Received May 9, 1990; revised March 26, 1991

Transcription

1 Foundations of Physics Letters, Vol. 4, No. 4, 1991 MOBILITY AND NON-SEPARABILITY Marek Czachor Institute for Theoretical Physics Polish Academy of Sciences AL Lotnik6w 32/46, Warszawa, Poland ~ Received May 9, 1990; revised March 26, 1991 A mobility phenomenon is typical for a large class of non-linear evolutions. It is shown that the mobility can be responsible for a "malignant" form of quantum mechanical non separability of two correlated systems. An explicit example based on Weinberg's non linear quantum mechanics is considered. We show that in such a theory a transfer of information without a transfer of energy is possible and discuss physical roots of this phenomenon. In Polchinski's approach this effect is not present. Key words: non-linear quantum mechanics, non-locality 1. INTRODUCTION Recently it has been shown in several papers that an introduction of non-linear terms into evolution equations can lead to a violation of separability we are accustomed to. In non-linear, classical electrodynamics light travels faster or more slowly than "ordinary" light in vacuum [1]. In Weinberg's non-linear quantum mechanics, Hamiltonians which correspond to energies of two separated systems may not commute [2]. A reduction of a wave packet in one arm of the EPR-like experiment leads to observable, statistically different measurements in the other arm provided some non-linearity is introduced [3]. A question of the role of non-linearity in the context of independent evolutions of separated systems was considered also by Biatynicki-Birula and Mycielski [4]; it lead them to a logarithmic form of non-linearity. The fact that the very appearance of some nonlinearity can be, in general, related to a "malignant EPR paradox" was discussed probably for the first time by Gisin [5]. Also Mielnik [61 in his analysis of "Schr6dinger's cat" pointed out the "malignancy '~ /91/ /0 199l Plenum Publishing Corporation

2 352 Czachor of its non-linear version. In what follows, I will approach the problem from a different point of view. The discussion goes along a l!ne proposed by myself in an unpublished lecture presented in ~dansk in 1989 (cf. [7,8]). A notion of a mobility phenomenon was introduced in the context of the non-linear evolution by Mielnik [9,10]. He noted that, once linearity is lost, a qualitative metamorphosis occurs. The resulting theory does not differ "little" from those based on linear evolutions, even though the non-linear terms are small in some reasonable sense. In particular, under suitable conditions a scalar product of two (pure) states is not conserved. This is what we call the mobility phenomenon. The role mobility plays in the question of separation of two non-interacting systems is the subject of this paper. In the following discussion I concentrate on a particuiar non-linear system taken from the Weinberg paper. The analysis is therefore explicit, and we can easily control the physical mechanisms that appear.. SPIN- 71 FIELD INTERACTING WITH ELECTRIC QUADRUPOLE IN WEINBERG'S NON-LINEAR QUANTUM MECHANICS In Weinberg's theory the evolution of states is given by Hamilton equations dck Oh. d ~ Oh - -~ - (1) z dt 0 ~' dt OCk' where the Hamiltonian function h satisfies a homogeneity condition Oh., Oh (summation convention) Let us consider now two states ~ and whose evolution is generated by the same Hamiltonian function h, and such that h(, *) = h(, '). (2) For a scalar product, defined in the usual way as (, ) = ~ k, (3) we find d(, ) _ i(ohck Oh..', dt \O~k - O-~k*k) ' (4)

3 Mobility and Non-separability 353 with.the.rhs of.()4 identically vanishing..in the case of bilinear Hamaltoman functions. Hence the ordinary hnear quantum mechanical evolution conserves a scalar product of states. In the general case, the homogeneity condition implies also the conservation of the norm (, >-~. The homogeneity condition per se does not guarantee the vanishing of the RHS of (4). In order to see this let us consider a case of a spin-~ field interacting with the electric quadrupole. For a specific choice of the electric field, the system can be described by the following Hamiltonian function [2,3]: h(, *) = E0<, > + (, > (5) The Schrgdinger equation which follows from (1) and (5) is Its solution is given by ~N =E0 + <,0) <~'~3 >2)~. (7) (, >2 (v/-ae-i{eo+e(a-b)(a+b) - 2(A+3 B) }t-i6~ (t) =k v~e-i{e -e(a-b)(a+b)-2(3a+b)}t ]" (8) Now, if is a solution of (1), then = icr2 is its other solution corresponding to the same mean value of the energy, and we get <~#, > = 2ix/-~sin(4e(A- B)(A + B)-lt + 6). (9) For e 0, AB 0, 6 = 0, and A B the states are orthogonal for = 0, but later lose their orthogonality, although the norm of each of them is conserved. This is an explicit example of the mobility phenomenon. Let us assume now that we have two separated systems, say I and II, described by Hamiltonian functions hl(~:~, ~*)-- EI(~, q~>, h~(x,x*)=e~<x,x>+ (x'~x>~ <x,x> " (lo) According to Weinberg, a Hamiltonian function of the whole I+II system is given by h(~, ~*) : E l hl(~9(l)'~l) ) + E h2(~(k)'x~k) )' k (11)

4 354 Czachor where 9~k(1) = Xl(k) = q2kl (an alternative proposal of Polchinski [8] shall be discussed in the final section). This shape of the Hamiltonian function means physically, that the energy of a subsystem is averaged separately for each subbeam collapsed ~y a remote observer in the other subsystem. It is not difficult to imagine a situation, where a beam of left- and right-handed photons reacts to the Weinberg's non-linearity, while the analogous one, consisting of x' and y- polarized ones, does not. The form and value of (11) is, therefore, dependent not only on, but also on a particular choice of bases. This observation raises new questions, but they will be discussed in a forthcoming paper. The general solution of (1), with h given by (11), in a basis diagonalizing as, is (1"1/11 kli12): ( Olx OLY )= -~- (12) /I) : ~k ~21 1I;22 OLX! ~y! where 6 = : (o),. : exp(- El ) : : are some solutions (8). Like in ordinary quantum mechanics, the state evolves here by means of separate evolutions of "collapsed states". To complete this part of the discussion let us note that the Hamittonian functions hl(ki/,(, l/*) = ~-'~hl(~(1),~l)) and h2(ko,~*)= E h2(x(k),x~k)) l k commute because h1(, *) = E1 (~, q~). Bearing in mind that there is no interaction part-in h, we conclude that there is no flow of energy between I and II. In the next section, I shall demonstrate a peculiar non-local property of states that do not factorize into a tensor product state; afterwards we shall return to the discussion of this section. 3. AN INTERFERENCE IN A MACH-ZEHNDER INTERFEROMETER Let us consider a polarizing Mach-Zehnder interferometer and a source of photons, as shown in Fig. 1. Kegs representing, respectively, right- or left-hand polarized photons travelling to the right along the x axis,evolve in the interferometer in the following way (the index y means that the photon travels along a y axis): I+, x).s~ i i +, y).~ il +,x) ~, ie ~" I+, x) Bs~ i~d%i I +,y)+ i +,z)), (13 a)

5 Mobility and Non-separability 355 Y / BS2 BS1 M2 Fig. 1 I-,x> ~-~s~ I-,x> ~-~/~ I +,x) -~1 +,y> BS , ~(~ I +,~>+ I +,y>). (13b) Here oe denotes an optical retarder which introduces a phase shift c~, A/2 is a half-wave plate, BS1 is a polarizing beam splitter, BS2 a non-polarizing one, and M1 and M2 are mirrors. A transmitted beam is always phase-shifted by 7r/2 with respect to the reflected one [11]. Let us consider now the following two cases: (1) A source produces a linear polarization state, say 1 I > = ~(I +,x>+ I -,=>). (14) The M-Z interferometer acts by means of (13), thus the whole state transforms into I '> = ~(i(el~' + 1) I +,x) + (1- J~) I +,y)). (15) We observe an interference between the two outgoing channels. In particular, for a = 0 the whole beam goes through the z channel.

6 356 Czachor (2) A source produces a singlet state [11] 1 = I +,x>+ I -,-x> I -,x>), (16) where the photons in a pair travel along the x axis but in opposite directions. Now the whole system consists of two separated subsystems, I and II, where in I the photons enter the M-Z interferometer. This subsystem is described by a reduced density matrix 1 1 col : Tr2 l= ~ I +,x><+,x I +~ [ -,x}<-,x! and is therefore in a mixed state. No interference should be observed, as there is no coherence between the incoming right, or left-handed photons. Indeed, the transformation (13) transforms the density matrix into, 1 1 l=~l-t-,x)(~,xl+~l+,y)(+,y], hence the intensities of beams in the outgoing channels are equal and independent from the optical retarder. This property of the singlet state was considered by Pykacz and Zukowski [11] as the proof that a superluminal signalisation is impossible in this experiment. This is obviously true in linear quantum mechanics. Let us, however, recalculate the problem in a different way. We must obtain an identical result if we first transform the whole singlet state according to (13), then form a density matrix of the whole system, and, finally, take the trace over II. The whole state transforms as follows: I+,-x>l+, x)-ei"l+,-x)l+, v> (17) Now we can explicitly see why there cannot appear interference in I: The linear dependence of the states which interfered in the linear polarization case is destroyed by orthogonality of their singiet state partners. The lack of the interference in I is therefore a non-local phenomenon. The two explanations of the lack of the interference are, unfortunately, experimentally indistinguishable. As long as ordinary linear devices are applied to the subsystem II no possibility of signalisation between I and II is possible, because the kets will always evolve according to some unitary transformation and will remain orthogonal.

7 Mobility and Non-separability 357 Let us, however, recall our discussion from the previous section. We have noted that there exist situations where the whole state evolves by means of independent evolutions of "collapsed kets" even if one of the subsystems interacts with a non-linear device. Eq. (9) shows that states which were orthogonal in the absence of nonlinearity, after the non-linearity is turned on (or after having entered the region with the non-linearity), may lose their othogonality because of the mobility phenomenon. This is the general property of non-linear evolutions [9,10] in Hilbert spaces (i.e. it takes place if the evolution in Schr6dinger's picture is non-linear; in non-linear quantum optics evolutions are non-linear in Iteisenberg's picture). So, what happens if we introduce in II an optical device that acts analogously to the quadrupole field described in See.2? According to a superposition principle, the source emits pairs which are superpositions of the states corresponding to results (+, +) or (-, -). Once the "left" photon enters the non-linearity region, the superposition starts to evolve like the state (12). The right- and lefthanded states start to rotate one around the other because of the mobility phenomenon (compare Eq. (9). Their partners do not interfere as long as the states are orthogonal, but the "less orthogonal" they are, the stronger is the interference of their partners. It follows that the intensities in both of the outgoing channels of the M-Z interferometer in I start to change. In the beginning of the process the intensities were equal. At the point of time in which the orthogonality is broken in the strongest way (let us accept, for inessential simplicity, that the two states are then proportional to each other; for reversible evolutions, the states can be arbitrarily close [101, but never proportional) the interference is strongest and, for a = 0, the intensity in the y channel is 0, while in the x one it is maximal. After this extreme point, the mobility starts to make the states "more and more ortho go ~ nnl", ~11u ~A ~ne,1 interference becomes weaker and weaker. At the moment when the states are again orthogonal the intensities are again equal. And so on. It is clear that the outgoing intensities oscillate with the mobility frequency. It means that in the subsystem I there exist some observables that exhibit oscillations with this frequency. By a modulation of parameters of the non-linearity in II, we can modulate the frequency of its corresponding mobility and send some information to I. All of this can happen even if the Hamiltonians of the two subsystems commute (in Weinberg's sense, for instance) and there is no transfer of energy between I and II. The actual reason of such a behaviour is the fact that the non-linearity introduces order (there is some analogy to Prigogine's chemical clocks [15]). The states interfere in a statistically observable way, provided the state of the subsystem is not "fully mixed". This means that in the extreme point, in which the states originally orthogonal become proportional to each other, the singlet state factorizes. The reduced density matrix describes in this pecular point of time a pure state, and it is at

8 358 Czaehor this moment that the interference is strongest. We, accordingly, expect a violation of the second law of thermodynamics: The entropy of I should oscillate. This conclusion is in agreement with Peres's analysis [12,13]. 4. "MALIGNANT NON-LOCALITY" IN WEINBERG'S THEORY With the physical background of the previous section, we can understand the behaviour of separated systems in the Weinberg theory. Any bilinear observable in the system I described in See. 2 can be written either as (~b I I X I ) or as Trl(XTr2 i ~b)( I)- Therefore, it is convenient to calculate the reduced density matrices, even if we assume that they no longer carry the whole statistical information about the non-bilinear observables. It is also interesting from the point of view of Polchinski's proposal [8]. So let us calculate the reduced density matrix of the linear subsystem I. With the notation of Eq. (12), we find (we take normalized states) ] (is) The off-diagonal elements vanish in the linear theory (e = 0) because of (9), and we get a "fully mixed" state. For e 7~ 0 these coherences oscillate with the mobility frequency given by (9). There therefore exist observables whose average values oscillate in this wa,y. For example, the components of spin satisfy (ch) = (c~3) = 0, but (~r2) = Im(, ). (19) Note that the RttS of (19) vanishes in eigenststes of h which correspond to A=0 or B=0 or A=B. This is a consequence of the fact that eigenstates evolve unitarily in Weinberg's theory. Note also that, quite remarkably, no malignancy occurs if initially the system is in the singlet state (cf. [3]). 5. FINAL COMMENTS In this paper I have tried to argue that the :'malignant nonseparability" of Weinberg's theory, as well as of all those other theories in which the mobility phenomenon occurs, is motivated phys= ically. What is indeed counter-intuitive is the explicitly non-local behaviour of the states that, like the singlet, do not factorize into tensor product states. There is, however, also another point of view which suggests that this phenomenon is physically reasonable.

9 Mobility and Non-separability 359 We know that the interference does not take place if one knows a route of a particle (say, a photon) in an interferometer. This observation inclined many people to propose the so-called delayed-choice experiments. It is sometimes believed that this interpretation means that the photons in one arm of the interferometer can interfere until a remote observer in the other arm makes an experiment which determines their route in the interferometer. Such an indirect measurement could be possible if there was a one-to-one relation between a trajectory in the interferometer and the results of the remote measurements. This, on the other hand, means that the state of the whole system takes a form --t ye4 I one route>+ i no> l another route>, where the kets I yes} and ] no} must be orthogonal since they correspond to different results of the measurements made by the remote observer. But our analysis from Sec. 3 proves that there is no interference in such a case. Therefore the question whether the experiment is delayed or not, is inessential. It is important that in principle one can make such an experiment. Now, what happens if the photons traverse some non-linear device which introduces the mobility of initially orthogonal states I yes) and I no} before they reach the remote observer? The stronger the violation of their orthogonality is, the less information one gets about the trajectories in question. In the extreme case of the mobility, say I yes) --+l yes), I no) -'-+1 yes), the result "yes" tells us nothing about the trajectories in the other arm, and then one obtains the strongest interference. Again, the "more orthogonal" the states are, the greater is the probability of a correct prediction of the route of the photons in the interferometer and the weaker should be the interference. The above discussed physical mechanisms make sense because the whole state of the composed system evolves by means of separate evolutions of the "collapsed kets". This is a straightforward consequence of the form of the Hamiltonian function (11). Recently a new proposal was formulated by Polchinski [8]. He assumes that the Hamiltonian function of the system I+II is given by h(, 9*) = h,(a) + h2(e2), (20) where th and t~2 are reduced density matrices of the subsystems. In the case discussed in Secs. 2 and 4 we have then the following Hamiltonian functions hl(i >< I) = E~Tr I >< I, h2(i X><X I) = E2Tr I x><x l+e t X><X!)) Tr I x><x I

10 360 Czachor With these explicit forms, as well as eqs. (20) and (1), we obtain a non-linear SchrSdinger equation whose solutions never take the form (12) with ~ and ~ "in mobility". This guarantees a peaceful coexistence between the non-unitarity of the evolution and the reduced density matrix formalism. A remarkable property of the Polchinski's proposal lies in the fact that any two separated observables commute; this, obviously, excludes any dependence of evolution in I on parameters of observables in II, and vice versa. Weinberg's Hamiltonian function for a subsystem leads to an average energy which is a sum of averages calculated independently for beams collapsed by a remote observer in the other subsystem. Physically this may correspond to single, collapsed particles entering the non-linearity region. The collapse is therefore implicitly present already in the form of the Hamiltonian function (11). In Polchinski's choice of the form of the average energy, the average energy of the non-linear interaction is a function of a density matrix of the subsystem. The non-linearity can represent then a reaction of medium to statistical properties of the whole beam, for example, to average spin as in the case discussed above. The whole ensemble is not sensible to a collapse [5,14], therefore the "EPR phone" cannot work in such a non-lineafity. The difference between the two approaches is very clear and is related to the question of the very existence of fundamental non-linearities, instead of non' linearities being a result of approximations. The problem we face is close to Mielnik's discussion of the "first impossibility principle" [14]. In Polchinski's approach one does not deal with reduced states entering II but only with a statistical ensemble of them as a whole. I think this approach is closer to a quantum mechanical orthodoxy [14] than the one advocated by myself, Gisin, and - implicitly - Weinberg in [2]. Anyway, the wave packet reduction problem and the ShrSdinger's cat paradox become reatly malignant in non-linear quantum mechanics. Summing up, we may conclude that there exist non-trivial arguments for both of the mentioned approaches. Each of them is based on a different philosophy of quantum theory and the different nature of non-linearities considered. It remains to hope that non-linear theories will finally help us understand the philosophical background and limitations of quantum mechanics, on the analogy of the role played by relativity theory in our understanding of Newtonian physics. ACKNOWLEDGEMENTS I am indebted to Drs. A. Posiewnik, J. Pykacz, M. Zukowski. K. Gdrny, and N. Gisin and to Profs. B. Mielnik, and K. Rz~ewsk{ for their interesting comments and other help. I express my grattitude to Z. Mielewczyk for a help with the TEX typesetting. Last but not least, I want to thank my wife Magda who, although not a physicist herself, was so patient in listening to my non,linear problems.

11 Mobility and Non-separability 361 REFERENCES 1 H. S. Ibarguen, A. Garcia and J. Pleba nski, d. Math. Phys.30, 2689 (1989). 2 S. Weinberg, Ann. Phys. (NY) 194, 336 (1989). 3 N. Gisin, Phys. Lett. 143A, 1 (1990). 4 I. Biatynicki-Birula and J. Mycielski, Ann. Phys. (NY) 100, 62 (1976). 5 N. Gisin, IIelv. Phys. Acta 62,363 (1989). 6 B. Mielnik, private communication and a lecture at "Problems in Quantum Physics, Gdafisk' 89", Gdafisk, Poland. 7 N. Gisin, Itelv. Phys. Acta 63, 929 (1990). 8 J. Polchinski, Phys. Rev. Left. 66, 397 (1991). 9 B. Mielnik, J. Math. Phys. 21, 44, (1980). 10 B. Mielnik, Comm. Math. Phys. 101,323 (1985). 11 J. Pykacz and M.Zukowski, Phys. Left. 127A, 1 (1988). 12 A. Peres, Phys. Rev. Left. 63, 1114 (1989). 13 S. Weinberg, Phys. Rev. Lett. 63, 1115 (1989). 14 B. Mielnik, "Is the reduction of a wave packet indeed necessary in quantum mechanics?, " in Problems in Quantum Physics- Gda/tsk'89 J. Mizerski et al., eds. ( World Scientific, Singapore, 1990). 15 I. Prigogine and I. Stengers, Order out of Chaos (Fontana Paperbacks, 1985) NOTE 1 Permanent address: Laboratory of Dielectrics and Organic Semiconductors, Technical University of Gdafisk, Gdafisk, Poland.