Quantum Nonlocality Does Not Exist : Bell ' s Theorem and the Many - Worlds Interpretation

Quantum nonlocality is shown to be an artifact of the Copenhagen Interpretation, which assumes that observers obey the laws of classical mechanics, while observed systems obey quantum mechanics. Locality is restored if this arti cial distinction between observed and observer is abolished, and both are assumed to be subject to quantum mechanics, as in the Many-Worlds Interpretation. Using the MWI, I shall show that the quantum side of the Bell's Inequality, generally believed non-local, is really due to a series of three purely local measurements. Finally, I shall justify the Principle of Indi erence of probability theory using quantum mechanics. PACS numbers: 03.67.Hk, 42.50.-p, 03.65.Bz, 89.70.+c 1 e-mail address: [email protected] 1 Nonlocality is the standard example of a quantum mechanical property not present in classical mechanics. A huge number of papers are published each year (in 1997, four in PRL alone [1, 2, 3]) purporting to clarify the meaning of \nonlocality". The phenomenon of nonlocality was rst discussed in the EPR Experiment [4]. We have two spin 1/2 particles, and the two-particle system is in the rotationally invariant singlet state with zero total spin angular momentum. Thus, if we decide to measure the particle spins in the up-down direction, we would write the wave function of such a state as j >= j ">1 j #>2 j #>1 j ">2 p2 (1) where the direction of the arrow denotes the direction of spin, and the subscript denotes the particle. If we decide to measure the particle spins in the left-right direction, the wave function would be written in a left-right basis as j >= j >1 j !>2 j !>1 j >2 p2 (2) Nonlocality arises if and only if we assume that the measurement of the spin of a particle "collapses the wave function" from the linear superposition to either j ">1 j #>2 or j #>1 j ">2 in (1). If such a collapse occurs, then measuring the spin of particle one would x the spin of particle two. The spin of particle two would be xed instantaneously, even if the particles had been allowed to separate to large distances. If at the location of particle one, we make a last minute decision to measure the spin of particle one in the left-right direction rather than the up-down direction, then instantaneously the spin of particle two would be xed in the opposite direction as particle one | if we assume that (2) collapses at the instant we measure the spin of particle one. The mystery of quantum nonlocality lies in trying to understand how particle two changes | instantaneously | in response to what has happened in the location of particle one. There is no mystery. There is no quantum nonlocality. Particle two doesn't know what 2 has happened to particle one when its spin is measured. State transitions are nice and local in quantum mechanics [5]. All these statements are true because quantum mechanics tells us that the wave function does not collapse when the state of a system is measured. In particular, nonlocality disappears when the Many-Worlds Interpretation [6,7,8] is adopted. The Many-Worlds Interpretation (MWI) dispels the mysteries of quantum mechanics. D.N. Page has prevously shown [9] how the EPR reality criterion is completely full lled by the MWI. Here I shall show how the quantum side of Bell's inequality [10], generally believed to be non-local, actually arises from local measurements. In the derivation of this quantum side, I shall also show how the Principle of Indi erence can be justi ed by quantum mechanics. To see how nonlocality disappears, let us analyze the measure of the spins of the two particles from the Many-Worlds perspective. Let Mi(:::) denote the initial state of the device which measures the spin of the ith particle. The ellipsis will denotes a measurement not yet having been performed. We can for simplicity assume that the apparatus is 100% e cient and that the measurement doesn't change the spin being measured (putting in a more realistic e ciency and taking into account the fact that measurement may e ect the spin slightly would complicate the notation but the conclusions would be unchanged). That is, if each particle happens to be in an eigenstate of spin, the e ect of measuring the ith particle changes the measuring device | but not the spin of the particle | as follows: UM1(:::)j ">1=M1(")j ">1; UM1(:::)j #>1=M1(#)j #>1 (3) UM2(:::)j ">2=M2(")j ">2; UM2(:::)j #>2=M2(#)j #>2 (4) where U is a linear operator which generates the change of state in the measurement apparatus, corresponding to the measurement. The operator U will actually be unitary, but this is not essential to the argument. What is essential is linearity. 3 In particular, if particle 1 is in an eigenstate of spin up, and particle 2 is in an eigenstate of spin down, then the e ect of U is UM1(:::)M2(:::)j ">1 j #>2=M1(")M2(#)j ">1 j #>2 (5) even if particles 1 and 2 are light years apart when their spin orientations are measured. Similarly, the result of measuring the ith particle in the eigenstate of spin left would be UMi(:::)j >i= Mi( )j >i, and for an eigenstate of spin right UMi(:::)j !>i= Mi(! )j !>i, which will generate equations for spins left and right analogous to eqs. (3) (5). Now consider the e ect of a measurement on the two particle system in the Bohm state, that is, with total spin zero. This state is (1) or (2) with respect to an up/down or left/right basis respectively. The result is completely determined by linearity and the assumed correct measurements on single electrons in eigenstates. For example, the e ect of measurements in which both observers happen to choose to measure with respect to the up/down basis is UM2(:::)M1(:::) j ">1 j #>2 j #>1 j ">2 p2 = UM2(:::) M1(")j ">1 j #>2 p2 M1(#)j #>1 j ">2 p2 = M2(#)M1(")j ">1 j #>2 p2 M2(")M1(#)j #>1 j ">2 p2 (6) It may appear from eqn. (6) that it is the rst measurement to be carried out that determines the split. This is false. In fact, if the measurements are carried out at spacetime events which are spacelike separated, then there is no Lorentz invariant way of determining which measurement was carried out rst. At spacelike separation, the measuring operators commute, and so we can equally well perform the measurement of the spins of the electrons in reverse order and obtain the same splits: 4 UM1(:::)M2(:::) j ">1 j #>2 j #>1 j ">2 p2 = UM1(:::) M2(#)j ">1 j #>2 p2 M2(")j #>1 j ">2 p2 = M1(")M2(#)j ">1 j #>2 p2 M1(#)M2(")j #>1 j ">2 p2 (7) the last line of which is the same as that of (6), (except for the order of states, which is irrelevant). The e ect of measurements in which both observers happen to choose to measure with respect to the left/right basis is UM2(:::)M1(:::) j >1 j !>2 j !>1 j >2 p2 = UM2(:::) M1( )j >1 j !>2 p2 M1(!)j !>1 j >2 p2 = M2(!)M1( )j >1 j !>2 p2 M2( )M1(!)j !>1 j >2 p2 (8) A comparison of (6)/(7) with (8) shows that if two spacelike-separated observers fortuitously happen to measure the spins of the two particles in the same direction | whatever this same direction happens to be | both observers will split into two distinct worlds, and in each world the observers will measure opposite spin projections for the electrons. But at each event of observation, both of the two possible outcomes of the measurement will be obtained. Locality is preserved, because indeed both outcomes are obtained in total independence of the outcomes of the other measurement. The linearity of the operators forces the perfect anti-correlation of the spins of the particles in each 5 world. Since the singlet state is rotationally invariant, the same result would be obtained whatever direction the observers happened to choose to measure the spins. In the EPR experiment, there is actually a third measurement: the comparison of the two observations made by the spatially separated observers. In fact, the relative directions of the two spin measurements have no meaning without this third measurement. Once again, it is easily seen that normalization of this third measurement by the e ect of eigenstates of spins and linearity implies that this third measurement will con rm the split into two worlds. In the Copenhagen Interpretation, this third measurement is not considered a quantum measurement at all, because the rst measurements are considered to transfer the data from the quantum to the classical regime. But in the MWI, there is no classical regime; the comparison of the data in two macroscopic devices is just a much a quantum interaction as the original setting up of the singlet state. Furthermore, this ignored third measurement is actually of crucial importance: it is performed after information about the orientation of the second device has been carried back to the rst device (at a speed less than light!). The orientation is coded with correlations of the spins of both electrons, and these correlations (and the linearity of all operators) will force the third measurement to respect the original split. These correlations have not been lost, for no measurement reduces the wave function: the minus sign between the two worlds is present in all eqns. (1) | (8). To see explicitly how this third measurment works, represent the state of the comparison apparatus byMc[(:::)1(:::)2], where the rst entry measures the record of the apparatus measuring the rst particle, and the second entry measures the record of the apparatus measuring the second particle. Thus, the third measurement acting on eigenstates of the spin-measurement devices transforms the comparison apparatus as follows: UMc[(:::)1(:::)2]M1(") =Mc[(")1(:::)2]M1(") 6 UMc[(:::)1(:::)2]M1(#) =Mc[(#)1(:::)2]M1(#) UMc[(:::)1(:::)2]M2(") =Mc[(:::)1(")2]M2(") UMc[(:::)1(:::)2]M2(#) =Mc[(:::)1(#)2]M2(#) where for simplicity I have assumed the spins will be measured in the up or down direction. Then for the state (1), the totality of the three measurements together | the two measurements of the particle spins followed by the comparision measurement | is UMc[(:::)1(:::)2]M2(:::)M1(:::) j ">1 j #>2 j #>1 j ">2 p2 = UMc[(:::)1(:::)2] M2(#)M1(")j ">1 j #>2 p2 UMc[(:::)1(:::)2] M2(")M1(#)j #>1 j ">2 p2 =Mc[(")1(#)2]M2(#)M1(")j ">1 j #>2 p2 Mc[(#)1(")2] M2(")M1(#)j #>1 j ">2 p2 where the middle equation shows the importance of linearity. Heretofore I have assumed that the two observers have chosen to measure the spins in the same direction. For observers who make the decision of which direction to measure the spin in the instant before the measurement, most of the time the two directions will not be the same. The experiment could be carried out by throwing away all observations except those in which the chosen directions happened to agree within a predetermined tolerance. But this would waste most of the data. The Aspect-Clauser-Freedman Experiment [11] is designed to use more of the data by testing Bell's Inequality for the expectation value of 7 the product of the spins of the two electrons with the spin of one electron being measured in direction n̂1, and the spin of the other in direction n̂2. If the spins are measured in units of h=2, the standard QM expectation value for the product is < j(n̂1 1)(n̂2 2)j >= n̂1 n̂2 (9) where j > is the singlet state (1)/(2). In particular, n̂1 = n̂2 is the assumed set-up of the previous discussion. Since the MWI shows that local measurements in this case always gives +1 for one electron and -1 for the other, the product of the two is always -1 in all worlds, and thus the expectation value for the product is -1, in complete agreement with (9). To show how (9) comes about by local measurements splitting the universe into distinct worlds, I follow [12] and write the singlet state (1)/(2) with respect to some basis in the n̂1 direction as j >= (1=p2)(jn̂1; ">1 jn̂1; #>2 jn̂1; #>1 jn̂1; ">2) (10) Let another direction n̂2 be the polar axis, with the polar angle of n̂1 relative to n̂2. Without loss of generality, we can choose the other coordinates so that the azimuthal angle of n̂1 is zero. Standard rotation operators for spinor states then give [12] jn̂1; ">2= (cos =2)jn̂2; ">2 + (sin =2)jn̂2; #>2 jn̂1; #>2= (sin =2)jn̂2; ">2 + (cos =2)jn̂2; #>2 which yields j >= (1=p2)[ (sin =2)jn̂1; ">1 jn̂2; ">2 + (cos =2)jn̂1; ">1 jn̂2; #>2 8 (cos =2)jn̂1; #>1 jn̂2; ">2 (sin =2)jn̂1; #>1 jn̂2; #>2] (11) In other words, if the two devices measure the spins in arbitrary directions, there will be a split into four worlds, one for each possible permutation of the electron spins. Just as in the case with n̂1 = n̂2, normalization of the devices on eigenstates plus linearity forces the devices to split into all of these four worlds, which are the only possible worlds, since each observer must measure the electron to have spin +1 or 1. The squares of the coe cients in (11) are the number of worlds in each class, relative to the chosen experimental arrangement. This is most easily seen using Deutsch's MWI derivation of the Born Interpretation (BI). DeWitt and Graham [7] originally deduced the BI using the relative frequency theory of probability [13, 14], and this derivation is open to the standard objections to the frequency theory [13,15]. Deutsch instead derives the BI using the Principle of Indi erence of the classical/a priori theory of probability [13, 14]. According to the Principle of Indi erence, the probability of an event is the number of times the event occurs in a collection of equipossible cases divided by the total number of equipossible cases. Thus, the probability is 1/6 that a single die throw will result in a 5, because there are 6 equipossible sides that could appear, of which the 5 is exactly 1. Deutsch assumes the Principle of Indi erence applies to any experimental arrangement in which the expansion of the wave function j > in terms of the orthonormal basis vectors of the experiment (the interpretation basis) give equal coe cients for each term in the expansion. For example, both (1) and (2) are two such expansions, because in both cases the coe cients of each of the two terms is 1=p2. The Principle of Indi erence thus says that each of the two possibilities is equally likely in either the experimental arrangement (1), or in the interpretation basis (2). Equivalently, there are an equal number of worlds corresponding to each term in either (1) or (2), since in the MWI \equally possible" means \equal number of worlds" (equal relative to a preset experimental arrangement). Deutsch shows [16] that if the squares of the coe cients in the interpretation basis of 9 an experiment are rational, then a new experimental arrangement can be found in which the coe cients are equal in the new interpretation basis. Applying the Principle of Indifference to this new set of coe cients yields the BI for the coe cients in the original basis. Continuity in the Hilbert space of wave functions yields the BI for irrational coe cients. In particular, the percentage of worlds with the value of a given basis vector is given by the square of the coe cient, if the total number of worlds is normalized to one. The Principle of Indi erence is open to three objections in classical probability theory: (1) how do we know the supposed equipossible cases are really equipossible? (how do we know a priori the die is honest?); (2) why is it always legitimate to split a given set of possibilities into equipossible set (the classical theory has to use the probability calculus to construct an arti cial set | see [17] for a discussion), and (3) how do we obtain irrational probabilities from rational ones (see [13] and [14] for a discussion of this di culty). These di culties disappear in quantum mechanics: rst, systems in the same quantum state really are identical [18], and this identity carries over to baeses like (1) and (2) where cos = 1; second, the consistency of the quantum Indi erence Principle for all interpretation bases requires the split into equipossible states to be legitimate; and (3) continuity requires extension of rational probabilities to be extended to irrational ones. In quantum computation, the Church-Turing Conjecture on the equivalence of all universal computers can be proven [19, 20], whereas it must remain a conjecture in classical computation theory. Similarly, in quantum theory the Principle of Indi erence can be justi ed, and its standard applications proven valid. The expectation value (9) for the product of the spins is just the sum of each outcome, multiplied respectively by probabilities of each of the four possible outcomes: (+1)(+1)P"" + (+1)( 1)P"# + ( 1)(+1)P#" + ( 1)( 1)P## (12) where P"# is the relative number of worlds in which the rst electron is measured spin up, 10 and the second electron spin down, and similarly for the other P 's. Inserting these relative numbers | the squares of the coe cients in (11) | into (12) gives the expectation value: = 2 sin2 =2 1 2 cos2 =2 1 2 cos2 =2 + 1 2 sin2 =2 = cos = n̂1 n̂2 (13) which is the quantum expectation value (9). Once again it is essential to keep in mind the third measurement that compares the results of the two measurements of the spins, and by bringing the correlations between the worlds back to the same location, de nes the relative orientation of the previous two measurements, and in fact determines whether there is a twofold or a fourfould split. The way the measurement of (9) is actually carried out in the Aspect-Clauser-Freedman Experiment is to let be random in any single run, and for the results of each xed from a series of runs be placed in separate bins. This separation requires the third measurement, and this local comparison measurement retains the correlations between the spins. The e ect of throwing away this correlation information would be equivalent to averaging over all in the computation of the expectation value: the result is R 0 < j(n̂1 1)(n̂2 2)j > d = 0; i.e., the measured spin orientations of the two electrons are completely uncorrelated. This is what we would expect if each measurement of the electron spins is completely local, which in fact they are. There is no quantum nonlocality. Bell's results [10, 15] lead one to think otherwise. But Bell made the tacit assumption that each electron's wave function is reduced by the measurement of its spin. Speci cally, he assumed that the rst electron's spin was determined by the measurement direction n̂1 and the value some local hidden variable parameters 1: the rst electron's spin is given by a function A(n̂1; 1). The second electron's spin is given by an analogous function B(n̂2; 2), and so the hidden variable expectation value for the product of the spins would not be (13) but instead 11 Z ( 1; 2)A(n̂1; 1)B(n̂2; 2) d 1 d 2 (14) where ( 1; 2) is the joint probability distribution for the hidden variables. By comparing a triple set of directions (n̂1; n̂2; n̂3), Bell derived an inequality showing that the hidden variable (14) was inconsistent with quantum mechanical (9). But (14) assumes that the spin of each particle is a function of n̂i and i; that is, it assumes the spin at a location is single-valued. This is explicitly denied by the MWI, as one can see by letting i be the spatial coordinates of the ith electron. Bell's analysis tacitly adopts the Copenhagen Interpretation, wherein the macroscopic world is the single-valued world of classical mechanics. Physicists' desire to keep both the single valued classical everyday world and the manyvalued atomic quantum world is reminiscent of 17th century philosophers' desire to retain separate fundamental substances for the Earth, and for the heavenly bodies. Galileo got far more opposition to his claim that there were mountains on the Moon | which would imply that the Moon was not made of quintessence, but rather was made up of the same sort of stu as the Earth | than to his claim that the Earth went around the Sun [21]. Accordingly, in the Dialogue on the Great World Systems, Galileo's spokesman Salviati made refuting this two-substance idea the very rst order of business [21]. Just as 17th century philosophers had to accept the fact that the entire universe is made up of the same material, so physicists must accept the fact that the entire universe, both microscopic and macroscopic, is a many-valued quantum universe, and completely local. Conversely, the automatic elimination of action at a distance by the MWI is an argument for the MWI, just the Galileo's astronomical observations were an argument for one substance. I thank R. Chiao, D. Deutsch, B. DeWitt, and D.N. Page for helpful discussions.