Foundations of Physics, Vol. 30, No. 6, 2000

The Zeno Effect in the EPR Paradox, in theTeleportation Process, and in Wheeler'sDelayed-Choice Experiment

D. Bar1

Received June 15, 1999; revised April 4, 2000

We treat here three apparently uncorrelated topics from the point of view of densemeasurement: The EPR paradox, the teleportation process, and Wheeler'sdelayed-choice experiment (DCE). We begin with the DCE and show, using itsunique nature and the histories formalism, that use may ascertain and fix thenotion of dense measurement (the Zeno effect). We show here by including theexperimenter (observer) as an inherent part of the physical system and using theAharonov�Vardi notion of dense measurement along a path, that knowledge ofcertain properties of the incoming system (after a measurement) is equivalent todense measurement. We reach the same conclusion by discussing the teleportationprocess, and the EPR paradox from, the same point of view.

1. INTRODUCTION

It is known that two quantum systems that were once correlated, evenmomentarily, continue to show a correlation(1) (though separated to a verylarge distance) in that knowing the state, for example, the spin, of anyone system gives us immediately the state of the other far removed system(the EPR paradox(1, 12)). It has been proved also that these systems arecorrelated not only over the large distance separating them (at the sametime), but also over any large time difference separating them (at the sameplace).(2)

We argue in this paper that even with respect to only one system theknowledge of the state of this system at some definite time (in the timeinterval which spans the total time along which the experiment is conducted)

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0015-9018�00�0600-0813�18.00�0 � 2000 Plenum Publishing Corporation

1 Department of Physics, Bar Ilan University, Ramat Gan, Israel.

causes this experiment to be different physically from the experiment inwhich this information is lacking. We make use of the delayed choiceexperiment (DCE), (3) the teleportation process, (8) the EPR paradox, (1) andthe histories formalism(4, 5) which deal with closed systems that include theobserver as an integral part of the physical system. This histories formalismis not identical to that of Gell-Man and Hartle(4, 5) in which their historieschain is an unreduced entity. That is, the separate parts of the chain cannot be regarded as separate entities on their own with assignedprobabilities. The histories chains to be used in this work are absolutelyreduced entities. In Sec. 2 we give a short review of the DCE and thenrepresent it as a histories chain comprised of five separate parts. We thenshow in Sec. 3 that a mere knowledge of the would be state of the systemcauses the outcome of the relevant experiment to be physically differentfrom the case in which this information is lacking. In Sec. 4 we show thatthis previous knowledge is the same as performing dense measurement(6, 11)

upon the system in the sense that in both cases the state of the systembecomes known to us. We make use of the Aharonov�Vardi method(7) ofmaking real and concrete any predetermined Feynman path(13) from alarge collection of possible ones by performing dense measurement alongthis predetermined path. We will explain this equivalence of knowledge anddense measurement and show that what makes the DCE so unique andapparently unlogical is that we do not take into account the whole relevantFeynman path, but only a part of it. We then proceed, in Secs. 5 and 6, andtreat the teleportation process proposed by Bennett, Brassard, Crepeau,Jozsa, Peres and Wootters (BBCJPW)(8, 10, 9) from the same point of viewof dense measurement along a definite Feynman path and get the sameconclusion. In Sec. 7 we do this for the EPR paradox, using the Bohm'sformulation.(12)

2. THE DCE FROM THE POINT OF VIEW OF THE HISTORIESFORMALISM

We give now a short review of Wheeler's DCE.(3) In the two figuresbelow we see an electromagnetic wave (quantum particle) coming in at 1to a half silvered mirror marked 1

2 s. This mirror splits the incoming waveinto two beams 2a and 2b with equal intensity. These beams are reflectedby the mirrors A and B to the junction point marked 4. Two detectors aresituated to the right of this junction point. The only difference between thetwo figures is that in Fig. 1 we leave the junction point unoccupied, whereasin Fig. 2 the junction point is occupied by another half silvered mirrormarked 1

2 s. From the experiment 1 (see Fig. 1) we conclude that the initial

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Figure 1

particle has passed either through the route (2a+3a) or through the route(2b+3b), whereas from experiment 2 (see Fig. 2) we conclude that in thiscase the initial particle has to pass at the same time through the two routesso as to have the constructive or destructive interference marked in Fig. 2.We remark here that the correct alignment of the experiment 2 is to haveonly one detector to the right of the half silvered mirror situated at thejunction point 4. This detector will click when we have constructive inter-ference and will not click in the opposite case. There is no need for asecond detector in experiment 2. Here we put two detectors to the right ofthe junction point 4 so that this experiment will he symmetric with theexperiment in Fig. 1. The second detector will click in the case of destruc-tive interference. We follow here also the same deployment of experiment2 as in Wheeler's article.(3) Now if we begin our experiment by letting firstthe initial wave (particle) pass through the system approaching the junc-tion point 4 and only at the very last instant we decide if we put the halfsilvered mirror at the junction point 4 or leave it unoccupied we get thefollowing retroactive results: (1) in the first case, where we put the halfsilvered mirror at the junction point 4, we actually cause the particle topass the two routes at the same time. (2) in the second case, where we leavethe junction point 4 unoccupied, we essentially influence the initial particleto pass through only one of the two possible routes.(3) These influences, in

Figure 2

815The Zeno Effect

both cases, are retroactive, that is, by some act which we do now we haveinfluenced not only the future but also the past. From the point of view ofthe histories formalism we represent this DCE by the following chain oftime ordered projection operators.(4)

CDCE=P5:5

(t5) P4:4

(t4) P3:3

(t3) P2:2

(t2) P1:1

(t1) (1)

From Eq. (1) we see that our representation of the DCE is a coarse grainedone(4) since the P 's are not projections onto a basis (a complete set ofstates), and they are not set at each and every time. P1

:1(t1) is actually a set

of projections at time t1 into two possible different states: (1) The observerhas decided to begin the experiment by sending the initial particle to thefirst half silvered mirror. (2) The observer has decided not to begin theexperiment at all. P2

:2(t2) signifies the projections at time t2 into three dif-

ferent possible states: (1) The initial particle passes through route 2a. (2)passes through route 2b. (3) passes through both routes. P3

:3(t3) are the

projections at time t3 into three different possible states: (1) The initialparticle passes through the upper route 3a. (2) passes through the lowerroute 3b. (3) passes through both routes. P4

:4(t4) signifies the projections at

time t4 into two different possible states: (1) the observer has decided attime t4 to let the junction point 4 remain unoccupied. (2) This observer hasdecided at time t4 to insert the half silvered mirror at the junction point 4.P5

:5(t5) are the set of projections at time t5 into two different possible states:

(1) The upper detector has clicked. (2) The lower detector has clicked.According to our discussion thus far the chronologically later P4

:4(t4)

decides the earlier P2:2

(t2).

3. THE KNOWLEDGE OF THE INTERIM STATES AS A FACTORWHICH DETERMINES SPECIFIED CONCRETE HISTORIES

We want now to check the probabilities of the possible realizations ofthe chain (1) into specified histories. There are four separate possiblehistories that can be reduced from the chain (1). The first history is

CCDE1=P5

lower(t5) P4unoccupied(t4) P3

3a(t3) P22a(t2) P1

begin(t1) (2)

This concrete history means that at time t1 the observer has decided tobegin the experiment by sending the initial particle through the first halfsilvered mirror. At tine t2 this particle passes through the upper route 2a,from there it continue through the upper route 3a at time t3 . At time t4 the

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junction point 4 is unoccupied, and at time t5 the lower detector hasclicked. The second possible history is the following one

CCDE2=P5

upper(t5) P4unoccupied(t4) P3

3b(t3) P22b(t2) P1

begin(t1) (3)

Here our initial particle passes through the lower routes in which case theupper detector is clicking (see Fig. 1).

As for the computation of the probability of the history (2) we can seewith the help of Fig. 1 and the special character of the DCE that if at t4

the junction point 4 is unoccupied then we know that at t2 the particle pass2a or 2b with equal probability of 1

2 (it can not pass both routes whichhappen only when the junction point 4 is occupied). Also we know that ifthe particle at the time t2 passes the upper route 2a (with probability 1

2)then it certainly must pass at time t3 the upper route 3a (with probability 1)in which case the lower detector must click (with probability 1). If weassume now that the observer decides at the tines t1 and t4 , with equalprobabilities of 1

2 for each case, upon the possible decisions he may takethen we arrive at the following probability for the history (2)

p(CCDE1)=1 V 1

2 V 1 V 12 V 1

2= 18 (4)

The same simple considerations lead us to the same identical probabilityfor the history (3)

p(CDCE2)=1 V 1

2 V 1 V 12 V 1

2= 18 (5)

The third possible history is the following

CDCE3=P5

upper(t5) P4occupied(t4) P3

3a+3b(t3) P22a+2b(t2) P1

begin(t1) (6)

in which at time t4 the junction point 4 was occupied, and at the times t2

and t3 our initial particle pass through both routes (2a+2b) and (3a+3b)respectively. Here we see from Fig. 2 that if at time t4 the junction point 4is occupied by the half silvered mirror then at the times t2 and t3 our initialparticle must pass both routes (2a+2b) and (3a+3b) respectively (withprobability 1 for each). Also in this case any one of the two detectors mayclick with equal probability of 1

2 . If we assume now that the observerdecides at the times t1 and t4 , with equal probability of 1

2 for each case,upon the possible decisions he may take, we get the following probabilityfor the history (6)

p(CDCE3)= 1

2 V 12 V 1 V 1 V 1

2= 18 (7)

817The Zeno Effect

We get the same probability also for the fourth following equivalent history

CDCE4=P5

lower(t5) P4occupied(t4) P3

3a+3b(t3) P22a+2b(t2) P1

begin(t1) (8)

that is

p(CDCE4)= 1

2 V 12 V 1 V 1 V 1

2= 18 (9)

We can consider other histories like, for example, the following

CDCE5=P5

lower(t5) P4occupied(t4) P3

3a(t3) P22a(t2) P1

begin(t1) (10)

CDCE6=P5

upper(t5) P4occupied(t4) P3

3b(t3) P22b(t2) P1

begin(t1) (11)

From the character of the DCE and Fig. 2 we conclude that the probabilitymeasures of the histories (10) and (11) are zero, because when the junctionpoint 4 is occupied then the relevant routes are (2a+2b) and (3a+3b).The only possible histories of the chain (1) (except those in which theobserver has decided at the time t1 not to begin the experiment) are thosefrom Eqs. (2), (3), (6), and (8).

We want now to reduce the number of the possible histories of thechain (1) from 4 to 3. We do this by representing a variant of this historieschain. We get this variant by using a coherent light source. The initial elec-tromagnetic wave at the point 1 in Figs. 1 and 2 above comes from thissource. This change in the deployment of the experiment causes the twopartial waves that pass the half silvered mirror at the junction point 4 inFig. 2 to be in constructive interference, so that the lower detector willsurely click. As can be seen this change in the experiment will not changethe possible histories (2) and (3) and their related probabilities of 1

8 foreach (see Eqs. (4) and (5)), but it does change the history (8), since nowwe know from the start that at the time t5 the lower detector clicks (withprobability 1). Thus the appropriate history is

CDCE7=P5

slower(t5) P4occupied(t4) P3

3a+3b(t3) P22a+2b(t2) P1

begin(t1) (12)

The suffix ``slower'' in P5 means that the lower detector will surely click(with probability 1). Thus the related probability of the history (12) is

p(CDCE7)=1 V 1

2 V 1 V 1 V 12= 1

4 (13)

We can see that, for this case of coherent light source (where only thelower detector clicks) the history (6) is not possible, and so its probabilitymeasure is zero as are the probability measures of the histories (10) and

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(11). We consider now the following version of (12). In this version theobserver decides even before the initial time t1 that at the time t1 he willbegin the experiment (with a probability 1). He also decides, before hebegins the experiment, that at the time t4 the junction point 4 will beoccupied by the half silvered mirror. That is, unlike the original DCE inwhich the observer do not know until the time t4 if he would put at thistime the half silvered mirror at the junction point 4 or leave it unoccupied,in this variant of (12) he knows all the time that at the time t4 he will putthe half silvered mirror at the junction point 4. The probability for thisversion of (12) is obviously 1 (compare with (13)). We move now one stepforward and take the original DCE itself, that is, we assume now that theobserver does not know until the very last minute if he would put at thetime t4 the half silvered mirror at the junction point 4 or leave it unoc-cupied. In this case as we have already shown, the history (12) is only onepossible realization with a probability 1

4 (see Eq. (13)). Other possibilitieswith the probability of 1

8 for each are the histories (2), (3) (see Eqs. (4) and(5)). We see, therefore, that knowledge of the would be state of the systemat the times t1 and t4 cause the relevant history and the probability of theexperiment to he different physically from the case in which this informationis lacking.

4. THE FEYNMAN PATH TREATMENT OF THE DCE

We therefore see with respect to the DCE, once the observer hasdecided to begin the experiment at the time t1 (with a probability of 1

2),that it is represented by the chain of histories from (2), (3), and (12). Thesethree alternatives are the three possible Feynman paths(13, 7) (of the DCEin the coarse grained version we adopt here) between the states of thesystem at the times t1 and t5 . Now according to Aharonov and Vardi(7) itis possible to realize any predetermined Feynman path from a large collec-tion of probable Feynman paths of states that begin at some initial givenstate and end at another final given state. This realization of any definiteFeynman path is performed through dense measurement that begins withthe given initial state and goes on continuously along the predeterminedpath till the given final state. If we take, for example, the definite path ofthe history (12) of the DCE then it can be realized, as we have just said,by doing dense measurement along this concrete path. But as we haveshown in Sec. 3 this same realization of the history (12) is achieved (for thiscase of the DCE) by knowing beforehand the latter would be state of thesystem at some definite points (t1 and t4) along the path that comprisesthe relevant history. Thus we see that dense measurement along a definite

819The Zeno Effect

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Figure 3

Feynman path, from a large collection of probable Feynman paths, whichrealizes it, is equivalent to knowing beforehand the would be state of thesystem at some moments (for the DCE it is the beginning of the experimentat t1 and the presence of the half silvered mirror at the junction point 4 atthe time t4) which also realizes and makes concrete the history (12). Werepresent now the three alternative histories from Eqs. (2), (3) and (12) inthe form of Feynman paths that begin at the time t1 and end at the timet5 (see Fig. 3). The upper path denoted by (2) represents the history (2),and likewise the other two paths denoted by (3) and (12) represent thehistories (3) and (12), respectively.

We show now, by using the formalism in Ref. 11 that the apparentparadoxical traits seemed inherent in the DCE and the EPR paradoxes areonly apparent ones, and really there is no paradox in both cases. We referto Fig. 4 which shows that to the third Feynman path denoting the history(12) there are secondary Feynman paths appended to it. For this primarypath we plot 2 secondary Feynman paths between the points at t1 and t2

which represent the two possible decisions of the observer to initiate theexperiment of sending the initial wave at the time t1 or not begin theexperiment. We also plot two other secondary Feynman paths between thepoints t4 and t5 which represent the two possible decisions of the observerto put the half silvered mirror at the point 4 at time t4 , or leave it unoc-cupied. Between the times t2 and t3 , and between t3 and t4 we plot nosecondary Feynman paths because the initial wave proceeds between thesepoints through the routes (2a+2b) and (3a+3b) with probability unity.The three given points t2 , t3 , and t5 through which pass all the secondaryFeynman paths are ``sure'' states.(11) That is, these states have probabilitiesunity for each as can be seen from Eqs. (12) and (13). We remark that for the

Figure 4

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Figure 5

other two primary paths (2) and (3) which have probability of 18 for each,

and two sure points (states) at the times t3 and t5 , the corresponding figureappropriate for them is Fig. 5 in which we see two secondary Feynmanpaths between the points t1 and t2 which represent the two possible deci-sions of the observer at the time t1 to begin or not begin the experiment.Figure 5 also shows two other secondary Feynman paths, between thetimes t2 and t3 , which represent the two possible paths of the initial wavethrough 2a or 2b. Two other secondary Feynman routes are shownbetween the times t4 and t5 and these represent the two possible decisionsof the observer to put the half silvered mirror at the point 4 at time t4 , orleave it unoccupied. (see Eqs. (2)�(5), especially (4) and (5))). We discussnow the history from Eq. (12) in terms of the corresponding Feynmanpaths graphed in Fig. 4. The amplitude for the initial wave to propagatefrom the initial state at time t1 to the first ``sure'' state at time t2 is(13, 11)

91st=7i,i (14)

On the left side of the last equation we have the mentioned amplitude. Onthe right ,i denotes the amplitude of a certain path from the initial stateto the first ``sure'' state. The sigma sign denotes summation over the twopossible paths between these states (see Fig. (4)). Equation (14) can beunderstood in the following way: if we denote the parameter that specifiesthe initial state by h and that specifying the first ``sure'' state by j, then theamplitude to go from the initial state to the first ``sure'' state is:(13)

,hj=7i,hi,ij . If we compare the last equation with (14) we find 91st=,hj ,,hi,ij=,i . In the same manner we can understand the followingEqs. (15)�(17). As can be seen from Fig. 4 the amplitude of the initial waveto propagate from the point t2 to t3 is 1 (between these two points we haveno secondary Feynman paths). Thus the amplitude to propagate from theinitial state to the second ``sure'' state at the time t3 via the first ``sure'' stateis the following conditional probability

92nd=7i,j,i (15)

,j on the right side of (15) is the amplitude of a certain path from the first``sure'' point (state) to the second ``sure'' point. As we have just remarkedthis amplitude has the value of 1, so that we have 91st=92nd . Continuing

821The Zeno Effect

the same procedure we get, for the amplitude of the initial wave to cometo the third ``sure'' state at t5 via the first two ``sure'' states.

93rd=7k 7i ,k, j,i (16)

,k is the amplitude of the initial wave to propagate between the points 4and 5 (between the times t4 and t5), and 7k is the appropriate sum overthe two possible paths as shown in Fig. 4.

Now, the essence of the paradox of the DCE is that we assume, thatbefore the observer decides at the time t4 if he would put the half silveredmirror at the junction point (4) or leave it unoccupied, the initial wave hasalready propagated through one of the possible Feynman paths. That is,for this wave this possible path has been concretized and becomes real, andas we have assumed this happened before the time t4 . If we refer to Fig. 3then just before the time t4 (when the observer has finally decides if hewould put the half silvered mirror at the point 4, or leave it unoccupied)our initial wave is propagating, according to our assumption, through onlyone Feynman path from the existent three. But this concretization of onlyone Feynman path is achieved according to Aharonov and Vardi onlythrough dense measurement along this path. Suppose, for example, thatthis path is the Feynman path (12) in Fig. 3 (our following arguments andequations will be valid also if we take the path (2) or (3) in Fig. 3 with thecorresponding changes) in which there are 3 ``sure'' points (states). In orderto make real, in the sense of Aharonov and Vardi, (7) the path (12) we dodense measurement along it in the following manner:(11) Fig. 4 shows usthat at the point t1 we have two secondary Feynman paths that denote thetwo possible decisions of the observer at the time t1 to begin or not beginthe experiment. Also at the point t4 we have two other secondary Feynmanpaths which, likewise, represent two possible decisions of the observer atthe time t4 to put the half silvered mirror at the point 4 or leave it unoc-cupied. Dense measurement is accomplished (by the observer who performsthe relevant experiment) by repeating the same experiment a large numberof times over an overall finite time. Equation (16) gives us the amplitudeof the initial wave to come to the third ``sure'' state at t5 via the first two``sure'' states at t2 and t3 . When this experiment is performed twice alongthe same time that was allocated to performing it once (so that each ofthese two identical experiments takes half the time it has taken when it wasperformed once) we obtain for the amplitude of the initial wave to arrivetwice to the third ``sure'' state via the previous two ``sure'' states

9twice=(7k 7i,k, i,j )(7m 7n,m,n,p) (17)

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The expression in the first parentheses is the probability amplitude of thefirst experiment, and the second one is the corresponding probabilityamplitude of the second identical experiment. The suffix of twice on the lefthand side denotes that we deal with an experiment performed twice. ,j inthe first parentheses, and ,p in the second denote the amplitudes of theinitial wave to propagate from t3 to t4 . These amplitudes have each thevalue of 1 as we have already remarked. Thus these two amplitudes havenot 7 signs to sum over which, as ,k and ,i have in the first parentheses,and ,m and ,n have in the second. The different ,'s in each parentheses arenow the probability amplitudes of the initial wave to propagate betweenthe different given ``sure'' points in half the times it takes it to propagatebetween the same ``sure'' points when the experiment is performed onlyonce. We can write Eq. (17) as

9twice=7k 7i 7m 7n,k ,i,m,n ,j,p (18)

It is not hard now to generalize to n repetitions of the same experimentover the same total time that was allocated when this experiment has beenperformed once. We obtain

9n=(7k 7i,k ,i, j )(7m7n,m,n,p) } } } (7r7s,r,s,t)

=7k7 i7m7n } } } 7r7s,k ,i,m,n } } } ,r,s ,j ,p } } } ,t (19)

On the right hand side of the last equation we have 2n sums, where eachsum is over only two possibilities, and where each , is the probabilityamplitude of the initial wave to propagate between the different ``sure''points in t�n time, where t is the overall finite time that takes when thisexperiment is performed once.

In the limit of dense measurement when n � � we write (19) as

limn � �

9n=7k 7i } } } ,k,i } } } (20)

where the number of the sums and the ,'s tend also to �. In this limit the,'s that were the probability amplitudes for the different paths between thedifferent ``sure'' points (states), become the probability amplitudes for thesure states themselves, since as n � � the times t�n, that are the times ittakes the initial wave to propagate between the sure states, tend to zero. Inother words, when n � � all the different secondary Feynman pathsactually vanish and we remain with only the primary one, which is now the``sure'' realized Feynman path. Thus the ,'s become the probabilityamplitudes for states and not for paths, so that we can represent them inDirac's notation as |,). Now, in order to obtain the probability of the

823The Zeno Effect

Feynman path from Eq. (12) in the limit of dense measurement, we multiplyEq. (20) by its conjugate and obtain, using Dirac's notation

limn � �

(9n | 9n) =(,k | ,k)(,i | , i) } } } =1 (21)

The result of 1 is obtained because as we have remarked the ,'s, in the limitn � �, are the probability amplitudes for the ``sure'' states which hasprobabilities of 1.(11) Also we can assume orthogonal wave functions. Insummary, we see that the probability of the appropriate Feynman path, inthe limit of dense measurement, is 1. That is, this path has been realized byperforming dense measurement along it. We must remark that in order todo the above dense measurement along a path we take the maximum finegrained histories chain in which the projection operators are considered ateach and every time (compare with the coarse grained representation inEq. (1)). From (21) we get the known result that the amplitude of the``sure'' Feynman path is 1 (which entails that the amplitudes of the otherFeynman paths are zero), and this is valid immaterial of what theobserver's decision will be later at the time t4 . That is, the path (12) ismade ``sure'' irrelevant of any later decision of the observer at the time t4 .We, therefore, see that the first assumption that the initial wave propagates(before the time t4) through one concrete Feynman path has deprived theobserver of its capability of deciding later the wave route in contrast to thenature and essence of the DCE. Thus we conclude that this assumption isfalse and we can not speak of any concrete Feynman path through whichthe initial wave propagates before the time t4 . Only after the act of theobserver at t4 we can begin to speak about the route of the wave, and thisis in complete accord of what we have already remarked about the equiv-alence of the knowledgment of the observer and dense measurement. Inother words, and as we have remarked, we cannot consider only a part ofa Feynman path as we do here when we falsely assume that the incomingwave has propagated along a definite Feynman path before the time t4 . Wecan speak of a Feynman path only between two previously given knownstates (``sure'' points). There is no Feynman path that begins at a ``sure''point (state) but did not end at such a point.

5. THE FEYNMAN PATHS OF TELEPORTATION

We present a short review of the BBCJPW teleportation proposal forteleporting the quantum state |,) of a spin 1

2 particle. The two componentparticles that play the central role in this teleportation process are two spin

824 Bar

12 particles : and ;, that are supposed to originate, by some decaying pro-cess, from an initial physical system (atom, molecule etc). We denote by|u:(+)) and |u:(&)) the eigenvectors that correspond to the eigenvalues+1 and &1 respectively of the Pauli matrix _3(:) which represents thethird component of the spin angular momentum for :. Likewise we denoteby |u;(+)) and |u;(&)) the corresponding eigenvectors of the Paulimatrix _3(;) for ;. These two particles are prepared in the EPR singletstate

|9 (&):; )=

121�2 [|u:(+)) |u;(&)) &|u:(&)) |u;(+))] (22)

The subscripts : and ; signifies the particles of this EPR state. We keep thenicknames used in Ref. 8 and call the teleporter Alice, and the receiver ofAlice's information Bob. We designate Alice's particle, whose unknownstate she wishes to teleport to Bob by #. Alice keeps also the particle :,while Bob keeps the particle ;. Now Alice performs a measurement, on thetwo particles : and #, in the Bell operator basis, (10)

|9 (&)#: )=

121�2 [|u#(+)) |u:(&))&|u#(&)) |u:(+))]

|9 (+)#: )=

121�2 [|u#(+)) |u:(&))+|u#(&)) |u:(+))]

(23)

|8 (+)#: )=

121�2 [|u#(+)) |u:(+))+|u#(&)) |u:(&))]

|8 (&)#: )=

121�2 [|u#(+)) |u:(+))&|u#(&)) |u:(&))]

The unknown state of # is written generally as

,#=a |u#(+))+b |u#(&)) (24)

where |a|2+|b| 2=1. Before Alice's measurement the complete state of thethree particles :, ; and # is

|9#:;) =a

21�2 ( |u#(+)) |u:(+)) |u;(&))&|u#(+)) |u:(&)) |u;(+)) )

+b

21�2 ( |u#(&)) |u:(+)) |u;(&)) &|u#(&)) |u:(&)) |u;(+)) )

(25)

825The Zeno Effect

where use is made of Eq. (22). We write the last equation in terms of theBell operator basis vectors from Eqs. (23) as

|9#:;)= 12 [|9 (&)

#: ) (&a |u;(+))&b |u;(&)) )

+|9 (+)#: ) (&a |u;(+))+b |u;(&)) )

+|8 (&)#: ) (a |u;(&)) +b |u;(+)) )

+|8 (+)#: ) (a |u;(&)) &b |u;(+)) )] (26)

After Alice's measurement she will receive as a result one of the four Bellvectors from Eq. (23) as representing the state of her two particles :and #. Each one of these states is received with a probability of 1

4 . After themeasurement of Alice the particle ; of Bob will be in one of the four purestates multiplying the Bell basis vectors in Eq. (23), depending on theAlice's measurement result. These possible states of Bob's particle are

|u;)=_ab& , _&1

001& |u;) , _0

110& |u;) , _0

1&10 & |u;) (27)

Each of the four outcomes for the state of Bob's particle is connected in asimple fashion to the original state of the # particle. That is, Bob eitherreceives for the state of his ; particle an exact replica of the unknown #particle, except for a phase factor (in case the outcome of Alice's experi-ment is the singlet 9 (&)

#: ), or, in the other three cases all he has to do, inorder to receive an exact replica of #, is some unitary operation that causesa 180 degrees rotation around the z, x, or y axes.

In order to be able to discuss the specific BBCJPW process in termsof Feynman paths we first represent a more general one which take theobservers Alice and Bob as integral parts of our physical system. Thus theappropriate chain of the proposed teleportation process will be

CBBCJPW=P2:2

(t2) P1:1

(t1) P0:0

(t0) (28)

The :'s subscripts in the last equation do not denote the particle :. P0:0

(t0)is a set of projections at the time t0 into four possible different states: Thetwo EPR particles # and : has been prepared in one of the three tripletstates, or these two particles has been prepared in the singlet state. Assum-ing that Alice's experiment has been performed at the time t1 , P1

:1(t1)

denotes the projections at the time t1 into four possible different statescorresponding to the four different outcomes of Alice's experiment (seeEq. (23)). Assuming also that Bob receives the information about Alice'sexperiment result at the time t2 , P2

:2(t2) are the projections at the time t2

into four possible different states corresponding to the four possible pure

826 Bar

states into which his ; particle might have been projected (see Eq. (24)),depending on the four possible results of Alice's experiment. We thus seethat the chain (28) is actually composed of 64 different Feynman paths.Now, if we take into account only the BBCJPW scheme that deals, specifi-cally, with the singlet state of the two initial EPR particles, then accordingto our discussion the chain of Eq. (28) is reduced into 16 possible differenthistories

(1) P21(t2) P1

9 #:(+)(t2) P0

singlet(t0)

(2) P22(t2) P1

9 #:(+)(t2) P0

singlet(t0)

(3) P23(t2) P1

9 #:(+)(t2) P0

singlet(t0)

(4) P24(t2) P1

9 #:(+)(t2) P0

singlet(t0)

(5) P21(t2) P1

9 #:(&)(t2) P0

singlet(t0)

(6) P22(t2) P1

9 #:(&)(t2) P0

singlet(t0)

(7) P23(t2) P1

9 #:(&)(t2) P0

singlet(t0)

(8) P24(t2) P1

9 #:(&)(t2) P0

singlet(t0)(29)

(9) P21(t2) P1

8#:(+)(t2) P0

singlet(t0)

(10) P22(t2) P1

8#:(+)(t2) P0

singlet(t0)

(11) P23(t2) P1

8#:(+)(t2) P0

singlet(t0)

(12) P24(t2) P1

8#:(+)(t2) P0

singlet(t0)

(13) P21(t2) P1

8#:(&)(t2) P0

singlet(t0)

(14) P22(t2) P1

8#:(&)(t2) P0

singlet(t0)

(15) P23(t2) P1

8#:(&)(t2) P0

singlet(t0)

(16) P24(t2) P1

8#:(&)(t2) P0

singlet(t0)

The subscripts of 1 through 4 in P2(t2) denote the four possible outcomesfor Bob's particle ; as is written in Eq. (24) from left to right. Now fromEqs. (22) and (23) we realize that from the 16 different possible Feynmanpaths only the paths 2, 5, 12 and 15 have a high probability to be realized,as compared to the other 12 paths that are known to have vanishingprobabilities of realization. The reason being that the initial two EPR par-ticles have been prepared in the singlet state which is characterized bypredicting opposite results for measurements of the components along n̂ ofthe spins of the particles : and ;, where n̂ is an arbitrary unit vector.(12)

Moreover, this piece of information (that the two EPR particles have been

827The Zeno Effect

prepared in the singlet state) must have been known to Bob when hereceives Alice's message. Because only by using this knowledge he can atmost do only a unitary operation in order to get a replica of the initialstate of the particle #. This is shown a explicitly in Eq. (25) which has beenget directly by using Eq. (22), which gives the singlet state of the two EPRparticles. In other words, Bob knows before he get Alice's message that theparticles : and ; have been prepared in the singlet state. Otherwise, as wewill now show, no teleportation process can at all be accomplished. Inorder not to encumber our discussion and equations with all the 64possible Feynman paths of Eq. (28) we assume that the set of projectionsP0

:0(t0), which refer to the possible states of the initial : and ; particles,

contains only two states: the singlet state from Eq. (22) which has beendealt with up to now, and the following triplet state

|9 (+):; )=

121�2 [|u:(+)) |u;(&)) +|u:(&)) |u;(+))] (30)

In the following discussion we take into account, that the triplet state fromEq. (30) also has the same property of the singlet state from Eq. (22) ofpredicting opposite results for the measurements of the components, alongan arbitrary unit vector n̂, of the spins of the particles : and ;.(12) As forthe experimental part of Alice (on the two particles # and :) in the Belloperator basis (see Eq. (23)), where the two initial particles : and ; hasbeen prepared in the triplet state (30), we can see that the former Eq. (25)(for the case of the initial singlet state of : and ;) which expresses the com-plete state of the three particles :, ; and # before Alice experiment becomesnow (where the unknown state of # is written as in Eq. (24))

|9#:;) =a

21�2 ( |u#(+)) |u:(+)) |u;(&))+|u#(+)) |u:(&)) |u;(+)) )

+b

21�2 ( |u#(&)) |u:(+)) |u;(&)) +|u#(&)) |u:(&)) |u;(+)) )

(31)

Now, as for Eq. (25) we express each product |u#) |u:) as a linear com-bination of the Bell operator basis vectors from Eq. (23) and get

|9#:;) = 12 [ |9 (&)

#: (a | u;(+))&b |u;(&)) )

+|9 (+)#: )(a | u;(+))+b |u;(&)) )

+|8 (&)#: )(a | u;(&)) &b |u;(+)) )

+|8 (+)#: )(a | u;(&)) +b |u;(+)) )] (32)

828 Bar

Now, the possible states of Bob's particle ;, corresponding respectively toAlice's possible results |9 (&)

#: ), |9 (+)#: ) , |8 (&)

#: ) and |8 (+)#: ) , are then (com-

pare with Eq. (27))

|u;)=_10

0&1& |u;) , _1

001& |u;) , _0

1&1

0 & |u;) , _01

10& |u;)

(33)

From the last equation we see that now Bob has to perform differentunitary operations in order to get a replica of the initial state of #, as com-pared to the unitary operations he does for the former case of an initialsinglet state of the particles : and ;.

Suppose now that Bob does not know the state of the two EPR par-ticles ; and : at the time t0 . All he knows is that these two particles maybe at the time t0 in either the singlet state from Eq. (22) or the triplet statefrom Eq. (30). But this ignorance of the exact initial state of : and ; doesnot prevent Alice from doing her experiment on the two particles # and :in the Bell operator basis from Eqs. (23). As before, she gets one of the fourresults from these equations. But now when Alice sends her result to Bobhe will not be able, by any unitary or nonunitary operation, to get a replicaof the unknown state of #. This is because for any result of Alice, Bob doesnot know which unitary operation to perform, that from Eq. (27) in casethe two particles : and ; were prepared in the singlet state at the time t0 ,or the corresponding one from Eq. (33) where these particles has beenprepared in the triplet state (30). Thus we see that when the initial state of thetwo particles : and ; is not known then no teleportation process is possible.

In summary, from our discussion, we see that in order to enable Bobto perform his unitary operations so as to get a replica of the initial stateof # he must know in advance the exact EPR state of the two particles :and ;. Thus in order to have any kind of teleportation the exact state ofthe EPR pair must be known to Bob.

6. THE DENSE MEASUREMENT OF TELEPORTATION

In the former section we have discussed a coarse grained version of thehistory chain of the BBCJPW proposed teleportation scheme (see Eqs. (28)and (29)). This coarse grained treatment of the BBCJPW scheme separatesthe whole process into, actually, three parts: (1) the preparation of theEPR pair, (2) Alice's experiment in the Bell operator basis, (3) Bob'sunitary operations after receiving Alice's results. We have also show that aformer knowledge of the exact EPR state amounts to realizing the four

829The Zeno Effect

Feynman paths 2, 5, 12, and 15 (see Eq. (29)) out of the 64 possible Feynmanpaths of the chain from Eq. (29), and when Bob obtains the result of Alice'sexperiment then we would be left with only one Feynman path realized.Thus a premature knowledge of the would be result of the experiment(compare with the DCE case) realizes from all the possible Feynman pathsonly one of them. This realization of one Feynman path, from, generally,a large number, due to previous knowledge of the result of an experimentdraw again our attention to the Aharonov�Vardi method(7) of realizing anypredetermined Feynman path, from a large number of possible ones, byperforming dense measurement(6) along it.

Now, in order to be able to discuss any Feynman path in terms ofdense measurement along it we must treat this path in its most fine grainedversion. That is, instead of separating it into the three separate parts of theonce performed experiment, as we had done in the coarse grained treat-ment above, we divide it into a very large number of tiny fragments, eachone of which represents the same process of teleportation described in theprevious section. We begin by first realizing that our physical closed systeminclude the three particles :, ; and #, and the two observers Alice and Bob.But since the teleportation process depends only upon Alice's measure-ment, Bob does not have the same importance as that of Alice (the onlyaction of Bob is some unitary operation that does not influences theteleportation process). Following Feynman's formalism(13) we can write forthe probability amplitude of finding the particle ; in the same state as thatof the particle # (up to some unitary operation)

91a(:;) c;=:

b

,a(:;) b(:#),b(:#)c;

(34)

91a(:;)c;is the mentioned probability. ,a(:;) b(:#)

is the probability amplitudethat if the two initial particles : and ; has been prepared as an EPR pair,then the two particles : and # will be found, after a measurement of Alice,in some state corresponding to one of the states of the Bell basis. ,b(:#) c;

isthe probability amplitude that if the two particles : and # are found to bein a state corresponding to one of the states of the Bell basis, then the par-ticle ; will be found to be in the same state as that of #. The �b is the sum(or integration) over all the mutually exclusive alternatives of Alice'sresults. The 1 in the subscript of 9 denotes that we are considering onlyone measurement of Alice. But since we treat here the Zeno effect, we haveto deal with a large number of measurements of Alice and not only one.Thus if Alice repeats her experiment a second time we get

92a(:;) c;=:

b

:d

,a(:;) b(:#),b(:#)d(:#)

,d(:#) c;(35)

830 Bar

The 2 in the subscript of 9 denotes that we deal now with twoexperiments. On the right hand side of (35) we insert a second �d over thesame alternatives as those of �b since Alice repeat the same experiment(note that the subscripts of d are as those of b). The generalization to nrepetitions is obvious and we get

9na(:;) c;=:

b

:d

:e

} } } :i

:j

,a(:;)b(:#),b(:#) d(:#)

,d(:#) e(:#)} } } ,i(:#) j(:#)

,j(:#)c;(36)

Again the meaning of the subscript n at the left hand side of the last equa-tion means that we deal now with n repetitions of the same measurementof Alice. We take n sums on the right hand side of Eq. (36), all of themover exactly the same alternatives as can be seen from all the subscriptsof :#. Equation (35) can be understood as the probability amplitude to gofrom the initial state of preparing the EPR state to another state, whichdenotes performance of Alice's measurement, via another similar statewhich also denotes performance of the same measurement. Likewise,Eq. (36) signifies the probability amplitude to arrive from the initial stateto the state that denote the nth measurement of Alice via all the previous(n&1)st measurements. Each of the ,'s in the Eqs. (35)�(36) (except thefirst and the last) which has subsubscripts :# denotes the amplitude of acertain path from a measurement in the Bell operator basis to anotheridentical measurement. Equation (36) is the sum over all paths that beginin the initial state of preparing the EPR state, go through n consecutiveidentical measurements in the Bell vector basis, and end in finding the par-ticle ; in the same state (up to some unitary operation) as the initial stateof #. When the number n of these points (states denoting performances ofAlice's experiments in the Bell vector basis) goes to infinity (n � �), thenall these different paths that denote the possible evolutions of the systembetween these identical measurements become, actually, one path. Thereason is that each two close different points, in the limit of n � �, becomealmost the same point so that the secondary Feynman paths between themvanish, and the ,'s in Eq. (36) become the conditional amplitudes for states(not for paths). That is, the conventional wave functions | ,#:) (in Dirac'snotation). In such case we can write Eq. (36) in the limit of n � � as

limn � �

9na(:;) c;= lim

n � �:b

:d

:e

} } } :i

:j

,a:; b(:#),b(:#)d(:#)

,d:#e(:#)} } } ,i(:#) j(:#)

,j(:#) c;

= limn � �

:b

:d

:e

} } } } } } :i

:j

|,a(:;) b(:#)) |,b(:#) d(:#)

) |,d(:#) e(:#))

} } } |,i(:#) j(:#)) |,j(:#) c(;)

) (37)

831The Zeno Effect

In order to find the probability of the Feynman path of teleportation, inthe limit of dense measurement, we multiply Eq. (37) by its conjugate andfind (in Dirac's notation), assuming orthonormalized wave functions.

limn � �

(9na(:;)c;| 9na(:;) c;

)

=(,a(:;) b(:#)| ,a(:;) b(:#)

)(,b(:#) d(:#)| ,b(:#)d(:#)

)(,d(:#) e(:#)| ,d(:#) e(:#)

) (38)

} } } (,i(:#) j(:#)| , i(:#) j(:#)

)(, j(:#) c(;)| ,j(:#)c(;)

)

=1

Thus we get that in the limit of dense measurement the Feynman path ofteleportation, from the initial state of preparing the EPR pair to the finalstate of obtaining a replica of the initial state of the # particle, is realizedwith probability 1. We see, therefore, that prior knowledge of the corre-sponding state is equivalent to performing dense measurement along theappropriate Feynman path, and both lead to the realization of the telepor-tation process as in the DCE case. We must remark here that in our treat-ment of the DCE case in the previous section we have summed twice overeach performance of the experiment, once over the two possible decisionsof the observer at the time t1 , and second time over the two possible deci-sions of the observer at the time t4 (see Eq. (16)). Here in the teleportationprocess, for each performance of which, we sum only once over the possibleresults of Alice's measurement. That is the reason that the right hand sideof Eq. (36) contains n sums, and the right hand side of (19) contains 2nsums.

7. DENSE MEASUREMENT AND THE EPR PARADOX

We discuss now the EPR paradox, and show that the same falseassumption mentioned in Sec. 4 is actually what causes the EPR paradoxto seem so paradoxical. In the following we represent a short review of theBohm's formulation(12) of the EPR paradox (which seems to me the bestformulation of this paradox). We begin with an initial physical system(atom, molecule, etc) that decay into two spin 1

2 particles : and ;. Wedenote by u:(+) and u:(&) the eigenvectors that correspond to the eigen-values +1 and &1 respectively of the Pauli matrix _3(:) which representsthe third component of the spin angular momentum for :. In the samemanner we denote by u;(+) u;(&) the corresponding eigenvectors of thePauli matrix _3(;) for ;. As for the teleportation case we discuss also here

832 Bar

the two physically different states that can be composed from the aboveeigenvectors: the singlet state.

'0=1

21�2 [u:(+) u;(&)&u:(&) u;(+)] (39)

and the triplet state

'1=1

21�2 [u:(+) u;(&)+u:(&) u;(+)] (40)

Adding and subtracting the states (39) and (40) we get

121�2 ('0+'1)=u:(+) u;(&)

(41)

121�2 ('1&'0)=u:(&) u;(+)

respectively. In his treatment Bohm consider only (:, ;) pairs with thefollowing wave function

9(x1 , x2)='09:(x1) 9;(x2) (42)

'0 is the singlet state, 9:(x1) and 9;(x2) are the space dependent parts ofthe wave functions for : and ; respectively. 9:(x1) is assumed to be someGaussian function with modulus different from zero only in a region R1 ofwidth 21 centered around the point x10 . Likewise, 9;(x2) is anotherGaussian function localized in the region R2 of width 22 centered around x20 .In order to leave the systems : and ; separate from each other the followingcondition must be fulfilled |x20&x10 |>>21 , 22 . We suppose now the twoparticles : and ; to move to the right and to the left so that the distancebetween them increases linearly with time such that all interactionsbetween them are vanishingly small. Suppose now we have a large ensembleE of (:, ;) pairs in the state (42). If we measure _3(:) at time t2 on all :'sof a subset E1 of E and +1(&1) is found, then a future (t3>t2) measure-ment of _3(;) will certainly (with probability 1) give &1 (+1) (because ofthe singlet state coefficient in (42)). Taking into account the EPR realitycriterion(1) we assign to the ; 's of E1 an element of reality *1(*2) that fixa-priori the result &1 (+1) of the _3(;) measurement. Now the complete-ness assumption of quantum mechanics treats an object ; with a predeter-mined value _3(;) by assigning it the state u;(&) u;(+) so that the ensembleE1 has to be described in spin space by the second of the combination (41)

833The Zeno Effect

even for t1<t2 . If we exclude the possibility that *1(*2) is created by themeasurement of _3(:) we must conclude that *1(*2) belongs to all ; 's of Eand not only to those of E1 . Thus we conclude from the completenessassumption that the second of the combination (41) applies to all the pairsof E in contradiction to (42). This is the EPR paradox. We meet onceagain the same paradoxical characteristics we have already encountered inthe DCE case. That is, from a measurement performed on : at the time t2

we actually draw concrete conclusions about the physical description of the; particle at the earlier time t1<t2 . We now show, as we have done for theDCE case, by using the Feynman path formalism in the dense measure-ment limit that the paradox is only an apparent one. In this case we havethe following chain of time ordered projection operators:

CEPR=P2:2

(t2) P1:1

(t1) P0:0

(t0) (43)

The :'s in (43) do not denote the : particles. P0:0

(t0) is the set of projectionsat the time t0 into two possible different states: (1) The initial system hasdecayed into the two spin particles : and ;. (2) The initial system has notdecayed. P1

:1(t1) denotes the projections at the time t1 into the two possible

different states: (1) the spin state of the particle ; is (+). (2) the spin stateis (&). P2

:2(t2) are the following projections at the time t2 into the two

possible different states: (1) the spin state of the particle : is (&). (2) itsspin state is (+1). The chain (43) is reduced into the following twopossible different histories (we take into account only the decaying casealternative)

CEPR1=P2

:+(t2) P1

;&(t1) P0

decay(t0) (44)

CEPR2=P2

:&(t2) P1

;+(t1) P0

decay(t0) (45)

The suffixes ;& , ;+ , :& , :+ denotes the spin states of (&), (+) for theparticles ; and : respectively. From the histories (44) and (45) we see thatwhen the spin of the particle : is (+) with probability 1

2 then the corre-sponding spin state of the particle ; at the earlier time t1 (and at all times)is (&) (with probability 1), and vice versa. If we assume that the probabil-ity of decay is 1

2 then the probability of the histories (44) and (45) are equalboth to 1

4 :

p(CEPR1)= p(CEPR2

)= 12 V 1 V 1

2= 14 (46)

If we plot the corresponding Feynman paths as in Fig. 4, then from the lastdiscussion we can see that for each of the two histories (44) and (45) wehave one ``sure'' point (state) at the time t1 (``sure'' in the sense of Sec. 4)

834 Bar

and two secondary Feynman paths between the points t0 and t1 denotingthe two possible initial states of the system. That is, to decay to the twoparticles : and ; or not to decay. We have two other secondary Feynmanpaths between the points t1 and t2 that denote the two possible results ofmeasuring the spin of the : particle. We can take, for example, one of thetwo above histories (as we took the history (12) in Sec. 4) and proceed inexactly the same manner as for the DCE and the teleportation cases. Theonly two differences between either one of the histories (44) or (45) and thehistory (12) is that here we have three separate parts in contrast to the fiveseparate parts of (12). Also here we have one ``sure'' point compared to thethree ``sure'' points of (12). Now as we see from the last discussion the coreof the EPR paradox is the assumption that the particle ; is and was alwaysat some definite state, so that only one of the two histories (44) or (45)may be ``sure'' and real for him, and according to Aharonov and Vardi this``sureness'' is obtained through doing dense measurement along the rele-vant path. Thus we can take the dense measurement limit of either of thetwo histories (44) or (45) and obtain similar equations to these obtainedfor the DCE and the teleportation cases. That is, by the dense measure-ment limit we can make real and concrete any Feynman path we choosefrom the probable ones, and this is immaterial of what the result ofthe experiment performed on : at t2 . Thus the assumption of EPR that thestate of the particle ; is and was always clear and definite even before thetime t2 is erroneous since we can not speak before the measurement at t2

of any definite state of either the particle : or ;. In other words, we mustconsider the whole Feynman path from t0 to t2 and not part of it, as wehave already remarked for the DCE case. It has been proved in Ref. 11 thatwhat we call here the ``sure'' states are equivalent to the universal wavefunctions of Everett, (14) that is, these states which become ``sure'' as a con-sequence of dense measurement have the same characteristics as those ofthe everett's wave functions. Now the Everett's wave functions are obtainedfrom a superposition over the memory sequences of the observers. Thesememory sequences denote the knowledge and information of all their pastexperiments and experiences. Thus we see, as we have remarked here, thatprevious knowledge and dense measurement are equivalent to each other.To be more exact, knowledge follows dense measurement in the senseexplained in the following concluding remarks.

8. CONCLUDING REMARKS

The correlation we find in the previous sections between the knowl-edge of the observer and dense measurement brings us to the conclusion

835The Zeno Effect

that dense measurement is actually the means and apparatus throughwhich physical knowledge is obtained for the first time. Suppose, for exam-ple, that we want to establish and find out for the first time the propertiesof some new observable not thought of before. Since we deal with quantummechanics all the knowledge we can hope to obtain about this unknownobservable is its expected values. Now, each of these values can not beobtained not only from any single measurement, but also from any largenumber of automatically and unintentionally repeated unconnected measure-ments. Only when all these many identical experiments are performed in aconsistent and connected way with the intention to obtain a final result(from all these repeated experiments) after a finite time that we get the rele-vant expected value of this new observable for the first time. That is, wehave to do dense measurement in order to establish and discover theproperties of any unknown observable.

Thus the knowledge obtained about any physical observable becomesaccessible to us for the first time only after performing dense measurement,where by this term we do not necessarily mean, as is usually believed, thatthese dense measurements are performed in a serial and consecutivemanner without interruption (where each separate measurement is performedin a very short time) until some definite time where these measurementsend. Our notion of dense measurement is that its sole role is to cause somepreviously unknown physical knowledge to become known to us at the endof these measurements. These measurements can be conducted nonseriallyand nonconsecutively. Moreover, each separate measurement can be per-formed even for a long time, so long as this time is very short comparedto the overall time of all these measurements. The only required conditionfor these measurements to be called dense is that through them we acquirenew physical knowledge. We can see from the following fundamental Zenoeffect relation n= T

t (where T is the overall finite time of all the repeatedidentical experiments, and t is the time of a single experiment) that if T isvery much large, then t does not have to be infinitesimal in order to get theZeno effect. Moreover, these identical experiments may be performedsimultaneously by a large number of experimenters in which case we getalso the Zeno effect as has been shown in [11], without having to assumethat each such observer conducts his experiment in an infinitesimal time. Inother words, our description of the physical world becomes more exact andmore certain when the experiments and the measurement, establishing andcorroborating this description, are repeated a large number of times. Andthe denser these repetitions become the more ``sure'' this description turnsout to be.

The mathematical formalism used here is that of the histories for-malism [4, 5], but as has been remarked we did not adhere strictly to the

836 Bar

rules of this formalism, especially that which denied any assignment ofprobabilities to the separate parts of the histories chains. As has beenstated explicitly in [4] the reason for regarding the histories chains asunreduced entities are that in the quantum mechanics of closed systems[4] there is no fundamental division of a closed system into measuredsubsystems and measuring apparatus, and no fundamental reason for theclosed system to contain classically behaving measuring apparatus in allcircumstances. Here in our work we encountered no such problems inthe discussions of the Wheeler's DCE, the teleportation process, and theEPR paradox. The measurement part of each one of these physicalphenomena is sharply defined and confined to its specific time and spaceregion. For example, in the Wheeler's DCE the measurement part con-sists in inserting (or not inserting) at the time T4 the half silvered mirrorat the junction point 4, and in the teleportation process the measure-ment part is comprised of finding out at a definite time the exact state(from the four possible Bell's states) of the two given particles : and #(Alice's experiment, see Section 5). Thus we have no reason to ignorethe strong probabilistic nature of the separate parts of the historieschains of these three physical phenomena.

In conclusion, by discussing the EPR paradox, the teleportationprocess, and the Wheeler's DCE from the point of view of densemeasurement, we show that previous knowledge of the states of thesystem is equivalent to dense measurement.

ACKNOWLEDGMENT

I wish to thank L. P. Horwitz for discussions on the subject of thispaper and for his review of the present manuscript.

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837The Zeno Effect

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