Causality Joint Probability 2014 Slide

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Causality, Locality and Joint

Probabilities in Quantum

Mechanics

S. M. Roy, HBCSE, TIFR, Mumbai

HBCSE,TIFR, 10 June 2014


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From paradoxes to useful quantum ef-fects . The EPR paradox, Schr odingersCat paradox and Zeno paradox have longfocussed attention on counter-intuitive fea-tures of quantum foundations: superposi-tion principle ,entanglement and measure-ment theory. These striking features are nowelevated from paradoxes to useful quan-tum effects, and account for the strikinggains of quantum computation with respectto classical computation. I survey quantumfoundational issues illuminated by the EPR


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paradox and some applications of the as-

sociated non-classical features, e.g. quan-tum violations of Bell inequalities followingfrom Einsteins local reality principle and of the much more stringent inequalities follow-

ing from quantum separability or absence of quantum entanglement. The Bell-CHSH in-equalities and their multiparticle generaliza-tions bring out violation of Einstein local re-

ality principle by a factor 2( N

1) / 2

by certainentangled states of N particles. For quan-tum computation, the more relevant prop-erty is entanglement or its opposite, viz.


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separability. Quantum separability implies

inequalities exponentially stronger than Bellinequalities for large N (S. M. Roy). Theyare violated by a factor 2 N 1 by some en-tangled states; these states are natural can-

didates to exploit in seeking improvementsover classical computation. They also enableexponentially enhanced accuracy in measur-ing physical quantities such as time , and

interaction strengths (Roy-Braunstein).

Introduction: Quantum Versus ClassicalWorld View The beautiful and the grotesque


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indices i can specify different particles or dif-

ferent spatial components for the same parti-cle.The time evolution is then specied uniquelyas trajectories in phase space in terms of the laws of motion (e.g. relativistic Newtonslaws for the particles and relativistic waveequations ). Classical mechanics has the fol-lowing fundamental characteristics:

Objective Reality .The phase space co-ordinates are objectiveproperties of the system independent of theirobservation.


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Causality/ Determinacy .Different results (viz. phase space co-ordinates)arise from different causes (Jauchs formula-

tion of Causality).

Locality .

The phase space co-ordinates are indepen-dent of, i.e. not inuenced by, actions per-formed at a spacelike separation.


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In the quantum worldview a unied descrip-

tion of waves and particles is achieved . Theparticular disposition of experimental appa-ratus can bring out the particle (photon) na-ture of electromagnetic waves (as in photo-

electric effect), or the wave nature of parti-cles (as in neutron interference experiments).The wave and particle attributes which can-not be revealed simultaneously may be called

complementary (Bohrs complementarity prin-ciple). This unication is achieved at anenormous philosophical cost, e.g. loss of causality or determinism, loss of a unied


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description of the world due to its arbitrary

division into quantum object (or system) andclassical subject (or observer/apparatus), andas shown by Einstein and Bell, a consequentloss of local reality.

John Bell gave a seminar at TIFR in 1980October which brings out the surprise of quan-tum mechanics vis a vis classical mechanics

beautifully. To make it more accessible, Iquote the rst transparency of his seminarentitled, What in the world is quantum me-chanics about exactly ? :


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QM is about: Wave functions , Opera-

tors H , Schr odinger Eqn,i /t = H (1)

and how to solve it. But what in the world? The wave function is not like the world:

= + (2)

would be unrecognizable. So: Statistical In-terpretation. Statistics of what? measure-

ment results.

A quantum mechanics based on the statis-tics of measurement results is fundamentally


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ambiguous because it requires an unden-

able boundary between the quantum objectand the classical measuring apparatus. No-body knows what quantum mechanics saysexactly about any situation. For nobody

knows where the boundary really is, betweenwavy quantum system and the world of par-ticular events. This is the problem of quan-tum mechanics . It is no problem in practice-

because practice is not accurate enough-andmay be never will be. Bell considered thisfundamental drawback far more serious thanthe loss of determinism.


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Einstein in 1933 [0] expressed dissatisfaction

with quantum theory being merely a set of rules about the statistics of measurement re-sults :I still believe in the possibility of giv-ing a model of reality which shall represent

events themselves and not merely the prob-ability of their occurence, and more speci-cally, I am, in fact, rather rmly convincedthat the essentially statistical character of contemporary quantum theory is solely tobe ascribed to the fact that this (theory)operates with an incomplete description of physical systems .


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Solvay Conference 1927


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There have been attempts to dene con-

sistent histories in quantum mechanics of open systems 1 . Apart from detailed featuresfound unattractive by some 2 , there is thebasic proposition by the authors themselves

that only very special sets of histories areconsistent, and only these can be assignedprobabilities. For example, in a double slitinterference experiment we cannot assign aprobability that the particle reached a regionof the screen having earlier passed throughslit 1 (except in the case of vanishing inter-ference).


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Lack of Causality in Ordinary

Quantum Mechanics . One of the deni-tions of causality (e.g. that advocated byJauch) is that Different results should havedifferent causes. Consider a quantum su-perposition

|z+ +

|z

for a spin-1/2 par-

ticle, where |z+ and |z are eigenstates of z with eigenvalues +1 and -1 respectively.When the same state is prepared repeatedlyand passed through a Stern-Gerlach appara-

tus to measure z ,

|z+ + |z STERN GERLACH (3)


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quantum mechanics cannot predict whether

the result + or the result will occur ina given trial; it merely says that a fraction| |2 of the particles ends up at the detec-tor corresponding to z = +1, and a frac-

tion | |2

goes to the detector correspondingto z = 1. Thus different results (goingto one detector or the other) arise from ex-actly the same cause (the same initial state).Of course this lack of causality might be re-stored in a theory in which the wave functionis not a complete description of the state of the system (ref. Einstein quote).


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Schr odingers Cat Paradox . Due to the

linearity of the Schr odinger eqn, the jointstate of the particle and the detectors D+ , D is transformed as,

(

|z+ +

|z

) D + D

|z+ D+ D+ |z D + D , (4)which is a superposition of macroscopicallydifferent states D+ D and D+ D with Ddenoting excited states of the detectors.(Schr odingers Cat Paradox: if a cat locatedat D+ is killed by the arriving particle, wehave a superposition of dead and alive cats).


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Context Dependence of Quantum Real-

ity . In quantum mechanics,

|( x, t ) |2 dxis the probability of observing position tobe in

dx if position were measured. It is not

the probability of position being in dx in-dependent of observation. In fact, the samestate vector also yields

| ( p, t ) |2 d pwhich is the probability of observing momen-tum to be in the interval d p if momentum


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John S. Bell


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Einsteins Principle of Local Reality and

the EPR Paradox Bohr asserted that quan-tum reality of a system is contingent uponthe disposition of the apparatus in the entireworld, not just in the nearby part of it. This

was stated most forcefully by Bohr in thenow famous Bohr-Einstein debates on quan-tum mechanics [0]. In conict with Bohrsassertion, Einstein proposed a principle, nowcalled the Principle of local reality. This prin-ciple led to the EPR paradox, Bell inequal-ities and experiments which proved that lo-cal reality is violated in nature. It continues


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however to provide insights into the nature

of quantum entanglement, and via Bell in-equalities, a quantitative measure of entan-glement.

It is convenient to consider Einsteins localreality principle as a combination of the de-nitions of locality and reality spelt out be-low.

Einstein Locality . Suppose two systems S 1and S 2 which have interacted in the pasthave now separated and are experimented


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upon by two observers in spatially separated

regions. Observed properties of the two sub-systems can ofcourse be correlated due topast interactions. Einstein insisted, But onone supposition we should,in my opinion, ab-solutely hold fast: the real factual situationof the system S 2 is independent of what isdone with the system S 1 , which is spatiallyseparated from the former.

Physical Reality . The meaning of real fac-tual situation in Einsteins formulation wasclaried by the denition of physical reality


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in the landmark paper of Einstein, Podol-

sky and Rosen [0]: If without in any waydisturbing a system, we can predict with cer-tainty (i.e. with probability equal to unity)the value of a physical quantity, then there

exists an element of physical reality corre-sponding to this physical quantity.

Einstein, Podolsky and Rosen applied the

principle of local reality to argue that Quan-tum Mechanics is incomplete. Consider atwo particle system. In quantum mechanics,


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the position observable qi and the momen-

tum observable pi cannot be simultaneouslyspecied sharply; but the observables q1 q2and p1 + p2 are commuting observables andthere is a quantum state

|q1 q2 = q0 | p1 + p2 = p0

in which they are specied arbitrarily sharply,and have values q0 and p0 . Ignore momen-tarily the difficulty that such states are not


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normalizable, a difficulty removed later by

Bohm and Aharonov [0] by considering spinobservables. Suppose observers A and Bare spacelike separated. In such a state,if B chooses to measure q2 she predicts q1with certainty (without disturbing that parti-cle since it is spatially separated), and henceq1 must have physical reality; equally, if shechooses to measure p2 she predicts p(1) tohave reality. By the principle of local reality,reality for particle 1 must be independent of choices made by observer B. Hence, both q1and p1 must have physical reality for particle


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Bohm-Aharonov EPR Experiment, Lo-

cal Hidden Variables and Bells Theorem

Bell formulated the EPR local reality idea us-ing the Bohm-Aharonov [0] example of twospin-half particles or two qubits (quantumbits) prepared in a singlet state

| = |/ 2 = | / 2(5)

at a source S and then ying apart, to be de-tected by two spacelike separated observers,each equipped independently, (e.g. with ro-tatable Stern-Gerlach magnets) , to measure


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any arbitrarily chosen component of the par-ticle spin.

EPR-Bohm-Aharonov Experiment

Since the singlet state is rotationally sym-metric, the spin components measured alongany direction by the two observers must be


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opposite. Observer 1 can predict with cer-tainty the result of measurement of any com-ponent of (2) . a by observer 2 by previouslymeasuring the same component (1) . a forparticle 1.Since Einstein locality implies thatthe choice of magnet orientation made by

the remote observer 1 does not affect the re-sult obtained by observer 2, the result of anysuch observation must be predetermined.Theinitial quantum wave function does not spec-ify the result of an individual measurement;thereforethe predetermination requires a more com-plete specication of the state , say by addingadditional variables (hidden variables) .


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EPR Experiment: Perfect AnticorrelationHidden Variables

Hidden variables achieving perfect anti-correlation

(in each individual shot) between + and results along the same direction for two discsshot off along opposite directions is easy to


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visualize classically. We may even allow prob-

abilistic rather than deterministic hidden vari-ables. Suppose the hidden variables , withprobability distribution ( ) ,determine theprobability pr1 ( , a ) of observing the value r =

1 of the observable A( a ) = (1) . a and theprobability ps2 ( , b) of observing the value s =1 of the observable B ( b) = (2) . b. Einsteinlocality implies that pr1 ( , a ) and p

s2 ( , b) are

independent of the orientations of the re-mote measuring apparatus.Then the prob-ability pr,s ( a, b ) of observing A( a ) = r and


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B ( b) = s is,

pr,s ( a, b ) = d ( ) pr1 ( , a ) ps2 ( , b) (6)What would the predictions of such a lo-cal hidden variable theory (LHV) be inthe context of Einstein-Podolsky-Rosen typemeasurements of P QM ( a, b ) = < 1 a 2 b >,when we consider four possible measure-ments, with two orientations a, a on one side

and two orientations b, b on the other side ?Wigners proof of Bells Theorem . No-tice that Bells hypothesis actually allows the


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construction of a joint probability

pr,r ,s,s ( a, a ,b ,b ) = d ( ) pr1 ( , a ) ps2 ( , b) pr1 ( , a ) p

s2 ( , b )(7)

for A( a ) , A ( a ) , B ( b) , B ( b ) to have the valuesr, r , s , s respectively, such that the result of any of the four feasible experiments can beobtained as a marginal. E.g.

pr,s

( a, b ) = r = 1 ,s = 1 pr,r ,s,s

( a, a ,b ,b ) .(8)

The experimental correlation A( a ) B ( b) is


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then,

P ( a, b ) = r = 1 ,s = 1 rspr,s ( a, b )

= r = 1 ,s = 1 ,r = 1 ,s = 1 rspr,r ,s,s ( a, a ,b ,b ) ,

with similar expressions for P ( a , b) , P ( a, b ) , P ( a , b ).Hence, using the positivity of probabilities,

|P ( a, b ) P ( a, b ) |=

|

r,s,r ,sr ( s

s ) pr,r ,s,s ( a, a ,b ,b )

| r,s,r ,s |r ( s s ) | pr,r ,s,s ( a, a ,b ,b ) ,


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The choice of coplanar vectors such that

a.

b =

a.

b =

a .

b =

a .

b = 1

/ 2 yields the

value 2 2 for the left-hand side of the Bell-CHSH inequality in violation of local reality!It shows that even if the measurements of 1

a and 2

b are made at spacelike separa-

tion, statistical predictions of quantum me-chanics are inconsistent with the assumptionthat the measured value of 1 a has a realitythat is independent of whether 2

b or 2

b

is measured together with it. Experimentsof Alain Aspect and others support quantummechanics, and disprove the existence of a


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A B

Bellstate

Measurement Classical Communication U

e eporState

System 1 2Entangled

EPR Source

Usual Bennett et al 5Protocol ForTeleportation.


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| 1 = |z+ |z 1| 1 = |z |z+ 1

| 23 = |23 / 2

| = |/ 2|+ 1 | 23 =

12 |

+12 | 3

+ | 12 |+ 3 | + 12 | 3 | 12 |+ 3

Published Online May 29 2014 Science Ex-press Index


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Unconditional quantum teleportation betweendistant solid-state quantum bits

W. Pfaff1,*, B. Hensen1,H. Bernien1,S. B.van Dam1,M. S. Blok1, T. H. Taminiau1,M.J. Tiggelman1, R. N. Schouten1,M. Markham2,D.J. Twitchen2,R. Hanson1,

+ 1Kavli Institute of Nanoscience Delft, DelftUniversity of Technology, P.O. Box 5046,2600 GA Delft, Netherlands. 2Element Six,Ltd., Kings Ride Park, Ascot, Berkshire SL58BP, UK.


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Abstract

Realizing robust quantum information trans-fer between long-lived qubit registers is a keychallenge for quantum information science

and technology. Here , we demonstrate un-conditional teleportation of arbitrary quan-tum states between diamond spin qubits sep-arated by 3 m. We prepare the teleporter

through photon-mediated heralded entangle-ment between two distant electron spins andsubsequently encode the source qubit in a


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single nuclear spin. By realizing a fully deter-ministic Bell-state measurement combined withreal-time feed-forward quantum teleportationis achieved upon each attempt with an av-erage state delity exceeding the classicallimit. These results establish diamond spinqubits as a prime candidate for the realiza-tion of quantum networks for quantum com-munication and network-based quantum com-puting.

Causal Quantum Mechanics and JointPosition-Momentum Probabilities . The


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search for a causal quantum mechanics ledto joint position-momentum probabilities (DeBroglie-Bohm) much before the discovery of Bell inequalities for the EPR-Bohm-AharonovExperiment.Subsequently phase space Bellinequalities have led to the theorem that forgeneral states in 2 N dimensional phase spacemore than N + 1 marginals cannot agreewith corresponding quantum probability den-sities (Auberson-Mahoux-Roy-Singh).( N +1-marginal theorem).


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De Broglie and Bohm 4 (dBB) proposed atheory with position as a hidden variableso that {x, | }, i.e., the state vector supple-mented by the instantaneous position is thecomplete description of the state of the sys-tem. Here x = ( x1 , , xN ) denotes the con-guration space co-ordinate which evolvesaccording to

dx idt

= 1m i

iS ( x( t ) , t ) , (11)

where m i denotes the mass of particle i, andthe Schr odinger wave function is given by,

x| ( t ) R exp iS, (12)


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with R and S real functions of ( x, t ). DBBshow that if we start at t = 0 with an ensem-ble of particles whose position density coin-cides with |( x, 0) |2 at t = 0, then such timeevolution leads to a position density that co-incides with

| ( x, t )

|2 at any arbitrary time t.

Thus, the phase space density is

dBB ( x, p, t ) = |( x, t ) |2 p S ( x, t )(13)

whose marginal at arbitrary time reproducesthe position probability density,

dBB ( x, p, t ) d p = | ( x, t ) |2 . (14)


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Bell: The De Broglie-Bohm picture dis-poses of the necessity to divide the worldsomehow into system and apparatus

(because the position co-ordinates exist with-out the necessity of measurement). As faras the position variable is concerned the dBBtheory restores history and causality withoutaltering the statistical predictions of quan-tum mechanics. The lack of Einstein localityis an essential feature of quantum mechan-ics. It is enshrined in the dBB velocity of thei th particle depending on the instantaneous


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position of all the particles however far theymay be.

The momentum and other variables besidesposition do not have the same favoured sta-tus as position however. As Takabayasi 5

pointed out the dBB phase space densitydoes not yield the correct quantum momen-tum density, i.e.,

dBB ( x, p, t ) dx = |( p, t ) |

2.

For position, the value observed can be thesame as the preexisting value; not so for mo-


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mentum. Momentum therefore has not thesame reality as Position.

Causal Quantum MechanicsSymmetric in Position and Momentum. We [S. M.Roy and V. Singh] 6 ,7 asked thequestion , is it possible to remove this asym-metrical treatment of position and momen-tum and build a new causal quantum me-chanics in which momentum and position

can have simultaneous reality? We spelt outa non-unique but affirmative answer in onedimensional conguration space and later (with


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G. Auberson and G. Mahoux) 8 generalisedit to arbitrary dimensions. We recall rst theoriginal one dimensional construction with amonotonic dependence of momentum on po-sition.

The point of departure is to seek a phasespace density of the form

( x,p,t ) = |( x, t ) |2 ( p p( x, t )) , (15)where p( x, t ) is not given by the dBB formula.Rather, p( x, t ) is to be determined by therequirement

( x,p,t ) dx = |( p, t ) |2 . (16)


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If we assume that p( x, t ) is a monotonic func-

tion of x (non-decreasing or non-increasing),

( p p( x, t )) = ( x x( p, t ))

p( x,t )x

(17)

and

( x,p,t ) = | ( x, t ) |2 p( x,t )x

( x x( p, t )) (18)

If we determine p( x, t ) such that

|( x, t ) |2 = p( x, t )

x |( p, t ) |2 , (19)


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we obtain

( x,p,t ) = | ( p, t ) |2

( x x( p, t )) (20)which obeys the desired Eq. (6). Two ex-plicit solutions to Eq. (9), corresponding tonon-decreasing ( = 1) and non-increasing

( = 1) functions p( x, t ) are given by, p( x,t ) dp | ( p , t ) |

2

=

x

dx

| ( x , t )

|2 . (21)

Instead of Eq. (5) or Eq. (10), the phasespace density may now be written in the sym-


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metric form,

( x,p,t ) = |( x, t ) |2 |( p, t ) |2

p

dp

|( p , t )

|2

x

dx

|( x , t )

|2 .(22)

To compare with previous results , note thatthe Wigner distribution 9 , in 2 N dimensional

phase space,


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W ( q, p, t ) = 1

(2 ) N d y exp( i p. y) q y/ 2 |( t ) ( t ) | q + y/ 2 (23)also reproduces

| ( q, t )

|2 and

|( p, t )

|2 , as

marginals. However, unlike our phase spacedensity, that distribution cannot have a prob-ability interpretation as it is not positive def-inite.E.g. for two states , ,

d qd p W ( q, p, t ) W ( q, p, t )= | , |2 / (2 ) N , (24)


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b bl d


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commuting observables x1 , x 2 and approxi-mate values of system position and momen-tum are extracted from accurate observationof x1 , x 2 . The Von Neumann-Arthurs-Kellyinteraction during the time interval (0 , T ) is,

H = K ( qp1 + pp2 ) (25)

where K is a const, with KT = 1 and theother symbols denote the respective opera-tors. During interaction time, H is so strong

h h f H il i f h d


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that the free Hamiltonians of the system andapparatus are neglected. Arthurs and Kellystart with the system-apparatus initial state,

( q, x 1 , x 2 , t = 0) = ( q) 1 ( x1 ) 2 ( x2 ) (26)

where,

1 ( x1 ) = 1 / 4 b1 / 2 exp( x21 / (2 b2 )) (27)

2 ( x2 ) = 1 / 4

(2 b)1 / 2

exp( 2b2

x22 ) , (28)

and b/ 2 is the uncertainty of x1 in the ini-tial apparatus state. The uncertainty of x2


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d th di i i


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and x2 = p , the dispersions in x1 , x 2 arelarger than for the system,

( x1 ) 2 = ( q) 2 + b2 ,

( x2 ) 2 = ( p) 2 + 14b2

. (31)

By varying b we obtain the measurement ornoise uncertainty relation , (units = 1 ),

x1 x2 1 (32)where the minimum uncertainty is twice the

preparation uncertainty. Arthurs and Good-man 13 have given a beautiful proof of thisfundamental uncertainty relation, which is

independent of any particular choice of the


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independent of any particular choice of theHamiltonian. Further,for x1 distribution toapproximate q distribution closely, we needb q; for x2 distribution to approximate p distribution closely, we need b ( p) 1. Hence, for good approximation of both qand p distributions, we also need,

q p 1 . (33)

Testing a phase space density against theArthurs Kelly (AK) distribution is meaning-ful only in this region. Even when the phase


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tum mechanics gives


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tum mechanics gives,

pRS ( q, t ) = + p q( q t/m ) . (35)

qp+ pqRS

= ( t/m )[2 2 +

1 + ( m/ ( t )) 2 ].

(36)This agrees with the Arthurs-Kelly result,

2x1 x2 AK = ( t/m )[2 2 + ], (37)

when ( m/ ( t )) 1, i.e. q p 1. In con-trast, another phase space density which re-produces quantum q, p probability densities,


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viz.

| ( q, t )

|2

|( p, t )

|2 gives 2( t/m ) 2 for the

above correlation in disagreement with theArthurs-Kelly result. We know that for

( p) 1 b q,the separate q and p probability densities givenby AK and RS agree. Therefore the crit-ical test of the R S distribution against theA K distribution will be to compare thepositions of the momentum peaks pRS ( q)and pAK ( q). This has been performed by


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Arunabha Roy 14 .

pAK ( q) = ( pRS ( q) )[(1 + ( b/ q) 2 ) 1 (1 (1 / 4)( q p) 2 )] (38)


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The close agreement of the q, p correlationsin the two distributions in the relevant re-gion ( p) 1 b q is very striking.It in-


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dicates the possibility of a causal description

of single slit diffraction exactly in the region q p 1 highlighted by Einstein.

The non-uniqueness of such a description

even for a xed choice of the canonical pairq, p opens up the question of optimisation of phase space probability densities. Recently Inoticed (S. M. Roy, Physics Letters A, 377

(34-36). pp. 2011-2015 (2013)) that a con-vex combination of the two R-S phase space


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densities, = 1( q, p) C = + + ( q, p) + ( q, p) ,

= 12

12 1 (2 q p) 2

(39)

reproduces the quantum correlation < qp + pq > 2 < q >< p > exactly.The possibilityof experimental tests via Arthurs-Kelly mea-surements in quantum optics is an incentiveto look for this miracle for more general wavefunctions.


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Remote tomography and entanglement


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g p y gswapping via von NeumannArthursKellyinteraction S. M. Roy, Abhinav Deshpande,and Nitica Sakharwade, Phys. Rev. A 89,052107 (2014).

INTERACTION TRACKER 1

TRACKER 2

SYSTEM

PUMP PHOTONSSTRONG

DISTANT

STATION

TOMOGRAPH

STATION AB

PHOTON

APPARATUS PHOTON 1

APPARATUS PHOTON 2

SYSTEM PHOTON P(ENTANGLED WITH P)

REGIONP

99


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99

J. S. Bell, Physics 1 (1964) 195; J. F. Clauser,M. A. Horne, A. Shimony, and R. A. Holt,Phys. Rev. Lett. 26 (1969) 880.

A. Einstein, B. Podolsky and N. Rosen, Phys.Rev. 47 (1935) 777.

Stapp Phys. Rev.

D3 (1971) 1303; Nuovo

Cim. B29 (1975) 270; Found. of Phys. 9(1979) 1 .

A. Einstein, Phiolosopher Scientist , P.A. Schilp


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A. Einstein, Phiolosopher Scientist , P.A. SchilpEd.. Library of Living Philosophers, Evanston,Ill. (1949). See esp. Einsteins autobio-graphical notes and replies by N. Bohr hereand in N. Bohr, Phys. Rev. 48 (1935)696. See also, reprints of related articles in,Quantum Theory and Measurement , J. A.Wheeler and W. H. Zurek Eds., Princeton(1983).

D. Bohm and Y. Aharonov, Phys. Rev. 108(1957) 1070.

N. D. Mermin, Phys. Rev. Lett. 65 (1990)


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1838.

S. M. Roy and V. Singh, Phys. Rev. Lett.67 (1991) 2761.

M. Ardehalli, Phys. Rev. A 46 (1992) 5375;A. V. Belinskii and D. N. Klyshko, Phys.Usp. 36 (1993) 653; N. Gisin and H. Bechmann-Pasquinucci, Phys. Lett. A 246 (1998) 1.

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