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PHYSICAL REVIEW A
VOLUME 60, NUMBER 6
DECEMBER 1999
Contradiction of quantum mechanics with local hidden variables for quadrature phase
measurements on pair-coherent states and squeezed macroscopic superpositions
of coherent states
A. Gilchrist
Physics Department, University of Waikato, Hamilton, New Zealand
P. Deuar and M. D. Reid
Physics Department, University of Queensland, Brisbane, Australia
共Received 22 September 1998兲
We demonstrate a contradiction of quantum mechanics with local hidden variable theories for continuous
quadrature phase amplitude 共‘‘position’’ and ‘‘momentum’’兲 measurements. A contradiction is shown possible
for two quantum states: a pair-coherent state, and a superpositions of two coherent states, where the superposition state has been squeezed by the action of a two-mode squeezing operator. In one case a contradiction is
still possible for states of increasing photon number, though the effect becomes smaller and more difficult to
observe. The high efficiency of the homodyne method of measurement of quadrature phase amplitudes may
open a way for a loophole-free test of local hidden variable theories, and the effect of detection loss on the
contradiction with local hidden variables is calculated. 关S1050-2947共99兲02208-8兴
PACS number共s兲: 03.65.Bz
I. INTRODUCTION
Einstein, Podolsky, and Rosen 共EPR兲 关1兴 in 1935 presented an argument for the incompleteness of quantum mechanics. The argument was based on the validity of two premises: no action at a distance 共locality兲 and realism. The
original argument of EPR considered position and momentum measurements which could be performed on each of two
particles at spatially separated locations. Bell 关2兴 later
showed that the predictions of quantum mechanics are incompatible with the premises of local realism 共or local hidden variable theories兲. Experiments 关3兴 based on Bell’s result
indicate the failure of local hidden variable theories.
Bell’s original result, and subsequent theoretical 共for example, Refs. 关2,4–6兴兲 and experimental work which test local hidden variable theories against quantum mechanics,
considered measurements which have discrete outcomes,
such as measurements of spin or photon number. By this we
mean that the eigenvalues of the relevant system Hermitian
operator, which represents the measurement in quantum mechanics, are discrete. The more successful experimental tests
to date have involved photon counting measurements, for
which the results of the measurement, a photon present or
not, are discrete and only microscopically different. Associated with such experiments are relatively low detection efficiencies, which currently make a test of Bell’s original inequality not feasible. These experiments test weaker
inequalities 关7兴 for which one needs to make additional auxiliary assumptions, preventing local hidden variable theories
from being ruled out conclusively 关8兴.
In this paper we expand on our initial results published
previously 关9兴. We show how the predictions of quantum
mechanics for certain entangled quantum superpositions of
coherent states are in disagreement with those of local hidden variable theories for a situation involving continuous
quadrature phase amplitude 共‘‘position’’ and ‘‘momentum’’兲
1050-2947/99/60共6兲/4259共13兲/$15.00
PRA 60
measurements. By this we mean that the quantum predictions
for the probability of obtaining results x and p for position
and momentum 共and various linear combinations of these
coordinates兲 cannot be predicted by any local hidden variable theory.
This result is of fundamental interest since the original
argument 关1兴 of Einstein, Podolsky, and Rosen was given in
terms of position and momentum measurements. It was
pointed out by one of us 关10兴 that quadrature phase amplitude measurements performed on the spatially separated outputs of the nondegenerate parametric amplifier could potentially be an example of the EPR paradox. A criterion was
established to test for an EPR paradox even where correlations are not perfect, and ‘‘elements of reality’’ deduced using the premises of ‘‘local realism’’ 共as defined originally by
EPR兲 have an indeterminacy 关11兴 in their values. Such EPR
correlations, for continuous variables, were generated experimentally by Ou et al. 关12兴 in a high efficiency experiment
using homodyne detection. The original state considered by
Einstein, Podolsky, and Rosen, however, and that produced
experimentally in the realization by Ou et al., give probability distributions for x and p completely compatible with local
realism. This is so because, as discussed by Bell 关2兴, the
associated quantum Wigner distribution is positive in these
cases and can thus provide a local hidden variable theory.
We also note that the homodyne method of measurement
关13兴 of the quadrature phase amplitude employs a second
‘‘local-oscillator’’ field which combines with the original
field to provide an amplification prior to photodetection.
Large field fluxes fall incident on highly efficient photodiode
detectors. This high intensity limit has not been indicated by
previous works 关14兴 which showed contradiction of quantum
mechanics with local hidden variables using homodyne detection, since these analyses were restricted to a very low
intensity of ‘‘local-oscillator’’ field. The possible macroscopic nature of such experiments has been discussed previ4259
©1999 The American Physical Society
A. GILCHRIST, P. DEUAR, AND M. D. REID
4260
FIG. 1. Schematic representation of the proposed test of local
hidden variables. Balanced homodyne detection allows measurement of the quadrature phase amplitudes X A␪ and X ␾B , to give results
x and y, respectively.
ously 关11兴 in the context of the Ou et al. realization of the
Einstein-Podolsky-Rosen experiment. The high efficiency of
detectors may also provide a way to test local hidden variables without the use of auxiliary assumptions which weaken
the conclusions of the former photon counting measurements. We therefore calculate the effect of detection loss on
the predicted contradiction with local hidden variables.
We stress that recent independent work by Yurke, Hillery,
and Stoler 关15兴 has also shown an incompatibility of quantum mechanics with local realism for quadrature phase amplitude measurements performed on certain quantum systems. There have also been further recent calculations by
Munro and Milburn 关16兴.
with continuous variable outcomes 共corresponding to a continuous eigenvalue spectrum in quantum mechanics兲.
If we now postulate the existence of a local hidden variable theory 关2兴, we can write the probabilities P ␪ , ␾ (x,y) for
getting a result x and y, respectively, upon the simultaneous
measurements X ␪A and X ␾B in terms of the hidden variables ␭
as follows:
P ␪ , ␾ 共 x,y 兲 ⫽
A
P⫹
共 ␪ 兲⫽
The Bell inequality is a consequence of the assumptions
of locality and of realism. Consider an initial nondegenerate
two-mode state 兩 ⌿ 典 . Each mode of the state is directed to
two physically separate locations A and B as indicated in Fig.
1.
Measurements are made of the field quadrature phase amplitudes X A␪ at location A, and X ␾B at location B. Here we
define
X ␾B ⫽b̂
冕␳
共1兲
exp共 ⫺i ␾ 兲 ⫹b̂ exp共 i ␾ 兲 .
B
P⫹
共 ␾ 兲⫽
冕␳
冕␳
冕␳
AB
P ⫹⫹
共 ␪,␾ 兲⫽
共2兲
A
共␭兲p⫹
共 ␭, ␪ 兲 d␭,
共3兲
B
共␭兲p⫹
共 ␭, ␾ 兲 d␭,
共4兲
A
B
共␭兲p⫹
共 ␭, ␪ 兲 p ⫹
共 ␭, ␾ 兲 d␭,
共5兲
A
( ␪ ,␭)⫽ 兰 x⭓0 p Ax ( ␪ ,␭)dx, and
where we have simply set p ⫹
B
similarly for p ⫹ ( ␾ ,␭). It is well known that one can now
deduce 关2兴 the following ‘‘strong’’ Clauser-Horne-Bell inequality 共no auxiliary assumptions 关7兴 have been made兲:
†
Where our system is a harmonic oscillator, we note that the
angle choices ␪ 共or ␾ ) equal to zero and ␲ /2 will correspond
to momentum and position measurements. The result for the
amplitude measurement X A␪ is a continuous variable which
we denote by x 共or sometimes x ␪ ). Similarly the result of the
measurement X ␾B is a continuous variable denoted by y 共or
sometimes x ␾ ).
We formulate a Bell inequality test for the experiment
depicted by making the simplest possible binary classification of the continuous results x and y of the measurements.
We classify the result of the measurement to be ⫹1 if the
quadrature phase result x 共or y) is greater than or equal to
zero, and ⫺1 otherwise. With many measurements we build
A
( ␪ ) for obtainup the following probability distributions: P ⫹
B
ing a positive value of x; P ⫹ ( ␾ ) for obtaining a positive y;
AB
and P ⫹⫹
( ␪ , ␾ ) the joint probability of obtaining a positive
result in both x and y. While this coarse-grain classification
may not give as sensitive a test as a possible alternative Bell
inequality derived for the continuous variables x and y directly, a violation found for the coarser treatment is still firm
confirmation of failure of local realism for measurements
共 ␭ 兲 p Ax 共 ␪ ,␭ 兲 p By 共 ␾ ,␭ 兲 d␭.
The ␳ (␭) is the probability distribution for the hidden variable state denoted by ␭, while p Ax ( ␪ ,␭) is the probability of
obtaining a result x upon measurement at A of X ␪A , given the
hidden variable state ␭. The p By ( ␾ ,␭) is defined similarly for
the results and measurement at B. The independence of
p Ax ( ␪ ,␭) on ␾ , and p By ( ␾ ,␭) on ␪ , is a consequence of the
locality assumption that the measurement at A cannot be influenced by the experimenter’s choice of parameter ␾ at the
location B 共and vice versa兲. It follows that the final measured
AB
A
B
( ␪ , ␾ ), P ⫹
( ␪ ), and P ⫹
( ␾ ) can be written
probabilities P ⫹⫹
in a similar form,
II. GENERAL FORMALISM
X ␪A ⫽â exp共 ⫺i ␪ 兲 ⫹â † exp共 i ␪ 兲 ,
PRA 60
S⫽
AB
AB
AB
AB
P ⫹⫹
共 ␪ , ␾ 兲 ⫺ P ⫹⫹
共 ␪ , ␾ ⬘ 兲 ⫹ P ⫹⫹
共 ␪ ⬘ , ␾ 兲 ⫹ P ⫹⫹
共 ␪ ⬘, ␾ ⬘ 兲
A
B
P⫹
共 ␪⬘兲⫹ P⫹
共␾兲
⭐1.
共6兲
III. VIOLATION OF THE CLAUSER-HORNE-BELL
INEQUALITY USING QUADRATURE PHASE
MEASUREMENTS FOR A PAIR-COHERENT STATE
We consider the following two-mode entangled quantum
superposition state, discussed originally by Agarwal and
Tara and Agarwal 关17,18兴 and also Reid and Krippner 关19兴:
兩 ⌿ 典 m ⫽N
冕
2␲
0
e ⫺im␵ 兩 r 0 e i␵ 典 a 兩 r 0 e ⫺i␵ 典 b d␵.
共7兲
Here N is a normalization coefficient. Here 兩 . . . 典 a and
兩 . . . 典 b are coherent states in modes â and b̂, where the â †
and â, and b̂ † and b̂ are the usual boson operators for the two
spatially separated systems 共for example, field modes兲 at locations A and B, respectively. In many optical systems the â
and b̂ are referred to as the signal and idler fields, respectively. This state, originally defined and considered by Agar-
PRA 60
CONTRADICTION OF QUANTUM MECHANICS WITH . . .
wal and termed a pair-coherent state, is an eigenstate of the
photon number difference between signal and idler modes,
with eigenvalue m:
共 â † â⫺b̂ † b̂ 兲 兩 ⌿ 典 m ⫽m 兩 ⌿ 典 m .
共8兲
a具 x ␪兩 ␣ 典 a⫽
冉 冊
1
2␲
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1/4
再
冎
1
1
1
⫻exp ⫺ x ␪2 ⫹ ␣ e ⫺i ␪ x ␪ ⫺ ␣ 2 e ⫺2i ␪ ⫺ 兩 ␣ 兩 2 ,
4
2
2
共15兲
Also it is an eigenstate of the operator âb̂,
âb̂ 兩 ⌿ 典 m ⫽r 20 兩 ⌿ 典 m .
共9兲
For our purposes we shall concentrate on the m⫽0 case
in Eq. 共7兲 which when normalized is
兩 ⌿ 典 ⫽B 1/2
冕
2␲
0
兩 r 0 e i␵ 典 a 兩 r 0 e ⫺i␵ 典 b d␵,
where 兩 x ␪ 典 a is the eigenstate of X A␪ ⫽âexp(⫺i␪)⫹â†exp(i␪),
and we use similar definitions for mode b̂. We have
a 具 x ␪ 兩 b 具 x ␾ 兩 ⌿ 典 ⫽B
⫽
共10兲
1/2
冕
冉 冊
B
2␲
2␲
0
d␵ a Šx ␪ 円b 具 x ␾ 兩 r 0 e i␵ 典 a 円r 0 e ⫺i␵ ‹b
1/2
exp兵 ⫺r 20 其
再
再
1
⫻exp ⫺ 共 x ␪2 ⫹x ␾2 兲
2
with
2
B ⫺1 ⫽4 ␲ 2 e ⫺2r 0 I 0 共 2r 20 兲 ,
共11兲
and where I 0 is a modified Bessel function. Expanding out
each of the coherent states into number states 兩 n 典 for each
mode we can write the state as
⬁
兺
n⫽0
兩 ⌿ 典 ⫽ 关 I 0 共 2r 20 兲兴 ⫺1/2
共 r 20 兲 n
n!
兩 n 典 a兩 n 典 b ,
共12兲
as originally introduced by Agarwal.
The quantum state given by Eqs. 共10兲 and 共12兲 is potentially generated, from vacuum fields, by the interaction modeled by the following Hamiltonian, in which coupled signalidler loss dominates over linear single-photon loss.
H⫽iបE 共 â † b̂ † ⫺âb̂ 兲 ⫹âb̂⌫ˆ † ⫹â † b̂ † ⌫ˆ .
⫻exp ⫺
共14兲
⫺1
„where ␣ is real and N ⫾
⫽2 关 1⫾exp(⫺2兩␣兩2)兴… which are
generated 关21,22兴 by the degenerate form 共set â⫽b̂) of the
Hamiltonian equation 共13兲 and which have been recently experimentally generated 关23兴. These states are of interest in
that they resemble, for large ␣ , ‘‘Schröd̈inger-cat’’states
关24兴.
The calculation of the quantum prediction for S for the
quantum state 共10兲 is straightforward in principle. We use
0
d␽
冎
共 e 2i( ␽ ⫺ ␹ ) ⫹e ⫺2i ␽ 兲 ,
P ␪ , ␾ 共 x ␪ ,x ␾ 兲 ⫽円A 具 x ␪ 兩 B 具 x ␾ 兩 ⌿ 典 円2
⫽
共16兲
共17兲
再
B
1
exp ⫺ 共 x ␪2 ⫹x ␾2 兲
2␲
2
冎冕 冕
2␲
0
d␽
2␲
0
d␽⬘
⫻exp兵 ⫺ 冑2 关 A 共 r 0 兲 x ␪ ⫹B 共 r 0 兲 x ␾ 兴 ⫹C 共 r 0 兲 其 ,
共18兲
with the factors
A 共 r 0 兲 ⫽⫺
1/2
N⫾
共兩␣典⫾兩⫺␣典)
2
2␲
where ␽ ⫽␵⫹ ␾ , ␹ ⫽ ␪ ⫹ ␾ . The probability distribution
P ␪ , ␾ (x ␪ ,x ␾ ) becomes
共13兲
This interaction is achievable in principle by nondegenerate
parametric oscillation 关19兴 in a limit where uncorrelated
single-photon loss in each of the signal and idler fields becomes negligible. Here E represents a coherent driving parametric term which generates signal-idler pairs, while ⌫ˆ represents reservoir systems which give rise to the coupled
signal-idler loss. The Hamiltonian preserves the signal-idler
photon number difference â † â⫺b̂ † b̂, of which the quantum
state 共10兲 is an eigenstate, with eigenvalue zero. We note the
analogy here to the single-mode ‘‘even’’ and ‘‘odd’’ 关20兴
coherent superposition states
r 20
冎冕
B 共 r 0 兲 ⫽⫺
C 共 r 0 兲 ⫽⫺
r0
冑2
r0
冑2
r 20
2
共 e i( ␽ ⫺ ␹ ) ⫹e ⫺i( ␽ ⬘ ⫺ ␹ ) 兲 ,
共 e ⫺i ␽ ⫹e i ␽ ⬘ 兲 ,
共 4⫹e 2i( ␽ ⫺ ␹ ) ⫹e ⫺2i( ␽ ⬘ ⫺ ␹ ) ⫹e ⫺2i ␽ ⫹e 2i ␽ ⬘ 兲 .
It is evident from this expression that P ␪ , ␾ (x ␪ ,x ␾ ) is a function only of the angle sum ␹ ⫽ ␪ ⫹ ␾ so we can abbreviate
AB
AB
( ␪ , ␾ )⫽ P ⫹⫹
( ␹ ). Also we see that on making the variP ⫹⫹
able change ␩ ⫽⫺ ␽ and ␩ ⬘ ⫽⫺ ␽ ⬘ in the integrations 共18兲,
we obtain the same form for the expression P ␪ , ␾ (x ␪ ,x ␾ ) but
AB
(␹)
replacing ␹ with ⫺ ␹ . That is, we have P ⫹⫹
AB
⫽ P ⫹⫹ (⫺ ␹ ).
We note that the probabilities P ␪ , ␾ (x,y) for the paircoherent state could also be evaluated using the expression
共12兲. Here one uses the result
具 x ␪兩 n 典 ⫽
冉 冊
1
n
冑2 n! 2 ␲
1
1/4
2
e ⫺x ␪ /4e ⫺ni ␪ H n
冉 冊
1
冑2
x ␪ . 共19兲
A. GILCHRIST, P. DEUAR, AND M. D. REID
4262
PRA 60
FIG. 3. Plot of S versus r 0 , for the angle values indicated in the
text: ␦ 1 ⫽ ␦ 2 ⫽ ␦ ⫽ ␲ /2 and ␴ ⫽5 ␲ /4.
B
P⫹
共 ␾ 兲⫽
FIG. 2. Representation of the probability P ␪ , ␾ (x,y) for getting a
result x and y, respectively, upon the simultaneous measurements
X A␪ and X ␾B , where ␪ ⫽⫺ ␾ . 共a兲 r 0 ⫽1.1; 共b兲 r 0 ⫽2.5 showing the
increasing separation of peaks and the interference fringes characteristic of quantum superposition states.
This approach was used by Tara and Agarwal 关18兴, who
calculated P ␪ , ␾ (x,y) in a paper to establish the existence of
correlations between the quadrature phase amplitudes X ␪A and
X ␾B at the locations A and B, respectively. Tara and Agarwal
关18兴 showed the correlation to be sufficient to satisfy the
criterion developed by one of us 共Reid 关10兴兲 for a demonstration of the Einstein-Podolsky-Rosen paradox. As discussed in the Introduction and explained elsewhere, the existence of such correlations does not in itself imply a failure
of local realism.
Figure 2 shows the distribution P ␪ , ␾ (x,y) for selected
choices of r 0 and ␪ ⫹ ␾ . The strong correlation for ␪ ⫽⫺ ␾ is
evident.
We proceed to calculate the event probabilities needed for
the Clauser-Horne- 共CH兲 Bell inequality 共6兲. We have
AB
P ⫹⫹
共 ␪,␾ 兲⫽
A
P⫹
共 ␪ 兲⫽
冕
冕 冕
⬁
0
dx ␾
⬁
⫺⬁
dx ␾
⬁
0
冕
⬁
0
dx ␪ P ␪ , ␾ 共 x ␪ ,x ␾ 兲 ,
dx ␪ P ␪ , ␾ 共 x ␪ ,x ␾ 兲 ,
共20兲
冕
⬁
⫺⬁
dx ␪
冕
⬁
0
dx ␾ P ␪ , ␾ 共 x ␪ ,x ␾ 兲 .
These integrations could be evaluated by direct numerical
integration 关use Eqs. 共16兲 and 共17兲 directly兴. An analysis allowing for a much quicker numerical evaluation is presented
in Appendix A.
AB
( ␪ , ␾ ):
We note certain properties of the distribution P ⫹⫹
AB
AB
AB
AB
AB
P ⫹⫹ ( ␪ , ␾ )⫽ P ⫹⫹ ( ␹ ); P ⫹⫹ ( ␹ )⫽ P ⫹⫹ ( ␹ ⫹2 ␲ ); P ⫹⫹
(␹)
AB
⫽ P ⫹⫹ (⫺ ␹ ); and the marginals satisfy, as proved rigorously
A
B
in Appendix A, P ⫹
( ␪ )⫽ P ⫹
( ␾ )⫽0.5. It then becomes apparent that the value for S involves only three independent
angles 关which we specify by ␦ 1 ⫽ ␪ ⫺ ␪ ⬘ , ␦ 2 ⫽ ␾ ⬘ ⫺ ␾ , ␴ ⫽
⫺( ␪ ⫹ ␾ ⬘ )兴.
Results for S are shown in Fig. 3, for the choice of measurement angles giving ␦ 1 ⫽ ␦ 2 ⫽ ␦ ⫽ ␲ /2 and ␴ ⫽⫺3 ␲ /4 共or
the negative values ␦ ⫽⫺ ␲ /2 and ␴ ⫽3 ␲ /4). This choice
corresponds to ␪ ⫹ ␾ ⫽ ␪ ⬘ ⫹ ␾ ⬘ ⫽⫺( ␪ ⬘ ⫹ ␾ )⫽ ␲ /4, ␪ ⫹ ␾ ⬘
⫽3 ␲ /4 共for example set ␪ ⫽0, ␾ ⫽⫺ ␲ /4, ␪ ⬘ ⫽ ␲ /2, and ␾ ⬘
AB
⫽⫺3 ␲ /4) so that the simplification S⫽3 P ⫹⫹
( ␲ /4)
AB
⫺ P ⫹⫹ (3 ␲ /4) can be made. We have shown that for small
r 0 共less than about 1.5) this angle choice maximizes S. Figures 4 and 5 illustrate the variation in the value of S with
variation in choice of angles.
Violations of the Bell inequality, and hence contradiction
with the predictions of local hidden variables, are indicated
for 0.96ⱗr 0 ⱗ1.41, the maximum violation of S⬇1.0161 being around r 0 ⬇1.12. We note that this approximate value of
r 0 was also found by Tara and Agarwal 关18兴 to be optimal
for the demonstration of EPR correlations, and corresponds
to the greatest amount of two-mode squeezing. We have
mentioned previously in the Introduction, however, that the
existence of such correlations does not imply necessarily a
violation of local realism for the experimental arrangement
we consider in this paper.
We note that the violations are lost at larger coherent
amplitudes r 0 . It is possible to obtain 共Appendix B兲
asymptotic 共large r 0 ) analytical forms for the probability distributions which allow a complete search for all angles. Results indicate no violations of the Bell inequality 共6兲 are possible.
PRA 60
CONTRADICTION OF QUANTUM MECHANICS WITH . . .
FIG. 4. For r 0 ⫽1.1 we show a contour plot of S as a function of
␴ and ␦ . The inset has units of ␲ on its axis and shows a closeup of
the region of violation denoted in the dashed square. Note the same
violations occur for the lower square at ␦ ⫽3 ␲ /2 and ␴ ⫽3 ␲ /4.
In the larger r 0 limit the probability distributions for x and
y begin to show two widely separated peaks 共as indicated by
Fig. 2兲. For our particular choice of quantum state 共the pair
coherent state兲 the ⫹1 and ⫺1 results will never be truly
macroscopically distinct because there is always a nonzero
probability for values of x near zero. Nevertheless we hypothetically consider a situation where the ⫹1 and ⫺1 results
of the measurement correspond to macroscopically distinct
outcomes, resembling the ‘‘alive’’ and ‘‘dead’’ states of the
‘‘Schrödinger cat.’’ This is the truly macroscopic limit
stressed by Leggett and Garg 关25兴. In fact it can be demonstrated that, for any quantum state, the incompatibility with
local hidden variables must become increasingly small, for
the case where the quadrature phase amplitude results x and
y only take on values increasingly macroscopically distinct.
In this limiting case, the addition of a noise term of order the
standard quantum limit 共this corresponds to a variance ⌬ 2 x
⫽1) to the result of quadrature phase amplitude measurement will not alter the ⫹1 or ⫺1 classification of the result.
Yet it can be shown that the quantum predictions for the
results of such a noisy experiment are given by the convolution of the quantum Wigner function W(x A0 ,x ␲A /2 ,x B0 ,x ␲B /2) for
the state 共10兲, with the Gaussian noise term (1/4␲ 2 )
A 2
B2
B 2
⫻exp(⫺关xA2
0 ⫹x␲/2 ⫹x 0 ⫹x ␲ /2 兴 /2). This new Wigner function is always positive 共see, for example, Ref. 关26兴兲 and can
then act as a local hidden variable theory which gives all the
4263
quantum predictions in the truly macroscopic ‘‘dead’’ or
‘‘alive’’ classification limit.
Accurate quadrature phase amplitude measurements have
been performed in a now significant number of squeezedstate experiments, in which the measurement is performed
using homodyne detection. This technique employs a localoscillator field to create an amplification prior to detection,
which means that fields of large intensity fall on the photodetectors. The method employs photodiode detectors and is
more highly efficient than detection 共see, for example, Ref.
关27兴兲 in the photon counting experiments which have characterized tests of Bell inequalities so far. These previous experiments are limited by detection efficiencies to the extent
that no strong Bell inequality test has been performed to
date. Given the efficiency of the homodyne detection method
then, the smallness of the violation for the experiment proposed in this paper is not necessarily an indication of a relative lack of feasibility.
We then proceed to examine the effect of loss, such as
from nonideal detectors, on the violations of the strong Bell
inequalities.
IV. EFFECT OF DETECTION INEFFICIENCIES
AND LOSS
It is well known that a nonideal photon detector can be
modeled by a beam splitter followed by an ideal detector
关13兴. The attenuating beam splitter mixes our input signal
mode operator â with the vacuum operator â vac to give two
outputs ĉ A and d̂ A at location A:
ĉ A ⫽ 冑␩ â⫹ 冑1⫺ ␩ â vac ,
共21兲
d̂ A ⫽ 冑1⫺ ␩ â⫺ 冑␩ â vac,
where ␩ is the overall efficiency factor. The measured
quadrature phase operator is now that of ĉ A . With two spatially separated beam splitters modeling loss at each detector
we may write our total input state as
兩 in典 ⫽B 1/2
冕
2␲
0
兩 r 0 e i␵ 典 a 兩 0 典 a vac兩 r 0 e ⫺i␵ 典 b 兩 0 典 b vacd␵,
共22兲
where the 兩 0 典 represent the vacuum inputs for input modes
a vac and b vac to the two beam splitters. Using techniques
outlined in Yurke and Stoler 关28兴, one writes the output state
as
兩 out典 ⫽B 1/2
冕
2␲
0
兩 冑␩ r 0 e i␵ 典 c A 兩 冑1⫺ ␩ r 0 e i␵ 典 d A
⫻ 兩 冑␩ r 0 e ⫺i␵ 典 c B 兩 冑1⫺ ␩ r 0 e ⫺i␵ 典 d B d␵.
FIG. 5. Plot of the variation of the maximum value of S 共optimizing with respect to ␴ ) versus ␦ 1 and ␦ 2 . Here r 0 ⫽1.1.
共23兲
The final probability of observing results x ␪ and x ␾ for the
quadrature phase amplitude measurements in attenuated
modes c A and c B is
4264
A. GILCHRIST, P. DEUAR, AND M. D. REID
P ␪ , ␾ 共 x ␪ ,x ␾ 兲 ⫽
冕
⫹⬁
⫺⬁
dx vac, ␪
冕
⫹inf
⫺inf
dx vac, ␾
⫻円c A 具 x ␪ 兩 d A 具 x vac, ␪ 兩 c B 具 x ␾ 兩 d B 具 x vac, ␾ 兩 out典 円2 ,
共24兲
where we use Eq. 共15兲 to calculate the probability amplitude.
Calculation of the Gaussian integrals 关use 兰 d 2 x exp(⫺␭兩x兩2
⫹␮x*⫹␯x)⫽(␲/␭) exp(␮␯/␭) which is valid for any ␮ or ␯
and Re ␭⬎0兴 in x vac, ␪ and x vac, ␾ and simplification gives the
following modification of Eq. 共18兲:
P ␪ , ␾ 共 x ␪ ,x ␾ 兲 ⫽
冉
B
1
exp ⫺ 共 x ␪2 ⫹x ␾2 兲
2␲
2
冊冕 冕
2␲
0
d␽
2␲
0
d␽⬘
⫻exp兵 ⫺ 冑2 关 A 共 冑␩ r 0 兲 x ␪
„where ␣ is real and N ⫺1 ⫽2 关 1⫹exp(⫺2兩␣兩2)cos(␸)兴…
through a beam splitter with a second vacuum input such that
the output modes are given by Eq. 共21兲. Yurke and Stoler
关28兴 have considered this situation to show that the output
state is the two-mode cat state
兩 out典 ⫽N 共 兩 冑␩ ␣ 典 c 兩 冑1⫺ ␩ ␣ 典 d ⫹e i ␸ 兩 ⫺ 冑␩ ␣ 典 c 兩 ⫺ 冑1⫺ ␩ ␣ 典 d ).
共29兲
We evaluate the relevant probability distributions
P ␪ , ␾ (x,y) for measurement of X ␪A and X ␾B for the two-mode
superposition state 共26兲,
P ␪ , ␾ 共 x,y 兲 ⫽
A
2␲
共25兲
Following the calculation in Appendix A finally gives the
marginals 1/2 as before and the expression P ␪ , ␾ (x ␪ ,x ␾ ) unchanged except that k⫽⫺ 冑␩ r 0 / 冑2 in Eq. 共A15兲.
Calculations reveal violations to be negligible for ␩
⬃0.95. Such high efficiencies may be achievable with homodyne detection. However, the sensitivity to loss, also noticed in the observation of fringes due to quantum
‘‘Schrödinger-cat’’ states 关28兴, indicates that the limiting factor may well be the difficulty in the preparation of the quantum state.
V. TWO-MODE ‘‘SCHRÖDINGER-CAT’’ STATES
In this section we look at the possibility of violating the
inequality 共6兲 with the following superposition of two-mode
coherent states 共dropping explicit reference to the two modes
for brevity兲:
共26兲
where 円␣0‹⫽ 兩 ␣ 0 典 a 兩 ␤ 0 典 b and 兩 ␣ 0 典 a is a coherent state in the
mode â. The normalization is given by
A ⫺1 ⫽2 共 1⫹e ⫺2( 兩 ␣ 0 兩
2⫹兩␤ 兩2)
0
cos ␸ 兲 .
共27兲
Where the values of ␣ 0 , ␤ 0 are large this state becomes a
superposition of states macroscopically distinct in phase
space and thus resembles the Schrödinger-cat state. There
has been much discussion of the single-mode versions, Eq.
共14兲, of this state, and recent experimental developments
关23兴. Multimode even-odd coherent states were studied in
关29兴.
Both the EPR argument and the Bell inequalities make an
assumption of locality and hence we need to make measurements at two distinct locations, distant from each other. To
this end we have generalized the single-mode cat state 共14兲
to two modes which are separated as indicated in Fig. 1.
We note that the two-mode cat state can be generated in
principle by passing a single-mode ‘‘cat’’ state 共14兲,
N 1/2共 兩 ␣ 典 ⫹e i ␸ 兩 ⫺ ␣ 典 )
兺 K LR exp
L,R⫽⫺1
再
再
1
A, ␪ 2
⫺ 关 x ␪ ⫹ ␨ LR
兴
2
冎
共28兲
冎
共30兲
where in the above equation L and R index four terms with L
and R taking on the values ⫾1 and K LR is defined as
K LR ⫽exp兵 共 LR⫺1 兲共 兩 ␣ 0 兩 2 ⫹ 兩 ␤ 0 兩 2 ⫹Ri ␸ /2兲 其
共31兲
A, ␪
and the variable ␨ LR
⫽⫺(R ␣ 0 e ⫺i ␪ ⫹L ␣ 0* e i ␪ ) has been inB, ␾
for the
troduced to simplify the notation. The variable ␨ LR
other mode can be obtained simply by replacing ␣ 0 and ␪ by
␤ 0 and ␾ , respectively.
The event probabilities 共20兲 can then be written in terms
of error functions, for example, as
AB
P ⫹⫹
共 ␪,␾ 兲⫽
兩 cat典 ⫽A 1/2共 円␣0‹⫹e i ␸ 円ⴚ␣0‹兲 ,
⫹1
1
B, ␾ 2
⫻exp ⫺ 关 x ␾ ⫹ ␨ LR
兴 ,
2
⫹B 共 冑␩ r 0 兲 x ␾ 兴 ⫹C 共 冑␩ r 0 兲 其
⫻exp兵 2 共 1⫺ ␩ 兲 r 20 cos␽ ⫺ ␽ ⬘ 其 .
PRA 60
A
4
⫹1
兺
L,R⫽⫺1
A
B
E LR
共 ␪ 兲 E LR
共 ␾ 兲 K LR .
共32兲
A
A, ␪
B
Here E LR
( ␪ )⫽erfc( ␨ LR
/ 冑2) 关and similarly for E LR
( ␾ ) but
replacing A with B兴 has been introduced as shorthand for the
complementary error function of a particular argument.
Using techniques 关30兴 allowing calculation of the error
functions with imaginary arguments we conduct a numerical
search over all angles for a violation of the inequality 共6兲.
We found no violation, however.
VI. CONTRADICTION WITH LOCAL REALISM FOR
QUADRATURE PHASE AMPLITUDE MEASUREMENTS
ON A ‘‘SQUEEZED TWO-MODE CAT’’ STATE
In order to improve chances of observing both EPR correlations and contradiction with local hidden variables we
consider the two-mode superposition state evolved under the
action of a two-mode squeezing operator, corresponding
physically to interaction with a nondegenerate parametric
amplifier or equivalent system generating photon pairs. Thus
we consider interactions given by the interaction Hamiltonian,
H I ⫽iប ␬ 共 â † b̂ † ⫺âb̂ 兲 .
共33兲
A. EPR correlations for the squeezed cat state
It can be shown that such an evolution will generate EPR
correlated beams â and b̂, in the sense of the original EPR
argument 关1,10兴. We define the following particular quadra-
CONTRADICTION OF QUANTUM MECHANICS WITH . . .
PRA 60
4265
ture phase amplitudes for modes â and b̂: X̂ a ⫽â⫹â † and
P̂ a ⫽(â⫺â † )/i, and similar definitions for the mode b̂. Now
the solutions for the operators after a time t are
â 共 t 兲 ⫽â cosh共 ␬ t 兲 ⫹b̂ sinh共 ␬ t 兲 ,
b̂ 共 t 兲 ⫽b̂ cosh共 ␬ t 兲 ⫹â sinh共 ␬ t 兲 ,
共34兲
and for the quadrature phase operators
X̂ a 共 t 兲 ⫽X̂ a 共 0 兲 cosh共 ␬ t 兲 ⫹X̂ b 共 0 兲 sinh共 ␬ t 兲 ,
X̂ b 共 t 兲 ⫽X̂ b 共 0 兲 cosh共 ␬ t 兲 ⫹X̂ a 共 0 兲 sinh共 ␬ t 兲 ,
P̂ a 共 t 兲 ⫽ P̂ a 共 0 兲 cosh共 ␬ t 兲 ⫺ P̂ b 共 0 兲 sinh共 ␬ t 兲 ,
共35兲
P̂ b 共 t 兲 ⫽ P̂ b 共 0 兲 cosh共 ␬ t 兲 ⫺ P̂ a 共 0 兲 sinh共 ␬ t 兲 .
As ␬ t increases X̂ a (t) becomes increasingly correlated with
X̂ b (t) and P̂ a (t) becomes increasingly correlated with
⫺ P̂ b (t), with the correlation becoming perfect in the limit
␬ T→⬁. With modes â and b̂ spatially separated after interaction, this has been shown by one of us 关10兴 to give a direct
example of EPR correlations.
We can make two spatially separated measurements of the
correlated quantities in each mode. The results can be subtracted, yielding an estimate of the error in inferring the
value at A from a measurement at B. That is, we calculate
␦ x ⫽X̂ a (t)⫺ ␥ X̂ b (t) and ␦ p ⫽ P̂ a (t)⫹ ␥ P̂ b (t) 关10兴. The factor
␥ is a simple amplification factor which we shall modify to
give the best estimate possible 共the minimum error兲. Over an
ensemble of measurements we can calculate the variances
associated with our inference of X̂ a from X̂ b , and P̂ a from
P̂ b : ⌬ 2x ⫽ 具 ␦ 2x 典 ⫺ 具 ␦ x 典 2 and ⌬ 2p ⫽ 具 ␦ 2p 典 ⫺ 具 ␦ p 典 2 .
The minimum variance will occur for a particular value of
␥ . Hence finding the local turning point with ␥ yields
2
⫽
⌬ x,min
⌬X̂ a 共 T 兲 2 ⌬X̂ b 共 T 兲 2 ⫺ 关 具 X̂ a 共 T 兲 ,X̂ b 共 T 兲 典 兴 2
⌬X̂ b 共 T 兲 2
FIG. 6. The product of the minimum inference variances
2
⌬ x,min
⌬ 2p,min for the ‘‘squeezed’’ two-mode ‘‘cat’’ state. The EPR
incompleteness argument can be formulated when this product
drops below 1/g 4 共dashed line兲. The parameters are ␣ 0 ⫽ ␤ 0 ⫽0.9,
␸ ⫽0, and g⫽ 冑2.
␳ 共 t 兲 ⫽N̂ 兩 cat典具 cat兩 N̂ † .
共37兲
We write this as, recalling L and R take the values ⫹1 or
⫺1,
⫹1
␳ 共 t 兲 ⫽A
兺
L,R⫽⫺1
e (LR⫺1)Ri ␸ /2N̂D̂ 共 R ␣0 兲 円0‹Š0円D̂ † 共 L ␣0 兲 N̂ † ,
共38兲
where
†
D̂ 共 ␣0兲 ⫽exp共 ␣ 0 â † ⫺ ␣ *
0 â⫹ ␤ 0 b̂ ⫺ ␤ *
0 b̂ 兲 ,
N̂⫽exp共 ⫺iĤ I t/ប 兲 .
The symmetric characteristic function is
⫹1
, 共36兲
␹ s 共 ␩ 1 , ␩ 2 兲 ⫽Tr兵 ␳ 共 t 兲 e
␩ 1 â † ⫺ ␩ 1* â⫹ ␩ 2 b̂ † ⫺ ␩ 2* b̂
兺
其 ⫽A
L,R⫽⫺1
␹ LR ,
共39兲
⌬ 2p,min
and the covawhere we have a similar expression for
riance is 具 x,y 典 ⫽ 具 xy 典 ⫺ 具 x 典具 y 典 . We can calculate the necessary averages for the two-mode ‘‘cat’’ state using the equations of motion. It is then easy to calculate the minimum
variance product and this is illustrated in Fig. 6. As can be
2
⌬ 2p,min
seen from the figure, we predict that the product ⌬ x,min
2 2
4
drops below the quantum limit (⌬ x ⌬ p ⬍1/g ) illustrating
EPR correlations.
B. Contradiction with local realism
In order to search for a violation of the Bell inequality, we
must calculate the probability distributions for the results of
the two quadrature phase amplitude measurements at locations A and B. One may use the same techniques as used
above for the pair-coherent state. To give some visual information, however, we choose here to perform the calculation
by first calculating the Wigner function, which is easily
evaluated. The density operator for the system is
where
␹ LR ⫽ŠL␣0円N̂ † exp共 ␩ 1 â † ⫺ ␩ 1* â⫹ ␩ 2 b̂ † ⫺ ␩ 2* b̂ 兲 N̂円R␣0‹
⫻e (LR⫺1)Ri ␸ /2.
共40兲
Now, since N̂ is a unitary operator
†
␹ LR ⫽ŠL␣0円exp关 ␩ 1 â 共 t 兲 † ⫺ ␩ *
1 â 共 t 兲 ⫹ ␩ 2 b̂ 共 t 兲
⫺ ␩ 2* b̂ 共 t 兲兴 円R␣0‹e (LR⫺1)Ri ␸ /2,
共41兲
where the operators â(t) and b̂(t) are given by the equations
of motion 共34兲. Normally ordering the products in the exponential yields
再
冎
1
␹ LR ⫽exp ⫺ ␩ † A1␩ ⫺ ␩ † XLR⫹XRL† ␩ K LR ,
2
共42兲
A. GILCHRIST, P. DEUAR, AND M. D. REID
4266
PRA 60
where
␩ ⫽ 共 ␩ 1 ,⫺ ␩ 2* 兲 T ,
XLR⫽
冉
R cosh共 ␬ t 兲 ␣ 0 ⫹
L sinh共 ␬ t 兲 ␤ 0*
L cosh共 ␬ t 兲 ␤ *
0⫹
R sinh共 ␬ t 兲 ␣ 0
A1⫽
冉
cosh共 2 ␬ t 兲
sinh共 2 ␬ t 兲
sinh共 2 ␬ t 兲
cosh共 2 ␬ t 兲
冊
冊
,
.
The Wigner function is the Fourier transform of the characteristic function:
⫹1
兺
W⫽A
L,R⫽⫺1
W LR ⫽
冕
␲
1
4
共43兲
W LR ,
†␩⫹␩†X
d 2 ␩ e ⫺X
␹ LR ,
共44兲
with X⫽( ␣ , ␤ * ) T . We can use the result for Gaussian integrals
I⫽
冕
d 2 ␩ exp共 ⫺␭ ␩ † A␩ ⫹ ␩ † x1 ⫹x†2 ␩ 兲
⫽ 兩 A兩 ⫺1
冉冊 冉
␲
␭
D
共45兲
冊
1
exp x†2 A⫺1 x1 ,
␭
where D is the dimension of the vectors to evaluate the integrals, yielding
W LR ⫽
4K LR
␲2
exp兵 ⫺2 共 X⫺XRL 兲 † A2 共 X⫺XLR 兲 其 ,
共46兲
FIG. 7. Wigner function plotted with x a ⫽x b ⫽x and p a ⫽p b
⫽p and ␸ ⫽0 for an initial two-mode ‘‘cat’’ state with ␣ 0 ⫽ ␤ 0
⫽2.
where
A2 ⫽A⫺1
1 ⫽
冉
cosh共 2 ␬ t 兲
⫺sinh共 2 ␬ t 兲
⫺sinh共 2 ␬ t 兲
cosh共 2 ␬ t 兲
冊
.
共47兲
Now, we can introduce new variables x and p such that
␣ ⫽(x a ⫹ip a )/2 and ␤ ⫽(x b ⫹ip b )/2. Using the vectors x
⫽(x a ,x b ) T and p⫽(p a ,⫺p b ) T , together with x0 ⫽(XLR
* )/g and p0 ⫽(XLR ⫺XRL
* )/gi, Eq. 共46兲 then becomes
⫹XRL
W LR ⫽
g 4 K LR
4␲2
⫺
再
exp ⫺
g2
共 x⫺x0 兲 T A2 共 x⫺x0 兲
2
冎
g2
共 p⫺p0 兲 T A2 共 p⫺p0 兲 .
2
共48兲
Setting ␬ T⫽0 gives the Wigner function for the twomode cat state, plotted in Fig. 7. We observe the presence of
fringes and note the function is negative. The figure illustrates the effect of increasing ␬ T. Note that the Wigner function remains negative, preventing the interpretation of the
Wigner function as a direct hidden variable theory.
In the limit of large interaction time where cosh(␬t)
⬇sinh(␬t) then
冉冊
x0 ⫽
1
2 cosh共 ␬ t 兲
,
Re共 R ␣ 0 ⫹L ␤ 0 兲
g
1
p0 ⫽
1
2 cosh共 ␬ t 兲
Im共 R ␣ 0 ⫺L ␤ 0 兲
,
g
1
冉冊
where here Re and Im refer to real and imaginary parts. Both
of these expressions are real, and hence there are no complex
terms in W LR that can lead to oscillations. Here W LR becomes a Gaussian and is everywhere positive, and can act as
a local hidden variable theory for quadrature phase measurements. We conclude therefore that no contradiction of local
realism for our proposed experiment will occur in this limit.
We introduce the following variables to include a homodyne measurement at an arbitrary phase angle. That is, we
start from Eq. 共46兲 but introduce the more general x ␪ , x ␾ ,
p ␪ and p ␾ : ␣ ⫽e i ␪ (x ␪ ⫹ip ␪ )/2 and ␤ ⫽e i ␾ (x ␾ ⫹i p ␾ )/2. We
define the vectors x⫽(x ␪ ,x ␾ ) T , p⫽( p ␪ ,⫺ p ␾ ) T . Now we
obtain
CONTRADICTION OF QUANTUM MECHANICS WITH . . .
PRA 60
W LR ⫽
g 4 K LR
4␲2
再 冉
exp ⫺
冉
g2
2
x⫹ip⫺ XRL
2
g
2
⫻A3 x⫹ip⫺ XLR
g
冊冎
冊
4267
†
共49兲
,
with
A3 ⫽E† A2 E⫽
冉
cosh共 2 ␬ t 兲
⫺sinh共 2 ␬ t 兲 e ⫺i( ␪ ⫹ ␾ )
⫺sinh共 2 ␬ t 兲 e i( ␪ ⫹ ␾ )
cosh共 2 ␬ t 兲
冊
共50兲
and
E⫽
冉
e i␪
0
0
e
⫺i ␾
冊
共51兲
.
We need to integrate out the p terms as this will yield
P ␪ , ␾ (x,y):
P ␪ , ␾ 共 x,y 兲 ⫽
冕
dpW,
共52兲
⫹1
P ␪ , ␾ 共 x,y 兲 ⫽
兺
L,R⫽⫺1
P ␪ , ␾ 共 x,y 兲 LR .
Now, expanding out Eq. 共49兲 and integrating over p ␪ followed by p ␾ leads to a messy expression which, after some
work, can be simplified as
P ␪ , ␾ 共 x,y 兲 LR ⫽
⫽
g 2 K LR
2 ␲ 冑C
g 2 K LR
2 ␲ 冑C
再
exp ⫺
g 2 T ⫺1
x A4 x⫹RT x⫹G
2
冎
exp兵 ⫺a 共 x ␪2 ⫹x ␾2 兲 ⫹2bx ␪ x ␾ ⫹R 1 x ␪
共53兲
where
C⫽cosh共 2 ␬ t 兲 2 ⫺sinh共 2 ␬ t 兲 2 cos共 ␪ ⫹ ␾ 兲 ,
冉
cosh共 2 ␬ t 兲
sinh共 2 ␬ t 兲 cos共 ␪ ⫹ ␾ 兲
sinh共 2 ␬ t 兲 cos共 ␪ ⫹ ␾ 兲
cosh共 2 ␬ t 兲
* 兲
R⫽g 共 A3 X̃LR ⫹AT3 X̃RL
⫺
G⫽
g
* 兲,
共 A ⫺AT3 兲 A4 共 A3 X̃LR ⫺AT3 X̃RL
2C 3
⫺1
* 兲 T A4 共 A3 X̃LR ⫺AT3 X̃RL
* 兲
共 A X̃ ⫺AT3 X̃RL
2C 3 LR
†
⫺2X̃RL
A3 X̃LR ,
X̃LR ⫽E⫺1 XLR ,
a⫽g 2 cosh共 2 ␬ t 兲 /2C,
b⫽g 2 sinh共 2 ␬ t 兲 cos共 ␪ ⫹ ␾ 兲 /2C.
Now, finally, we integrate over a quadrant of the twodimensional Gaussian to get the joint probability of detecting
a (⫹) result at A and a (⫹) result at B.
⫹1
AB
P ⫹⫹
共 ␪,␾ 兲⫽
AB
P ⫹⫹
共 ␪ , ␾ 兲 LR ⫽
⫹R 2 x ␾ ⫹G 其 ,
A4 ⫽
FIG. 8. The violation of the CH-Bell inequality found for a
squeezed two-mode ‘‘cat’’ state. A contour plot of S that contains
the maximum violation found. Parameters ␣ 0 ⫽ ␤ 0 ⫽0.9, ␬ t⫽0.6,
␸ ⫽0, and ␦ ⫽0.7␲ . The maximum achieved violation is S
⫽1.008 at ␪ ⫽ ␾ ⫽0.42␲ .
冊
,
兺
AB
P ⫹⫹
共 ␪ , ␾ 兲 LR ,
L,R⫽⫺1
冕 冕
⬁
0
dx ␪
⬁
0
dx ␾ P ␪ , ␾ 共 x,y 兲 LR .
共54兲
We may integrate this directly by numerical integration. Alternatively some work 共Appendix C兲 allows for expressions
which may be evaluated more readily by numerical techniques allowing for a search with a wide range of angles. The
main obstacle to finding a violation is the large size of parameter space that needs to be searched. The full CH-Bell
inequality depends on values of ␣ 0 , ␤ 0 , ␬ t, and ␸ . Again it
is useful do define the angles ␦ 1 ⫽ ␪ ⫺ ␪ ⬘ and ␦ 2 ⫽ ␾ ⬘ ⫺ ␾ . A
quick search reveals that the maximum value of our CH-Bell
inequality always seems to occur for ␦ 1 ⫽ ␦ 2 . In order to
reduce the size of the search of parameter space, we will take
␦ 1 ⫽ ␦ 2 in the following calculations. Given ␦ 1 ⫽ ␦ 2 ⫽ ␦ , the
CH-Bell inequality can be reduced to a three-angle form:
S( ␪ , ␾ , ␦ ). We shall also assume that ␣ 0 ⫽ ␤ 0 and further that
this value is a real number.
With these restrictions a preliminary search of parameter
space does indeed find a violation, with the ‘‘best’’ value
occurring for the parameters ␣ 0 ⫽ ␤ 0 ⫽0.9 and ␬ t⫽0.6.
Exploring the behavior of the maximum of S( ␪ , ␾ , ␦ )
with ␸ shows the behavior that seems independent of the
other parameters. In this search the domains of ␪ , ␾ , ␦ , and
␸ were divided into 50 points.
Henceforth we will choose ␸ ⫽0. Examining the behavior
with ␦ also gives a preferred value for this parameter, with
␦ ⫽0.7␲ .
Finally we are in a position to plot S( ␪ , ␾ ) and this is
performed in Figs. 8 and 9, which show a region where the
A. GILCHRIST, P. DEUAR, AND M. D. REID
4268
PRA 60
B
Similarly we can construct 关note that we must have P ⫹
(␾)
A
⫽ P ⫹ ( ␾ ) from the symmetry of the pair-coherent state under
interchange of â with b̂]
A
P⫹
共 ␪ 兲⫽
B
2
冕 ␽冕
2␲
2␲
d
0
0
d␽⬘
⫻exp兵 2r 20 关 cos共 ␽ ⫺ ␽ ⬘ 兲 ⫺1 兴 其 erfc共 A 兲 . 共A2兲
We will employ a power series expansion for the error
functions:
erf共 k 关 e ⫹e
ia
FIG. 9. The violation of the CH-Bell inequality found for the
squeezed two-mode ‘‘cat’’ state. The behavior of S with ␣ 0 共note
that ␣ 0 ⫽ ␤ 0 ). The parameters are ␬ t⫽0.6, ␸ ⫽0, ␪ ⫽ ␾ ⫽0.42␲ ,
and ␦ ⫽0.7␲ .
violation occurs. The maximum value reached is S⫽1.008 at
␪ ⫽ ␾ ⫽0.42␲ .
Now we can examine the behavior with ␣ 0 in more detail.
This is depicted in Fig. 9. Notice that S rapidly approaches
one so that the violation is for a small parameter range only.
Note also that S seems to approach one asymptotically from
above, seeming to imply that a macroscopically sized superposition state would also violate the CH-Bell inequality
共though by a tiny amount兲.
VII. CONCLUSION
We have expanded on our previous publication to present
two quantum systems which give a violation with local realism for experiments involving only quadrature phase amplitude 共position and momentum兲 measurements. A small but
conclusive deviation from a Bell inequality has been found
for a pair-coherent state and a suitably squeezed superposition of two two-mode coherent states. The effect of detection
inefficiency and loss on the violation has been calculated
共transmission of order 95% required兲. While such efficiencies may be obtainable by the homodyne detection procedure, this sensitivity to loss may hinder the generation of the
suitable quantum state.
ACKNOWLEDGMENTS
We are grateful to C. W. Gardiner and W. J. Munro for
many helpful suggestions.
Provided we can interchange the order of integration and
complete the square, we write, using the result 共18兲, Eq. 共20兲
as
B
4
冕 冕
2␲
0
d␽
2␲
0
兴 兲⫽
⬁
2
2n⫹1
兺 r⫽0
兺
冑␲ n⫽0
共 ⫺1 兲 n k 2n⫹1 共 2n 兲 !
n!r! 共 2n⫹1⫺r 兲 !
共A3兲
⫻e iu(2n⫹1⫺r) e ia(2r⫺2n⫺1) ,
where we have also used the binomial expansion and the
substitution a ⬘ ⫽a⫺u. Note the following result which
arises when integrating the above expression:
冕
2␲
0
da erf共 k 关 e ia ⫹e ⫺ia ⬘ 兴 兲 ⫽
⫽
共 . . . 兲冕
兺
n,r
0
2␲
dae ia(2r⫺2n⫺1)
共 . . . 兲 2 ␲ ␦ r,n⫹1/2⫽0,
兺
n,r
共A4兲
where ( . . . ) denotes the terms in Eq. 共A3兲 that are not explicitly written. This result follows since both r and n are
integral values and the delta function ␦ r,n⫹1/2 will always be
zero.
Now we have
2
A
P⫹
共 ␪ 兲⫽
Be ⫺2r 0
2
冕 ␽冕
2␲
2␲
d
0
0
再 冉
r0
⫻ 1⫺erf ⫺
冑2
d ␽ ⬘ e 2r 0 cos( ␽ ⫺ ␽ ⬘ )
2
关 e i( ␽ ⫺ ␹ ) ⫹e ⫺i( ␽ ⬘ ⫺ ␹ ) 兴
冊冎
.
共A5兲
With a change of variables to a⫽ ␽ ⫺ ␹ and a ⬘ ⫽a⫺u,
where u⫽ ␽ ⫺ ␽ ⬘ , it is evident that because of Eq. 共A4兲, the
integral over the error function vanishes, leaving an integral
that can be identified as a Bessel function:
2
A
P⫹
共 ␪ 兲⫽
Be ⫺2r 0
1
共 2 ␲ 兲 2 I 0 共 2r 20 兲 ⫽ .
2
2
共A6兲
AB
( ␪ , ␾ ) can be treated in a similar way.
Now, P ⫹⫹
APPENDIX A
AB
P ⫹⫹
共 ␪,␾ 兲⫽
⫺ia ⬘
2
AB
P ⫹⫹
共 ␪,␾ 兲⫽
Be ⫺2r 0
4
冕 ␽冕
2␲
2␲
d
0
0
d ␽ ⬘ e 2r 0 cos( ␽ ⫺ ␽ ⬘ )
2
⫻ 兵 1⫺erf共 A 兲 ⫺erf共 B 兲 ⫹erf共 A 兲 erf共 B 兲 其
d␽⬘
共A7兲
⫻exp兵 2r 20 关 cos共 ␽ ⫺ ␽ ⬘ 兲 ⫺1 兴 其 erfc共 A 兲 erfc共 B 兲 .
共A1兲
and it can be seen that the first term will give a value of 41 ,
the next two terms will vanish, and the last term is the only
one that will present any difficulty. Hence we can write
PRA 60
CONTRADICTION OF QUANTUM MECHANICS WITH . . .
1
AB
P ⫹⫹
共 ␪ , ␾ 兲 ⫽ ⫹F„erf共 A 兲 erf共 B 兲 …
4
共A8兲
where Xs is a vector of values indexed by s and given explicitly as
M
and F is the function left after dropping the first three terms
in Eq. 共A7兲. As before, each of the terms in the product
erf(A)erf(B) can be expanded in a power series, yielding
4k 2
erf共 A 兲 erf共 B 兲 ⫽
␲
⬁
⬁
4269
Xs ⫽
M
兺
兺 f
n⫽ 兩 s/2兩 ⫺1/2 m⫽ 兩 s/2兩 ⫺ 1/2
冉
n,m,
s⫹2n⫹1
2
⫻I n⫺m 共 2r 20 兲共 ⫺k 2 兲 n⫹m .
2n⫹1 2m⫹1
兺 兺 兺 p⫽0
兺
冊
共A16兲
Equation 共A15兲 is the definition of a discrete Fourier transform so by using the fast Fourier transform we can evaluate
this expression very quickly.
n⫽0 m⫽0 r⫽0
⫻ f 共 n,m,r, p 兲共 ⫺k 2 兲 m⫹n
⫻e iu(2n⫺r⫹p⫺2m) e 2i ␽ (r⫺n⫹m⫺p)
⫻e ⫺i ␹ (2r⫺2n⫺1) ,
APPENDIX B
共A9兲
共 2n 兲 ! 共 2m 兲 !
f 共 n,m,r,p 兲 ⫽
,
n!m!r!p! 共 2n⫹1⫺r 兲 ! 共 2m⫹1⫺ p 兲 !
共A10兲
where k⫽⫺r 0 / 冑2 and the variable change of ␽ ⬘ ⫽ ␽ ⫺u
has been utilized.
The integrals will now yield a Kronecker ␦ function and a
modified Bessel function:
In the limit of large r 0 the exponent in Eq. 共A1兲 becomes
the delta function ␦ ( ␽ ⫺ ␽ ⬘ ):
lim
r 0 →⬁
冕
2␲
0
d ␽ ⬘ exp兵 2r 20 关 cos共 ␽ ⫺ ␽ ⬘ 兲 ⫺1 兴 其 ⫽
⫻
冕
2␲
兺 f 共 n,m,r, p 兲 e ⫺i ␹ (2r⫺2n⫺1)
n,m,r,p
2
due 2r 0 cos(u) e iu(2n⫺r⫹p⫺2m)
0
冕
2␲
0
2
⫽4 ␲ Be ⫺2r 0 k 2
⫻e
兺
n,m,r,p
⫺i ␹ (2r⫺2n⫺1)
␦ m⫺p,n⫺r I 2(n⫺m)⫹p⫺r 共 2r 20 兲 .
共A12兲
Utilizing the ␦ function by setting p⫽m⫺n⫹r on the understanding that the factorial sequences terminate at zero, we
can write
2
⬁
F⫽4 ␲ Be ⫺2r 0 k 2
⬁
2n⫹1
兺 兺 兺
n⫽0 m⫽0 r⫽0
f 共 n,m,r 兲共 ⫺k 2 兲 m⫹n
⫻I n⫺m 共 2r 20 兲 e ⫺i ␹ (2r⫺2n⫺1) ,
共 2n 兲 ! 共 2m 兲 !
.
n!m!r! 共 r⫹m⫺n 兲 ! 共 n⫹m⫺r⫹1 兲 ! 共 2n⫹1⫺r 兲 !
共A14兲
We can now change variables in the summations to s
⫽2r⫺2n⫺1. Truncating the n and m summations at some
value M will yield
F⫽4 ␲ Be
2
2M ⫹1
⫺2r 0 2
k
兺
s⫽⫺(2M ⫹1),s odd
4r 0
A
P⫹
共 ␪ 兲⫽
冑␲ B
Xs e ⫺i ␹ s ,
共A15兲
冕
2␲
0
2r 0
冕
d ␽ erfc关 ⫺ 冑2r 0 cos共 ␽ ⫺ ␹ 兲兴
2␲
0
共B2兲
d ␽ erfc关 ⫺ 冑2r 0 cos共 ␽ ⫺ ␹ 兲兴 . 共B3兲
For large r 0 , erfc关 ⫺ 冑2r 0 cos(␽⫺␹)兴 acts like a step function
and hence we can evaluate the remaining integrals. Noting
that for large r 0 the normalization constant evaluates to
B ⫺1 ⫽2 ␲ 冑␲ /r 0 we get
AB
P ⫹⫹
共 ␪,␾ 兲⫽
冏
冏
1 共 ␪ ⫹ ␾ 兲 mod 2 ␲
⫺
,
2
2␲
A
B
P⫹
共 ␪ 兲⫽ P⫹
共 ␾ 兲 ⫽1/2.
共A13兲
f 共 n,m,r 兲
⫽
冑␲ B
⫻erfc关 ⫺ 冑2r 0 cos共 ␽ 兲兴 ,
d ␽ e 2i ␽ (r⫺n⫹m⫺p)
f 共 n,m,r, p 兲
␦共 ␽⫺␽⬘兲
in which case performing the ␽ ⬘ integrals we get
AB
P ⫹⫹
共 ␪,␾ 兲⫽
共A11兲
r0
共B1兲
2
Be ⫺2r 0 k 2
F⫽
␲
冑␲
共B4兲
共B5兲
APPENDIX C
AB
In order to evaluate the expression P ⫹⫹
( ␪ , ␾ ) we rotate
the axis by ␲ /4, then scale the axis, and finally change to
polar coordinates: (x ␪ ⫹x ␾ ) 冑(a⫺b)/2⫽r cos ␻ and (x ␪
⫺x ␾ ) 冑(a⫹b)/2⫽r sin ␻. With this transformation we arrive at
AB
P ⫹⫹
共 ␪ , ␾ 兲 LR ⫽
K LR
␲
冕 冕
⬁
dr r
0
␻0
⫺␻0
2
d ␻ e ⫺r e r(Acos ␻ ⫹Bsin ␻ ) ,
共C1兲
where ␻ 0 ⫽tan⫺1 关 冑(a⫹b)/(a⫺b) 兴 . Using a power series
expansion will give
A. GILCHRIST, P. DEUAR, AND M. D. REID
4270
AB
P ⫹⫹
共 ␪ , ␾ 兲 LR ⫽
K LR
␲
⫻
冕
兺冕
⬁
n⫽0
␻0
⫺␻0
d␻
⬁
0
r n⫹1 e ⫺r
dr
n!
PRA 60
AB
P ⫹⫹
共 ␪ , ␾ 兲 LR
2
冋冉 冊 冉 冊 册
A⫺iB i ␻
A⫹iB ⫺i ␻ n
e ⫹
e
,
2
2
⫽
A⫽ 共 R 1 ⫹R 2 兲 / 冑2 关 cosh共 2 ␬ t 兲 ⫹sinh共 2 ␬ t 兲 cos共 ␪ ⫹ ␾ 兲兴 ,
B⫽ 共 R 1 ⫺R 2 兲 / 冑2 关 cosh共 2 ␬ t 兲 ⫺sinh共 2 ␬ t 兲 cos共 ␪ ⫹ ␾ 兲兴 .
The r integral in Eq. 共C2兲 gives a gamma function ⌫, while
the ␻ expression can be further expanded as a binomial series, and upon integration gives
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再
⬁
n
兺兺
⌫ 共 n/2⫹1 兲 A r2 B n⫺r
2
共 n⫺r 兲 !r!
n⫽0 r⫽0
sin共 ␻ 0 关 2r⫺n 兴 兲
,
共 2r⫺n 兲
⫻
␻0 ,
共C2兲
where
K LR
␲
2r⫽n
共C3兲
2r⫽n
where A 2 ⫽(A⫺iB)/2 and B 2 ⫽(A⫹iB)/2.
A
B
( ␪ ) and P ⫹
( ␾ ) is a straightforward modiCalculating P ⫹
fication of the previous calculation. Performing the integrations yields, after some manipulation,
A
P⫹
共 ␪ 兲 LR ⫽
K LR
erfc
2
B
P⫹
共 ␾ 兲 LR ⫽
冉
冉
K LR
erfc
2
⫺1
冑2cosh共 2 ␬ t 兲
⫺1
冑2cosh共 2 ␬ t 兲
冊
冊
* ⫹E* XLR 兲 ,
uT1 共 EXRL
共C4兲
* ⫹E* XLR 兲 ,
uT2 共 EXRL
共C5兲
where u1 ⫽(1,0) and u2 ⫽(0,1) .
T
T
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关24兴
关25兴
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