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97/1475 CLNS Rede nitions of Histories by Measurements - An Explanation of \Nonlocality" Observed in EPR-Bohm Experiments Yuri F. Orlov July 23, 1999 Floyd R. Newman Laboratory of Nuclear Studies Cornell University, Ithaca, New York 14853 USA Abstract It is proved in the frame of standard quantum mechanics that selection of di erent ensembles emerging from measurements of an observable leads to identi cation of corresponding reductions of the initial, premeasured state. This solves the problem of \nonlocality\ observed in EPR-Bohm-type experiments. The phenomenon of \nonlocality\ is well-established in EPR-Bohm-type experiments (EPRB). For example, in entangled two-particle systems 1,2], a choice of measurement in one of the two causally separated channels in uences results of measurements in another channel. The observed correlations are predicted by quantum mechanics (QM); but since measurement procedures are not included in its formalism, their mutual, long-distance in uence remains inexplicable. This di culty is cleared up in this work by analyzing logical statements about quantum states in the frame of the standard QM formalism, and by clarifying the meaning of measurement procedures. This approach permits us to nd logical connections otherwise hidden. It is proved that in EPRB all such connections are local. In general, the choice of measurement in one of the channels locally de nes the reduction of a premeasured state, which in turn de nes connections between channels in the usual way. Let jkl i; l = 1; 2; ; be eigenstates of an observable(or of a number of commuting observables), K, considered here as discrete, with all kl 's di erent, and let j i be an arbitrary state of a physical system, j i = Pl al jkli. An exact measurement of K with result K = ki leads to the reduction j i ) jkii. It is generally accepted, without proof, that reduced states describe only developments of physical systems subsequent to measurements. Below, in Theorem 3, we prove that under special conditions a reduction of a state may act in both directions of time, so one can replace premeasured state j i by its reduced part, using the information received from the measurement. Lemma 1. The truth of a logical statement about a numerical value of a physical observable is itself an observable, and is represented in quantum mechanics by the pure state density matrix or by a sum of such matrices. Proof. Consider the logical statement kl : \The system is in state jkl i\ or, in short, \K = kl .\ kl can be either \true\ (the numerical value 1) or \false\ (the value 0); these are two values of the classical logical truth of statement kl . Statements corresponding to di erent kj 's (let us call them \elementary\) are mutually exclusive. kl is true if the measurement of K results in kl , and false otherwise. Therefore, kl is an observable and, like any other observable in QM, should be represented by a Hermitian operator, ^ kl , the \truth 2 ^ operator of statement kl ," ^ kl = ^ kl . K and ^ kl have a common set of eigenvectors, jkj i; j = 1; 2; ; ^ kl jkj i = lj jkj i. Here lj is an eigenvalue: kl =1 (\true\) if j = l, and kl =0 (\false\) otherwise. We conclude that ^ kl must be identi ed with the density matrix of the pure state jkli : ^ kl jkl ihklj. Since our choice of K was arbitrary, a density matrix of any pure state j i de ned in the Hilbert space of the physical system, ^ j ih j; ^ j i = j i; and ^ j i=0, where j i is any state orthogonal to j i, represents the logical statement, \The system is in state j i.\ The truth of such a statement depends on whether the physical system is really in state j i. Theorem 1. On the existence of exact locations. If the state of the system, j i; and its observable, K, are such that j i can be written as a superposition It is easy to show 3,4] that every nonelementary statement about numerical values of commuting physical observables is represented by a corresponding sum of density matrices. If there is a degeneracy such that the same state corresponds to kl1 ; kl2 ; , then the statement that the system is in this state is represented by ^ kl1 + ^ kl2 + 2 ln X l1 j i= cl jkli; (1) 2 then the system is located at some point ki of K-space, provided that jkii is represented in the superposition. K: Proof. Let us nd the operator of the following statement about the numerical value of ( ; K) : kl1 _ kl2 _ _ kln (\K = kl1 \) _ (\K = kl2 \) _ _ (\K = kln \); (2) in which only those kl 's that are in the superposition are represented. ( ; K) is symmetric relative to permutations of its constituent mutually exclusive elementary statements. Using Lemma 1, the condition of mutual exclusiveness in the operator representation, ^ kl ^ km = h i ^ kl , and equation ^ ( ; K) 2 = ^ ( ; K), we nd lm ^ ( ; K) = X ^ kl l1 ln jkl1 ihkl1 j + jkl2 ihkl2 j + ::: + jkln ihkln j: (3) ^ ( ; K)j i = j i; (4) i.e., when the system is in state j i; ( ; K) is a true statement. Thus, according to the meaning of ( ; K); this system is located in K-space at one of the points enumerated in (2). Corollary. The statement-disjunction, k1 _ k2 _ values of K, is true in every state-superposition (1). From this ; containing all possible numerical From Theorem 1 it follows, for example, that when a (nonrelativistic) system is in a state of a certain momentum, jpi, it is also certain that this system is located somewhere in the coordinate q-space. Such a statement does not contradict QM, but where this location is remains uncertain. The next theorem shows why such a question has no logical meaning. n > 1; is an eigenstate of operator L; Lj i = lj i, then there does not exist a logical statement, either true or false, about the exact location of this system in K-space. We will omit the formal proof of this theorem. Nor will we discuss here the origins of noncommutativity. If noncommutativity is granted, then Theorem 2 provides the basis for quantum indeterminism 4]: in the general case, we cannot describe with certainty the future of the system undergoing a measurement. The following theorem states that this can be incorrect for the past. 3 Theorem 2. If K and L are two noncommuting observables, and the state (1) of system, Theorem 3. On the rede nition of history by measurements. Let K be an observable such ^ ^ ^^ that, in Heisenberg's representation, the commutator, K(t t)K(t) K(t)K(t t) ! 0 ^ when the time interval t ! 0: (In particular, K may not depend on time at all.) Let the measuring procedure satisfy the condition of \ideal measurement\ formulated below. And let a single measurement at time t0, by de nition, result in some K=k only when statement ( k )t0 msr about that measured value is formulated, and the case separated from cases k 6= k; the corresponding postmeasured ensemble will be called \ensemble K = k.\ The following logical implication is true: If a measured state is a superposition, j i = Pl cl jkl i, and a single measurement results in K = kj at time t0 , then after this measurement, the statement: \Before t0 , the system belonging to the ensemble K = kj was in state jkj i\ is true. Proof. First we formulate a condition of an \ideal measurement,\ a postulate that is commonly assumed and does not contradict practice. We assume that an observable, K, is being measured. If the measured system is in state jkli; l = 1, or 2, or 3, , then the result of the measurement is K = kl . We will write this condition as the following logical implication (valid for every l): ( kl )t0 t ! ( kl )t0 msr (5) Statement (5) makes sense since its constituent statements, formalized being in QM, are assumed to commute, at least in the approximation t ! 0; otherwise, according to Theorem 2, it would be meaningless. From (5) it follows that if the premeasured system is in one of the states jkj i such that j 6= l, i.e., in any eigenstate of K but jkli, then the result of the measurement of K is a kj ; j 6= l. This can be written as k1 _ k2 _ _ kl 1 _ kl+1 _ t0 t ! k1 _ _ kl 1 _ kl+1 _ t0 msr (6) The premise is true if at least one of the elementary statements on the left side is true, and this is indeed the case; then according to (5), that very same elementary statement on the right side is true also, so the conclusion is also true. Now, according to the corollary of Theorem 1, the disjunction of all possible elementary statements kl ; l = 1; 2; ; about numerical values of K is always a true statement (a tautology). Therefore, the disjunction from which only one elementary statement is excluded, as it is in (6), is a logical equivalent (denoted by ) of the logical negation of this statement: k1 _ k2 _ kl 1 _ kl+1 _ kl (7) Indeed, if one of the statements on the left side of the equivalence is true, then the left side is true and all other statements about numerical values of K; kl included, are false; therefore, kl on the right side is true. If none of the statements on the left side is true, then the left 4 side is false and kl is true, as only possibility left to the physical system. Therefore, false, as is the left side. As a result, we can rewrite (6) as kl t0 t kl is ! kl t0 msr (8) (9) And nally, as logically follows from (8), ( kl )t0 msr ! ( kl )t0 t Implication (9) can be used to conclude whether the state of the system was really jkli, only when it is certain that ( kl )t0 msr is true, that is, when the results of the measurement formulated as \K = kl \ have been selected. Absent this procedure, the detector is not being used as a measuring device but only as a target. In such a case, some quantum state emerges as a result of the interaction of the system with this target, so premise ( kl )t0 msr in (9) is not true. For the ensemble selected as K = kl , the a posteriori conclusion from (9) and from the results of measurement is that the state of the system before the measurement was jkli 5,6]. 2 Corollary. If an observable K satis es the commutation conditions of Theorem 3, then when we select an ensemble corresponding to a result K=k of our measurement, we simultaneously select an ensemble corresponding to the value K=k of premeasured physical systems. (Note that the collapse of the premeasured state accompanying this selection is a purely informational e ect.) In EPRB 1,2,7-9] the observables are polarizations of particles, their operators do not depend on time, and t may be nite. In such an experiment, let a pair of particles be prepared in an entangled state j i at time t=0 (for the center of their wave packet), and the time-size of the wave packet, 4t h=4E, be much less than the ight time to either of two detectors. Let detector D1 measure observable K of particle 1 moving in channel 1, and L be an observable of particle 2 in channel 2 with its detector, D2. In the general case, j i= X ij aij jkii1jlj i2; (10) where 1(2) refers to particle 1(2), and ki; lj are eigenvalues of K,L. The phenomenon called \nonlocality\ is the in uence of random choices of noncommuting observables K; K 0; K 00, , to measure, say, particle 1 in channel 1, on the outcomes of independent measurements of 5 particle 2 in the other channel. The signi cance of Theorem 3 is that it permits establishing deterministic logical connections, as in classical physics, between postmeasured states of particle 1 selected in channel 1, and premeasured states of particle 2 that are thereby selected also. Let observable K be measured at time t0 , and the postmeasured ensemble K = kn selected. Then it can be concluded that the premeasured state of particle 1 in ensemble K = 0 kn is jkni1. But in common state (10), jkni1 is coupled one-to-one with jlmi2 = A Pj anj jlj i2, 0 where 1=jAj2 = (wkn )1 is the probability of a (K = kn) result in channel 1, and lm a value 0 of an observable L0 . Therefore, the state of coupled particle 2 is jlmi2 . By selecting the one-particle, postmeasured ensemble K = kn, then, we have also automatically selected the 0 two-particle, premeasured ensemble described by the reduced state jkni1jlm i2: Thus, there are no nonlocal physical in uences; \nonlocality\ in the sense of one-to-one correspondence 0 between selected states of the spatially separated particles, kn $ lm, is a result of the common history of the two particles. It is interesting that in EPRB we can even reconstruct the logical chain, with every in nitesimal link local, from the measurement in channel 1 to the state of particle 2 in channel 2. Since observables K and L in EPRB are constant before measurements, we conclude that at time t0 = N t the value of K is the same as at time t0 (N 1) t; N = 1; 2; . From Theorem 3 we know that at time t0 t; K = kn. Therefore, at time t = 0, when the pair of particles is born, the state of particle 1 in ensemble K = kn is jkni1. Therefore, the state of 0 particle 2 at that moment is jlmi2. Applying similar logical steps to channel 2, we conclude 0 that particle 2 conserves its state L0 = lm until some measurement in channel 2 is made either before or after t0. Let detector D2 now measure an observable, L. If L commutes with L0 , then according 0 to (5) the result of this measurement is deterministic and equals l = lm. If L does not 0 commute with L0, the transition jlmi2 ! jlj i2, where lj results from the measurement of L, is unpredictable. Applying Theorem 3, now, not to the measurement of K but to the measurement of L by detector D2, and selecting postmeasured ensemble L = lj , we will thereby automatically select a two-particle, premeasured ensemble di erent from the twoparticle ensemble connected with ensemble K = kn. Indeed, the result of measurement in 0 channel 2, L = lj , de nes the state of particle 1 in channel 1, jkpi1 = B Pi aij jkii1, where 0 kp is a value of an observable K 0. From this we can conclude that the premeasured reduced 0 state of the two-particle ensemble is jkpi1jlj i2 . Despite the di erences between ensembles, however, probability w (kn; lj ) = h j ^ kn ^ lj j i = janj j2 for measurements in both channels 0 0 does not depend on which of the two possible ensembles, jkni1jlmi2 or jkpi1jlj i2, is chosen. Our a posteriori conclusion about the collapse of initial state j i therefore has an uncertainty caused by noncommutativity, and depends on the information we choose to be our premise 6 | either \K = kn\ or \L = lj \ (or both). References 1] M. Lamehi-Rachti and W. Mittig, Phys. Rev. D. 14, 2543 (1976). 2] A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 49 , 91 (1982). 3] J. von Neumann, Mathematische Grundlagen der Quanten-Mechanik (Springer-Verlag, Berlin, 1932). 4] Y. F. Orlov, Annals of Physics 234, 245 (1994). 5] The \consistent history\ theory (see, for example, Robert B. Gri ths, Am.J.Phys. 55, 11 (1987), contains assumptions similar to Theorem 3 but is conceptually di erent from standard quantum mechanics, whereas our implication (9) lies within the frame of QM. 6] Nariman B. Mistry has proposed a hypothesis similar in some respects to Theorem 3 (personal communication). 7] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). 8] J. S. Bell, Rev. Mod. Phys. 38, 447 (1966). 9] Lucien Hardy, Phys. Rev. Lett. 71, 1665 (1993). 7

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