We show that quantum mechanics predicts a contradiction with local hidden
variable theories for photon number measurements which have limited resolving
power, to the point of imposing an uncertainty in the photon number result
which is macroscopic in absolute terms. We show how this can be interpreted as
a failure of a new premise, macroscopic local realism.Comment: 9 pages 3 figure

Reid, M. D.

English

arXiv

We show that quantum mechanics predicts a contradiction with local hidden variable theories for photon number measurements which have limited resolving power, to the point of imposing an uncertainty in the photon number result which is macroscopic in absolute terms. We show how this can be interpreted as a failure of a new premise, macroscopic local realism

Swinburne Research Bank

Incompatibility of macroscopic local realism with quantum mechanics in measurements with macroscopic uncertainties

The author defines the premise of 'macroscopic local realism' so that its failure is more surprising than failure of the local realism addressed previously. Macroscopic local realism is a premise stating the following: If one can predict the result of a measurement at A by performing a simultaneous measurement on a spatially separated system B, then the result of the measurement at A is predetermined but described by an Einstein-Podolsky-Rosen 'element of reality,' which has an indeterminacy in each of its possible values, so that only values macroscopically different to those predicted are excluded

Macroscopic local realism incompatible with quantum mechanics: failure of local realism where measurements give macroscopic uncertainties

University of Queensland eSpace

Macroscopic local realism incompatible with quantum mechanics: Failure of local realism where measurements give macrosopic uncertainties

PHYSICAL REVIEW
LETTERS
VOLUME 84 27 MARCH 2000 NUMBER 13Incompatibility of Macroscopic Local Realism with Quantum Mechanics in Measurements
with Macroscopic Uncertainties
M. D. Reid
Physics Department, University of Queensland, Brisbane, Australia
(Received 19 February 1999)
We show that quantum mechanics predicts a contradiction with local hidden variable theories for
photon number measurements which have limited resolving power, to the point of imposing an uncer-
tainty in the photon number result which is macroscopic in absolute terms. We show how this can be
interpreted as a failure of a new premise, macroscopic local realism.
PACS numbers: 03.65.BzBell [1] in 1966 showed that the premises of local real-
ism (or local hidden variable theories) were incompatible
with the predictions of quantum mechanics. Experiments
[2] support quantum mechanics, and the general viewpoint
is to reject the premise of local realism.
To date, theoretical and experimental effort has focussed
on situations where results of relevant measurements are
only microscopically separated. The measurements per-
formed are intrinsically microscopic, in that one requires
to clearly distinguish between results (eigenvalues of the
appropriate quantum operator) which are microscopically
distinct.
Theoretical work has shown a failure of local realism for
multiparticle (or higher spin) systems [3], where the sys-
tem and range of results can be macroscopic. There have
also been proposals [4] which show the failure of local re-
alism for quantum superpositions of two macroscopically
distinct states. However the violations are still apparently
indicated only where measurements at some point must
resolve microscopically different results, such as adjacent
photon numbers or spin values. While the results indicate
failure of local realism for macroscopic systems, it is not
clear whether one is testing a premise different from that
tested in the microscopic experiments.
Schrödinger [5] raised the issue of quantum mechan-
ics apparently predicting superpositions of states macro-
scopically distinct (“Schrödinger cat states”), questioning
the possibility of their true existence, based on the no-
tion that such states apparently violate a type of macro-
scopic realism. Recent progress [6] in the experimental0031-90070084(13)2765(5)$15.00generation of such superpositions highlights a need to test
objectively for a true incompatibility with a macroscopic
realism. Progress has been made by Leggett and Garg
[7] who predict an incompatibility of quantum mechanics
with the premise of “macroscopic realism and noninvasive
measurability.”
Here we define the premise of “macroscopic local re-
alism” in such a way that its failure is more surprising
than the failure of the local realism addressed in previous
Bell-type studies. This local realism becomes immediately
testable in experiments where the results of all relevant
measurements are macroscopically distinct, if a failure of
local realism in the usual way can be shown. Experiments
which still show a failure of local realism, even when un-
certainties in all relevant measurements are macroscopic,
will also show a failure of this type of macroscopic local
realism. In this paper we prove this result and present a
quantum state with this property, claiming therefore what
is, to our knowledge, the first reported predicted failure of
such macroscopic local realism.
In 1935 Einstein, Podolsky, and Rosen [8] defined “lo-
cal realism” in the following way. “Realism” is sufficient
to state that if one can predict with certainty the result of a
measurement of a physical quantity at A, without disturb-
ing the system A, then the results of the measurement were
predetermined and one has an “element of reality” corre-
sponding to this physical quantity. The element of reality
is a variable which assumes one of a set of values which are
the predicted results of the measurement. This value gives
the result of the measurement, should it be performed.© 2000 The American Physical Society 2765
VOLUME 84, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 27 MARCH 2000Locality states that the events at A cannot, instantaneously,
disturb in any way a second system at B spatially separated
from A. Taken together, local realism is sufficient to im-
ply that, if one can predict the result of a measurement of
a physical quantity at A, by making a simultaneous mea-
surement at B, then the result of the measurement at A is
described by an element of reality.
Macroscopic local realism [9] is defined as a premise
stating the following. If one can predict the result of a
measurement at A by performing a simultaneous measure-
ment on a spatially separated system B, then the result of
the measurement at A is predetermined but described by
an element of reality which has an indeterminacy in each
of its possible values, so that only values macroscopically
different to those predicted are excluded.
Macroscopic local realism is based on a “macroscopic
locality,” which states that measurements at a location B
cannot instantaneously induce macroscopic changes (for
example, the dead-to-alive state of a cat, or a change be-
tween macroscopically different photon numbers) in a sec-
ond system A spatially separated from B. Macroscopic
local realism also incorporates a “macroscopic realism,”
since it implies elements of reality with (up to) a macro-
scopic indeterminacy. Suppose our “Schrödinger’s cat” [5]
is correlated with a second spatially separated system, for
example, a gun used to kill the cat. The strength of macro-
scopic local realism is understood when one realizes that
its rejection in this example means we cannot think of the
cat as being either dead or alive, even though we can pre-
dict the dead or alive result of “measuring” the cat, without
disturbing the cat, by measuring the correlated spatially
separated second system.
In this paper we present a quantum state which violates a
Bell inequality even for coarse measurements with macro-
scopic uncertainties (in absolute terms), and show how this
implies a failure of macroscopic local realism as we define
it. Our proposed experiment is depicted in Fig. 1a, where
(I0 is a modified Bessel function and r0  1.1)
jc  I02r20 212jaa1 jbb1
√ X`
n0
r20 n
n!
jna2 jnb2
!
.
(1)
The aˆ6 and bˆ6 are boson operators for four outgoing
fields. Fields aˆ1 and bˆ1 are in coherent states jaa1 andjbb1 , respectively, and we allow a,b to be real and large.
jnk is a Fock state for field k. The fields aˆ2 and bˆ2 are
microscopic and are generated in a pair-coherent state [10].
Such states are the two-mode equivalent of the recently re-
alized [6] “even” and “odd” coherent superposition states
1 6 exp22jaj2212ja 6 j2ap2  and could po-
tentially be generated using nondegenerate parametric os-
cillation in a limit where one-photon losses are negligible.
(The coherent states for aˆ1 and b1 would be derived from
the laser pump for the oscillator.) We point out later other
possible choices of jc.2766FIG. 1. Our proposed test of macroscopic local realism. (a)
For large a,b, macroscopic fields a06 b06 are incident on
each measuring apparatus. (b) This measurement scheme for
a,b large corresponds to balanced homodyne detection of the
quadrature phase amplitudes XˆAu and XˆBf.
The fields aˆ6 are mixed using phase shifts and beam
splitters to give two new output fields aˆ02  aˆ2 2 aˆ1p
2 and aˆ01  iaˆ2 1 aˆ1
p
2 at the location A. Simi-
larly, the fields bˆ6 are mixed to give outputs bˆ06at
location B, spatially separated from A. The mixing is
incorporated into the experiment simply to provide the
nice feature that both fields, aˆ06 say at A, incident on each
measuring apparatus are macroscopic. We measure
simultaneously at A and B the Schwinger spin operators
SˆAu  cˆ
y
1cˆ1 2 cˆy2cˆ22 and Sˆ
B
f  dˆ
y
1dˆ1 2 dˆy2dˆ22.
The measurements are made through the transforma-
tions (achieved with polarizers or beam splitters with a
variable transmission) c1  aˆ01 cosu2 1 aˆ02 sinu2
and c2  aˆ01 sinu2 2 aˆ02 cosu2, at A, and
d1  bˆ01 cosf2 1 bˆ02 sinf2 and d2  bˆ01 sinf2 2
bˆ02 cosf2, at B, followed by photodetection.
In Fig. 1b we demonstrate how the measurement SˆAu can
also be performed directly from aˆ6 by introducing a rela-
tive phase shift u and mixing with a 5050 beam splitter
to produce cˆ06  aˆ1 6 aˆ2 exp2iu
p
2, followed by
photodetection to give SˆAu  cˆ
0y
1 cˆ01 2 cˆ0y2 cˆ022.
Our particular test of macroscopic local realism requires
noisy measurements. The result for the photon number dif-
ferences nˆAu  2SˆAu  cˆ
y
1cˆ1 2 cˆy2cˆ2 and nˆ
B
f  2Sˆ
B
f 
dˆ
y
1dˆ1 2 dˆy2dˆ2 is of the form n 1 noise, where n is the
result of the measurement in the absence of the noise. We
introduce noise distribution functions at each location A
and B, and define probabilities such as PAnoise $ x,
such that the noise at A is greater than or equal to the
value x. A probability PBnoise $ x is defined simi-
larly. Later we allow “noise” to be a random noise term
with a Gaussian distribution of standard deviation s.
Photon number measurements for macroscopic fields
VOLUME 84, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 27 MARCH 2000are performed with photodiode detectors, which already
introduce a limited resolution because of detection
inefficiencies.
The results of measurements are classified as 11 if the
photon number difference result is positive or zero, and
21 otherwise. We determine the following probability
distributions: PA1u for obtaining 1 at A, PB1f for
obtaining 1 at B, and PAB11u,f the joint probability of
obtaining 1 at both A and B.
As a first step we define the probability P0,ABij u,f
for obtaining results i and j, respectively, upon the joint
measurement of nˆAu at A, and nˆBf at B, in the absence of
the applied noise s. With noise present at the detectors,
the measured probabilities PAB11u,f become
PAB11u,f 
X`
i,j2`
P
0,AB
ij u,fPAnoise $ 2i
3 PBnoise $ 2j . (2)
Before presenting the quantum prediction for these prob-
abilities, we examine the prediction given by macroscopic
local realism.
Local realism as originally defined by Einstein-
Podolsky-Rosen, Bell, and Clauser-Horne [1] implies the
following well-known expression:
P
0,AB
ij u,f 
Z
rlpAi u,lp
B
j f,l dl . (3)
Local realism implies an underlying set of elements of
reality, or hidden variables l [with probability distribution
rl], not specified by quantum theory. The element of
reality is a variable which assumes one of a set of values
which are the predicted results of the measurement, nˆAu
say. For our experiment, a precise prediction of nˆAu is not
possible given a measurement at B, for any choice f at B.
The elements of reality then do not take on definite values,
and local realism is only sufficient to imply a probability
pAi u,l for the result i of the measurement nˆAu , for a given
l. The independence of pAi u,l on f is based on the
locality assumption.Now we consider the prediction given by macroscopic
local realism. With macroscopic local realism the locality
condition is relaxed, but only up to the level of M photons,
where M is not macroscopic, by maintaining that the mea-
surement at B cannot instantaneously change the result at
A by an amount exceeding M photons. The elements of
reality deduced using macroscopic local realism can give
predictions for the results of measurement which are mi-
croscopically (or mesoscopically) different, but not macro-
scopically different, to those predicted from the elements
of reality deduced using local realism. Where our pre-
dicted results at A is i0 using local realism, macroscopic lo-
cal realism allows the result to be i  i0 1 mA, where mA
can be any number not macroscopic. Importantly, while i0
is not dependent on the choice f at B, the nonmacroscopic
value mA can be. Where local realism specifies a (local)
probability distribution pAi0 u,l for obtaining i0 photons
at A, the prediction is only correct to within 6M photons.
The actual result at A is determined by a further nonlocal
perturbation term pAmAi
0,u,f,l, which gives the proba-
bility of a further change of mA photons. The macroscopic
local realism assumption then is that the conditional prob-
ability pAi u,l in Eq. (3) is replaced by [and similarly for
pBi f,l]
pAi u,f,l 
1MX
mA2M
pAmAi
0,u,f,lpAi0i2mAu,l .
(4)
The original local probability pAi0 u,l is convolved with a
microscopic nonlocal probability function pAmAi
0,u,f,l,
the only restriction being that the nonlocal distribution
does not provide macroscopic perturbations, so that the
probability of getting a nonlocal change outside the range
mA  2M, . . . ,1M is zero. Equivalently we must have
(and similarly for terms with B)
MX
mA2M
pAmAi
0,u,f,l  1 . (5)
We substitute the macroscopic locality assumption (4) into
the hidden variable prediction (3) to obtain the prediction
for the measured probabilities (2):pAB11u,f 
X`
i, j2`
Z
rl
"
MX
mA2M
pAmAi
0,u,f,lpAi0i2mAu,l
MX
mB2M
pBmB j
0,f,u,lpBj0j2mBf,l
#
dl
3 PAnoise $ 2iPBnoise $ 2j . (6)
Recalling i  i0 1 mA and j  j0 1 mB we change the i, j summation to one over i0, j0 to get
PAB11u,f 
X`
i0, j02`
Z
rlpAi0 u,l
"
MX
mA2M
pAmAi
0,u,f,lPAnoise $ 2i0 1 mA
#
3 pBj0 f,l
"
MX
mB2M
pBmB j
0,f,u,lPBnoise $ 2 j0 1 mB
#
dl . (7)2767
VOLUME 84, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 27 MARCH 2000We assume that the applied noise function is slowly varying over the microscopic (or mesoscopic) range 2mA 1 · · · 1
mA for which nonlocal perturbations are possible according to macroscopic local realism (and similarly at B):
MX
mA2M
pAmAi
0, u,f,lPAnoise $ 2i0 1 mA 	 PAnoise $ 2i0
MX
mA2M
pAmAi
0, u,f,l . (8)
This is valid only if s is macroscopic. Using (5), one simplifies to get the final form pAB11u,f 
P
i0, j0 3R
rlpAi0 u,lp
B
j0f,l dlPAnoise $ 2i0PBnoise $ 2j0. This prediction of the hidden variable theory is now
given in a (local) form like that of (3), from which Bell-Clauser-Horne inequalities [1] follow, for example,
S 
PAB11u,f 2 PAB11u,f0 1 PAB11u0,f 1 PAB11u0,f0
PA1u0 1 PB1f
# 1 . (9)The noise terms which add a macroscopic uncertainty
to the photon number result alter the premises needed to
derive the Bell inequality. With s macroscopic we need
only assume macroscopic local realism to derive the Bell
inequality (9).
The quantum prediction for state (1) is shown in Fig. 2.
Violations of the Bell inequality (9) in the absence of noise
are shown in curve (a). Violations are still possible [curve
(b)] in the presence of increasingly larger absolute noise
s, simply by increasing a  b. This violation of the Bell
inequality (9) with macroscopic noise s implies the failure
of macroscopic local realism.
The asymptotic behavior in the large a,b limit is
crucial in determining a violation of macroscopic local
realism, and is best understood by replacing the boson
operators aˆ1 and bˆ1 with classical amplitudes a and
b, respectively. We see that then SˆAu  aXˆAu 2 and
SˆBf  bXˆ
B
f2, where XˆAu  aˆ2 exp2iu 1 aˆy2 expiu
and XˆBf  bˆ2 exp2if 1 bˆy2 expif are the quadrature
phase amplitudes of fields aˆ2 and bˆ2. In fact, Fig. 1b
with a,b large shows the experimental setup for balanced
homodyne detection of the quadrature phase amplitudes
XˆAu and XˆBf, of fields aˆ2 and bˆ2. Homodyne detection has
been used experimentally to detect “squeezed” fields [11],
FIG. 2. (a) S versus a  b, for u  0, f  2p4, u0 
p2, and f0  23p4 for the quantum state (1) with no noise
present. (b) Maximum noise s still giving a violation of the
Bell inequality (9) versus a.2768where the fluctuation in XˆAu is reduced below the standard
quantum limit.
Violations of Bell inequalities for measurements XˆAu , XˆBf
on state (1) have recently been predicted [12], confirming
Fig. 2(a) in the large a limit. These violations vanish [13]
when Gaussian noise of standard deviation s0 $ 0.26 is
added to the measurements XˆAu , XˆBf. With a large, this
corresponds to a noise value of as0 in the photon number
measurement 2SˆAu , confirming Fig. 2(b). In fact, since
there is always a finite cutoff s0, any state jc which
shows a failure of local realism for measurements XˆAu and
XˆBf on fields aˆ2 and bˆ2 will also show a violation of
macroscopic local realism, provided a,b are large. Other
such states have been recently predicted [14], increasing
the scope for a practical violation of macroscopic local
realism.
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Incompatibility of Macroscopic Local Realism with Quantum Mechanics in Measurements with Macroscopic Uncertainties

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Macroscopic Local Realism Incompatible with Quantum Mechanics: Failure of Local Realism where Measurements give Macroscopic Uncertainties

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