.,
GREGG JAEGER AND SAHOTRA SARKAR
COHERENCE,ENTANGLEMENT, AND REDUCfIONIST EXPLANATION IN QUANTUM PHYSICS
Abstract The scopeand natureof reductionistexplanation in physicsis analyzed,with special attention being paid to the situation in quantummechanics. Five different sensesof "reduction" are identified. The strongestof these,called "strong reduction,"is the one that purportsto capturethe relationsbetween macro- scopic and microscopicphysics. It is shown that the criteria for strong reductionare violated by explana- tions in quantum mechanicswhich involve "entangledstates." The notion of "coherence" in physical systemsis also defined.It is shownthat, contraryto many currentviews, the invocationof coherencedoes not necessarilylead to the violation of strong reduction.However, entangled systems also exhibit coher- ence.Therefore, the subclassof coherentsystems that are entangledpresents problems for strong reduction.
INTRODUCfION In recent years, the problems surrounding the role of reductionism in science have beenextensively studied, mainly in the contextsof biology and psychology.I Philoso- phers seem to have generally assumedthat (i) instancesof reduction in the physical sciencesare ubiquitous; and (ii) theseinstances are straightforward in the sensethat they are trivially captured by at least one of the alternative models of reduction that have been developed by philosophersof science} These assumptionsseem to be basedon intuitive analysesof putative instancesof reduction such as the following: (i) there is a straightforward sensein which Newtonian mechanicscan be obtained from the special theory of relativity (for instance,by taking the c - 00 limit, where c is the speedof light in vacuum).Thus, Newtonian mechanicsis reducedto special relativity; (ii) similarly, the so-called "Newtonian" limit of general relativity yields Newton's theory of gravitation. Once again, the latter is thus reducedto the former; (iii) geometricaloptics is reducible to physical optics through Maxwell's electromag- netic theory in the sensethat the behavior of light waves,as predicted by Maxwell's theory, shows why geometricaloptics is correct to a good approximation; (iv) classical thermodynamicscan be derived from, or reducedto, statistical mechan- ics by the construction of kinetic models of gases.This putative reduction is particu- larly fascinating becausemechanics, the laws of which are invariant under time reversal,is supposedto give rise to the secondlaw of thermodynamicswhich embod- ies a direction of time;
523 A. Ashtekluet al. (eds.),Revisiting the Foundationsof Relativistic Physics,523-542. C 2003 Kluwer AcademicPublishers. Printed in the Netherlands. 524 GRroG JAEGER AND SAHOTRA SARKAR
(v) quantum mechanicsreduces classical chemical bonding theory, and accountsfor the various valency rules of chemistry that havebeen used since at least the early 19th century It is quite possible that a half-century ago most, though not all, physicists and chemists would have acceptedthese casesas successfulreductions. Most of them would also have endorseda positive and unproblematicassessment of the status of reductionism in the physical sciences.The developmentsin physics during the last fifty years,however, have called thesejudgments into question.Among the most star- tling have been scaling theory and renormalization"group" techniquesin condensed matter physics.A perusalof thesedevelopments has even led Leggett (1987) to sug- gest that the appropriaterelationship betweenmacroscopic and microscopic physics is only one of consistency:microscopic physics has no significant role in the explana- tion of macroscopicphenomena.3 Leggett's views are by no meansidiosyncratic. For instance,Fisher (1988), one of the foundersof scaling theory, has argued that there are aspectsof condensedmatter physicsfor which the underlying microphysics (viz., quantummechanics) is explanatorily irrelevant. Meanwhile, Frohlich (e.g. 1968, 1969) initiated a researchprogram that aims to prove that in several instancesbiological entities must be considered as quantum mechanicalsystems that exhibit "coherence"or phasecorrelations in their dynamics. Frohlich (1973) hasclaimed that biological explanationsthat invoke such phenomena are not "mechanistic," that is, reductionist in the usual senseof that term (see section 2). This claim has also beenendorsed by Ho (1989).4What is particularly philosoph- ically interestingabout Frohlich's conjectureis that. if it turns out to be correct, these attempts at physical explanation in biology would fail to satisfy the strictures of reduction not becauseof any peculiarity of biological systemsbut becauseof a failure of reduction (e.g., as it will be construedin section 2) in physics (more specifically, quantummechanics) itself. Finally, it hasbecome apparent that at leastsome of the ostensiblystraightforward casesof reduction, such as those mentionedabove, are not quite as unproblematicas they customarily havebeen taken to be. Returningto the casesmentioned above: (i) Though, from a strictly mathematicalpoint of view, Newtonian mechanicsdoes emergefrom special relativity in the c - 00 limit, that limit is counterfactual: the speedof light is finite; (ii) For the Newtonian limit of generalrelativity to exist and yield Newton's theory of gravitation, constraintshave to be imposedon generalrelativity (Ehlers 1981; Mala- ment 1986).In particular, the spatial part of space-timemust be flat; (iii) A series of approximations is required to obtain geometrical optics from Max- well's laws. (iv) The relation of thermodynamicsto statistical mechanicshas turned out to be even more complicated. It should consist of the derivation of the thermodynamic laws from statistical mechanicsin the so-called"thermodynamic limit," the one in which the number of particles (N) and the volume (V) of the system both go to 00 while the density (N IV) remainsconstant All thermodynamicparameters must approach well-defined values in this limit In the classical (non-quantum)realm, the existence COHERENCE,ENTANGLEMENT, AND RFDUcnONISf ExPlANATION. 525
of the thennodynamiclimit so far has beenrigorously proved for only very contrived and simple systems(fhompson [972). For classical systemswith only electrostatic interactions, it is even easy to show that the thennodynamic limit does not exist. If quantum mechanicsis invoked, however,the limit is defined (Ruelle 1969, 60-68). Perhapsthere is even some unexpectedinsight to be gleanedfrom this situation: that classical thennodynamics is reducible to quantum statistical mechanics,but not to. classicalstatistical mechanics; (v) To obtain the classicalrules of valencyfrom quantummechanics, approximations about the nature of the wave-functions,about the Hamiltonian for complex atoms, and assumptionsabout the convergenceof solutions must all be brought into play (Pauling 1960). These observationsshould indicate that the issuessurrounding reduction are far from settled in contemporary physics. Our immediate purpose in this paper is to examine whether explanations invoking the concept of "coherence" (including the claims of Frohlich) really do violate reductionism. We conclude that the use of ."coherence"does not necessarilypreclude an explanation from being reductionist. Nevertheless,there exists a class of quantum-mechanicalsystems exhibiting coher- ence, those having "entangled states,"for which explanationsare no longer clearly reductionist. This class of systems has long been of philosophical interest because one of its subclasses,that of entangledtwo-particle systemseach with an associated two-dimensional Hilbert space,includes the systemsthat violate Bell's inequality. It appearslikely that the resultsestablishing the relationshipbetween entanglement and Bell-inequality violations can be generalizedto the result that all entangledsystems are capableof violating Bell-type inequalities. Furthermore,it turns out that though the type of model that Frohlich hasinvoked involves coherence,any failure of reduc- tion will be due to entanglementrather than coherence.In the long run, we hope that our analysis will help reintroduce detailed discussionsof the place of reductionist explanationin physics. In section 2 we discusswhat we mean by a "reductionist explanation." In section 3 we give several examplesof classical and quantum systemsthat exhibit "coher- ence."We then attempta generaldefinition of that and relatedconcep~, which is nec- essary becausea sufficiently general definition has proved to be elusive in the past We are not fully satisfiedwith our definition; we hope, however,that it will spur fur- ther discussion by others. In section 4 we show that all entangled(quantum) states exhibit coherence,and that the coherentstates invoked by Frohlich are entangled.We observe that the quantum statesthat violate Bell's inequality are also entangled. In both sections3 and 4 our more technical conclusionsare presentedas relatively pre- cise theorems.We give proofs of thesetheorems in thosecases in which they are not explicitly available in the extant literature. In section5 we show that, while the useof coherencedoes not necessarily lead to an explanation failing to be reductionist, I entanglementleads to such a failure. We note someof the implications of our analysis in the concluding section(section 6). 526 GRroG JAEGFR AND SAHOTRA SARKAR
2. ON REDUCfION Systematicanalysis of the conceptof reduction in the natural sciencesbegan with the pioneeringefforts of Nagel (1949,1961) and Woodger(1952), who viewed reduction as a type of inter-theoretic explanation.This approach has since been significantly extended,especially by Schaffner(1967, 1994).All theseapproaches view explana- tion as deductive-nomological.Alternative accountsthat view reduction as a relation betweentheories, though not necessarilyone of explanation,have beendeveloped by Kemenyand Oppenheim(1956), Suppes(1957) and, more recently,Balzer and Dawe (1986, 1986a).Meanwhile analysesof reduction that view it as a form of explanation but not necessarilyas a relation betweentheories have also been developed (e.g., Kauffman 1971;Wimsatt 1976; Sarkar 1989, 1992, 1998).These analyses have been motivated primarily by the situation in molecular biology, where reductionist expla- nation seemsto be rampant, but the explanationsrefer to a variety of mechanisms rather than to theories. Using a set of distinctions introduced by Mayr (1982), Sarkar (1992) classifiedthese models of reductioninto three categories:(i) theory reduction- ism which consistsof those models that view reduction as a relation between theo- ries; (ii) explanatory reductionism which consists of those models that view it as explanation,but not as a relation betweentheories; and (iii) constitutive reductionism which consists of those models such as the various types of superveniencethat eschewboth theoriesand explanation.s Accounts in all three categories make both epistemological and ontological claims. In particular, all models of reduction sharethe rather innocuous ontological claim that what happensat the level of the reducedentities (theories or not) is not novel in the senseof being inconsistentwith what happensat the level of the reducing entities - otherwise thesewould not be modelsof reduction.Throughout our discus- sion we will continue to make this assumption.However, we.will ignore other onto- logical issuesthat have beencontroversial: whether the terms invoked in a reduction refer to "natural kinds"; whether reductionsestablish relations between"types" at the two levels or between"types" and "tokens", etc. Theseissues, though often regarded as philosophically important, are orthogonal to our present pUrpose.6We will also ignore those formal epistemological issuesthat have persistently been the focus of dispute: (i) whether the factors involved in a reductionistexplanation are codified into theories;(ii) whetherthe structureof the explanationis basically deductive-nomolog- ical (Schaffner 1994)or statistical (Wimsatt 1976),etc. Thus our presentanalysis will be consistentwith any of the models of reduction that view it as a type of explana- tion} Meanwhile, we will focus on three substantiveclaims that have often been implicit in modelsof reduction but have rarely beendiscussed in detail. Our analysis will reveal some rather surprising subtletiesabout reductionistexplanation. We assumethat what we have at hand is an explanation,i.e., it satisfies whatever strictures that one choosesto put on "explanation." Our problem is solely to specify additional criteria by which we can decide whether that explanation is reductionist. The reasonsfor this moveare to avoid.disputes about the explication of "explanation" (on which there is no consensus),and to focus attention precisely on thosefactors that makean explanation reductionist.8 COHERENCE, ENTANGLFMENf, AND REDUcnONIST ExPLANATION.. 527
We suggestthat, at the substantivelevel, the three most important criteria are: (i) Fundamentalism:the factors invoked in an explanationare warrantedby what is known, either from theoretical considerations(characteristically involving approxi- mations) or only from experimentsentirely at the level of the reducing theories or mechanismswhich are more "fundamental" than those at the level of the reduced entities in the sensethat their presumeddomain of applicability is greater.9Satisfac~ tion of this criterion is a matter of degree. If the demonstrationof such a warrant involves only theoreticalderivation, from first principles, with no approximation, and so on, then its satisfactionis most complete.Approximations, especially counterfac- tual or mathematically questionableapproximations, hurt the degree to which this criterion is satisfied; (ii) Abstract hierarchy: the complex entity whose behavior is being explained is rep- resentedas having a hierarchical structure (with identifiable levels and an ancestral relation betweenlevels) in which only the propertiesof entities at lower levels (of the hierarchy) are used to explain the behavior of the complex entity.10 Suchan abstract hierarchical representationcan involve any space,not necessarilyphysical space,that is used to model a system.For instance,it can be a hierarchy in any configuration or phasespace in (classical)analytical mechanicsor a Hilbert or Fock spacein quantum mechanics; (iii) Spatial hierarchy: the hierarchical structureof the entity (that is invoked in the explanation)must be realized in physical space,that is, entities at lower levels of the hierarchy must be spatial parts of entities at higher levels of organization. These criteria are not all independentof one another: (iii) can only be satisfied provided that (ii) is. Moreover, if (i) is not satisfiedat all, it is doubtful (at least) that an explanationshould be considereda reduction. It will be assumedhere that, for all reductionistexplanations, (i) is at leastapproximately satisfied. What_will distinguish the different types of reduction are the questions whether (i) is fully satisfied and which of the other two criteria, if either, is also satisfied.With this in mind, five dif- ferent sensesof "reduction" basedon thesecriteria can be distinguished.For eachof these senses,several illuminative biological examplesexist-these are discussedin Sarkar (1996). Here, we only mention putativeexamples from physics: (a) criterion (i) is (fully) satisfiedwhile none of the others are: examplesinclude the reduction of Newtonian mechanicsto special relativity, and of Newtonian gravity to generalrelativity. More controversially,the reduction of geometricaloptics to physi- cal optics is a reduction of this sort.I I (b) criterion (ii) is fully satisfied while (i) is approximately satisfied:this is clearly a very weak senseof reduction. Little more than a hierarchical structureis assumed.In the study of critical phenomena,explanations involving renormalizationin parameter spaceare of this type: the system(say, a ferromagnet)is given a hierarchical repre- sentationin parameterspace (though not in physical space)but the interactions pos- ited at each stage as the number of units of which the system is composed(in our example, magneticspins) is iteratively decreasedhave at best only approximatewar- rants from the underlying mechanisms;12 528 GRroG JAEGFR AND SAHOfRA SARKAR
(c) criteria (i) and (ii) are fully satisfiedbut (iii) is not: in thesereductions, the organi- zational hierarchy does not correspondto a hierarchy in the usual physical space.If quarks really are confined, in the sensethat free quarks do not exist, explaining the propertiesof hadronson the basisof propertiesof quarks would be an explanationof this type. There is a hierarchicalstructure: hadrons consist of quarks but this is clearly not a hierarchy in physical space.Examples of this sort aboundin particle physics; (d) criteria (ii) and (iii) are (fully) satisfiedwhile (i) is approximately satisfied.Once again, in the study of critical phenomena.real-space renormalization, that is, renor- malization with a representationin physical spaceinvolves a reduction of this type; (e) all three criteria are satisfied:this is obviously the strongestsense of reduction. It is the one that is invoked in the putative reduction of thermodynamicsto statistical mechanics.It is also the one that will be most relevantto the discussionin section 5. We will refer to this senseas "strong reduction." Criteria (ii) and (iii) togethercapture the intuition behind thosetypes of reduction which refer to a spatial whole being made up of identifiable constituent parts. When (i) is also satisfied,the explanatoryforce in a reduction, which comesfrom the prop- erties of the parts, is supposedto provide a deeper explanation entirely from the lower level. In general,this is the one which is usually assumedto capture the rela- tion betweenmacroscopic and microscopicphysics. In the particular caseof explana- tions involving the notion of coherence,this is the intuition that we will explore in this paper. Of course, if what we have already said about explanations involving renormalizationtheory in condensedmatter physicsis correct, there are other reasons for doubting that strong reduction describesthe relation between microscopic and macroscopic physics- we leave a full discussion of renormalization for another occasion.
3. COHERENCE We turn, now, to the concept(s)of coherence.Any attempt at an explication of this concept faces a peculiar quandary: though it is quite routinely used in a variety of areaswithin physics, it is almost never defined outside optics, where it has become associatedwith various measuresof correlation betweenfield variablesat two space- time points (cf. Mandel and Wolf 1995,chs. 4-8, 11). Usagein physics and biophys- ics is so varied that it is open to questionwhether there are non-trivial criteria that all uses share. This is so despite 75 years of very gradual development and several attemptsto broadenthe definition beyondquantum optics, where there is a fairly con- sistent pattern of use (Mandel and Wolf 1970, 1995). Becauseof this, we start with some remarks in the quantum optics literature regarding this definition. We then attempt to generalizeit to be generallyapplicable across physics. But we are less that sure that the result is non-trivial. So we proceedto avoid triviality by relativizing our definition to an independentlycharacterized class of physical systems.At the very least, we hope that our attempt will provoke not just criticism but other attempts at providing a generaldefinition of coherence. COHERENCE, ENTANGI.FMENT, AND RF1>UCflONIST ExPLANATION.
Section 3.1 discussesexamples of coherenceacross physics. Section 3.2 presents some illuminating commentsby others regarding usageof the term, as well as our explication of one trivial and one non-trivial conceptof coherence;we demonstrate the triviality of the former by an explicit (and itself somewhattrivial) theorem.
3.1 Examples Skepticismabout the possibility of reductionistexplanation in physics has often been based on the belief that the componentsof composite systems exhibit collective behavior that cannot be accountedfor by a straightforward examination of the prop- erties of theseparts taken in isolation. In physics,descriptions of such behavior often invoke the notion of "coherence."As we noted above, this notion is almost never given a definition when it is used outside optics. To motivate our suggesteddefini- tions to be given at in section 3.2, considerthe following four examples.The first is from classicalphysics. The other three involve quantumconcepts: (i) Coupled harmonic oscillators: Considera pair of identical oscillators, having the samespring constantk and massm, coupled by a third spring of spring constant k'. Such a pair of oscillators are capableof moving in a "high-frequency normal mode" of vibration in which the motion of each oscillator is of constantamplitude and the samefrequency 00 higher than their common "natural frequency," 000= (k/m)1/2 The equationsof motion for the two oscillators are:
Xl = Acosrot, X2 = A cosrot,
where x 1 and x2 are the displacements of each of the oscillators from its equili b- riurn position, t is the time, and A and B are the (constant) amplitudes of oscillation. There is a correlation in the position of the pair of oscillators because bo~h ~scillators vibrate with the same frequency. Note that, given t and knowing 00 = ~oo; + k'lm, we can infer X2 from Xl and vice versa; (ii) Rabi oscillators: Consider an electron capable of being in one of two coupled states 1<1>1)'and I~with energies E1 and E2 (respectively), of an atom, ion, or mol- ecule, so that its quantum state, which can be written
1'1'(/» = al (1)1+1)+ ~(/)I+J. (3.2)
where a;{t) (i = 1,2) are state probability amplitudes and t is the time, obeys the standard time-dependent Schrooinger equation.13It is convenient to decompose I'l'(t» using the eigenvectors, 1'lI+) and I'll-), of the Hamiltonian, H = Ho + W 12' where W 12 is the operator representing the "energy" of the "force" coupling the two states.Let 1'lI+) and I'll-) have the energies E+, E- (respectively). Taking
1'l'(O» = Icpu, we have I'l'(t» = l..e-iE+l/AI'lI+)+~e-iE_l/AI'll), where I.. and ~ are constants.The probability of the electron being in either of the two states(or in the other) varies periodically with a frequency that dependson the strength of the
~ 530 GRroG JAEGERAND SAHarRA SARKAR
coupling IW l~ betweenthem: assuming W 12 to be purely non-diagonal,the proba- bilities P2(t) and Pt(t) of respectivelybeing in states 1t1>~ and lt1>t)are
(3.3a)
. 2 . 2(E+- E-)' P.(t) = I-P2(t) = -sm 8sm -~t (3.3b)
where 6 = (~) is the oscillation period and Ii is h/(23t) (with h being
Planck's constant). There is a strict correlation betweenthe probabilities of the electron occupying the two states,thus of the atom, ion or molecule having the feature of one energy level or its alternative energy level occupied and the probabilities can be inferred from one another using eq. (3.3b). Note that this discussioncan be extendedbeyond two-state systemsto caseswith severalstates and different associatedenergy levels. The correlationsare then less trivial; (iii) Frohlich systems:(Frohlich 1968, 1975)claims that quantum statesof a macro- scopic system described by "macro wave functions" are neededto explain a wide range of biological phenomena.These wave functions are supposedto exist when there is "off-diagonal long-rangeorder" (ODLRO) in a system.Yang's (1962) defini- tion of ODLRO, which is a property of a "reduced density matrix," will now be used to show how coherenceis presentin such a system.14
Let a large system ~ containing a subsystem 0 be described by the density matrix p. The "reduced density matrix" representingsubsystem 0 is first obtained by taking a weighted average of the parametersspecifying the portion of ~ not including o. This averaging is achieved by "tracing out" the parametersfrom the density matrix P representing ~, yielding a reduced density matrix Po for o. ODLRO is a property of Po' illustrated here by the following two examples. The simplest and most interesting example in our context is the one-particle reduceddensity matrix PI describingsingle particles.This matrix has the elements
v1p.li} = Tr(ajpa;t), (3.4) where i and j are the labels of the possible statesfor the individual particles of the system, p is the density matrix for the entire system, Tr( .) is the trace of a matrix .. . ", the ai's are the annihilation operatorsfor single-particle statesand the ai's are the corresponding creation operators.The second example is the two-particle reduceddensity matrix describing pairs of particles,having the elements COHFRENCE,ENTANGLB4FNf, AND RFDucnONISTExPLANAll0N...
where the new labels k and I refer to particle statesjust as i and j do. (Note that ele- mentsof this reduceddensity matrix are labelled by four indices.) Other reducedden- sity matrices for yet larger subsystemsof particles, PlIOfor n- particle statescan be similarly defined.The pertinent subsystemsof many-particlesystems, which we will call "Frt)hlich states(F-states)" (following Pokorny 1982), are those whose single- particle reduceddensity matricesexhibit ODLRO. These consist of a large number, N, of bosons(either simple bosonsor bosonicquasi particles formed from fermions) that are said to exhibit ODLRO when they can be representedby single-particle reduceddensity matricesof the form
P.("', r~) = aNcI>(r')"(r~) + x(r', ,.-),
where ~(r) is the quantum "macro wavefunction" attributed to the subsystem, ~*(r) is its complex conjugate, x(r', r") is a positive operator, 0 ~ a ~ 1 and r' and r" representtwo (spatial) positionsof the subsystem. The reduceddensity matrix for the subsystemhas the spectralresolution
p.(r'. r-
where the ;(r) are energyeigenstates and t1; are weights. Most of the N particles in our system lie in the same state, ;n(r) = ~(r), for which t1n = aN (for some value n of i and a is a constant in [0, I] near 1) and the weightS t1; « I for all i - n. Thus
~ p.(r'. r-) = aNcI»(r')cI»*(r")+ . ~ l.Li~i(r' )~i (rW), i=l i-N
. ~ where }: t1/ = (l-a)N. With}: t1f;/(r');;(r-)-x(r',r"), which i=II-- 1=li." describesthat portion of PI describing individual particles of our subsystemnot in the state <1>.we have
p.(r', r-) = aNcI>(r')cI>*(r")+ x(r', r-),
wherex(r', r") is small com~red to aN exceptwhen r' - r". As a resultof the presenceof the first term, p.(r', r")..,. 0 even as Ir' - r"l- 00. Thus, for F-states, an identity of quantum state-and, therefore, perfect correlation-will persist over
~ 11
532 G REGG JAB} FR.AND SAHOTRA S ARKAR
largespatial distances. This is the sensein which ODLRO is a form of long-range order.
(3.10)
where, in each term, the subscripts refer to the Hil~rt sJXlCeof the corresponding particle, 1+) is the single-particle state ..spin up," and 1-) is the single-particle state ..spin down." The system is correlated in two ways: (a) there is a correlation of both quantum phases,the angles e of the complex exponential part of each complex probability amplitude, for the combined system(as explainedin theorem3.1 below); and (b) the spins are strictly anticorrelated, that is the spin statesof the electron and positron are such that if one has spin "up" then the other has spin "down"; knowing the spin state of either particle allows one to infer the spin stateof the other. The example will be relevantto section4. Such a systemis traditionally referredto as being in a "coherent" superpositionof states.Here this involves two quantum-mechanicalstates: one in which the electron hasthe relevantcomponent of its spin in the "up" stateand the positron has the same spin component"down" (i.e. it is describedby 1+)-I-) +), and one in which the elec- ~ ~ tron has spin "down" and the positron has spin "up" (describedby 1-)~ -1+) ~ +).
3.2 Tentative Explication COHERENCE, ENTANGLEMFNf, AND RFDUcnONIST EXPLANA11ON. 533
'coherent' is also widely used in physics to indicate conelation betweentwo or more functions of either spaceor time, (such as its use in the descriptionof two light beams obtainedby the splitting of a single beam),although the functions themselvesmay have somerandom properties. Klauder and Sudarshan(1968, 56) echo thesesentiments in their own attempt to characterize"coherence" broadly: ... we must come to tenDswith the vagueconcept of 'coherence..Being purposely gen- eral, let us say that a 'coherentfeature' of a statisticalensemble is an observableaspect held in common by eachmember of the ensemble.Different ensembleswill in general have different coherentfeatures depending on what collection of quantities is deemed 'observable..This suggeststhat we call a 'relative coherentfeature' one which fulfills the criteria for a coherentfeature for a subsetof the observables.That is, the criteria for a rel- ative coherentfeature are necessarybut not sufficient for a coherentfeature. ... 'Full coherence'can be said to exist if the membersof the ensembleare identical in all their observableaspects. This definition, which is explicitly intended as a general one (though motivated by Klauder and Sudarshan'sinterest in optics), is unnecessarilyrestricted to situations where a statistical ensembleexists. One should be able to speak of "coherence.' in systemsthat are not normally regardedas ensembles.This point has beenlong recog- nized (e.g by Hopkins 1952,263~ and Senitzky 1962,2865). It is also suggestedby our examples,all of which are systemsof this sort. Furthermoreeven in 1hecontext of ensembles,it seemsunreasonable to require that all propertiesof a system be corre- lated for a systemto be called "fully coherent."Similarly, in one of the most highly regardeddefinitions requiresthat a fully coherentfield be "defined as one whose cor- relation functions satisfy an infinite successionof statedconditions" (Glauber 1963), which also seemsunreasonably strict However, Klauder and Sudarshanare correct to indicate that .even a context- dependentdefinition of coherencecan invoke nothing more than the existenceof a correlation. Ho and Popp (1993) also come to the same conclusion.IS We might, therefore,attempt modify and extend such a putative definition in the following way: a systemis coherentif and only if, for at leasttwo of its features,A and B, there exists a correlation betweentheir values.16Then,either the value of B can be estimatedfrom that of A or the value of A from that of B. The trouble with this definition of coherenceis its weakness:almost all systems exhibit coherence.17 For the purposesof discussionwe will insteadcall this property "trivial coherence."In classical physics almost all systemsexhibit such coherence becauseof the fully generallaws suchas Newton's third law, which routinely provide the framework in which all modelsare constructed~each such mod-fl will ~ve coher- C-J1ceautomatically built into it In the caseof Newton's third law, Fij = -F ji (where Fij is the force of one object, i, on another, j), each time two objects interact, the f2rces involved will correlatetheir motions in accordancewith Newton's secondlaw, I F;j = mll;j, where ll;j is the contribution to the accelerationof each object due to the force F ij of one object, i, on the other j. In quantum mechanics,for all pure statesa similar situation arises becauseall systemsare representedby a ray in an associatedHilbert space.The following trivial theoremcan be proved:
~ 534 GREGG JAEGER AND SAHOTRA SARKAR
Theorem 3.1: All isolated nonstatistical (pure) quantumstates exhibit trivial coher- ence. Proof. Consider the casewhen the Hilbert spaceof the quantum-mechanicalsystem in question is countable.Every isolatedsuch a systemcan be written as a pure stateof n ~ 11;. 1 the following form: "') = L., AjIY ), whereA j E r je J,;j = -;; Ejt (E j being the j:l energyof the systemin state IY j)' t the time, r j real constants)and the eigenvectors
IY j) form an orthonormal basisfor the Hilbert spaceof the system.A sustainedcorre- lation will exist betweenthe propertiescorresponding to the eigenvectorsIy j) since the quantity ~kl E(llt)(;t-;I) = -~(Ek-El) is constantfor any k and I, for all times t; since, for isolated systems,the energiesE j areconstant in time, a correla- tion exists, representedby the energy difference ~ kl betweenthe values of ;k and ;1 at any time t. This correlation is observablein the fonn of quantum interference. (The proof in the caseof uncountablyinfinite Hilbert spacesis identical but with the sumsreplaced by integralsand discreteindices replacedcontinuous ones.) Every physical system will usually exhibit some such trivial coherencewhich makes it, therefore, not a very satisfying notion. We will, therefore, introduce the notion of (non-trivial) coherenceand relativize it to a class K of physical systems which fonns the backgroundagainst which new correlations become interesting.A systemis (non-trivially) coherentwith respectto K if and only if it exhibits a type of correlation betweentwo featuresthat is not exhibited by all membersof K. This defi- nition is context-dependentsince it is relativized to the class K. Neverthelessit remains,in a sense.largely context independentsince it is applicable to a wide vari- ety of situations. Non-trivial coherenceis most interesting when class K is almost universal. for instance,the class of all Newtonian systems (N) or the class of all quantum-mechanicalsystems (Q). Returning to our examples.it is obvious that the coupled oscillators exhibit non-trivial coherencewith respect to N and the other three casesexhibit non-trivial coherencewith respectto Q.
4. ENTANGLEMENT AND COHERENCE IN QUANTUM SYSTEMS In the quantum mechanicsof composite systems,entangled states are those states system that cannot be expressedas a product of statesof its individual subsystems. For simplicity, consider a system involving just two subsystems,each representable by a finite-dimensional Hilbert space.In this case,each possiblestate of the compos- ite systemis representedby a vectorin a Hilbertspace HI @ H2' where HI and H2 are the Hilbert spacesof each of the two subsystemsin isolation. The product states of such a system are those representedby vectors t-P) I + 2 E HI @ H2 such that !'i')1 + 2 = Itl)118)2' for a pairs of states, Itl)1 E HI and 18)2E H2. The entangled COHERENCE,ENTANGLEMENT, AND RFDUcnONIST ExPlANATION. 535
statesof sucha systemare thoserepresented by vectors /1IJ}. + 2 E H. @ H2 such that /1IJ}.+ 2 -!11}IIO}2 for any of the possiblepairs of states!11}1 E H} and IO}2E H2. Note that the state vector of any composite system of two objects (each having a finite-dimensional Hilbert space)can be written as a linear combination of unit vec- tors laJIJ3J,i = I, ..., n forming an orthonormal basisfor HI @ H2. This basis is known as the Schmidt basisand the linear combinationis called the "Schmidt decom- position." This basis makesmanifest the characterof a quantum state: if only one of these componentsis non-zero then the state is a product state,.otherwise it is an entangled state. (Note that the Schmidt decomposition cannot, in general, be achievedfor systemswith more than two parts. Nevertheless,in such cases,any n- dimensional systemcan still be rather artificially decomposedinto two subsystems, one with dimension k(O < k < n) and the other with dimension n - k.) It is hard to over-emphasizethe importance of entanglementfor understanding quantum systems:as SchrOdinger(1935, 555), who introduced the concept, put it: "1 would not call [entanglement]one but rather the characteristic trait of quantum mechanics,the one that enforcesits entire departurefrom classical lines of thought." However,what are most interestinghere are the following results.
Theorem 4.1 (a): All isolated composite systems (having finite-dimensional Hilbert spaces) in entangled states are coherent with respect to Q; (b): However, not all quantum-mechanical systems that are coherent with respect to Q are entangled.
Proof (8): Any entangled state for which the Schmidt decomposition can be per- formed will exhibit correlationsbetween at leasttwo pairs of propertiesof its compo- nent systems.namely whoseeigenvectors form the Schmidt basis.Such an entangled n state may be written as: ~} = }: JA.i(t)lajll~j2' where the JA.i(t) are complex i=1 numbersand laj and I~j are the basis vectors. For that basis, there are clearly n ordered pairs (ai' ~j) (i = 1, ..., n, where n> I) of values of the observables A (of system I) and B (of system2). ~hese n orderedpairs of propertiesare perfectly cor- related. Such a correlation is not found in all quantum systems.Thus a system in an entangledstate will be non-trivially coherentwith respectto Q; Proof (b): This is shown by exhibiting a counterexample:the systemabove exhibit- ing Rabi oscillations (seesection 3) exhibits coherencebut is not entangled.There are many other counterexamplesthat could be supplied hereas well. The discussionof entanglementthat we have given above refers only to systems with two subsystems.This definition of "entanglement" and theorem 4.1, can be straightforwardly extendedto systemswith countably infinite subsystems.However, entangled statesof two-particle systemshave long been of interest to philosophers because,to begin with:
~ 536 GREGG JAEGER AND SAHOTRA SARKAR
Theorem 4.2 (8): States of two-particle (quantum) systems each having Hilbert spaces of dimension 2 that violate the Bell inequality are entangled. (b): Moreover, states of two-particle (quantum) systems that violate the Bell inequality are non-trivi. ally coherent with respect to Q.
Proof (a): This haslong beenpart of the folklore of foundationsof quantum mechan- iCS.IS(b): This now follows directly from theorem4.1(a). Furthermore,these results are very likely extendableto apply to larger systems and systemsof higher dimensionality.Finally, returning to F-states,the coherenceof which has been a source of disquiet about reductionist explanation in biology, it is easyto prove that:
Theorem 4.3 (8): Some F-states exhibit entanglement: (b): Not all F-states exhibit entanglement. Proof (a): Yang (1962) has demonstratedthat ODLRO is present only in density matricesof systemsof bosonsor fermion pairs forming bosonic quasiparticles(via pair occupation of single-particle states by fermions). Being composedentirely of bosons,Frohlich systemsmust havewave functions 'lI( 1,2, ..., N) that are symmet- ric under exchangeof particle labels. Wheneverthe particles are not in the samesin- gle-particle state, cI>,the overall systemstate will exhibit entanglementbecause this symmetry requirement yields a state of the system that is unfactorizable, i.e. 'lI( 1,2, ..., N) - cI>(l)cI>(2)...cI>(N) where cI>(i) are wavefunctions describing individual particles. For example. if one particle is in a state E - cI>.the stateof the other N - 1 particles. 'lI is be a sum of N different terms eachcontaining one factor E(i), for exactlyone value of i E Z:. and N - I factorscI>(j), j - i. Proof (b): F-statesthat have reduceddensity matricesfor which x(r', r") = 0, in which case all particles are in the same single-particle state, are not entangled. For N them, Pl(r', rn) = N 5. THE QU~TION OF REDUCTION In the explanationof the coherentbehavior of all the systemsthat we havediscussed, our first criterion of reduction (fundamentalism)is easily satisfied.Thus, none of our examplesfail to satisfy the conditionfor thefirst sense(a) of reduction. It is important to note that this is a non-trivial claim and should not be confusedwith the ontological question whether any new processexists at the higher level. Explanationsfrom the COHBENCE, ENTANGLatENT, AND REDucnONlST ExPLANAll0N. 537 more fundamental assumptionsroutinely involve making approximations or taking limits and this can be philosophically problematic. The relevant proceduresmight well be contrived in the sensethat the particular way in which a limit is taken or an approximation made may be motivated specifically by the desired result (Sarkar 1996, 1998). In such a situation, the fundamentalistcriterion is "iolated. It is an open questionhow often this happensin physics. Turning now to strong reduction, the analysis in section 4 pennits the following three conclusionsto be drawn: (i) The existenceoj coherencedoes not always imply a Jailure oj strong reductionJor either classical or quantum systems.The coherentmotions of the (classical) coupled oscillators system can be simply explained by describing each oscillator separately and noting the correlationsbetween their positions.The coherentfeatures of the Rabi oscillations are explained by the coupling betweenthe states,an interaction between partsof a hierarchythat can be spatially instantiated(though, as in most quantumsys- tems, it cannot be easily and accuratelyvisualized); (ii) Explanationsoj coherencein quantumsystems that involve entangledstates vio- late strong reductionism because they violate the second condition (hierarchical structure) and. ipso facto. also the third condition (spatial instantiation). If a com- posite system is describedby an entangledstate, definite statesin generalcannot be attributed to its individual subsystems.As Schrtidinger(1935) pointed out. in such a statethe subsystemscannot be in generalbe individuated,and locutions such as "sub- system A" and "subsystem B" do not refer to any precise entity within the entire state.Consequently, often no hierarchicalrelation betweenthe subsystems'states and that of the composite system exists, let alone is instantiatedin physical space.This conclusion has some seriousconsequences. Consider, for example,a hydrogen atom which consistsof an electron and a proton interacting with eachother. In a.fully quan- tum descriptionof this atom, it would be representedby an entangledstate. Therefore it cannot be representedas a hierarchical structurewith identifiable individual states for the proton and electron. The situation is the samefor atoms other than hydrogen (Jaeger2 6. CONCLUSIONS We only have two major conclusions.First, we have shown that the invocation of quantum-mechanical entanglement violates the conditions for strong reduction within physics though coherencealone is not sufficient for that purpose. Entangle- ment destroysthe possibility of strong reduction becausean entangledsystem cannot be describedas being hierarchically organized This does not, of course, imply that this is the only way in which strong reduction can fail in physics.Leggett (1987) and fisher (1988), for instance, have argued that the fundamentalist assumption itself fails in sometypes of explanationin condensedmatter physics. In our discussionof examplesof types (b) and (d) of reduction we, too, have indicated that strong reduc- tion fails in those explanationsof critical phenomenain condensedmatter physics that involve renormalization.Whatever be the merit of thoseclaims, we hope that our analysis has drawn attention to the fact that not all the issuessurrounding the role of reductionistexplanation in physicshave beenresolved. Second, physicalist explanationsin molecular biology have so far satisfied the conditions for strong reduction (Sarkar 1989, 1992, 1996, 1998). However, if entan- gled F-states have to be invoked in such explanations,strong reduction will fail becauseof the failure of the hierarchyassumption. Whether F-stateshave any explan- atory role in molecular biology remainshighly controversial.20It suffices here simply to note that there is at present no plausible candidate for a biological mechanism involving an F-state. It remains an empirical question whether, eventually, such a mechanism will be discovered,and lead to a failure strong reduction in biology. Should that happen,what will perhapsbe most ironic is that it will fail not becauseof any special property of biological systems(as anti-reductionistshave usually held), but becauseof the natureof physicsitself. Thus, if the reductionof biology to "funda- mental" physicsis a problem,then what is problematicis not the relation of biology to physics,but the relation of macroscopicphysics to "fundamental;' that is microscopic physics.The full oddnessof this situation seemsnot to have been fully appreciated, despiteits having beenoccasionally pointed out at times before (e.g. Shimony 1978).~ Boston University University of Texasat Austin ACKNOWLEDGEMENTS We would like to thank Abner Shimony for initiating our interest in the problemsof reductionism in quantum mechanics.Besides Abner, we would like to thank Tian Yu Cao, Don Howard, Simon Saunders,John Stachel and Chuck Willis for comments and criticism of an earlier version of this paper.This researchwas partly supportedby an NIH grant (No.7 ROI HG00912-02)to SS. COHFRENCE, ENTANGLEMENT, AND REDUCfIONIST ExPlANATION... 539 NarES I. SeeWimsatt (1979), Sarkar(1992, 1998),and Schaffner(1994) for reviews. 2. See,for example,Nagel (1961).However, Specter (1978) and Shimony(1987) are notableexceptions. 3. SeeLeggett (1987, III -143). 4. Seealso Ho and Popp(1993). 5. For modelsof superveniencesee, e.g., (Davidson 1970;Rosenberg 1984) and, especially, Kim (1993). 6. A detaileddiscussion of ontological issuescan be found in Wimsatt (1995).Arguments against the philosophicalimportance of theseissues are developedby Sarkar(1998). 7. This analysisis not straightforwardlycompatible with modelsof reductionwhich do not view it as a form of explanation - e.g. Balzerand Dawe(1986, 1986a)and Ramsey(1995). However, it is unclear that thesemodels should at all be regardedas thoseof "reduction" in any usualphilosophical sense, given that this term hasalmost always beenused to refer to a particularmode of explanation. 8. The approachthat is summarizedhere is developedin Sarkar(1998). That work includesa discussion of the epistemologicalproblems posed by approximations,which is glossedover in this paper. 9. The provision of a "warrant" may be weakerthan logical deductionor evenderivation. A warrantcan potentially be a theoreticalclaim that is madeon the basisof experimentalfacts known at the reducing level (seeSarkar 1989).Note that the reducinglevel neednot be a lower level of any hierarchyas, e.g., in the reductionof Newtoniangravitation to generalrelativity or in the reductionof Newtonian mechanicsto specialrelativity. 10. The simplestkind of sucha hierarchicalstructure is a (graph-theoretical)tree. More complicated structureswould only requirea directedgraph with an identifiableroot and no cycles. II. That one must distinguishbetween reductions only satisfying(i) and thosethat satisfy the other crite- ria was pointedout by Nickles (1973) though,in his usage,the reductionsoccurred in the opposite direction than the one discussedhere (Wimsatt 1976). 12. For a particularly clear exposition,see Binney, Downick, Fisherand Newman(1993). Renormaliza- tion theory in condensedmatter physics has not receivednearly the kind of philosophicalattention that it deserves.(For discussionsof renormalizationtheory in the contextof quantumfield theory see (Brown 1993).) 13. Two quantumstates of the systemare said to be "coupled" if the systemis subjectto a force, with cor- respondingenergy (here W 12), capableof causingthe systemto changefrom one stateto the other. 14. See(Yang 1962;Penrose and Onsager1956; Penrose 1951; and Ginzburgand L;andau1950). 15. Ho (1993, 141)admits that "(c]oherencein ordinary languagemeans correlation, a sticking together, or connectedness;also, a consistencyin the system."He eschewsany attemptat explicit definition and later (pp. 150-151) only providestwo jointly sufficientconditions for coherence,but eventhese are restrictedto a quantumcontext 16. We mean"feature" quite generallyin the sensethat evenpossible states of a systemwill be regarded as "features." 17. For instance,a classicalsystem consisting of a free particle with mass m, energy E, and momentum p is coherent because E can be estimated from p(E =~) though ~ cannot be estimated from E. Thereforethis systemwill not be fully coherent 18. See,for example,the proof of Gisin (1991). Seealso (Capassoetal. 1973;Werner (unpublished, cited in Popescuand Rohrlich 1992);Gisin and Peres1992; and Popescuand Rohrlich 1992). 19. This doesnot, of course,mean that suchexplanations necessarily satisfy the criteria of strong reduc- tion-that will dependon whetherthe other explanatoryfactors involved also satisfy thesecriteria. 20. T. Y. 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