Indeed, if such algorithm existed prior to bytheno-signalingprincipleonly,thenthestrongestone
the experiment, it could be considered as a local hidden is P (A|X) = 0.7455±0.0057, achieved by the chained
∗
variable, and no violation of a Bell inequality can be ob- inequality with N =4 settings per party.
served with only local hidden variables. Note that in or- Note that in order to extract a truly random bit
dertodiscardanysuchalgorithm,thedetectionloophole string out of measured outcomes, classical key distilla-
shouldbe closedduring the experiment. Indeed, suchan tion techniques should be used [22]. The ratio between
algorithm could in principle inform detectors when they the number of measured bits and the number of truly
should click according to some detection loophole model random, uniformly distributed, bits that is produced by
[21]. Inordertoavoidthisissue,weassumefairsampling this procedure is given by the min-entropy of Alice’s
of the detected events: the state of the particles coming outcome A conditioned on her measurement choice X:
onto a detector does not affect the fact that a detector H (A|X)=−log max P (A|X). In our case we find
min 2 a ∗
fires or not, so that the detected pairs of particles fairly H =0.55±0.03fortheCHSHviolation,meaningthat
min
represent the ones produced by the source. approximately one random bit every two measurements
In order to quantify the amount of true randomness have been created.
thatcanbefoundinsomeexperimentalresults,onemust
consider all the marginal probabilities P(a|x) that Alice
findstheoutcomeawhenshemeasuresx,whicharecom-
patible with the observed Bell inequality violation Iexp. VI. CONCLUSIONS
We searched for the largest marginal probability P(a|x)
numericallyamongallpossiblequantumcorrelations,i.e.
Wehaveshownthatnontrivialnonlocalitytestsalter-
correlations that can be achieved by measuring a quan-
nativetoCHSHcanbeperformedevenwitharathersim-
tum state,as wellas amongallno-signalingcorrelations,
ple commercial source. In particular, we have measured
denoting this largest quantity by P (A|X).
∗ Bell inequalities with multiple measurement settings. In
The computedupper bounds P (A|X)forthe chained
∗ particular, we have reported violations of I , of two
inequalities with up to 6 settings per party are shown in 3322
I and of chained inequalities with up to 6 settings
figure5togetherwithourexperimentalresults. Thelow- 4422
per side. Violations of these inequalities have not been
shown before. Moreover, by using the chained inequali-
ties, we haveput anupper bound onthe localcontentof
1
thepreparedstateat0.491±0.012,meaningthatatleast
Quantum bound
halfofthephotonpairsproducedbythesourcehavenon-
No−signaling bound for all N
0.9 localcorrelations. Wehavealsoquantifiedtheamountof
true randomness created in the experiment. Therefore,
evenwithanextremelysimplesetup,itispossibletoim-
0.8
plement not trivial nonlocality tests and get interesting
*P(a|x) conclusions on the nonlocal properties of the source and
0.7 on the randomness produced in the experiment. This
work emphasizes the richness of nonlocality and the im-
portance of the no-signaling and could be of interest for
0.6 undergraduate laboratory courses.
N=2 N=3 N=4 N=5 N=6
0.5
2N−2 2N−3/2 2N−1 2N−1/2 2N
IN
VII. ACKNOWLEDGMENTS
FIG. 5. Upper-bound on the marginal probability distribu-
tionasafunctionoftheinequalitiesviolation. Theboundim- This work is supported by EU project Q-ESSENCE
plied by the no-signaling principle is identical for all chained andbythe Swiss NCCR-QuantumPhotonics. We would
inequalities. liketothankAntonioAcin,ValerioScaraniandHenning
Weier for valuable discussions, suggestions and remarks.
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