VQLUME 66 28 JANUARY 1991 NUMBER 4

Weinberg's Nonlinear Quantum Mechanics and the Einstein-Podolsky-Rosen ParadoxJoseph Polchinski

Theory Group, Department of Physics, University of Texas, Austin, Texas 78712and Institute for Theoretical Physics, University of California, Santa Barbara, California 93106

(Received 18 June 1990)

I show that Weinberg s nonlinear quantum mechanics leads either to communication via Einstein-Podolsky-Rosen correlations, or to communication between branches of the wave function.

PACS numbers: 03.65.Bz

Weinberg ' has proposed a formalism for testing non-linear extensions of quantum mechanics. In this frame-work, there are nonlinear observables in addition to theusual linear ones. Heuristically, the greater number ofobservables suggests that there is more information inthe wave function than in the usual linear theory. Thisin turn raises the possibility that the fictitious violation oflocality that occurs in the Einstein-Podolsky-Rosen(EPR) experiment in linear quantum mechanics mightbecome a real violation in the nonlinear theory. That is,the EPR apparatus might be used to send instantaneoussignals.

In this Letter I determine the constraints imposedupon observables by the requirement that transmissionnot occur in the EPR experiment. This leads to adifrerent treatment of separated systems than that origi-nally proposed by Weinberg. I find that forbidding EPRcommunication in nonlinear quantum mechanics neces-sarily leads to another sort of unusual communication:that between diff'erent branches of the wave function.Oisin and Czachor have also discussed EPR com-munication in nonlinear quantum mechanics; their re-sults are discussed at the end of the paper.

Consider two widely separated systems I and II. Thewave function is 0';j, where the indices i and j refer tothe two systems, running over 1, . . . , M and 1, . . . , %,respectively. At time t=O the wave function has beenprepared, with correlations between the two systems. Atsome later time, an observable aII will be measured inthe receiving system II. It is sufhcient to consider ordi-

nary linear observables in system II. In the formalism ofWeinberg, linear observables are bilinear functions of +and Il'*,

h (t, +;~,+kt ) = e (t )a ~ (+;~,et*,t ) .This gives the equation of motion'

da/dt = ija)t, hl,where the Poisson bracket is

(3)

aa ab aa aba~;, a~,*, a~,*, a~

(4)

It is suScient to consider a weak field, integrating theequation of motion to first order in e:

f% (at)(t) =a(((0) i fa))(0),a)(0)t,dt'e(t') .

In this framework, ' the expectation value of a11 is simplythe value of aII, and so the condition that the measure-ment made in system II not depend on the field applied

&II +ij +j I+il ~with A, t a Hermitian matrix (repeated indices aresummed). At the receiving end I will therefore assumethat the usual idea of measurement applies. In system I,a signal is sent by turning on a field E(t) which couplesto a (possibly nonlinear) observable ai(+;1,+kt). Forsimplicity, suppose that this is the whole Hamiltonian:

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VOLUME 66, NUMBER PHYSICAL REVIEW LETTERS 28 JANUARY 1991

in system I is

ja,a, ) =O.The Poisson bracket facet, J generates unitary rotationson the second index of +;~. By considering all linear ob-servables aIIthat is, all Hermitian matrices A~(thecondition (6) is precisely the statement that a~(+;~, +ki)is invariant under general unitary rotations of the secondindex. Thus, this index must always be contracted be-tween + and 0'*, and aI is a function only of the densitymatrix

(4) If the observer originally saw (ylcr'lvl) = ,again measures the spin with a Stern-Gerlach devicecoupled to the linear spin component (vflcr'ly); other-wise, he does nothing.

Now follow the evolution of the wave functionwhere the indices refer respectively to the ion and to theobserver. It is useful to focus on the two partial densitymatrices

I(+ )Pik = m +im+km,

mGS+(i 3)

[Pik ~+im +km ~ (7)

I(- )pik M +im+km ~mGS

Now it is possible to go back and enlarge the set of ob-servables in system II, subject to the no-signal condition(6). The density matrices for systems I and II commute,

where 5+ is the set of all observer states in which(yl cr I y) = ,' was seen, and 5 is the set of all observerstates in which (@la IVf) = ,' was seen. After step(i),

~p,', ,p, l'~ =o, 1+a. I( ) 1 a.3 3p = p4 ' 4 (14)II

pg( =Z +nylon(

Thus, the no-signal condition allows observation of anyfunction a II of p".

This is the first main result: Assuming only that theobservables include all the usual linear Hermitian ob-servables, the necessary and suNcient condition for anisolated system not to receive information via EPR corre-lations is that all observables in the system depend onlyon the density matrix for the system:

a I =~ I ~ mini +knr, m

The observables proposed by %einberg' for separatedsystems,

aI =g/t(('I;'Pk ),are not of this form (except when linear) and thereforeallow EPR communication.

I now show that nonlinear observables of the form(10) lead to unusual communication of another sort.Consider a process involving four steps.

(1) A spin- , ion enters a Stern-Gerlach device, whichcouples to the linear spin component (ylcr lvf). In thedevice the beam splits, a macroscopic observer notes thedirection taken, and then the two paths are rejoined.

(2) After the Stern-Gerlach device: If the observersaw &vflcr ly) = ,' he does nothing. If he saw (ylcr'ly&

he takes one of two actions: (a) nothing, or (b)rotates the spin into the + ja. direction with a magneticfield coupled to the linear observable (yl a I vl).

(3) The ion enters a region of field coupled to the non-linear observable

&v l~'I v &&v l~'I v &&v lv &

After step (2), action (a) leaves the density matrices(14) unchanged, while action (b) gives rise to

I(+) 1+G I( ) 1+0p = 4 ~ pTo study step (3) it is necessary to be precise about

the form of the nonlinear interaction. Here, the result ofthe EPR analysis enters, requiring that the nonlinear ob-servable (12), when extended to the two-system wavefunction, must be a function only of the total p' and notof the separate p' . Note that one could arrange forthe ion to propagate a long distance between steps (2)and (3), and then to be reiiected back for step (4), mak-ing it clear that the observer and ion are separated sys-tems. Knowing the form (12) of hi on pure states doesnot fully determine it as a function of density matrices.Rather,

IT I I T I I I~3 JI +J2

with f~+f2=f. Expanding out the sums in the densitymatrices, there are cross terms between states in 5 and5, which will lead to cross terms between the evolutionof diA'erent macroscopic states of the observer. It maybe that a particular ratio of f~ and fq is always present,or it may be that there are independent fields which cou-ple to each interaction.

It is convenient to analyze first the case that f ~ =f andf2=0. The time i spent in the field, and the strength fof the field, will be chosen to satisfy if = , z. The equa-tions of motion in step (3) are then

I(+ ) I IdP 2f) (( ) )) PTrp

This integrates easily to give the partial density matricesafter step (3). If action (a) was taken, these do not

VOLUME 66, NUMBER 4 PHYSICAL REVIEW LETTERS 28 JANUARY 1991

&(+) 1+ i( ) 1 +~P = P4while if action (b) was taken they are

r(+) 1 ~ t(3 1+~P = 4 P

(20)

(2i)Again the observation made in step (4) depends on theaction taken in the other branch in step (2).

The alternative formulation of separated systems, Eq.(11), allows the construction of an EPR phone, but at

change,

I(+) 1+a t( ) 1 QP =, P4 ' 4while if action (b) was taken they become

[(+) 1 0 [() 1+0P = P4 ' 4Finally, in step (4), it follows from Eqs. (18) and (19)

that if the observer measures the spin, he obtains + 2 ifaction (a) was taken, and ,' if action (b) was taken.But the action and observation are in two differentbranches of the wave function, in which the original spinwas measured to be 2 and + ,', respectively. Themethod by which the choice of action is to be made hasnot been discussed, and there are various interesting pos-sibilities to contemplate, including classical and quantumrandom-number generators. For the present purpose it issufhcient to imagine that the observer has made somefirm choice prior to the experiment. If he then sees theoriginal spin as + 2, he does nothing but make a secondmeasurement, but the outcome depends on what hewoul'd have done had he originally seen ,' . In effect,the apparatus reads the observer's mind.

Note that in steps (1), (2), and (4), in which the ob-server participates, only linear observables are involved,and so the usual quantum-mechanical idea of measure-ment has been used. The experiment has been designedto produce no further bifurcation of the wave function,so that by iterating steps (2), (3), and (4), and allowingthe two branches to switch roles, the observers in the twobranches of the wave function may exchange binary mes-sages of arbitrary length. This is an Everett phone, incontrast to the EPR phone designed above.

In the event that fq is nonzero, the coupling betweenbranches remains but the evolution and outcome are nolonger as simple for the particular setup above. It is con-venient to alter the conditions of the experiment so as toobtain a simple outcome. There are many ways to dothis. One is the following: action (a) is now to rotatethe field into the +2 direction, action (b) is now to ro-tate the field into the 2 direction, the time interval istf2=tr/W2, and in step (4) it is the 2 component that ismeasured. After step (3), if action (a) was taken at step(2), the partial density matrices are

first sight appears to be free of the Everett-phonephenomenon. This is because the evolution for diAerentstates of the observer, diAerent I values, separates.However, the form (11) is basis dependent, and it is notclear how a basis is to be singled out. It is necessary togive a prescription for choosing the basis in which Eq.(11) holds. In order to forbid the Everett phone, it isnecessary that this basis never involve superpositions ofmacroscopic systems (observers) in different states.

It is important now to note that the Everett phone maynot actually work in practice. I have ignored previousbranchings of the wave function, describing the macro-scopic observer and apparatus as though they started in adefinite state, as would be acceptable in the lineartheory. However, the analysis of the Everett phoneshows this assumption to be self-inconsistent: The evolu-tion of the wave function will be coupled to all other pos-sible states. Thus, while the analysis does show thatbranches are coupled to one another, practical communi-cation between branches may be drowned out by thecoupling to all the other branches of the wave function ofthe Universe.

Do the results imply that nonlinear quantum mechan-ics is inconsistent, and thus "explain" the linearity of thetheory? ' Communication between branches of the wavefunction seems even more bizarre than faster-than-lightcommunication and consequent loss of Lorentz invari-ance, but it is not clear that it represents an actual incon-sistency. It means that reduction of the wave functionnever occurs, so that the standard Copenhagen interpre-tation of quantum mechanics no longer applies. Themany-worlds interpretation of quantum mechanics be-comes the natural one, with communication between theworlds now possible. I do not know whether this allows acompletely consistent interpretation of the nonlineartheory this is a far-reaching question but can onlynote that in the present thought experiments the evolu-tion equations are mathematically consistent and allow aconsistent interpretation.

Gisin has argued that nonlinear quantum mechanicsleads to EPR communication, both in Weinberg'stheory and more generally. He assumes reduction ofthe wave packet that is, the projection postulate. Aswe have seen, without the projection postulate EPR com-munication need no longer occur, '' provided that the ob-servables (11) for separated systems are replaced withthe form (10). Also, Czachor shows by explicit calcula-tion, using the form (11), that EPR communicationoccurs in Weinberg's theory.

To what extent do the results apply to more generalnonlinear quantum theories? %'einberg's theory is rath-er general, carrying over two main assumptions from thelinear theory. The first is that the equation of motioncan be put in the form (3). I have used this assumptionextensively. In particular, "observables" have beendefined pragmatically, as any quantities which can be

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added to h(+, +*),Eq. (2). This assumption in any caseseems very well motivated, as it leads to the usual con-nection between symmetries and conservation laws, andaids in the interpretation in other ways. The second as-sumption, which may be less well motivated, is that ob-servables are homogeneous of degree (1,1) in + and 4'*.This played no role in the analysis of the EPR phone,and only a minor role in the analysis of the Everettphone (it allowed the separation of the space and spinwave functions').

Finally, what are the experimental implications ofthese results? It is important to note that communica-tion between branches of the wave function invalidatesmost previous attempts to analyze the experimentalconsequences of nonlinearities, such as those in Refs. 1,2, and 12. This is because these analyses ignore previousbranchings of the wave function and treat macroscopicsystems as though they begin in definite macroscopicstates. A complete analysis requires consideration of theentire wave function of the Universe and is thereforerather complicated. Naively, it would seem that non-linear eA'ects will be very much diluted by the enormousnumber of branches, since the amplitude for any givenbranch of the wave function is exceedingly small. Thisleads to the discouraging conclusion that nonlinearitiescould be of order 1 in a fundamental theory and yet theeff'ective nonlinearity measurable experimentally wouldstill be unobservably small.

If quantum nonlinearities are observed nonetheless,the thought experiments described herein would seem tobe simple enough to carry out in practice, thereby deter-mining which of EPR communication and Everett com-munication is actually realized.

I am grateful to S. Weinberg for extensive discussionsand suggestions, and to T. Jordan for comments on the

manuscript. I would also like to thank N. Gisin forsending his papers and comments. This research is sup-ported in part by the Robert A. Welch Foundation andby NSF Grants No. PHY86-05978 and No. PHY89-04035 (supplemented by funds from NASA).

'S. Weinberg, Ann. Phys. (N. Y.) 194, 336 (1989).~S. Weinberg, Phys. Rev. Lett. 62, 485 (1989).A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777

(1935).4This issue of information in nonlinear quantum mechanics

has been discussed from a diAerent (thermodynamic) point ofview by A. Peres, Phys. Rev. Lett. 63, 1114 (1989), and S.Weinberg, Phys. Rev. Lett. 63, 1115 (1989).

sThe Many Worlds In-terpretation of Quantum Mechant'csedited by B. S. DeWitt and N. Graham (Princeton Univ. Press,Princeton, 1973). In this interpretation, a branch of the wavefunction is a subspace in which a macroscopic system is in adefinite macroscopic state.

6N. Gisin, Helv. Phys. Acta 62, 363 (1989).7N. Gisin, Phys. Lett. A 143, I (1990).sM. Czachor, "Mobility in Weinberg's Non-linear Quantum

Mechanics" (unpublished).9The state ~Vtl refers only to the spin wave function. It is

shown in Ref. 1 that the nonlinear equation admits solutions inwhich the wave function is a product of spin and space wavefunctions. For simplicity we have implicitly assumed this instep (3).

' Jordan has recently discussed general grounds from whichthe linearity of quantum mechanics might be derived in T. F.Jordan, Am. J. Phys. (to be published).

''In the many-worlds interpretation (Ref. 5), projection is aderived consequence of the (linear) theory, rather than a pos-tulate.

'2J. J. Bollinger, D. J. Heinzen, W. M. Itano, S. L. Gilbert,and D. J. Wineland, Phys. Rev. Lett. 63, 1031 (1989).

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