Experimental violation of three families of Bell's inequalities

Abstract

Bell’s inequalities are important to our understanding of quantum foundations and critical to several quantum technologies. A recent work [E. Wolfe and S. F. Yelin Phys. Rev. A 86 012123 (2012)] derived three parametrized families of two-particle, two-setting Bell inequalities. These inequalities are important as they theoretically explore a larger volume of allowed quantum correlations over local hidden-variable models than previous results [A. Cabello Phys. Rev. A 72 012113 (2005)] by exploiting marginal, or single particle measurements. In this work we subject those predictions to experimental test using nonmaximally entangled photon pairs to optimize the expected violation. We find excellent agreement with the upper bounds predicted by quantum mechanics with violations of the limits imposed by local hidden-variable models as large as almost 30σ for the optimal parameters and a significant violation over a wide range of parameters.

Full text

AA10885 PRA March 1, 2013 6:13
PHYSICAL REVIEW A 00, 002100 (2013)1
Experimental violation of three families of Bell’s inequalities2
L. Vermeyden,1M. Bonsma,1C. Noel,2J. M. Donohue,1E. Wolfe,3and K. J. Resch1
3
1Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Canada N2L 3G14
2Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139-4307, USA5
3Physics Department, University of Connecticut, Storrs, Connecticut, 06269-2218, USA6
(Received 16 January 2013; published xxxxx)7
Bell’s inequalities are important to our understanding of quantum foundations and critical to several quantum
technologies. A recent work [E. Wolfe and S. F. Yelin, Phys.Rev.A86, 012123 (2012)] derived three parametrized
families of two-particle, two-setting Bell inequalities. These inequalities are important as they theoretically
explore a larger volume of allowed quantum correlations over local hidden-variable models than previous results
[A. Cabello, Phys.Rev.A72, 012113 (2005)] by exploiting marginal, or single particle measurements. In this
work we subject those predictions to experimental tests using nonmaximally entangled photon pairs to optimize
the expected violation. We find excellent agreement with the upper bounds predicted by quantum mechanics
with violations of the limits imposed by local hidden-variable models as large as almost 30σfor the optimal
parameters and a significant violation over a wide range of parameters.
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DOI: 10.1103/PhysRevA.00.002100 PACS number(s): 03.65.Ud, 42.50.Ex17
I. INTRODUCTION18
Quantum mechanics is one of the most well-tested and
1
2
19
precise predictive theories in physics. Many of the features20
predicted by the quantum mechanical formalism are strange,21
yet the rapidly growing field of quantum information science22
is finding application for them in powerful new technologies23
[1]. One of the strangest and most important features of
324
quantum mechanics was discovered by John Bell in that25
quantum predictions cannot be explained by local realistic26
descriptions of nature [2]. This discovery was expressed in27
the form of mathematical inequalities that were refined by28
subsequent work [3]. Bell’s inequalities can place bounds on29
correlations imposed by local and nonlocal realistic theories30
[4,5] and can be used as witnesses to detect entanglement [6].31
They are applied in emerging technologies such as quantum32
key distribution [7,8], quantum communication complexity33
results [9], certification of random number generation [10], and34
lowering error rate in classical communication schemes [11].35
Bell’s inequalities have been the subject of a wide variety36
of experiments, from early experiments using polarization-37
entangled photon sources from atomic cascades [12,13]to38
later experiments using bright sources from parametric down-39
conversion [14–18] and other physical systems [19,20]. The40
inequalities have also been extended theoretically to include41
larger numbers of particles [5,21–28], more measurement42
settings [29], and higher dimensional quantum systems [30].43
Three families of Bell’s inequalities were recently inves-44
tigated in Ref. [32] which have important consequences for45
comparing quantum correlations to those governed by local46
hidden-variable models. The most well-known examples of47
Bell inequalities and tests of nonlocality [2,3,5,21,22] achieve48
maximal violation using maximally entangled states, such49
as Bell states or Greenberger-Horne-Zeilinger (GHZ) states50
[21], which have unpolarized subsystems and hence zero51
marginal expectation values. In contrast, two of the families of
452
inequalities include contributions from marginal expectation53
values which in general require nonmaximally entangled states54
to achieve maximal violations; they are furthermore able to55
explore a larger volume of correlation, as quantified by the56
analysis in Ref. [31], providing new insights into the limits 57
of quantum correlations. In the present work, we subject these 58
families of Bell’s inequalities to experimental tests over a wide 59
range of parameters using entangled photons. 60
Consider a bipartite system with measurement settings Ai61
and Bjfor particles 1 and 2, respectively, with outcomes 62
ai=±1 and bj=±1, respectively. Expectation values of 63
joint, or two-particle, measurements can be expressed as 64
AiBj=P(ai=bj)−P(ai=−bj), i.e., the difference be- 65
tween the probability that the two outcomes are the same and 66
the probability that they are different. Expectation values of 67
marginal measurements can be expressed as Ai=P(ai=68
1) −P(ai=−1), which is the difference between the proba- 69
bility of measuring +1 and the probability of measuring −1. 70
Reference [32] introduced a two-particle Bell operator Z,71
Z≡c1A0+c2A1+c3B0+c4B1+c5A0B0
+c6A1B0+c7A0B1+c8A1B1,(1)
where the ciare real constants. Note that Zcontains both 72
two-particle and single-particle operators. Three families of 73
Bell’s inequalities, QB1, QB2, and QB3, were derived from 74
the operator Zusing parametrized sets of the coefficients, 75
ci, as shown in Table I. In each family, the ciare either 76
constant or depend on a single adjustable parameter x. Local 77
hidden-variable models (LHVM) result in upper limits for 78
the Bell parameter, Z, which were shown to be functions 79
of x. These Bell’s inequalities are violated by quantum 80
mechanics (QM) over specific ranges of xin each family. The 81
bounds derived in Ref. [32] for LHVM and QM are shown 82
in Table I.83
The Clauser-Horne-Shimony-Holt (CHSH) inequality [3]84
is perhaps the most well-known Bell inequality. Each family 5
85
of Bell’s inequalities considered here includes the CHSH 86
inequality as a special case; specifically QB1, QB2, and QB3 87
reduce to the CHSH inequality for parameter values x=1, 88
x=0, and x=0, respectively. Maximal violation of the 89
CHSH inequality can be achieved with a two-qubit maximally 90
entangled state, such as a Bell state. However, maximally 91
entangled states have completely unpolarized subsystems, 92
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TABLE I. Three families of Bell’s inequalities. The coefficients defining the Bell operators are shown. Also shown
are the maximum bounds for the Bell parameter, Z, using the local hidden variable model (LHVM) and the quantum
mechanical model (QM) [32].
Family c1c2c3c4c5c6c7c8LHVMmax QMmax
x+3∀x−1
3
QB10000111x|x+1|+2(x−1)3
x∀x<−1
3
|x|+2∀|x|2
QB2 x000111−1|x|+2√2x2+8∀|x|<2
3|x|−2∀x2
3|x|−2∀x2
QB3 xx−x0111−1|x|+2∀1|x|<2
|x|+2∀x<2√(2−x2)(4−3x2)−x2
1−x2∀x<1
and thus any single-particle, or marginal, expectation value93
Ai,Bjfor a maximally entangled state is 0.94
The inequalities QB2 and QB3 contain single-particle95
expectation values. For these marginals to have an impact96
on the Bell parameter Z, the particles must have partially97
polarized subsystems which arise in nonmaximally entangled98
states. It is also unclear aprioriif the unequal magnitude co-99
efficients of the joint expectation values in QB1 would require100
nonmaximally entangled states to optimize the Bell inequality101
violations. To investigate this, we consider entangled states of102
the form103
|ψ(α)=cos α|HH+sin α|VV,(2)
where the angle αis an adjustable parameter that determines
104
the amount of entanglement. We represent each measurement105
setting Aiand Bjwith an operator σ·ˆ
n, where σ=(σx,σy,σz)106
is a vector of Pauli matrices and the unit vector ˆ
n=107
(sin θcos φ,sin θsin φ,cos θ), where choice of the angles108
determines the setting. Using the MATHEMATICA function109
NMAXIMIZE, we find the state and measurement settings which110
maximize the Bell parameter Zfor each Bell inequality for111
each value of the parameter x.112
In Fig. 1we plot the amount of entanglement, as char-113
acterized by the tangle [33], required to maximize Zas a114
function of the parameter xfor QB1 (solid red line), QB2115
(dashed blue line), and QB3 (dotted green line). For the Bell116
inequality QB1, a maximally entangled state is needed to117
reach the maximum violation when x<−1/3; for x1/3118
there exist states with any amount of entanglement which119
can achieve the maximum expectation value of Z, which we120
depict as the red region, but no violation of Bell’s inequality121
is possible in this regime since the LHVM and QM maxima122
coincide. For QB2 and QB3, Bell inequality violations are123
possible when |x|<2 and |x|<1, respectively. In these124
regions nonmaximally entangled states are required to achieve125
the maximum Bell inequality violation. The only exceptions126
are the cases where x=0, reducing these inequalities to127
the CHSH inequality which requires a maximally entangled128
state for maximum violation. To effectively test these new129
inequalities over the largest possible parameter space requires130
a source of entangled photons with variable entanglement.131
II. EXPERIMENT 132
Our experimental setup is shown in Fig. 2. In the state- 133
preparation part of the experiment, polarization-entangled 134
photon pairs are generated using parametric down-conversion 135
in a polarization-based Sagnac interferometer [34–36]. A 136
5 mW cw diode pump laser with center wavelength 404.8nm 137
is focused on a 20 mm PPKTP crystal, phase matched to 138
produce photon pairs at 809.8 nm and 809.3 nm from type-II 139
nearly degenerate down-conversion. The polarizing beam 140
splitter (PBS) in the source splits the pump beam into two 141
components, one circulating in a clockwise direction and the 142
other in a counterclockwise direction. HWP2 inside the inter- 143
ferometer ensures that the pump beam enters the crystal with 144
horizontal polarization in both directions. The photon pairs 145
FIG. 1. (Color online) Optimal tangle [33] required to achieve
the maximum Bell parameter. These plots show the numerically
calculated tangle τof a state of the form of Eq. (2) which achieves
the maximum Bell parameter as a function of xfor QB1 (solid red),
QB2 (dashed blue), and QB3 (dotted green). (A state with τ=1
is maximally entangled, while τ=0 indicates no entanglement;
nonmaximally entangled states have a value 1 >τ >0.) QB2 and
QB3 in general require nonmaximally entangled states to achieve the
largest Bell parameter. For QB1 the optimal tangle is τ=1 until
x<−1/3. For x−1/3 the maximum value of Zcan be reached
with any amount of entanglement, which we represent by a shaded
region, but there can be no Bell inequality violation since QM and
LHVM models predict the same bound. The tangle can be expressed
in terms of the angle αin Eq. (2) by the relation τ=sin2(2α)[33].
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FIG. 2. (Color online) Experimental setup. Pairs of polarization-
entangled photons were generated via spontaneous-parametric down-
conversion and measured using two polarization analyzers. The pump
beam was rotated using a half-wave plate (HWP) and quarter-wave
plate (QWP), then focused on a periodically poled KTP (PPKTP)
crystal. The down-converted photons traveled though long-pass (LP)
and band-pass interference (IF) filters to remove any remaining
pump signal before coupling into single-mode fibers. Projective
measurements were taken, and photons were detected with silicon
avalanche photodiodes (APDs) and analyzed with coincidence logic.
are thus created in a superposition of propagation direction146
(clockwise or counterclockwise) in the interferometer and are147
converted into polarization entanglement upon recombination148
at the PBS. After emerging from the PBS, the light is coupled149
into single-mode fibers containing polarization controllers to150
direct the light to the polarization analyzers. The polarization151
of the pump laser is controlled using HWP1 and a quarter-wave152
plate (QWP) oriented at 0◦and tilted. In conjunction with the153
polarization controllers, these wave plates enable production154
of the nonmaximally entangled states of the form shown155
in Eq. (2).156
The photon pairs emerge from fibers in the measurement157
part of the experiment. We used the angles of first two wave158
plates, a HWP followed by a QWP, to determine the basis159
for the measurement. The final HWP was set to either 0◦or160
45◦to measure the +1 and −1 outcomes, respectively, for a161
given measurement setting. After the analyzer wave plates,162
the photons in the transmitted ports of the PBSs were coupled163
into single-mode fibers and directed to single-photon counting164
detectors (Perkin-Elmer SPCM-AQ4C), labeled Alice and165
Bob, from which coincidence events were recorded with a166
3 ns window. Typical rates were about 25 kHz for singles167
counts and 1.4 kHz for coincidence counts.168
We begin by characterizing the states produced by169
our source. We use |Hand |Vto describe the hor-170
izontal and vertical polarization states; we use the no-171
tation |D= 1
√2(|H+|V), |A= 1
√2(|H−|V), |R=172
1
√2(|H−i|V), and |L= 1
√2(|H+i|V). We performed173
quantum state tomography on the coincidence counts from174
an overcomplete set of polarization measurements |H,|V,175
|D,|A,|R, and |Lfor each photon and a measurement176
time of 3 s per setting. The experimental density matrix was177
FIG. 3. (Color online) Reconstructed density matrices. Quan-
tum state tomography was performed on two sample states, (a)
where the target was the maximally entangled state |ψ(45◦)=
1
√2(|HH+|VV)and (b) where the target was the nonmaximally
entangled state |ψ(22◦)=cos 22◦|HH+sin 22◦|VV. The den-
sity matrices were reconstructed from our experimental data using the
maximum likelihood method in Ref. [37]. Fidelities with the target
states were (a) 0.982 and (b) 0.978.
reconstructed using the maximum likelihood iterative method 178
from Ref. [37]. Sample reconstructed density matrices are 179
shown in Figs. 3(a) and 3(b) for the maximally entangled 180
target states |ψ(45◦)and the nonmaximally entangled state 181
|ψ(22◦). The fidelity [38,39] of these states with their target 182
states are 0.982 and 0.978 for (a) and (b), respectively. These 6
183
fidelities were representative of those states measured for other 184
values of α.185
To experimentally test the Bell inequalities, we chose the 186
state and measurement settings at the analyzer to obtain the 187
maximum theoretical value of the Bell parameter as predicted 188
by our numerical optimization. To extract each joint expec- 189
tation value, we configured the measurement settings using 190
the first HWP and QWP in each analyzer, then measured four 191
coincidence counts corresponding to the four combinations 192
of ±1 outcomes on each side using the final HWPs. The 193
inequalities QB2 and QB3 require marginal expectation values 194
and these were extracted from coincidence measurements 195
by averaging results from those cases where the relevant 196
operator appeared in a joint expectation value. For example, 197
our measurement of the marginal A0was extracted from the 198
coincidence counts taken when measuring A0B0and A0B1.199
We collected Bell measurements with a measurement time of 200
1sfor each wave-plate setting. 201
Our experimental results can be found in Fig. 4.The 202
measured Bell parameter is plotted as a function of the 203
parameter xfor (a) QB1, (b) QB2, and (c) QB3. The maximum 204
theoretical values (see Table I) possible according to quantum 205
mechanics are shown as a solid line, while the maximum value 206
from a local hidden-variable model are shown as a dotted line. 207
Any experimental data points with a value greater than the 208
LHVM dotted line violate the corresponding Bell inequality. 209
It is clear from these plots that our data indeed violate these 210
families of inequalities over a large range of parameters. 211
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(a) (b) (c)
(d) (e) (f)
FIG. 4. (Color online) Experimentally measured Bell parameters for three families of Bell’s inequalities. The measured values for the Bell
inequalities (a) QB1, (b) QB2, and (c) QB3 are shown as a function of the parameter x. The experimentally measured values are shown as
circles, and the solid line shows the theoretical quantum limit. The theoretical bounds imposed by local hidden-variable models (LHVM) are
shown as a dotted line. Error bars were not included for parts (a)–(c) because they were smaller than the size of the data points. In panels (d)
and (e), we plot a function N, which, when positive, is the number of standard deviations of violation of the Bell inequality versus x.The
shaded areas in panels (a)–(e) indicate regions where a Bell inequality violation is possible, and the vertical lines inside this region show the
range over which our experiment achieved a 3σor greater violation. For QB1, we found a 3σor greater violation between x=−3 (largest
negative value tested) and x=−0.6. For QB2, we found a 3σor greater Bell violation between x=−0.9andx=0.9, which represents 49%
of the total possible range according to quantum mechanics. For QB3, the range of 3σviolation or greater was between x=−0.6andx=0.7,
representing 74% of the range possible according to quantum theory.
To explore the range of parameters where we observe a212
violation, we plotted the value of the function N=(ZExpt −213
ZLHVM)/Zversus x, where ZExpt is the experimentally214
measured Bell parameter, Zis the standard deviation in that215
result, and ZLHVM is the maximum value allowed by LHVM.216
If the value of the function Nis positive, then it quantifies the217
size of the violation in terms of the number of standard devia-218
tions over the maximum local hidden-variable prediction. If N219
is negative, then no violation was observed. The results of these220
measurements are shown in Figs. 4(d)–4(f). The maximum221
violation, in terms of the number of σ, occurred for QB2 and222
QB3 at x=0, and for QB1 at x=−1. In these three particular223
cases each inequality reduces to the familiar CHSH inequality.224
Theoretically it is possible to observe a violation over a specific225
range of parameters; these ranges are depicted as the shaded226
gray regions in each plot. However, comparing these regions227
to the theoretical lines in Figs. 4(a)–4(c), we see that near228
the borders the maximum QM and LHVM values converge229
making Bell violation experimentally very challenging. Using230
Poissonian errors for the experimentally measured counts, we231
calculated Nas a function of xin Figs. 4(d)–4(f). We estimated232
the range of xover which significant, >3σ, violation of233
the inequalities was achieved. We found x0.6,|x|0.9,234
and −0.6x0.7 for QB1, QB2, and QB3, respectively.235
For QB2 and QB3 this corresponds to a range which is 49%236
and 74%, respectively, of the theoretically possible range. (We 237
do not include such a measure for QB1 since the range of x238
over which theory predicts a violation is semi-infinite.) 239
III. CONCLUSIONS 240
We have experimentally tested the three families of Bell’s 241
inequalities derived in Ref. [32]. We identified the specific 242
relationship between maximal violation of each Bell inequality 243
and the entanglement of the associated quantum state. We have 244
measured very good agreement with theory and shown strong 245
violations of the inequalities, up to 29σ, over a wide range of 246
the parameters. We have demonstrated significant violation of 247
the inequalities QB2 and QB3 over 49% and 74%, respectively, 248
of the theoretically possible range of parameters. Our results 249
serve as a benchmark for these Bell inequalities, as extending 250
the range violations significantly will require substantive im- 251
provements in source fidelity and photon pair production rates. 252
ACKNOWLEDGMENTS 253
We thank D. Hamel, K. Fisher, and J. Lavoie for valuable 254
discussions and assistance in the laboratory. We are grateful for 255
financial support from NSERC, QuantumWorks, MRI ERA, 256
Ontario Centres of Excellence, Industry Canada, and CFI. 257
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