We present a potential realization of the Greenberger, Horne and Zeilinger ALL or NOTHING contradiction of quantum mechanics with local realism using phase measurement techniques in a simple photon number triplet. Such a triplet could be generated using nondegenerate parametric oscillation

Milburn, Gerard J.

Munro, W. J.

University of Queensland eSpace

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Characterizing GHZ Correlations in Nondegenerate Parametric Oscillation via Phase
Measurements
W. J. Munro and G. J. Milburn
Center for Laser Science,Department of Physics, University of Queensland, QLD 4072, Brisbane, Australia
(December 6, 2002)
We present a potential realization of the Greenberger, Horne and Zeilinger “all or nothing”
contradiction of quantum mechanics with local realism using phase measurement techniques in a
simple photon number triplet. Such a triplet could be generated using nondegenerate parametric
oscillation.
Many of the traditional tests of quantum mechan-
ics (using the Bell Inequalities [1,2]) have used paramet-
ric amplifiers or oscillators to generate correlated pho-
ton number states [3–5]. When these correlated pho-
ton pairs (the signal and idler) are passed through po-
larisers or beamsplitter/phase-shifters and, measured by
single photon detectors, a test of the Bell inequality
can be achieved. Such tests however require auxiliary
conditions [2] that lessen (or call into doubt) the con-
tradiction. Such contradictions are very microscopic in
nature as they involve single photon detection. Multi-
particle tests of the Bell inequality have also been pro-
posed using parametric amplification [6,7]. No multi-
particle test of the Bell inequality has ever been experi-
mentally considered.
The quantum states described by Greenberger, Horne
and Zeilinger [8–12] (GHZ states) give predictions con-
trary to those of all classical theories based on the
Einstein-Podolsky-Rosen [13] (EPR) premises of local re-
alism. The spin GHZ state is an entangled state of three
spins specified by stating that all spins are in the same di-
rection. As this correlation can be realized in two ways,
the state is the sum of the two amplitudes represent-
ing each way separately. The resulting interference be-
tween these two amplitudes ensures that there is a partic-
ular result for triple product spin measurements that can
never occur. In contrast, a classical local hidden variable
state exhibiting the same correlation (that is all spins the
same) will produce this forbidden result with a non-zero
probability. If this forbidden result were ever observed in
a perfect experiment, the quantum prediction would be
incorrect. On the contrary, never observing the forbidden
result would verify quantum mechanics. Unfortunately
not observing an event is a difficult way to test a theory
experimentally. Detector inefficiencies may also lead to
the non-observation of the forbidden result for reasons
that have nothing to do with quantum entanglement.
The GHZ paradox can be formulated as follows: Con-
sider three spin 12 particles in a state | ↑〉| ↑〉| ↑〉 + | ↓
〉| ↓〉| ↓〉 where the ↑ or ↓ specifies spin up or down
along the appropriate z axis. These particles originate in
a spin conserving gendanken decay and fly apart along
three different straight lines in the x − y plane. Now
because the spin vectors of distinct particles commute
component by component, we can simultaneously mea-
sure the x component of one particle and the y compo-
nents of the remaining two. In fact for the given initial
state the product of the results of the three spin mea-
surements Sx1Sy2Sy3, Sy1Sy2Sx3, Sy1Sx2Sy3, where Sxi
and Syi represent the spin along the horizontal and ver-
tical directions, has to be +1 according to both quantum
mechanics and local realism. According to local realism
the spin product Sx1Sx2Sx3 must also be unity. Such
a product can also be calculated quantum mechanically
and in fact is found to be minus the product of all the
three of them. To account for experimental situations
where the spin product predictions are not unity in size,
Mermin [14] derived the following inequality based on lo-
cal realism arguments
F = |Sx1Sx2 Sx3−Sy1Sy2Sx3
− Sy1Sx2Sy3 − Sx1Sy2Sy3 | ≤ 2. (1)
To date there have been no tests of the GHZ inequality
given by (1), due mostly to the difficult nature of generat-
ing a triple spin state of the form | ↑〉| ↑〉| ↑〉+ | ↓〉| ↓〉| ↓〉.
Recent developments by Laflamme et. al [15] have seen
the generation of a GHZ state using the proton and
carbon spins of trichloroethylene in NMR spectroscopy.
They have shown using state tomography techniques
that a 95% construction of the triple spin state can be
achieved. Because such an experiment was done using a
molecule no significant separation of the photon/carbon
spins could be achieved and hence a test of the local-
ity condition implicate in the GHZ paradox could not be
made.
In this letter we propose a novel use of phase mea-
surements to test the GHZ correlations. We will show
how a simple correlated photon number triplet could be
used to provide a definitive test. Such a state could be
produced via nondegenerate parametric oscillation where
we have signal, idler and pump modes. Discrete phase
measurements are however difficult to realize experimen-
tally and hence we consider how a homodyne quadra-
ture phase amplitude measurement can provide a more
1
realisable test. In a homodyne measurement the signal
field is coupled to a strong local oscillator, hence pro-
viding very efficient detection. Current homodyne detec-
tion efficiency [16] can exceed 99%, thus providing a more
stringent test by removing potential detection efficiency
loopholes [17–19]. Also the use of the strong local oscilla-
tor field means that large intensities are incident on the
highly efficient detectors.
A quantum entangled state shows correlations that
cannot be explained in terms of the correlations between
local classical properties of the subsystems. In this letter
we will describe a pure entangled state of three modes
in which the correlations are in photon number. More
specifically, the nature of the correlation can be suc-
cinctly stated by saying that there are equal number of
photons in each mode. As there are many different ways
to realize this fact, the total state is the sum over am-
plitudes for all possible ways in which this correlation
can be realized. This kind of sum over amplitudes for
correlations is characteristic of an entangled state.
The question now arises as how best to see the quan-
tum nature of the correlation. Obviously it is not enough
to measure photon number as this would not distinguish
a mixed state with equal photon numbers in each mode,
from the equivalent entangled pure state. In some sense
we need to measure an observable which carries as lit-
tle information as possible about photon number in or-
der to see the interference between all the possible ways
in which the correlation in photon number can be real-
ized. We conjecture that the best choice is the observable
canonically conjugate to photon number: the canonical
phase.
Pegg and Barnett [20,21] have shown that a set of s+1
orthonormal phase states, with values of θ differing by
2pi/(s+ 1), can be generated from
|θµ〉 = exp
[
iNˆµ2pi/(s+ 1)
]
|θ0〉, µ = 0, . . . , s (2)
where |θ0〉 is the reference (or zero) phase state, Nˆ is the
number operator and µ the particular discrete phase we
are interested in. The values for θµ are given by
θµ = θ0 +
2µpi
(s+ 1)
(3)
which are spread evenly over the range θ0 ≤ θµ ≤ θ0+2pi,
where θ0 is the initial (or reference) phase.
The probability of finding a generalized system |Ψ〉 in
a particular phase state |θµ〉 is
Pµ(θ0) = |〈Ψ|θµ〉|2 (4)
where µ labels the particular phase state, and θ0 is the
choice of initial phase.
We require large s to describe an arbitrary phase for
a general system. However, in the case of the measure-
ment schemes required for various quantum violations of
classical inequalities such as the Bell [1] and GHZ [8–11]
(or Mermin higher spin [14]) inequalities, all that is re-
quired and necessary is a binary result. Thus a discrete
phase measurement with s = 1 suffices, that is two phase
states are sufficient. If more phase states are chosen, for
example s = 3, a binary result is still required for these
particular quantum inequalities, which could be achieved
by dividing or binning the phase states into two discrete
distinct sets. However this will not be ideal as to get
this binary result we must discard information. Such a
process must lessen (or destroy) our potential GHZ vio-
lation.
Production of a state of the form
|Ψ〉 = 1√
2
| ↑〉| ↑〉| ↑〉+ 1√
2
| ↓〉| ↓〉| ↓〉 (5)
where ↑, ↓ represent the spin of the particle, has been
difficult to achieve experimentally. Reid and Munro [23]
have considered previously a photon triplet state
|Ψ〉 = 1√
2
|0〉|0〉|0〉+ 1√
2
|1〉|1〉|1〉 (6)
which can also be used to test the GHZ inequality. Pro-
duction of this triplet has yet to be realized. Potential
for similar photon triplet state production exists in para-
metric oscillation. The ideal nondegenerate parametric
oscillator may be specified by an interaction Hamiltonian
of the form
Hint = ih¯χ
[
c†ab− ca†b†] (7)
where aˆ, bˆ and cˆ are the boson annihilation operators for
the signal, idler, and pump modes, respectively and χ
is the parametric coupling constant. Initially preparing
the pump mode in a single Fock state |1〉, with the sig-
nal and idler modes initially in vacuum states, it can be
easily shown that
|Ψ〉 = c0|0〉|0〉|1〉+ c1|1〉|1〉|0〉 (8)
can be generated where normalization requires |c0|2 +
|c1|2 = 1. The state (8) is also a stable soliton
solution [22] when the system is driven by a classical
pump field coupled to mode cˆ.
Given the state (8) one can calculate the probability of
obtaining the phase states θµ1 , θµ2 , θµ3 (where the labels
µ1, µ2, µ3 corresponds to the aˆ, bˆ and cˆ modes respec-
tively). For s = 1, a choice of only two phase states, we
have
Pµ1µ2µ3 (θ0,1, θ0,2, θ0,3) = |〈Ψ|θµ1〉|θµ2 〉|θµ3〉|2
=
1
8
+
1
4
c0c1 cos [(µ1 + µ2 − µ3)pi + ψ0] (9)
where µi is zero or one, and ψ0 = θ0,1+θ0,2−θ0,3. We ex-
plicitly note that our initial phases for the three particles
2
θ0,i can be expressed as one ψ0. To classify our binary
result we say that µi = 1 corresponds to a “1” measure-
ment, while µi = 0 corresponds to a “0” measurement
(for each of the particles). If we consider a single par-
ticle, then there is a probability of detecting it in the
“1” (labeling the probability P1) or the “0” state (label-
ing this P0). Hence the probability of all three particles
being in a “1” state is
P111 (ψ0) =
1
8
− 1
4
c0c1 cos [ψ0] (10)
Similarly we can calculate the probability P000 of all par-
ticles being in a “0” state
P000 (ψ0) =
1
8
+
1
4
c0c1 cos [ψ0] (11)
Other probabilities such as P001 can be calculated in an
identical manner. It is necessary to point out that prob-
abilities such as P001 and P010 are not identical due to
the asymmetric initial state (8).
We define the spin of a single particle i as
Si(θ0,i) = P1(θ0,i)− P0(θ0,i) (12)
where we are explicitly indicating that the spin depends
on the initial reference angle choice θ0,i. The spin prod-
uct of the three particles is then the product of each of
the spins. Hence the triple spin product is
S1S2S3(ψ0) = −2c0c1 cos [ψ0] (13)
where we use the label Si to represent the spin of the i
th
particle and the angle ψ0 to represent the total simplified
initial phase choice. Given a triple spin product it is now
possible to examine the GHZ paradox.
Generally, previous authors [14,23] have considered a
GHZ inequality of the form (1). However because of our
asymmetric initial state, we will consider the following
inequality
F = |Sy1Sy2 Sx3−Sx1Sx2Sx3
− Sy1Sx2Sy3 − Sx1Sy2Sy3 | ≤ 2 (14)
which can be derived in an identical way to (1). We note
that according to local realism
Sy1Sy2Sx3 = Sx1Sx2Sx3 × Sy1Sx2Sy3 × Sx1Sy2Sy3 (15)
provided the magnitude of each spin product is one.
Now according to local realism, the triple spin product
Sy1Sy2Sx3 has the same sign as the product of the other
three triple products. It can be shown that the three
triple spin products Sx1Sx2Sx3 , Sy1Sx2Sy3 , Sx1Sy2Sy3 all
have the same negative sign and hence Sy1Sy2Sx3 should
be negative. Hence adding all four spin products together
according to (14) will give F ≤ 2.
Next we need to relate these Sx and Sy to our S(θ0,i).
We specify that Sx = S(0) and Sy = S (pi/2). It can be
easily shown using (13), our quantum mechanical triple
spin product result, that
F = 8c0c1 (16)
Therefore F > 2 if c0c1 > 1/4 and a violation is
possible. For the equal superposition in (8) we have
c0 = c1 = 1/
√
2. Therefore F = 4 giving a maximal
violation. If we have instead used the inequality given
by (1), then F = 4c0c1 ≤ 2 for all c0, c1.
The scheme presented here requires a discrete phase
measurement, which has yet to be experimentally real-
ized in the ultra high detector efficiency limit. How-
ever, recent work by Gilchrist et. al [24], and Yurke and
Stoler [25] has suggested how quadrature phase amplitude
measurements may be used to test the Bell inequality in
the high detector efficiency limit. A homodyne based
scheme is considered next to provide a feasible phase
measurement.
A quadrature phase-amplitude homodyne measure-
ment X(θ) can achieved by combining a signal field (say
aˆ) with a local oscillator field (say bˆ) to form two new
fields given by cˆ± =
[
aˆ± bˆ exp (iθ)
]
/
√
2. Here θ is
a phase shift which allows the choice of particular ob-
servable to be measured, for instance choosing θ as 0
or pi/2 allows the measurement of the conjugate phase
variables X(0) and X(pi/2) respectively. The homodyne
measurement is achieved by measuring using photodec-
tors the intensities of both the beams c+ and c−, and
then subtracting them to give a photocurrent difference
as Id = c
†
+c+ − c†−c−. Using the definition for c± the
photocurrent difference can be rewritten in terms of the
original signal and oscillator modes as
Id = bˆ
†aˆe−iθ + bˆaˆ†eiθ. (17)
In the limit of a large oscillator field we can make a re-
placement of the b mode by a real classical field . Hence
Id = ||
(
aˆe−iθ + aˆ†e−iθ
)
= ||X(θ) (18)
Thus performing a measurement on the quadrature
phase amplitude X(θ) yields a result x(θ) which ranges
in size and sign. For our state (8), the probability of ob-
taining x1(θ1), x2(θ2), x3(θ3) (abbreviated as x1, x2, x3)
for the three particles measured by individual homodyne
measurements is
Px1x2x3 (ψ0) = |〈x1|〈x2|〈x3|Ψ〉|2 (19)
For a given quadrature measurement xi, we classify the
result as “1” if xi > 0 and “0” if xi < 0. The probability
of obtaining the result “1” for all three particles is then
3
P111 (ψ0) =
∫ ∞
0
∫ ∞
0
∫ ∞
0
dx1dx2dx3Px1x2x3 (ψ0)
=
1
4
− c0c1
4
√(
2
pi
)3
cos [ψ0] (20)
Other probabilities such as P001 can be calculated in a
similar fashion.
Defining the spin Si in terms of P1 and P0 as before,
we can show that the triple spin product is given by
S1S2S3(ψ0) = −2c0c1
√(
2
pi
)3
cos [ψ0] (21)
and hence F given by (14) reduces to
F = 8c0c1
√(
2
pi
)3
(22)
We maximize the discrepancy between quantum mechan-
ics and local realism by choosing an equal superposition
(c0 = c1 = 1/
√
2, and hence F = 8
√
2/
√
pi3 ∼ 2.0318 >
2. Though this is a small violation, it is still a violation of
the GHZ inequality in the high detection efficiency limit.
A fundamental question that needs to be considered is
why the magnitude of the triple spin product in (21) is
not one as it is in the discrete phase case (for the case
c0 = c1 = 1/
√
2)? The answer is quite simple. Our
homodyne measurement, while it may have perfect de-
tection efficiency, is not an accurate (or efficient) mea-
surement of the discrete phase. This leads to a signifi-
cant lessening of the size of the violation of the potential
GHZ violation. The homodyne measurement does how-
ever have its advantages. First and foremost, current
homodyne measurement technology allows detection ef-
ficiencies in excess of 99%. Our model for homodyne as-
sumes perfect efficiency detectors. However, because of
our small potential violation, the homodyne detection ef-
ficiency would have to exceed 99.5% in a real experiment
provided the initial state could be produced accurately.
A second advantage is that as the homodyne measure-
ment involves a strong local oscillator via Id = X(θ)
(with  being the strength of the local oscillator), the
potential GHZ inequality violation could have a macro-
scopic nature.
To summarize, we have investigated a triple photon
correlated state (that may be able to be produced by
nondegenerate parametric oscillation) that can be used to
test the GHZ inequality proposed by Mermin. We have
proposed how discrete phase measurements could pro-
vide an effective test of the inequality. In fact, a binary
phase measurement could provide a maximal violation of
the GHZ inequality. As an approximation to the binary
phase measurement, we consider homodyne quadrature
phase amplitude measurements. Again a violation of the
GHZ inequality is possible although it is significantly re-
duced because it is an insensitive binary phase measure-
ment. An advantage of the homodyne method however
is that because it involves a strong local oscillator the
detection efficiencies are extremely high.
W. J. M would like to thank P. D. Drummond,
M. D. Reid, S. Schneider and M. J. Gagen for their valu-
able discussions and assistance.
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4


Characterizing GHZ Correlations in Nondegenerate Parametric Oscillation via Phase Measurements

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Characterizing GHZ Correlations in Nondegenerate Parametric Oscillation via Phase Measurements

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