the experimental demonstration of quantum mechanical non-locality and its consequences for ...
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The Experimental Demonstration of Quantum Mechanical Non-Locality and its Consequences for
Philosophy
Rowan G. TepperGoucher CollegeMay 3, 2004
PHL 290 Internship in Philosophy
Prof. John M. Rose – Goucher CollegeDr. Philip J. Adelmann – The Johns Hopkins University
Applied Physics Laboratory
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Table of Contents
I. Introduction 3
II. Historical Review 4
III. The Aspect Experiment 9
IV. Philosophical Implication 14
V. References 17
Appendix A 18
2
I. Introduction.
The revolution in physics dating to the early part of the
twentieth century has already had and is due to have more serious
implications for the practice of philosophy. Indeed, examples from
quantum mechanics have been used in philosophical texts and arguments,
albeit without the accompanying mathematical demonstrations. In this
paper, I plan to give a physically informed and mathematically
demonstrable examination of the results of the research team headed by
Alain Aspect into quantum entanglement of polarized photons;
specifically with regard to their achieved experimental violation of
Bell’s inequality. Furthermore, my aim is to show the richness of the
field of quantum mechanics for philosophical research and in the
pursuit of this overall aim, the philosophical implications of the
Aspect team’s results. Particularly intriguing are the results that
indicate a non-deterministic correlation between entangled photons, at
distances at which any signal between the photons must necessarily
propagate at a rate exceeding the speed of light by at least two
thousand times. Briefly, the possible implications of this scenario
include an observed violation of both special and general relativity.
The data collected by the Aspect team further bolsters claims to the
completeness of the quantum mechanical description of reality and lends
support to Niels Bohr’s quip to Albert Einstein regarding Einstein’s
famous statement that “God does not play dice with the universe”. To
this, Bohr replied, “don’t tell God what he can and cannot do.”
3
II. Historical Review
The paper that sparked the debate over the communication between
correlated particle pairs, and the resultant questions regarding the
completeness of quantum mechanics, was the 1935 paper Can Quantum-
Mechanical Description of Physical Reality be Considered Complete, by
Albert Einstein, Boris Podolsky and Nathan Rosen1 The conclusions drawn
by this paper is that
Either (1) the quantum-mechanical description of reality given by
the wave function is not complete or (2) when the operators
corresponding to two physical quantities do not commute the two
quantities cannot have simultaneous reality. [However] Starting
then with the assumption that the wave function does give a
complete description of the physical reality, we arrived at the
conclusion that two physical quantities, with non-commuting
operators, can have simultaneous reality. (EPR)
Their resulting conclusion, that because the two quantities can have
simultaneous reality, then, there must be something included in the
original wave function that determines from the outset the results of
the correlation. This is to say, that the EPR paper pits causal
determinism against quantum mechanics while at the same time it takes
as uncontested the limiting velocity of light. The apparent
superluminal ‘communication’ required by quantum mechanics, by the
means of which one particle affects the other, was later characterized
by Einstein as spooky action at a distance. The fundamental assumption
supporting the conclusions of the EPR paper is that there is no
possibility of communication beyond velocity = c.
Fundamentally, the conclusions of EPR serve to preserve locality in 1 This paper will henceforth be referred to as EPR
4
reality, that is, that the limiting velocity of light constitutes the
absolute limit of causal interaction.
The next contribution to the debate, in the mid-1950’s, was David
Bohm’s hidden variable theory. The hidden variable hypothesis has been
the one used by the most strident opponents of the non-deterministic
implications of quantum theory. The hidden variable theory in its
essence states that in a given quantum entangled system there are in
addition to the observables of a quantum system there exists
unobservable parameters denoted by λ, which allow the measured states to
be determined in advance of the measurement. However, hidden variables
are not in any way observable and do not in any way affect observables
of the system beyond that determination. Stated mathematically the
joint probability of A and B, E(a,b,λ)2 denotes the expectation value
for the joint measurement, with hidden variable λ determining at the
outset the state of the particle, the following statement must hold
true:
∫= λλρλλ dbBaAbaE )(),(),(),( (1)
Moreover, according to John S. Bell, this should equal the
quantum mechanical expectation value3. Thus, the hidden variable would
only effect the actualization of the particle’s state properties, but
could not be directly or indirectly observed. Thus the result would be
to preserve locality and causal determinism by including variable λ
which while determining a particle’s state in advance, it should not
introduce a change into the observable expectation value E(a,b). This
not only would preserve the integrity of causal determinism, locality
and special relativity but also quantum mechanics. However, this theory 2 Here we are using the notation used by Alain Aspect et al .3 J.S. Bell “On The Einstein Podolsky Rosen Paradox” in Physics Volume 1, Number 3, pp 195-200, 1964, pg. 196
5
carries a number of implications, some of which violate the postulates
of quantum mechanics.
This hidden variable theory and its accompanying difficulties led
to J.S. Bell’s famous inequality. Before stating the conclusions to
which Bell came, we must discuss precisely what the problems associated
with a local hidden variable theory. The primary problem encountered in
a hidden variable theory of quantum mechanics is that it violates the
first postulate of quantum mechanics as stated by Jim Baggott in The
Meaning of Quantum Theory, that is the state of a quantum mechanical
system is completely described by the wavefunction.4 That is, hidden
variable λ is not part of the wavefunction, yet it plays a determinant
role in the quantum mechanical system thus violating this first
postulate. Furthermore, in the context of the EPR Gedankenexperiment,
this difficulty is compounded by the fact that according to EPR, the
two entangled particles do not comprise wavefunction, Ψ ,but rather two
distinct wavefunctions, 'Ψ , and ''Ψ . This is to say that EPR’s
definition of physical reality requires that the two particles are
considered to be isolated from each other, i.e. they are no longer
described by a single wavefunction at the moment a measurement is made.
The reality thus referred to is sometimes called ‘local reality’. [or
separability]5 Thus, according to the hidden variable theory’s
interpretation of the EPR Gedankenexperiment, hidden variables λ and λ’
corresponding to local wavefunctions Ψ and 'Ψ are complementary and
determined at the moment of the emission of two correlated particles.
The correlation is determined at the outset and the wavefunctions Ψ
4 Jim Baggott The Meaning of Quantum Theory, (Oxford and New York: Oxford University Press, 1992), pg 435 Ibid, pg. 102
6
and 'Ψ corresponding to the emitted particles are entirely local and
the measurement of which are predetermined at the moment of emission.
The fundamental insufficiency of the hidden variable theory stems
from these difficulties, namely that (1) The system is no longer
described by a single wavefunction, but by two. Furthermore (2) this
complication is amplified by the introduction of hidden variable λ,
which is not included in the wavefunctions of the particles, yet serves
to correlate them and preserve locality. Furthermore, the hidden
variable theory is subject to the restriction that the hidden variable λ
may not alter the expectation value E(a,b), and thus, a hidden variable
theory of quantum mechanics must necessarily experimentally agree with
the results predicted by quantum mechanics.
It is at this point that J.S. Bell published his landmark paper on The
Einstein Podolsky Rosen Paradox in 1964. Stated simply, Bell’s
inequality sets the parameters within which a hidden variable theory
must operate. Bell’s inequality states that that for a hidden variable
system the following inequality must hold true:6
2 |)d,c(E)b,c(E||)d,a(E)b,a(E| ≤++− (2)
It will be shown that equation 2 is clearly violated by both the
predictions of quantum mechanics and by experimental results. Where
this inequality is violated, it can be concluded that the state of the
system at measurement is not determined by additional hidden variable
λ. Furthermore, quantum mechanics predicts that expectation values of
the form E(a,b)to be
)ab(2cos)ab(sin)ab(cos)b,a(E 22 −=−−−= 7 (3)
where a, b, c, and d are the angles at which the measuring polarizers
6 See Clauser et al 7 See Appendix A
7
are set. Thus, Bell’s inequality states that the sum of the absolute
value of the expectation values will yield a value less than or equal
to two. However, the quantum mechanical prediction for this relation of
expectation values yields a maximum value of 2.828 before adjusting for
the precision of measurement and the quality of equipment used.
Furthermore, Bell writes in the conclusion to his article that
“in a theory in which parameters are added to quantum mechanics to
determine the results of individual measurements, without changing the
statistical predictions, there must be a mechanism whereby the setting
of one measuring device can influence the reading of another instrument
however remote. Moreover, the signal involved must propagate
instantaneously8, so that such a theory could not be Lorentz
invariant.”9 Thus, if the hidden variable theory is rejected, Bell’s
paper contains a dilemma for EPR: either the system is local and
separable and a signal propagates at a velocity exceeding that of
light, a clear violation of special relativity, or the system is non-
local and non-separable and despite being space-like separated, the two
particles are still described by a single wavefunction, and
observation and measurement of one particle of the system
instantaneously determines the outcome of the observation of the other
space-like correlated particle.
III. The Aspect Experiment
In 1981 and 1982, Alain Aspect et al published a series of papers
in Physical Review Letters reporting the results of an experimental
8 Recent experimental research has demonstrated that this signal propagation not only violates the limiting quality of the speed of light, but does so in an extraordinary manner approaching instantaneity. The observed velocity by which the signal must propagate has been shown to be 2.0 x 104 the speed of light. 9 Bell, pg. 199
8
setup which approximated the EPR Gedankenexperiment. Rather than use
electrons as suggested by EPR (particles with spin 21± ), Aspect et al
used an apparatus that measured the polarization of two entangled
photons (bosons, whose spin is 91) emitted in a cascade by a Ca40 atom
excited by a dye-laser and a krypton laser returning to ground state.
This cascade simultaneously emits two entangled photons differentiable
by their different wavelength. The cascade emits photons of wavelength
551.3 nm (green) and 422.7 nm (blue). These two photons are correlated;
at any given time, one photon has spin 1 and the other has spin -1.
Because these photons are of differing wavelengths it is possible to
separate these photons and then detect correlations. The experimental
apparatus used by Aspect et al is represented schematically below.
Figure 1 – Aspect’s Experimental Setup
It should be noted that the photons emitted are emitted in all
directions, and as such, in a given period of time, only a certain
number of correlated photons will be emitted in opposite directions and
are properly aligned with respect to the experimental apparatus. The
blue and green photons are separated out by filters, one of which only
allows the passage of blue photons whereas the other only allows green
photons to pass. After passing through the filters, the photons
continue to the polarizers whose angles, while the photon is in motion,
have been randomly set among certain values. The angles used by Aspect
et al were 22.5°, and 67.5° (“Angles that cause the greatest conflict
between quantum mechanical predictions and the inequalities”10) the
difference between which determines the expectation values and measured
values described in equations 2 and 3.
10 Aspect et al, 1982, pg. 93. see Figure 2
9
The photons emitted are initially in a state of circular
polarization, which is composed of both a vertical and horizontal
component. Circular light polarization is considered either right
handed or left handed, depending on the direction of rotation, which is
correlated with the magnetic spin number ±1 of the photon. Angular
momentum is always conserved. Thus because angular momentum of the
excited state is zero and the angular momentum of the ground state is
also zero the angular momentum of the two photons emitted in the
cascade must cancel to zero. Thus, one photon is in a state of spin +1
while the other is in a state of spin -1.
The circular polarized photons reach the polarizers and depending
upon the relative angle between the polarizers becomes planar polarized
in either the horizontal or vertical plane. If the photon passes
through the polarizer it emerges vertically polarized and is counted as
a positive, whereas if the photon is deflected, it emerges horizontally
polarized and is counted as a negative. After this, the photon enters a
photomultiplier tube, which amplifies the photon into a measurable
signal, which then goes to a coincidence counter. The coincidence
counter detects correlated pairs of photons that arrive within 17ns of
each other. A correlation is counted whenever two signals arrive
simultaneously. This experiment is repeated numerous times (enough for
the results to be described probabilistically) with the polarizer
angles being changed while the photons are in flight. According to
Bell’s inequality, the relations of expectation values for the
different sets of angles should give a value that is less than or equal
to two for this experiment, while quantum mechanics predicts otherwise.
The quantum mechanical expectation value for this experiment is 2.828.
For the angles used by Aspect et al, E(a,b) is predicted by
quantum mechanics to be (neglecting measurement and mechanical
10
inaccuracy):
)ab(2cos)b,a(E −= (4)
This is the general form for the expectation value of a particle in
quantum, mechanics and holds true of any relationship between all
particles and their frame of reference. Thus for Aspect et al, Bell’s
inequality becomes:
S|)cd(2cos)cb(2cos||)ad(2cos)ab(2cos| =−+−+−−− (5)
According to Bell’s inequality, with A = 0, C = 45, B=22.5, and D =
67.5, a deterministic result would conform to this inequality:
2|)d,c(E)b,c(E||)d,a(E)b,a(E| ≤++− (2)
However, if E(a,b) is accurately predicted, there are values for S that
violate the above inequality. Furthermore, given the angles of 0 and 45
degrees, for polarizer 1, any setting of polarizer 2 should give a
violation of Bell’s inequality. The graph below (figure 2) and the
following data chart demonstrate that the maximum possible violation of
Bell’s inequality occurs with polarizer 2 is set to 22.5 or 67.5.
2.8279963422.827996342
0
0.5
1
1.5
2
2.5
3
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86
S Values for angles A = 0 andC = 45|cos2(B-a)-Cos2(d-a)
|Cos2(B-C)+Cos2(D-C)
Figure 2
11
Polarizer 1 Polarizer 2 Angle Expectation
0 22.5 22.5 cos[2(22.5)] = 0.707
0 67.5 67.5 cos[2(67.5)] = 0.707
45 22.5 -22.5 cos[2(-22.5)]= -0.707
45 67.5 22.5 cos[2(22.5)] = 0.707| 0.707 – (-0.707) | + | 0.707 + 0.707| = 2.828
Thus, the quantum mechanical prediction shown above, in which S =
2.828, clearly violates Bell’s inequality, and if confirmed would
experimentally rule out hidden variable theories of quantum mechanics.
In Aspect et al’s July 1982 paper in which they published the results
of this experiment, they report that adjusting for the efficiency of
their equipment, the expectation value predicted is 2.70 ± 0.05, and
that their results yielded a value of 2.697 ± 0.015, where “the
impressive violation of inequalities is 83% of the maximum violation
predicted by quantum mechanics with ideal polarizers.”11 This paper ends
with the conclusion that “we are thus led to the rejection of realistic
local theories if we accept the assumption that there is no bias in the
detected samples: Experiments support this natural assumption.”12
These results had a number of implications. First and foremost,
they imply a manner of non-locality. Two distant entangled objects must
be considered to still constitute the same wavefunction unless one were
to admit that the speed of quantum information transit vastly exceeds
the liminal velocity of light, approaching instantaneity. In a recent
paper, Valerio Scarani et al have experimentally determined that the
speed at which quantum information must be transmitted, relative to a
neutral frame of reference (Cosmic Background Radiation). “The
conservative bound that we obtained for the ‘speed of quantum
11 Aspect et al, July 1982, pg 9312 Ibid, pg 94
12
information’ in that frame = cx 4102 , is still quite impressive… the
present authors will not be astonished if further experiments provide
an even higher value.”13
IV. Philosophical Implications
The experimental violation of Bell’s inequality has far ranging
philosophical implications. Most importantly, this conclusively
demonstrates that the world operates differently depending upon scale.
Moreover, this implies that there is not one, but two coexistent and
operative systems of physics; one system that is valid for systems
above the atomic level and one system that is valid for systems at the
subatomic level. In most cases, these two physical systems agree
regarding the same macroscopic system, however, on the microscopic
level, predictions based upon classical (Newtonian) physics radically
diverge from those of quantum mechanics. When experimental verification
is conducted, the results, as seen above are in agreement with quantum
mechanical predictions. Often this distinction between the subatomic
realm, which is governed by quantum mechanical rules, and the larger
physical reality, which is governed by classical or relativistic
physics, forces us into uncomfortable positions regarding concepts that 13 Valerio Scarani et al, pg 6
13
are ordinarily taken for granted. This discomfort is made especially
acute because the threshold at which systems switch between being
governed by quantum mechanics and classical mechanics is ill-defined at
best.
The first concept that becomes confused is the nature of reality
and the possibility of truth. To put it strongly, we can under no
circumstances any longer refer truth to correspondence with the
empirically observable world without first specifying both the vantage
point of the viewer and the scale at which observation is conducted.
For example, we may say correctly that light is both a wave and a
particle. Light exhibits properties of both waves and particles. What
then is a photon? Unless we are to take the epistemologically weak
position of saying that an object may possess, in reality, both
supposedly exclusive properties, we must abandon the possibility of
founding an epistemology on the basis of correspondence with an actual
world that is impassive to the observers’ observation. This difficulty
is compounded by the fact that according to DeBroglie, it is not only
light that is both a wave and a particle. Moreover, everything has both
wave and particle properties, and their behavior depends upon their
size relative to the apparatus with which they are measured. Either
truth becomes indeterminate with respect to the world, or we must
introduce a value that is neither true nor false, but the
superimposition of both. Otherwise, we are forced to completely abandon
the relevance of correspondence theories of truth. A conception of
truth that is more in accordance with the findings of quantum mechanics
would be a coherence theory, or something akin to the Heideggerian
notion of truth as un-concealment. This latter hypothesis would be
particularly relevant because for truth to be un-concealed, there must
be an observer to which truth is un-concealed. And it is truth that is
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un-concealed, rather than reality itself. In effect, the Heideggerian
idea of aletheia would allow for truth in a world in which quantum
mechanics is in force. That is to say, that which is unconcealed is the
state after the wave function collapses into a determinate state. Thus,
we have modes in which reality may be un-concealed to the observer, and
the observation itself determines the mode of un-concealment.
Another important casualty of this particular experiment is the
idea of causality. In light of these results, the concept of causality
must be critically reconsidered. If, as the results of Aspect et al’s
experiment suggest, two apparently distinct particles compose a single
wavefunction, observation or action upon one portion of the spatially
extended wavefunction does in fact create “action at a distance.” This
is to say, an action on one particle causes an effect at such a
distance that conventional conceptions of causality cannot explain
without recourse to a violation of special relativity. This is to say
that in order to preserve the concept of causality, we must abandon the
idea that the speed of light constitutes the maximum velocity at which
causality can operate. This would create the appearance that the effect
preceded the cause from the perspective of the distant particle
affected by the wave-function collapse. Scarani et al have demonstrated
that if, in fact, there is a transmission of causality between one part
of the system and the other, that transmission occurs at a speed over
two thousand times that of light. They hypothesize also, that this is
the lower bound on the possible range of velocities. The speed at which
causality operates on the quantum mechanical level thus approaches
instantaneity. This would remain confined to the realm of quantum
mechanics if not for the development of practical applications such as
quantum computing and quantum cryptography. In the latter, an apparatus
similar to the one used by Aspect et al, is used to generate a one-
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time-use encryption key that is simultaneously generated at remote
locations. Due to quantum uncertainty, if one were to eavesdrop on the
generation of this encryption key, uncertainty would be introduced into
the resulting key and the eavesdropper readily discovered. The
apparatus by which this is demonstrated experimentally is analogous to
the apparatus used by Aspect et al. See Figure 3:
Figure 3: Thanks to Dr. Bryan Jacobs for the photo, tour of
cryptography lab and discussion.
A final reflection: Deleuze & Guattari, in A Thousand Plateaus discuss
the fact that systems, human or otherwise, display radically different
behavior when observed on the macroscopic or ‘molar’ scale as opposed
to the microscopic or ‘molecular’ scale. Quantum mechanics seems to
provide demonstration that this view is in principle valid and fruitful.
16
17
References
Alain Aspect, “Bell’s Inequality test: more ideal than ever” in Nature, Volume 398, March 1999, pp. 189-190
Alain Aspect, Jean Dalibard, and Gerard Roger, “Experimental Test of Bell’s Inequality Using Time-Varying Analyzers” in Physical Review Letters, Volume 49, Number 25, December 1982, pp. 1804-1807.
Alain Aspect, Philippe Grangier and Gerard Roger,“Experimental Realization of Einstein-Podolsky-Ros
en-Bohm Gedankenexperiment: A New Violation of Bell’s Inequalities” in Physical Review Letters, July 1982, pp. 91-94.
--------------,“Experimental Tests of Realistic Local Theories via Bell’s Theory” in Physica-l Review Letters, August 1981, pp. 460-463.
Jim Baggott The Meaning of Quantum Theory, A Guide for Students of Chemistry and Physics (Oxford & New York: Oxford University Press, 1992).
J.S. Bell “On the Einstein Podolsky Rosen Paradox” in Physics, Volume 1, Number 3, 1964 pp. 195-200.
Clauser, John F. and Shimony, Abner (1978). Bell's theorem: experimental tests and implications. Reports on Progress in Physics, 41, p. 1881.
Albert Einstein, Boris Podolsky and Nathan Rosen, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” in Physical Review Volume 47, May 1935, pp. 777-780
Malcolm R. Forster “A Local-Collapse Interpretation of Quantum Mechanics in the GHZ Example” Department of Philosophy, University of Wisconsin, Madison.
Valerio Scarani, Wolfgang Tittel, Hugo Zbinden, Nicolas Gisin “The speed of quantum information and the preferred frame analysis of experimental data.” In Physics Letters A 276 (2000) 1-7
Sol Wieder The Foundations of Quantum Theory (New York: Academic Press, 1973)
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APPENDIX A
The quantum mechanical prediction for the Aspect et al, experiment is derived in this
section, and follows closely from The Meaning of Quantum Theory, Chapter 4. An
abbreviated version of Aspect’s experimental setup is shown in Figure A-1.
Figure A-1 Experimental Setup
The entangled pairs of photons move from the source toward each of the polarization
analysers P1 and P2. Although the photons approaching P1 and P2 are both circularly
polarized, P1 and P2 can only measure photon states in a plane-polarized direction
composed of vertical and horizontal components as shown in figure A-2.
Figure A-2 Axis Convention Applied to Analysers P1 and P2
The analysers P1 and P2 are oriented at some relative angle, (b-a) = φ . Both analyzers
transmit maximum light when the vertical component of circular polarized light is
parallel to the axis of the analyser and reflect light in a perpendicular direction when the
19
P1 * P2vh
A B v’h’
Orientationa
Orientationb
SOURCE
v’ h’v
h(a-b)
axis is 90 degrees relative to the vertical component. From the classical theory of light,
Malus’s law predicts the intensity of light transmitted through the analysers varies as
)(cos2 φ . According to quantum theory we can assign each photon transmitted through P1
a state vector of vertical polarization (denoted as a ket vector in Dirac notation) as,
>Ψ v| or a state vector of horizontal polarization, >Ψ h| . As analyser P2 is rotated
relative to P1 each photon is projected into a new state, >Ψ 'v| or >Ψ 'h| . The projection
probability is the modulus-squared of the corresponding projection amplitude. It is seen
from Figure A-2, that the possible projection amplitudes and the associated projection
probabilities can be computed from the inner products of the four linear polarization
states as,
)(cos ||| and )cos( | 22vv'v'v φ=>ΨΨ<φ=>ΨΨ<
)(sin ||| and )-sin()2cos( | 22vv'h'v φ=>ΨΨ<φ=φ+π=>ΨΨ<
)(cos ||| and )cos( | 22vv'h'h φ=>ΨΨ<φ=>ΨΨ<
1 ||| and 1 | 2v'v''v'v =>ΨΨ<=>ΨΨ<
1 ||| and 1 | 2h'h''h'h =>ΨΨ<=>ΨΨ< Eqns. (1A)
1 ||| and 1 | 2v v v v =>ΨΨ<=>ΨΨ<
1 ||| and 1 | 2 h hh h =>ΨΨ<=>ΨΨ<
0 ||| and 0 | 2 h vh v =>ΨΨ<=>ΨΨ<
0 ||| and 0 | 2h'v''h'v =>ΨΨ<=>ΨΨ<
As was discussed in the main text, the net angular momentum carried away by the two
photons must sum to zero in order to satisfy the conservation of angular momentum.
Photons have a magnetic spin quantum number of either +1 or –1, corresponding to states
of right and left circular polarization. The net angular momentum of the emitted photon
pair can only sum to zero if their respective angular momentum vectors are of opposite
direction. Because the experiment measures only photon pairs that travel in opposite
20
directions, either both photons must be left circular polarized (LCP) or both must be right
circular polarized (RCP), in order to conserve angular momentum. Therefore, the two
detectors P1 and P2 will ‘see’ the same direction of circular polarization. Two LCP
photons emitted in opposite directions or two RCP photons emitted in opposite directions
have the required net zero angular momentum because their magnetic spin angular
momentum vectors cancel. The two-particle normalized state vector can be written as a
linear superposition of the product states >Ψ AL| (photon A in a state of left polarization
traveling to the left toward P1) and >Ψ BL| (photon B in a state of left polarization
traveling to the right toward P2), or >Ψ AL| >Ψ B
L| and the product states >Ψ AR|
(photon A in a state of right polarization traveling to the left toward P1) and >Ψ BR|
(photon B in a state of right polarization traveling to the right toward P2), or >Ψ AR|
>Ψ BR| as,
)( BR
AR B
L AL | | ||
21 | >+> Ψ>Ψ>ΨΨ=>Ψ Eqn. (2A)
Because the photons are indistinguishable, the assignment of A or B is arbitrary and we
could have written the two-particle state vector as,
)( AR
BR A
L BL | | ||
21 | >+> Ψ>Ψ>ΨΨ=>Ψ
The LCP and RCP state vectors can be defined as a linear combination of the
measurement eigenstates: the vertical, >Ψ B,Av| and horizontal, >Ψ B,A
h| plane
polarized states for either photon A or B, respectively as,
21
)( BA,h
B,Av
B,A |i - |2
1 | L >Ψ>Ψ=>Ψ Eqns. (3A)
)( BA,h
B,Av
B,A |i |2
1 | R >Ψ+>Ψ=>Ψ
The measurement eigenstates >Ψ v| , >Ψ 'v| , >Ψ h| and >Ψ 'h| refer to the directions
imposed on the quantum system of the apparatus (i.e. the analysers P1 and P2) during the
measurement of a single photon. The experimental design (i.e. the coincidence counters)
associates a (+1) expectation value for photons that are measured in either the >Ψ v| or
>Ψ 'v| state and a (–1) expectation value for photons that are measures in the >Ψ h| or
>Ψ 'h| state. Defining the measurement operator as, M, we can express the
measurement process in terms of an eigenvalue problem for analyser P1 or P2 and photon
A or B as,
>Ψλ=>ψ Av
Av
Av | |M Eqns. (4A)
>Ψλ=>ψ Bv
Bv
Bv | |M
>Ψλ=>ψ Ah
Ah
Ah | |M
>Ψλ=>ψ Bh
Bh
Bh | |M
>Ψλ=>ψ Av'
Av'
A'v | |M
>Ψλ=>ψ Bv'
Bv'
B'v | |M
>Ψλ=>ψ Ah'
Ah'
A'h | |M
>Ψλ=>ψ Bh'
Bh'
B'h | |M
where the eigenvalues, are 1B,A'v
B,Av +=λ=λ and 1B,A
'hB,A
h −=λ=λ .
Four eigenstates can be defined for the two-particle joint measurement. They are given in
terms of product states composed of the four single photon eigenstates defined in Eqns.
4A as,
22
>Ψ>Ψ=>Ψ>Ψ=>Ψ ++A'v
Bv
B'v
Av || || | Eqns. (5A)
>Ψ>Ψ=>Ψ>Ψ=>Ψ −+A'h
Bv
B'h
Av || || |
>Ψ>Ψ=>Ψ>Ψ=>Ψ +−A'v
Bh
B'v
Ah || || |
>Ψ>Ψ=>Ψ>Ψ=>Ψ −−A'h
Bh
B'h
Ah || || |
where we have denoted the two-particle states with + +, - +, + - or - -, subscripts to
associate them with corresponding measurement expectation values of +1 or – 1 defined
in Eqns 4A. In order to find the probabilities of measuring the two-particle system in any
one of these states we must express the initial state vector, >Ψ| , given in Eqn. 2A, in
terms of the four two-particle joint measurement states. The basis of >Ψ| can be
changed to the measurement basis by use of the projection operators, || P
j ij ij i Ψ> <Ψ= ,
where i, j are either/or + - and ||
j ij i Ψ> <Ψ is the matrix formed by the outer product of
the two-particle eigenstates. Thus >Ψ| written as,
>ΨΨ> <Ψ+>ΨΨ> <Ψ+>ΨΨ> <Ψ+>ΨΨ> <Ψ=>Ψ
−−−−+−−+
+−+−++++
||| ||| ||| ||| |
Eqn. (6A)
The probabilities of measuring the two-photon system in any one of the four eigenstates
listed in Eqns. 5A are given by the square of the associated projection amplitudes. The
four required projection amplitudes are, >ΨΨ< ++ | , >ΨΨ< +− | , >ΨΨ< −+ | and
>ΨΨ< −− | . Because the measurement eigenstates of Eqns. 5A are mutually orthogonal
we have,
>ΨΨ<Ψ<=>ΨΨ< ++ || | B'v
Av Eqns. (7A)
>ΨΨ<Ψ<=>ΨΨ< −+ || | B'h
Av
>ΨΨ<Ψ<=>ΨΨ< +− || | B'v
Ah
23
>ΨΨ<Ψ<=>ΨΨ< −− || | B'h
Ah
Now recalling the expression for the initial state, >Ψ| from Eqn. 2A we have,
)|| || ( |21 | BR
AR
BL
AL
B'v
Av >Ψ>Ψ+Ψ>ΨΨ<Ψ<=>ΨΨ< ++ Eqn. (8A)
)|| || ( |21 | BR
AR
BL
AL
B'h
Av >Ψ>Ψ+Ψ>ΨΨ<Ψ<=>ΨΨ< −+ Eqn. (9A)
)|| || ( |21 | BR
AR
BL
AL
B'v
Ah >Ψ>Ψ+Ψ>ΨΨ<Ψ<=>ΨΨ< +− Eqn. (10A)
)|| || ( |21 | BR
AR
BL
AL
B'h
Ah >Ψ>Ψ+Ψ>ΨΨ<Ψ<=>ΨΨ< −− Eqn. (11A)
We now compute the projection amplitudes given above starting with Eqn. 8A.
)|| | |(21 | BR
B'v
AR
Av
BL
B'v
AL
Av >ΨΨ> <ΨΨ<+ΨΨ> <ΨΨ<=>ΨΨ< ++ Eqn.12A)
1 2 3 4
Evaluating the inner products labeled 1-4 above, and using Eqns. 1A and 3A we have,
1: 21 )|i - |( 21 | Ah
Av
Av
Av
AL
Av =>ΨΨ<>ΨΨ<=>ΨΨ<
=1 = 0 2: )iexp(21 ) |i - |( 21 | A
hAv'
Av
Av'
AL
A'v φ=>ΨΨ<>ΨΨ<=>ΨΨ<
)cos(φ= )sin(φ−=
where we have used the Euler formula )isin( )cos( )iexp( φ±φ=φ± .
3: 21 )|i |( 21 | Ah
Av
Av
Av
AR
Av =>ΨΨ<+>ΨΨ<=>ΨΨ<
=1 = 0 4: )iexp(21 )|i |( 21 | A
hAv'
Av
Av'
AR
A'v φ−=>ΨΨ<+>ΨΨ<=>ΨΨ<
)cos(φ= )sin(φ−=
Collecting the inner products 1 – 4 above we have,
2)cos( ))iexp(21 )iexp(21( 21 | φ=φ−+φ=>ΨΨ< ++
Similar calculations for the other three projection amplitudes given in Eqn. 9A – 11A
yields,
2)sin( | φ=>ΨΨ< −+
2)sin( | φ=>ΨΨ< +−
2)cos(- | φ=>ΨΨ< −−
24
The probabilities for each of the four possible joint results are given by the square of the
projection amplitudes as,
2)(cos | || P 22 φ=>ΨΨ<= ++++ Eqns. (13A)
2)(sin | || P 22 φ=>ΨΨ<= −+−+
2)(sin | || P 22 φ=>ΨΨ<= +−+−
2)(cos | || P 22 φ=>ΨΨ<= −−−−
The expectation value, )b,a(E , predicted by quantum mechanics for the relative angle
between analysers P1 and P2 given by (b-a) = φ ,
B'h
Ah
B'v
Ah
B'h
Av
B'v
Av P P P P )b,a(E λλ+λλ+λλ+λλ= −−+−−+++ Eqn. (14A)
or using the results of Eqns. 4A and Eqns. 13A we have
−−+−−+++ +−−= P P P P )b,a(E
or finally,
b)-cos2(a b)-(asin )ba(cos )b,a(E 22 =−−=
25
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