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

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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

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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

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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

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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.

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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

26