We analyze the class of nonlocal realistic theories that was originally
considered by Leggett [Found. Phys. 33, 1469 (2003)] and tested by us in a
recent experiment [Nature (London) 446, 871 (2007)]. We derive an
incompatibility theorem that works for finite numbers of polarizer settings and
that does not require the previously assumed rotational symmetry of the
two-particle correlation functions. The experimentally measured case involves
seven different measurement settings. Using polarization-entangled photon
pairs, we exclude this broader class of nonlocal realistic models by
experimentally violating a new Leggett-type inequality by 80 standard
deviations.Comment: Published versio

A. Aspect

Alessandro Fedrizzi

Anton Zeilinger

C. Branciard

J. S. Bell

Marek Żukowski

Markus Aspelmeyer

Simon Gröblacher

Thomas Jennewein

Tomasz Paterek

English

arXiv

We analyze the class of nonlocal realistic theories that was originally considered by Leggett and tested by us in a recent experiment. We derive an incompatibility theorem that works for finite numbers of polarizer settings and that does not require the previously assumed rotational symmetry of the two-particle correlation functions. The experimentally measured case involves seven different measurement settings. Using polarization-entangled photon pairs, we exclude this broader class of nonlocal realistic models by experimentally violating a new Leggett-type inequality by 80 standard deviations

Paterek, Tomasz

Fedrizzi, Alessandro

Groblacher, Simon

Jennewein, Thomas

Zukowski, Marek

Aspelmeyer, Markus

Zeilinger, Anton

University of Queensland eSpace

Experimental Test of Nonlocal Realistic Theories Without the Rotational Symmetry Assumption
Tomasz Paterek,1,2 Alessandro Fedrizzi,1 Simon Gro¨blacher,1 Thomas Jennewein,1 Marek Z˙ ukowski,3,4
Markus Aspelmeyer,1 and Anton Zeilinger1,3
1Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences,
Boltzmanngasse 3, A-1090 Vienna, Austria
2The Erwin Schro¨dinger International Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria
3Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
4Institute of Theoretical Physics and Astrophysics, University of Gdan´sk, ul. Wita Stwosza 57, PL-08-952 Gdan´sk, Poland
(Received 13 August 2007; published 21 November 2007)
We analyze the class of nonlocal realistic theories that was originally considered by Leggett [Found.
Phys. 33, 1469 (2003)] and tested by us in a recent experiment [Nature (London) 446, 871 (2007)]. We
derive an incompatibility theorem that works for finite numbers of polarizer settings and that does not
require the previously assumed rotational symmetry of the two-particle correlation functions. The
experimentally measured case involves seven different measurement settings. Using polarization-
entangled photon pairs, we exclude this broader class of nonlocal realistic models by experimentally
violating a new Leggett-type inequality by 80 standard deviations.
DOI: 10.1103/PhysRevLett.99.210406 PACS numbers: 42.50.Xa, 03.65.Ud
It is the essence of Bell’s theorem [1,2] and its many
experimental tests, for example [3–6], that no theory based
on the joint assumptions of locality and realism can serve
as an alternative underlying explanation of quantum phe-
nomena (further relevant references can be found in [7,8]).
Recently, a new type of incompatibility theorem was in-
troduced by Leggett [7] that allowed for the first time a test
of a specific and intuitive class of nonlocal realistic hidden-
variable theories. The experimental exclusion of this class
has been reported in Ref. [8]. Since the original inequality
required infinitely many measurement settings, it was nec-
essary to supplement it by an assumption of rotational
symmetry of the correlation functions in each measure-
ment plane. The main point of the present Letter is that this
assumption is no longer needed. We prove a theorem which
allows an experimental test of a broader class of nonlocal
realistic theories. In contrast to the original derivation
[7,8], the correlation functions of measurement outcomes
are no longer constrained to be rotationally symmetric. We
test the resulting new inequality by using a high-efficiency,
high-fidelity source of polarization-entangled photon pairs
[9].
Generalized incompatibility theorem.—The theories
under investigation aim to uphold realism, while allowing
for nonlocal influences. It is additionally assumed that
particles of a defined property behave locally in concord-
ance with quantum laws. Specifically, we consider polar-
ization measurements on pairs of photons. Additionally to
realism—the assumption that measurement outcomes are
well defined prior to and independent of the measure-
ments—the theories attribute polarization to single pho-
tons. Particles with the same polarization build up
subensembles in which Malus’ law is assumed to hold.
All observationally accessible expectation values are com-
puted as statistical mixtures over such subensembles. In
particular, the theories provide a model for all experiments
in which a Clauser-Horne-Shimony-Holt inequality [2] is
violated, and they reproduce all perfect correlations of the
Bell singlet state [10].
Let us introduce the notation, which follows Ref. [8].
Alice and Bob perform measurements on individual pho-
tons of the pairs. All observables are parameterized by
vectors on the Poincare´ sphere. Alice (Bob) performs
measurements along direction ~ak ( ~bl). The corresponding
correlation function is denoted by Ekl;’, where the
angles  and ’ parameterize the position of the vectors
~ak and ~bl within the plane spanned by them (’ is the angle
between ~ak and ~bl, and  describes the orientation of the
vector bisecting the angle ’).
The inequality of the original derivation makes use of
the averaged correlation functions
 
E kl’  12
Z 2
0
Ekl;’d: (1)
In order to get well approximated values of the averaged
correlation functions, we would have to perform a large
number of measurements. Following the earliest experi-
mental tests of local realism [3,12,13], we resorted to the
assumption that the observed correlation functions must
have the property of rotational symmetry, i.e., that the
correlation functions depend only on the angle ’. This is
experimentally well-established. Various checks of this
symmetry were performed, e.g., in Refs. [5,14–18].
The basic idea of the new incompatibility theorem is to
replace the averaged correlations (1) by the finite sum of
nonaveraged correlation functions [19]. A detailed deriva-
tion of the set of new inequalities, that essentially parallels
the one in [8], is provided in the Appendix. In the experi-
mentally most easily realizable case, this leads to the
PRL 99, 210406 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending23 NOVEMBER 2007
0031-9007=07=99(21)=210406(4) 210406-1 © 2007 The American Physical Society
following inequality:
 
S
Ekl0;’Ek0l0


2
;’

Emn0;0Em0n0


2
;0


E?qp0;’E?q0p0


2
;’

E?rs0;0E?r0s0


2
;0

82
sin’2
: (2)
It is a necessary consequence of the derivation that settings
of the correlation functions in the second modulus have to
lie in any plane orthogonal to that defined by the settings of
correlation functions in the first modulus. Thus, they have
an additional superscript ? .
Again, quantum predictions violate the new inequality
(2). For example, the two-particle singlet state yields the
quantum correlation function E’   cos’. For this
state, S  4j1 cos’j. Maximum violation is obtained
for angle ’max  14:6	 at which the bound equals 7.746,
in contrast to the quantum value of 7.871 at the left-hand
side of the inequality. The ratio of the bound of Eq. (2) for
’max and the quantum value is 0.984. As a consequence,
the minimal visibility of the two-particle interference that
is required to unambiguously observe a violation at the
optimal difference angle must be larger than 98.4%.
In order to test the inequality, it is sufficient that Alice
and Bob choose among seven different pairs of settings.
Alice’s setting vectors are
 ~a 1  1;0;0; ~a2  0;1;0; ~a3  0;0;1: (3)
Bob’s setting vectors are
 
~b1  cos’max; sin’max; 0;
~b2   sin’max; cos’max; 0;
~b3  0; cos’max; sin’max;
~b4  0; sin’max; cos’max;
~b5  ~a1;
~b6  ~a2;
~b7  ~a3: (4)
All vectors are depicted on the Poincare´ sphere in Fig. 1.
For these settings, inequality (2) reads
 jE11  E22  E15  E26j  jE23  E34  E26  E37j  8 2j sin’max2 j  7:746: (5)
Experiment.—We tested inequality (5) by measuring
polarization correlations on polarization-entangled photon
pairs for the settings (3) and (4) with ’max  14:6	. We
used a high-efficiency, high-fidelity pair source based on
spontaneous parametric downconversion (SPDC) in peri-
odically poled KTiOPO4 (PPKTP) inside a polarization
Sagnac interferometer (Fig. 2). This configuration was
originally demonstrated in Ref. [20]; the setup used here
is explained in more detail in Ref. [9]. In our experiment,
the source was aligned to emit the singlet state j i 1
2
p 

jHiAjViBjViAjHiB, where jHViAB denotes a hori-
zontally (vertically) polarized photon that leaves the inter-
ferometer towards Alice (Bob).
Polarization measurements were performed by passing
each photon of a pair through a combination of quarter-
wave plates and polarizers and by subsequently detecting it
by single-photon avalanche photodiodes. In that way, ar-
bitrary setting directions on the Poincare´ sphere can be
FIG. 1 (color online). The Poincare´ vectors corresponding to
measurement settings of Alice and Bob for the maximal viola-
tion of inequality (2).
FIG. 2 (color online). Scheme of the experiment: A 405 nm
diode laser is focused into a PPKTP crystal inside a polarization
Sagnac loop comprised of a dual-wavelength polarizing beam
splitter (PBS), a dual-wavelength half-wave plate (HWP), and
two laser mirrors. In the crystal, orthogonally polarized photon
pairs are created in modes C and D. At the PBS, these photon
pairs are coherently superposed, the horizontally polarized pho-
tons are transmitted, and the vertically polarized ones are re-
flected such that the resulting two-photon state in modes A and B
is polarization-entangled. The polarization states of the photons
are analyzed and measured by a combination of a quarter-wave
plate (QWP), a polarizer (POL), and an Silicon avalanche
photodiode (APD).
PRL 99, 210406 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending23 NOVEMBER 2007
210406-2
realized (see, e.g., [21]). Joint detection events between
Alice and Bob were registered within a coincidence time
window of 4.4 ns by analyzing the individual events via a
field-programmable gate array (FPGA). Without polarizers
in place, we typically observed coincidence count rates of
20 000 per second with single count rates of 80 000
(the same for Alice and Bob) at an optical pump power of
0.6 mW. Polarization correlation measurements along di-
rections jH=Vi (corresponding to correlation function
E37), ji  12p jHi  jVi (E15), and jR=Li  1~~2p jHi 
ijVi (E26) reveal an actual visibility of 99:70 0:02%,
99:47 0:02%, and 99:25 0:03%. The average two-
photon interference visibility of 99:47 0:01% clearly
surpasses the required limit of 98.4%.
We measured the expectation values of inequality (5) by
using the optimal settings described above. The resulting
value of its left-hand side, S, is compared to the bound of
7.746 valid for the generalized class of nonlocal realistic
models. The result is summarized in Table I. Error analysis
is performed by taking into account both Poissonian count-
ing statistics and the inaccuracy in setting the measurement
direction with the quarter-wave plates and polarizers. In
summary, we observe a violation of inequality (5), as given
by the minimum distance between the measured S-value
and the theoretical bound, by 80 standard deviations.
Conclusions.—Based on the recent works by Leggett [7]
and by us [8], we derive a new incompatibility theorem that
does not require the previously assumed rotational sym-
metry of the two-particle correlation functions and hence
puts to test a broader class of nonlocal realistic hidden-
variable theories. We demonstrate an experimental viola-
tion of the resulting new inequality by 80 standard
deviations.
We acknowledge discussions with Cˇ . Brukner and A. J.
Leggett. The work was supported by the Austrian Science
Fund (FWF), by the Austrian Research Promotion Agency
(FFG), by the EC funded project QAP and by the
Foundational Questions Institute (FQXi).
Note added.—Recently, we were informed of a closely
related work by Branciard et al. [22].
Appendix.—We derive an infinite set of inequalities,
the simplest case of which is the inequality (2). We fol-
low exactly all the steps as in the Supplementary
Information (SI) of Ref. [8] up to the formula (27). In an
abbreviated notation of the integration, which utilizes
the fact that explicit integration boundaries play no role
in the derivation, it reads (for notation and symbols see
[8])
 Eklkl; ’kl  1 2
Z
d ~ud ~vF ~u; ~v
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
n22cos
2

’kl  uv
2

 n21sin2

’kl  uv
2
s
j coskl   uv  j: (A1)
In the original proof, Eq. (A1) was averaged over all possible setting angles in one plane. Here, we avoid this entirely by
considering N  2 correlation functions for settings from a single plane: Eknlnknln  n=N;’knln, with n 
0; 1; 2; . . . ; N  1. We set the angle between the setting vectors which enter all correlation functions to be the same,
i.e., ’  ’knln . The settings of the nth correlation function are rotated by N with respect to the settings of the (n 1)th
correlation function. We sum up inequalities (A1) for all N correlation functions:
 
XN1
n0
Eknln

n
N
;’

 N  2
Z
d ~ud ~vF ~u; ~v
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
n22cos
2

’ uv
2

 n21sin2

’ uv
2
s XN1
n0
cos

n

N
  uv  
: (A2)
We utilize the following inequality
 
XN1
n0
j cosn=N   uv  j  cot 2N ; (A3)
and obtain
TABLE I. Comparison of ideal quantum theoretical and ex-
perimental expectation values, Eij, that enter inequality (5).
Below, the theoretical S-value [the left-hand side of (5)] is
compared with the experimental one. The discrepancy between
theory and experiment is explained by taking into account an
average two-photon interference visibility of 99:47 0:01%
and the inaccuracy in setting the local measurement direction of
approximately 0:5	. The standard deviations E and S are
the relevant errors for calculating the violation (see text) and are
essentially due to Poissonian count statistics of the measured
coincidence rates.
Etheory Eexperiment E
E11 0:9677 0:9749 0.0005
E22 0:9677 0:9733 0.0005
E15 1 0:9947 0.0002
E26 1 0:9925 0.0003
E23 0:9677 0:9601 0.0007
E34 0:9677 0:9662 0.0006
E37 1 0:9970 0.0002
Stheory Sexperiment S
7.8708 7.8511 0.0013
PRL 99, 210406 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending23 NOVEMBER 2007
210406-3
 1
N
XN1
n0
Eknln

n
N
;’

 1 2KN
N
Z
d ~ud ~vF ~u; ~v
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
n22cos
2

’ uv
2

 n21sin2

’ uv
2
s
; (A4)
where KN  cot 2N . This inequality is valid for any choice of observables within a single plane. We introduce N new
observable vector pairs in this plane, again rotated by n N with respect to an initial pair. The correlation functions of these
new observables will be denoted byEinjn0injn  nN ; ’0injn, where the angles 0injn can be relative to another axis than knln .
Again, we set ’0injn  ’0 and arrive at the analog of inequality (A4). The sum of these inequalities reads
 
1
N
XN1
n0
Eknln

n
N
;’

 XN1
n0
Einjn

n
N
;’0

 2 2KN
N
Z
d ~ud ~vF ~u; ~v
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
n22cos
2 ’ uv
2
 n21sin2
’ uv
2
r

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
n22cos
2 ’
0  uv
2
 n21sin2
’0  uv
2
s 
: (A5)
To estimate the bound, we use the manipulation involving the triangle inequality, which follows exactly relations (31)–(33)
of the SI, to get
 
1
N
X
N1
n0
Eknln

n
N
;’

 XN1
n0
Einjn

n
N
;’0
 2 ~~2p KNN
sin’ ’
0
2
Z d ~ud ~vF ~u; ~v
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
sin2u  sin2v
q
: (A6)
This formula replaces (35) of the SI. As we see, the net change on the right hand side is that 2
~~
2
p
 , which is equal to
limN!1

2
p
KN
N , is replaced by

2
p
KN
N . For settings within a plane orthogonal to the initial one, we get a similar
inequality. After adding the inequalities for the two orthogonal planes we set angles ’ in those planes to be equal, and
angles ’0 to zero. Next, we utilize the fact
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
sin2u  sin2v
p  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~sin20u  sin20vp  2p , see (37)–(43) of the SI, which
leads us to the following set of inequalities:
 
1
N
X
N1
n0
Eknln

n
N
;’

 XN1
n0
Einjn

n
N
; 0
 1N
X
N1
n0
E?qnpn

n
N
;’

 XN1
n0
E?rnsn

n
N
; 0
 4 2KNN
sin’2
:
(A7)
In the limit N ! 1, since limN!12 KNN  4=, we re-
cover the inequality published in [8]. The simplest one, for
N  2, has the explicit form of inequality (2) and uses only
seven pairs of measurement settings.
[1] J. S. Bell, Physics (Long Island City, N.Y.) 1, 195 (1964).
[2] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt,
Phys. Rev. Lett. 23, 880 (1969).
[3] S. J. Freedman and J. F. Clauser, Phys. Rev. Lett. 28, 938
(1972).
[4] A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49,
1804 (1982).
[5] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and
A. Zeilinger, Phys. Rev. Lett. 81, 5039 (1998).
[6] M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M.
Itano, C. Monroe, and D. J. Wineland, Nature (London)
409, 791 (2001).
[7] A. J. Leggett, Found. Phys. 33, 1469 (2003).
[8] S. Gro¨blacher, T. Paterek, R. Kaltenbaek, Cˇ . Brukner,
M. Z˙ ukowski, M. Aspelmeyer, and A. Zeilinger, Nature
(London) 446, 871 (2007). Also at arXiv:0704.2529v2.
[9] A. Fedrizzi, T. Herbst, T. Jennewein, and A. Zeilinger,
Opt. Express 15, 15377 (2007).
[10] Note that such nonlocal realistic models do not follow the
Furry-Schro¨dinger hypothesis [11].
[11] W. H. Furry, Phys. Rev. 49, 393 (1936).
[12] J. F. Clauser, Phys. Rev. Lett. 36, 1223 (1976).
[13] E. S. Fry and R. C. Thompson, Phys. Rev. Lett. 37, 465
(1976).
[14] A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 47,
460 (1981).
[15] Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and
C. O. Alley, Phys. Rev. A 50, 23 (1994).
[16] P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V.
Sergienko, and Y. Shih, Phys. Rev. Lett. 75, 4337 (1995).
[17] A. Aspect, in Quantum [Un]speakables—From Bell to
Quantum Information, edited by R. A. Bertlmann and
A. Zeilinger, (Springer, New York, 2002).
[18] Note that this does not imply a pre-assumption of the
validity of quantum theory, because only assumptions
about the phenomenologically defined quantities ‘‘corre-
lation functions’’ are made.
[19] T. Paterek, Ph.D. thesis, University of Gdan´sk, 2007.
[20] T. Kim, M. Fiorentino, and F. Wong, Phys. Rev. A 73,
012316 (2006).
[21] P. Walther, K. Resch, T. Rudolph, E. Schenck, V. Vedral,
H. Weinfurter, M. Aspelmeyer, and A. Zeilinger, Nature
(London) 434, 169 (2005).
[22] C. Branciard, A. Ling, N. Gisin, C. Kurtsiefer, A. Lamas-
Linares, and V. Scarani, following Letter, Phys. Rev. Lett.
99, 210407 (2007).
PRL 99, 210406 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending23 NOVEMBER 2007
210406-4


Experimental test of nonlocal realistic theories without the rotational symmetry assumption

Crossref

Experimental Test of Nonlocal Realistic Theories Without the Rotational Symmetry Assumption

We analyze the class of nonlocal realistic theories that was originally considered by Leggett [Found. Phys. 33, 1469 (2003)] and tested by us in a recent experiment [Nature (London) 446, 871 (2007)]. We derive an incompatibility theorem that works for finite numbers of polarizer settings and that does not require the previously assumed rotational symmetry of the two-particle correlation functions. The experimentally measured case involves seven different measurement settings. Using polarization-entangled photon pairs, we exclude this broader class of nonlocal realistic models by experimentally violating a new Leggett-type inequality by 80 standard deviations.</p

Groeblacher, Simon

Heriot Watt Pure

Experimental test of nonlocal realistic theories without the rotational
  symmetry assumption

Experimental test of nonlocal realistic theories without the rotational symmetry assumption

Abstract

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University of Queensland eSpace

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Heriot Watt Pure

Heriot Watt Pure

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