m. hossein partovi

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Volume 137, number 9 PHYSICS LETTERS A 5 June 1989

QUANTUM THERMODYNAMICS

M. Hossein PARTOVI

DepartmentofPhysics, California State University, Sacramento, CA 95819, USA

Received 28 February 1989; acceptedfor publication 6 April 1989

Communicated by J.P. Vigier

Equilibrium and nonequilibrium thermodynamics ofmicroscopic systems are derived from quantum mechanics. An example

ofa particle interacting with a heat reservoir is considered, and its rate ofapproach to equilibrium is calculated.

In a series ofpapers [13]concerned with a rig- quantum level and is not the exclusive domain of

orous formulation of quantum measurements we macroscopic systems.have emphasized the fact that a pure quantum state We shall start by introducing certain preliminary

is in general an idealization and that a fundamental notions. The state of a quantum system is deter-

understanding of the underlying physics must be mined by a measurement/preparation process

basedon a statistical treatment. Such a treatment was whereby a large number of copies ofthe system is

developed in ref. [2] on the basis ofa maximumun- produced, a fraction ofthese is subjected to inter-certainty/entropy principle rather as the statistical action with measuring devices, and the information

mechanics oflarge aggregates is based on some ex- so obtained is used to define the state ofthe re-

tradynamical hypothesis such as ergodicity, equal a maining copies. Since in general no finite measure-priori probability or maximum entropy. The intro- ment processcan be exhaustive [2], the state sopre-

duction of a statistical hypothesis into microscopic pared is only incompletely specified and must

dynamics, however, immediately resurrects the cen- therefore be represented by a density matrix, ~3,cor-tury-old question ofwhether the extra assumption is responding to a mixed state.

a necessary consequence ofthe underlying dynamics The introduction of a statistical mixture directly

or whether it is in fact an additional law ofnature. raises the question of a measure ofuncertainty, orA survey ofthe recent literature [4,5] indicates that entropy, for the ensemble. In many ways, the issue

in spite of a massive body of research since Boltz- of the choice (or even the universal existence) of themanns pioneering work, the problem of the foun- entropy function lies at the heart ofthe irreversibil

dations of statistical mechanics and the origin of ir- ity problem [5]. However, as shown in ref. [2], thereversibility is still basically unresolved. Thus, choice for a quantum system is (up to a choice of

whereas the first and the third laws of thermody- scale) essentially unique, being given by the von

namics are well-understood consequences ofdynam- Neumann formula tr , ln~5;we shall refer to thisical principles, the second law is not. It is the pur- quantity as the BoltzmannGibbsShannon (BGS)

pose of this Letter to resolve this issue by showing entropy [5]. Note that the BGS entropy is a constant

that for microscopic systems, irreversibility and the ofthe motion for a closed system (which evolves

second law follow from the dynamics and do not have unitarily according to a Hamiltonian), a fact whichan independent status. By so restricting our consid- is often considered to contradict the principle ofen-erations to microscopic systems we are able to avoid tropy increase and invalidate the BGS entropy as ainessential complications associated with large ag- universal (i.e., nonequilibrium) entropy function.

gregates and at the same time underscore the fact that Forquantum systems, however, we shall consider this

thermodynamic behavior is already present at the factas paramount and take it to imply that theorigin

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ofirreversible behavior must be sought in interac- in the final state. To see this, consider the initial state

tions with external systems. The idea that entropy ~ the final statef~,and the respectiveincrease can only occur for open systems was em- entropies S[f~J=S~~and S[f~]=S~. Sincephasized in the fifties ~, although the formulations t~~andf(~are unitarily related,S~=S~.On thethat were based upon it have not proved conclusive other hand, the entropies of the individual (or par-

[4]. The essential uniqueness, at the quantum level, tial) systems, Sa and Sb , must be defined in terms of

of(a) the choice ofthe entropy function and (b) the the individual density matrices Pa =trbF and Pb =origin ofirreversibility is a significant aspect ofthe tra I, where tra denotes the trace operation relativepresent formulation. to the Hilbert space of system a. Thus S~ab

A thermodynamic interaction for quantum sys- S[15~],and similarly for the initial states. Using

tems is in general a collision, i.e., a state for which these definitions, one can readily verify that

the interacting systems are initially and finally out of 5(f) +5~f)5~)5~)each others interaction range. This characterization

is basically the microscopic rendition ofthe standard = trf~[ln(fr~4j3~,)In~definition for macroscopic systems. For a closed, or

This last quantity is known to be non-negative as afree, system we define a stationary state as one whose result ofa convexity property of the entropy func-

density matrix is constant, implying that p= i [J~,tion [5], and it can only be zero if ~ ~~ Pa Pb

i5 ] =0, where Ris the Hamiltonian operator. Among i.e., ifthere is no correlation in the final state. Thus,the stationary states, a special class known as Gibbs

in an obvious notation, L~Sa+L~Sb~ 0.states will play a distinguished role. The Gibbs states

The general result just stated has an importantare ofthe formZ

1 exp(flui), where fland Zare

real constants (to insure the self-adjointness of j); consequence when one of the two systems, say b,is initially in a Gibbs state. With ~

the normalization constant Zwill be omitted where x exp ( flAb), then, let us consider the quantityinessential ~t2

To lend substance to this definition, we consider 1.\Sbfl~Ub,where Ub =trpbHb is (the mean valueof) the energy of system b, and z~denotes a change

an interaction between systems a and b which are from the initial to the final state. We haveinitially in the Gibbs states A =exp( flRa) and/b=exp( flAb). Since the S-matrix ~ fora collision ~5b fl~Ub=tr[~5~,In ~3~/)~ In ~commutes with the sum ofthe free Hamiltonians

fi(i~/5~)Ab]Ha +Hb, it follows that [S,PaPb] =0, implying thatno change will result from the collision no matter Using ln ~ +flHb = In (Z), one can reduce thewhat the interaction (as long as it is sufficiently reg- right-hand side ofthe above equation to tr~5~ular to allowthe existence ofan ~ matrix). Thus, sys- x (ln ~3~0 ln ~ The convexity property statedtems in Gibbs states with the same value of/3 , when previously, on the other hand, shows that this last

allowed to interact, will emerge fromthe interaction combination is non-positive, implying thatwith no change. It is not hard to see that in general E s (Sb flUb) ~ 0, with equality holding only if

the Gibbs states are the only ones with this property. ~ =,3~,),i.e., in general only when the two systemsWith the preliminaries out ofthe way, we consider are in equilibrium. It is worth remarking that the

the interaction of two systems in arbitrary (but un- changes contemplatedhere are not necessarily small,

correlated) initial states ~ and ~5~/). It is well- and that therefore E s (SbflUb) is not in general the

known [51 that the sum ofthe BGS entropies will change in the Helmholtz free energy ofsystem b, sinceingeneral increase as a result ofsuch an interaction, there is no reason tosuppose that/3~is a Gibbsstate.

the difference arising from the correlations present The inequalityjust derived, when combined with~a +EsSb ~ 0 established before and conservation ofenergy EsUa+EsUbO, implies that

~ See ref. [6]. This paper wasin turninspired by the pioneering

workofBergmannandLebowitz [7]. 1~(SaflUa)>~0. (1)~2 This shows the distinguished role thatthe canonical ensemble

plays in microscopic statistical mechanics. This is an inequality that governs the interaction of

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system a, initially in any state, with another (un- equivalent states, and an equilibrium state is not so

correlated) system initially in a Gibbs state char- equivalent to any nonequilibrium state. Clearly, the

acterized by the parameter f l . inequality expressed in (1) indicates that an inter-

Inequality (1) implies the zeroth and the second action with a Gibbs state drives a system toward

laws of thermodynamics. The first is obtained by equilibrium. But thesame interaction drives the other

considering both systemsa and b tobe inGibbs states system initially in a Gibbs state out ofequilibrium.with corresponding parameters P a and f i b . Then (1) To circumvent the latter difficulty, we will define animplies that E s (5a fit, Ua) ~ 0 as well as ideal heatreservoiras an infinite aggregate ofmicro-

E s (Sb P a Ub)?~0. Since E sS a +E sSb ~ 0, it follows that scopic systems in equilibrium at some temperature(P a f i t , ) E s U a ~ 0. Thus when two Gibbs states inter- fl~.When a microscopic system in any state is al-act, the state with the higher value offl~loses en- lowed to interact with such a heat reservoir, it willergy to the other, while, as established earlier, with undergo a series ofinteractions with the constituents

equal values off i no flow ofenergy and no change of ofthe reservoir with the probability ofa reencounterstate occurs. This establishes the zeroth law and the of being vanishingly small (essentially proportional

existence oftemperature, the latter may be taken to to the inverse ofthe number ofconstituents in the

be fl~as usual. reservoir). Clearly in each interaction eq. (1) mustTo establish the second law, consider a cyclic be satisfied, i.e., the quantity SflU must increase,

transformation ofany system a brought about by a where S and Uare properties ofthe system whilefl~series of interactions with an array of systems b~ is the temperature ofthe reservoir. Given a suffi-which are initially in equilibrium at temperatures ciently large number ofinteractions, the system will

fl~respectively. Then eq. (1) applies to each in- approacha limitingstate for which SflUis as large

teraction, implying that EsSan ?~finEs Uan. Since for a as it can be. But this maximum property is precisely

cyclic transformation ~,, EsS~=0, we have the result the recipe for the canonical ensemble, and the lim-that iting state is none other than a Gibbs state with the

temperature parameterequal to that of the reservoir.~ finEs Uan ~0, (2) The argumentjust concluded tacitly assumes thatI

the interactions among the constituents ofthe res-

which is a precise statement ofthe Clausius pnnci- ervoir do not cause correlations to be established be-

pie ~. Observe that the fl~are the initial temper- tween the system and the as yet unencountered con-

atures ofthe systems with which system a interacts, stituents ofthe reservoir. Stated simply, the argumentso that the theorem expressed in (2) is in no way essentially relies on the infinite size ofthe reser-

limited to processes where temperature is defined for voir to guarantee that it will not be corrupted by a

system a, as often stated in textbooks. few interactions with the system. Clearly, if the

Thus far we have established the two laws ofther- constituents of the ideal reservoir are required to be

modynamics which are statistical in nature. To- sufficiently weakly interacting, the problem will not

gether with conservation ofenergy and the existence arise. It canbe shown, however, that even in the case

of a unique ground state (certainly true ofsystems ofa stronglyinteracting reservoir, the correlation en-

with a few degrees offreedom), they constitute the tropy ~ of the system relative to a constituent mem-

laws ofthermodynamics. The remaining task is to ber ofthe reservoir is on the average no greater than

show how and why a system not in equilibrium can (EsS f lEsU) /N, where Nis the number ofthe con-approach equilibrium (i.e., tend toward a Gibbs stituents in the reservoir, and EsS and EsU referto the

state). system as before. Clearly, forlargeNthe correlationsApproachto equilibrium is a resultofinteractions, in question are vanishingly small.

since a closed system can only evolve into unitarily To see how these ideas work in practice, we shall

1 1 3 Note that in the absence ofchanges in the Hamiltonian ofthe ~ The difference between thesum oftheindividual entropiesof

system (which would give rise to exchange ofwork), IsU is thetwo systems and theentropyoftheir combination is known

justwhat we would identifyastheheatabsorbedby thesystem. as their correlation entropy.

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Volume 137, number 9 PHYSICS LETTERSA 5 June 1989

consider an example ofa system interacting with a As mentioned above, we shall takeheat reservoir in some detail. To simplify matters as p~,~(p)=N(flb/21tmb)

312exp(flbp2/2mb)

much as possible (without making the example

unrealistic), we shall consider a particle (e.g.,a mol- corresponding to a reservoir at temperature fl~.Forecule) of mass ma, initially in the state ~ inter- the system, we shall take

acting with a reservoir composed of Nparticles of pi~j~)(2it).2) 3/2 exp[ (pp0)

2/2A2]

mass mt, in the Gibbs state /5~=Zj

xexp ( f i t , At,). All particles are confined to a vol- and consider two limiting cases. For case I, we takeume Vwith perfectly reflecting walls, withNand V Po= 0, corresponding to a system initially in equilib-

sufficiently large and N/Vsuitably small, and they riumat temperaturefly =22/me, and for case II, wewill be taken to be without internal structure and consider the opposite limit of p~>>A2,p~>>m~

moving nonrelativistically. The problem is to con- mt,/lt,, corresponding to a system with a narrowly de-

sider a collision between the system and the particles fined momentum P o , withp0/m11 much larger than

of the reservoir, and to calculate the change the rms speed ofthe reservoir particles. With this in-E s (Sfit, U) that results for the system. formation substituted in eq. (3) and the integrals

The initial state for the combined system is given carried out, we finally arrive at the following expres-

by f(i) =/3~1~3~i), and the final state by sions for the two cases I and II:1/2~ where ~ is the S-matrix for the col- 2~2+2~ \ 18)3N

lision. The final state ofthe system is then given by ~i~i, (392(1 +p)~) i7~ (4)~ =trb ~(, and in general it depends on the de-tails ofthe initial states. However,just as in collision

sf~t,U

1I 3p9(l+1u)+3u

2(l9)2theory, one can proceed to the (improper) limit of 9(9-I- u)(l +11)2

plane waves and still extract the desired informa-

2Ntion. In that limit, the density matrices are diagonal X (va/ti) ~v0, (5)

in the momentum representation, e.g.,

= (pp )p~I) (p), etc. Moreover, the relevant where 9 flt,/fla ( flbA2/ma) for case I (II), p=ma/

quantity tn this limit is A, the rate ofchange ofthe m~,v= (3/mbflb)I /2 is the rms velocity ofthe par-state ofthe system; cf. the derivation of transition tidesofthe reservoir, and v

0 =p0/m is the velocity

probabilities in collision theory. Once A is obtained, ofthe system in case II. Moreover,

the rate ofchange ofSfl,, U can be calculated fromit as an expectation value.

~(v)= jdss5exp(_s2/2)Whenthe steps outlined above are carried out one

finds, after some rearrangement,

xJdQe(l_coso)If(sv,O)12,~ ~ ~=JdQo(l_cosO)If(po/(l+p),e)I2,

xIf(q, 0) J2p (p+q)p,~,~[(ab/aa)pq]

wherex{ln[p~(p+q)] +fl,,(,p+q)2/2m

5= [~u(1u+)/3(l +p)

2]2m,,ti. ln [p~) (p+q )] fit,(p+q )2/2m

11} , (3)Note that a and ffare basically weighted cross sec-

wherefis the scattering amplitude, q and q are the tions, favored by large-angle scattering as would beinitial and final momenta ofeach particlein the cen- expected physically. Notice also that the bracketed

ter-of-mass frame, cos 0=qq/q2, and aa,b=ma,b/ combinations involving these cross sections in eqs.

(ma+mt,). Note that eq. (3) gives the steady-state (4) and (5) are roughly of the order ofthe collisionrate ofchange ofSAUat a momentwhen the states frequencies for the two cases.

ofthe system and reservoir particles are specified by To interpret eqs. (4) and (5), it is useful to first~ and ~ respectively, recall the underlying physical situation, say for case

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Volume 137, num ber 9 PHYSICS LETTERS A 5 June 1989

II: particles of mass m11 (the system) are prepared in point out here that eq. (1) is perfectly consistent with

the state ~~i) (p) (this may be accomplished by pre- time reversal invariance, as are eqs. (4) and (5) al

paring a sufficiently dilute gas of them in equilib- though their time reversed versions would correrium at temperature )~

2/mawithin an enclosure that spond to extremely improbable situations. One such

is moving with velocity P a /ma), and are subse- situation would in fact occur ifthe system is conquently allowed to enter the reservoir and interact fined to the enclosure ofvolume V(with perfectly

with it for a time E st, after which they are collected reflecting walls, not real walls made ofatoms) and

and momentum analyzed (assuming this to be pos- forced to interact with the reservoir for a sufficien

sible in practice). Then the changes ASand EsU are length oftime (expectedto be longer than the age o

calculated and the quantity (EsS fl,4U)/E st is corn- the universe for a macroscopic reservoir); the syspared to the prediction of eq. (5) for sufficiently tern would then reverse its approach to equilibrium.

small E s t . With this picture in mind, first note that It is clear that the practical impossibility ofisolating

eqs. (4) and (5) have been organized as products of macroscopic systems from external influence prethree factors, as in F,F

2F3. In both cases I and II, F, vents such manifestations ofreversibility ~.vanishes as p= m a/m b* O, reflecting the factthat an In this paper we have only dealt with microscopic

infinitely massive reservoir particle becomes a per- systems. However, we expect no major modification

fectly reflecting wall, and will not drive the system in extending the present treatment to the macro-

toward equilibrium. Note also that 9-+0 represents scopic domain. Indeed as it may be verified, the ara (relatively) very cold system in case I, and a very guments leading to eq. (1) and the Clausius prim-

sharply collimated system incase II, and that in both ciple basically hold for systems ofany size. It may

cases F~behaves like 9, indicating a very large m i- also be recalled that the results given in eqs. (4) andtial rate of approach toward equilibrium. On the (5) remain unchanged in the classical limit, mdi-

other hand 8 1100 represents a (relatively) very cold dating that the second law is not a specifically quan

reservoir, wherefore F1F2 behaves like 93~2in case turn mechanical result.

I, again indicating a very large value forSfl,, U. By

contrast, as 9i~1 the rateofapproachto equilibrium This work was supported by the National Sciencevanishes for case I, confirming the fact that two sys- Foundation under Grant No. PHY-85 13367 and by

terns in equilibrium at the same temperature will not a grant from CaliforniaState University, Sacramento.

change upon interaction. The factor (v0/IY)2 which

is stipulated tobe much larger than unity for case II However, see ref. [8], and ref. [61for a discussion ofspin-echo experiments.on the other hand shows that a system moving very

fast through the reservoir will rapidly lose energy and

move toward equilibrium. The third factor, F3, de-

pends on the systemreservoir interaction potential References

and has the structure of a weighted collision fre-quency, as remarked earlier. Remarkably, in spite of [11M.H. Partovi, Phys. Rev.Lett. 50 (1983)1883.

their quantum mechanical derivation, eqs. (4) and [2] R. Blankenbecler and M.H. Partovi, Phys. Rev. Lett. 54(1985) 373.

(5) do not involve Plancks constant. Finally, note [3] M.H. Partovi and R. Blankenbecler, Phys. Rev. Lett. 57

that the quantity S fi t, U is manifestly positive for (1986) 2887, 2891.both equilibrium (case I) and nonequilibrium (case [4] 0. Penrose, Rep. Prog. Phys. 42 (1979) 1937.II) situations, in accordance with eq. (1) and the [5] A. Wehrl, Rev. Mod. Phys. 50 (1978) 221.

[6]J.M. Blatt, Prog. Theor. Phys. 22 (1959) 745.second law ofthermodynamics. [7] P.G. Bergmann and J.L. Lebowitz, Phys. Rev. 99 (1955) 578;As already remarked, the idea that irreversible be- J.L. Lebowitz and P.G. Bergmann, Ann. Phys. (NY) 1

havior is associated with openness has been exten- (1955) 1.

sively discussed in the literature [6,4]. Suffice it to [8] E.L. Hahn, Phys. Rev. 80 (1950) 580.

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Volume 137, number 9 PHYSICSLETTERS A 5 June 1989

IRREVERSIBILITY, REDUCTION, AND ENTROPY INCREASEIN QUANTUM MEASUREMENTS

M. Hossein PARTOVIDepartmentofPhysics, California State University, Sacramento, CA 95819, USA

Received 28 February 1989; accepted for publication 6 April 1989

Communicatedby J.P. Vigier

Using the strong subadditivity property ofentropy, it is shownthat the interaction ofthe measuring device with the environ-ment brings aboutthe reduction processcharacteristic ofa quantum measurement. This entails an entropy increase whichhasan

inviolable lower limit. This limit is found and explicitly calculated for illustrative examples.

There are two long standing unresolved problems scopic systems cannot be prevented from interactionin the foundations ofphysics, themeasurementprob- with their environment, and that the loss ofinfor-lem of quantum mechanics and the reversibility mation which is at the heart ofboth the reduction

problem ofstatistical mechanics, with notable par- process in quantum mechanics and the irreversible

allels between them. Each dates back to the incep- entropy increase in statistical mechanics is simply ation ofthe corresponding field, the latter to the in- consequence ofthis unavoidablecoupling to the largetroduction ofBoltzmanns H-theorem and the former number ofdegrees offreedom ofthe (unobserved)to the advent ofmodern quantum theory. Both have environment. The ideas ofopennes and interactionbeen the subject ofintense debate and activity ever with the environmentare not new andoccur in manysince [1,2], resulting in a vast and often bewildering writings, although only a few authors have seriouslyliterature which, among other things, includes a con- pursued the consequences. Among these, the contri-siderable amount ofqualitative, nearly metaphysi- butions of Bergmann, Lebowitz, and Blatt in the

cal, discourse. Both problems are largely ignored by context ofstatisticalmechanics [3], andthose ofZeh,most physicists since there is in each case a more or Zurek, and Peres in the context ofmeasurement the-less standard formalism which adequately accounts ory [461should be mentioned.for the practical aspects of the problem. These are In the preceding Letter [7] we have shown in a

the reduction postulate for quantum mechanics and rigorous manner how the second law of thermody-the ergodic hypothesisor the extremum principles of namics, irreversibility, and approach to equilibrium

ensemble theory for statisticalmechanics. However, for microscopic systems follow from the laws ofthese hypotheses are additional to the known laws of quantum mechanics. The purpose ofthis paper is todynamics, serving only to explain the one problem present a rigorous prooffor the reduction process as

they are dedicated to, rather like the aether ofpre- a consequence ofinteraction with the environment.relativity electrodynamics. Naturally, many physi- Furthermore, it will be shown that a given quantum

cists have found this state ofaffairs unsatisfactory, measurement necessarily entails an entropy increase

seeking instead to find the underlying physical which has an absolute lower bound ~. The analysismechanism responsible in each case and from that follows methods and ideas introduced in a series ofa quantitative resolution in terms ofthe known laws papers [10] concerned with the foundations ofofdynamics. quantum mechanics, with entropy asthe central con-

These efforts now appear to have reached a suc-

cessful conclusion on the recognition that macro- ~ This was first considered by Szilard [8], see also ref. [9].

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cept. Together with the preceding Letter [7], the generality, one can represent ~ as an assembly ofNpresent analysis provides a unified and rigorous res- independent counters C

1 , one for each bin a1, so

olution ofthe irreversibility and measurement prob- that the state ofthe measuring device can be rep-lems at the microscopic level, resented by the product density matrix r = ~, f, ~.

In the followingwe shall use the notation and def- Now for ~ to function as a measuring device, each

initions introduced in ref. [101. The state ofquan- counter C, must possess a pair of(in general, mixed)turn system is in general represented by a density states t~,to be referred to as thefiredandquiescentmatrix /3 corresponding to a mixture. A measure- states respectively, withthe following properties: (a)ment is a process whereby a fraction ofthe members t,-~are stable against interactions with the environ-of a similarly produced ensemble is subjected to in- ment, (b) a quantum system with its value for the

teraction with a measuring device ~ designed to observableA known to be in the interval a,will, uponmeasure a physical observableA. Implicit in the con- interaction with the counter C 1, cause the latter to

struction of ~ is a partition ofthe spectrum ofthe evolve from r1 tor~,i.e., will cause it to fire, andoperatorA (which represents the observableA) into (c) the states f,~are macroscopically distinguish-a (necessarily finite) number ofsubsets a,, i= 1, 2, able, i.e., stable against, and resolvable by means of,

N. With no loss in generality, one can take these macroscopic observation processes (typically in-

to be disjoint intervals, or bins, with the correspond- volving an exchange ofa large number ofquanta).

ing projection operators ft, and the associated reso- The last condition in particular implies the orthog-lution ofthe identity >~ ,f t , =1. The end result ofthe onality ofthe pair ofcounter states, namely the re-measurement is a set ofrelative frequencies,f, show- quirement that tr f~=0, which is a far weakering how often the value of the observable A turned condition than macroscopic distinguishability. Con-up in the interval a1. Thesefrequencies then serve to dition (a), on the other hand, simply states that two

approximate the corresponding probabilities ,~, (mixed) states ofthe counter are in equilibrium withwhich are in turn given by the set ofexpectation val- the environment and will notundergo a macroscopic

ues tr/3 ft 1 . In addition to the quantum system and change ofstate as a consequence of interaction with

the measuringdevice there is a third system that plays it. Condition (b) simply characterizes the C asa crucial role in the process ofmeasurement, namely counters that detect and fire exactly as required.the environment, or that part ofthe rest ofthe uni- While the measuring device is in interaction withverse which, without being directly observed, will the environment throughout the measurement pro-

unavoidably interact with the device while the mea- cess, it is convenient to consider (following vonsurement is in progress [46].Often the inessential Neumann) two distinct stages. During the first ofparts ofthe measuring device, that is the parts which these the quantum system interacts with the meas-do not directly interact with the quantum system uring device, with the environmentas a spectator ~.

being measured, play the role ofthe environment. Then in the second stage all systemdevice interac-While the experimental arrangement is designed to tions cease while the device is once again subject to

isolate the quantum system from such spuriousin- interactionwith the environment. The first stage es-teractions with the environment, the measuring de- tablishes systemdevice correlations that serve tovice cannot be so isolated on account ofits large in- yield the desired information about the system. An

teraction cross section with the environment interaction Hamiltonian that can accomplish this is(resultingfrom its enormous number ofmicroscopic ~, i v , j2~,where V~operates on the Hilbert space ofconstituents); herein lies the reason why the meas- counter C only. Thus the systemdevice Hamilto

uring device acts as a classical system. nian appears asThenext step isto describe the dynamical arrange-

ment for the measurement process. The measuring i2 This can be realized by spatially separating the subensembles

device ~ is in general an enormously complex sys- corresponding to different binsand directing them toward thecorresponding counter. Examples are the SternGerlach cx-

tern on the quantum scale. However, certain aspects periment, the momentum analyzer andscattering experiments.

of this complexity are irrelevant to the measurement ~ This simplifying assumption does not change the physics inprocess and may be ignored. Thus with no loss in anysignificantway.

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ofcorrelations. Intuitively, it is reasonable that theH=R~+~ 0, +~ i v , ~ deviceenvironment interaction will tend to degrade

the systemdevice correlations present in ~(T) 56where0. and0, are the self-Hamiltoniansofthe sys- However, to establish this result in general, we needtern and the counter C, respectively, with the com- to use a highly nontrivial property ofentropy knownmutator [ft

1,0~Jvanishing~.

as strong subadditivity.Theevolution operator which results from (1) has Consider three systems A, B, and C, initially in the

the following form in the interaction picture: state /3ABC(0) =/3AB (0 )/3c(0). The notation implies

U t ( t)=~ ft~i2 J (1) , (2) that systems A and B are initially correlated whilei neither is initially correlated with C. Startingat t=0,

where J (t)=exp [ i(R~+t~)t] is the (interaction systems B and C interact fora time t,withA as spec-picture) evolution operator that acts on counter c tator, culminating in the state PABC(t). The partialand, for sufficiently large t, say t~ T, causes it to fire. (or reduced) state I3AB ( t) is defined as usual byThus z 2 J ( T)T~-i~( T)=[7. It is also necessary that /3AB( t)=trc /3ABC(t), wherethe trace is taken withre- J ( T)tT be traceless and orthogonal to t;, i.e., that spect to the Hilbert space ofsystem C; other partialtr J ( T)t; =trt~~( T)t~=0 ~ states are defined analogously. Next consider the

The dynamics of the first stage can now be de- (BoltzmannGibbsShannon) entropy S, defined forscribed. The quantum system is initially in the state any state / 3 by the von Neumann formula S=

tr/ 3 ln/3. Now strong subadditivity ~ states that/ 3 while the counters are all in the quiescent states

t~the combined state is then described by ~= SABC+58 ~SAB +SBc, (4)/3H,t~.Under the action of (I), ~evolves to~(T)=U(T)~U~(T)by the end ofthe first stage. where SABC= tr/3ABc lfl/3ABc, SB= tr /3B ln /58. etc.

Using (2), one arrives at Note that strong subadditivity implies subadditivity

(i.e., the inequality SAB~SA+SB) for a pair ofsys-~(T)=fl ft

1j3ft,t,~fJ t7 tems A and B. Moreover, subadditivity makes it pos-

sible to define the correlation entropy CA B as the dif-ii k~,.J ferenceSA+SB

5AB Note that CA B~ 0, withequality+~ft

1i3ft~j2~(T)t1trj~t(T)~ (3) holding only if/5AB=jSA/SB, i.e., only ifthe two sys-1 , 1 k

tems are uncorrelated.

The desired correlations are evident in the first Consider applying (4) to /3ABC (t). The result is(diagonal) sum ofeq. (3), while the second (off-di- SABc(t)+SB(t)~SAB(t)+SBc(t). Since /SABc(t) isagonal) sum consists ofcorrelations which must be unitarily related to /

5ABC (0), it follows that

stripped away before the final results can be ob- SABc ( t)=SABc(0). On the other hand, becausetamed by means of macroscopic observations car- /3ABC (0) =/3AB(0 )/3~(0), it follows that SABC (0)=ried out on the measuring device. This selective re- SAB (0) +Sc(0). Furthermore, the factthat system A

moval ofunwanted correlations, or reduction ofthe is a spectator while B and C interact implies the re-systemdevice density matrix, constitutes the sec- lations SA (t)=SA (0) and SBC (t)=SBC (0). The lackond stage ofmeasurement and is the essence ofthe ofcorrelations between systems B and C at time t=0,

measurement problem. To show that the deviceen- on the other hand, implies that SBC( 0) =

vironment interactions bring about this reduction,

we must first establish a general result on the decay 56 Model calculations confirm this expectation and provide an

estimate ofthe time scale involved; see ref. [9].

~ The condition [*, 1? ] =0 ensures that the measurement does ~ AccordingtoWehrl [2], D. Robinson and D. Ruelle (1967)not change thevalue ofthe observableA.Pauli referred to this first recognized the importance ofstrong subadditivity andkind ofmeasurement as thefirst kind; see ref. [11]. proved it for the classical case. 0. Lanford and D. Robinson

~ This condition together with the previously stated ones are (1968) then conjectured it for the quantum case. Aftersome

generalizations ofthose commonly assumed for the idealized partiallysuccessfulattempts, theconjecture was finally proven

case ofpure system and device states, and in fact they are by E. Lieb and M. Ruskai (1973); see ref. [2] for references

weaker than the latter (see ref. [1]). to theoriginal papers.

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SB(O) +Sc(O). Combining all these relations, onearrives at the important inequality CAB(t) ~ ~R +~

I,]

vi CA B (0). Thus strong subadditivity implies, amongother things, that the correlation entropy ofa pair of where1~,,which only involvesthe counter variables,

evolves and changes under the action ofthe envisystems in general decreases when one ofthem in-teracts with a third system (which it is initially un- ronment. By contrast, tr f,~remains aconstant, sincecorrelated with either one ofthe pair) ~ Further- it is an invariant quantity under the unitary evolu

more, ifthis process is repeated a large number of tion of the deviceenvironment complex. This tracetimes (each time with a new, uncorrelated system), can therefore be calculated from eq. (3), at t=T,the pair will be driven toward a state ofminimum whereupon it is found to be equal to zero. On thecorrelation entropy consistent with the prevailing other hand, since tr ft,,5*~=c 5 , ) tr iv

1 /3 (no sum on i)conditions. This establishes the intuitively reasona- while f,~is nonvanishingonly when i ~j, it followsble result concerning the decay ofcorrelations. that the trace ofthe second contribution to ~with

The conditions just stated are precisely those of respect to the system variables also equals zero. This

the quantum system (A), the device (B) and the en- result together withthe vanishingoftrf,~shows thatthe two partial traces of~are the same as those of

vironment (C) during the second stage ofmeasure- ~. Thus they are found to be equal toment as the device is subjectedto a large number of

interactions with various uncorrelated constituents /3 =trD~=~of the environment. Thus the final systemdevice

state QR evolves from ~(T)under the conditions that and(i) the counter states t~do not change (the sta- ~,,. Ibility condition (a) stated above) and (ii) the sys- P =tr5 ~=~ ~ fl tj,terndevice correlation entropy is a minimum. Re-

ferring to eq. (3), we note that the first (diagonal) where, it will be recalled, ~ =tr ft 1 /3 . Note thatj 5 andsum will not undergo any change during the reduc- P stay fixed during the second stage.tion stage owing to the stability of the counter states With the partial traces thus fixed, the systemdet~.The second (off-diagonal) sum, on the other vice correlation entropy C~Dcan now be calculated;hand, involves the combinations ~(T)t~(and its

CSD=s(/3~)+s(r)s(S~). (6 )adjoint) which are traceless and orthogonal to the

stable states. Under these conditions it follows (see Clearly, CSD is minimized when S(~) is maxibelow) that the minimum correlation entropy state mized, since the other two entropies in eq. (6) staycorresponds to the vanishing of the second sum in fixed during the second stage. The question then iseq. (3). In other words, the second stage reduces what choice off,~maximizesS(~)? To answer this~(T) to question, we can use a well-known property ofthe

BGS entropy: If & and ~ are density matrices, thenj,~i

=~ ft1/3ftit,~fl [j. (5) tr & ln vi tr& in ~.I I

Applying this inequality to ~and ~R, we findTo establish the resultjust stated, we first observe

tr~ln~vitr~ln~R=tr~RlnS~R, (7)that the most general form ofthe systemdevice stateduring the second state is where the last equality follows from the fact that

tr ( ~ ~R) in ~R vanishes (owing to the fact that~R is diagonal with respect to the ~t, whereas

6 8 This intuitively obvious result is a highly nontrivial theorem ~ ~ is off-diagonal).

that follows from strong subadditivity. It can be generalized Inequality (7) now shows that the conditionto the smmetric case where system Ainteracts with a (fourth) f~=0, or ~=~R, is the one that maximizessystem Dwhile Binteracts with C. Then CAB(t)~ CAB(O) fol-

lows from

5ABCO+SA+SB~ SAB+SBc+SAD, which in turn S(~).Thus the final systemdevice state is defollows from a repeated application ofstrong subadditivity. scribedby ~R;the environmenthas removed the un

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wanted, off-diagonal correlations. This completes the To illustrate the entropy increase associated with

proofofthe reduction process (or the collapse) dur- reduction, we shall considertwo examples. The firsting the second stage ofmeasurement. is the measurement ofa spin-i/2 system initially in

The subensemble consisting ofthe quantum sys- the state p= (1 +~p)/2by means ofa SternGer-

tems that cause the counter C, to fire is described by lach apparatus set up in the direction n ;& is the vec-/3 , =ftI/3ftI /tr ftj5ft

1. Since tr ft 1 1 S it , =.~, one can write tor formed ofPauli matricesandpis the polarization c - ~ ~[(i) vectorofthespinsystem;p=~pIvilandlnI=1.The

it corresponding spin-up and spin-down projectionop-erators are =(1 ~n)/2.Clearly, this is an ex-

where ample of a two-bin measurement. The entropy in-crease can now be calculated from eq. (8);

1 1 p2

i~S=ln 2describes the state ofthe device with counter C

1 fired 2 1 p,,

and all others quiescent. Clearly, a macroscopic 1 ____

measurement (or reading) ofthe counters, repeated + ln L ~ii) ~~ ] (9)a sufficiently large number oftimes, will yield the set

ofprobabilities,~.

where p,, =p n. SinceIp~vi I, vi 1, it is evident thatThe systemdevice evolution, ~r~.~(T) ~, is ~ 0, with equality holding only when p =p,,, i.e.,

unitary during the first stage and (in general) non- only ifthe system is initially polarized in the direc-unitary during the second stage. The entropy change tion is (including the trivial case of zero polariza-

for the process, which occurs during the second stage, tion). In such an event there are no off-diagonal ele-is given by zS.S=S(4~)S(~).Using the fact that ments to be removed during the reduction stage and

.3 1 , 1 c s c ,1 j5 1 (no sum on i), we find i~.S=0 as can be seen from eq. (9).As a second example consider a momentum mea-

S(~R)= ~ tr ~p[~ln ei~p,t~~. .surement (in one-dimension) by means ofan ana-lyzer whose channels (or bins) cover uniform, non-

This expression in turn simplifies to overlappingintervals ofwidthz\p centered atp~.Thus

S(~R)=S(t) trM/3~ln ~./3. the operator ir1 projects onto the interval

with S(t) equal to the initial entropy ofthe mea-suring device. The entropy change (per event mea- Let the system to be measured be ina Gaussian statesured) is then found to be 10 , ,u>_(/~)_I/2exp(p2/2j~2)Let ,(p) denote the restriction of0(p) to the in-

A reasoning similar to the one leading to eq. (7) terval a,. Then

will show that zS.S~0, reflecting the fact that the en- .~ I d I ~ ( ) 2vironment removes information from the system J ~device complex during the reduction process. On the andother hand, E~.S=0 is possible only when/ 5 is diag-

onal with respect to the f t1, i.e., when ft1j5ft~=0for ~i#1. This is so precisely because in this case there are The entropy increase (per event analyzed) is thenno off-diagonal elements to be removed. It is im- found to beportant to realize that E~Sis an inviolable lower limitwhich is far exceeded in actual experiments, since Li.S= ~ in ~, (10)

counters and other equipment used in such experi-ments are far from reversible, a quantity which was named measuremententropy

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in ref. [10]. Note that in this example the initial state the measurement. This three-way arrangement thusis pure (an idealization) and has zero entropy. makes it essentially impossible for those systemde

What are the limiting values ofLs Sfor ji > Ap? In the former case, nearly all systems will the stable states to exist as macroscopically observ

turn up in one ofthe channels, say channel k, so that able states. It is useful to recall here that this selec1 , ~ 0 for i ~ k, and 4S~0, exactly as ex- tivity in the states ofa system in interaction with an

pected. In the latter case, one has environment is a rather commonplace fact in thecontext ofthermodynamics and statistical mechanics.

~p ( ji~/~)exp( ~2/~2) As stated at the outset, the measurement and the

so that irreversibility problems are intimately related andcan be understood in terms ofknown, fundamental

~(3+ln n)+ln~u/~p), Cu>> i\p). (11) dynamics. We believethe present worktogether with

Thus the minimum entropy increase necessary for the preceding Letter provide a rigorous resolution ofcollapsing the wavepacket in this case grows indef- these problems at a fundamental, microscopic level.initely as the resolution ofthe momentum analyzer

is increased. This is ofcourse exactly as expected, This work was supported by the National Science

and clearly points to the impossibility ofproducing Foundation under Grant No. PHY-85 13367 and by

pure states by means of a finite preparation agrant from California State University, Sacramento.procedure.

Althoughwe have been careful to preserve the fea-tures essential to the reduction process in the de- Referencesscription of the measurement process, we havenevertheless introduced idealizations in describing [1] J.A. Wheeler and W.H. Zurek, eds., Quantum theory and

the measuring device. Forexample, the stability con- measurement (Princeton Univ. Press, Princeton, 1983);

ditions on r,~imply, unrealistically, that the two M. Jammer, The philosophyofquantum mechanics (Wiley,New Y o r k , 1 9 7 7 ) .

counter states are inperfect equilibrium with the en-[210.P e n r o s e , R e p . P r o g . P h y s . 4 2 ( 1 9 7 9 ) 1 9 3 7 ;vironment. In practice, one has a range of (mixed) A . W e h r l , R e v . Mod. Phys. 50 (1978) 221.

states close to t,~which are macroscopically equiv- [3] P.G. Bergmann and J.L. Lebowitz, Phys. Rev. 99 (1955)alent and among which the counter state will fluc- 5 7 8 ;

tuate under the influence of the environment. Under J.L. Lebowitz and P.G. Bergmann, Ann. Phys. (NY) I( 1 9 5 5 ) 1 ;typical conditions, the associated entropy fluctua- J.M. Blatt, Prog. Theor. Phys. 22 (1959) 745.tions are far greater than the quantity calculated in [4]H.D. Zeh, in: Foundations ofquantum mechanics, ed. B .

eq. (8) owing to the macroscopic nature of the d E s p a g n a t (Academic Press, New York, 1971).measuring device. This observation againshows how [5] W.H. Zurek, Phys. Rev. D 26 (1982) 1862.

difficult it would be to realize the theoretical mini- [6] A. Peres, Am. J. Phys. 54 (1986) 688.[7] M.H. Partovi, Phys. Lett.A 13 7 (1989) 440.

mum E~.Sin actual measurements.[8] L. Szilard,Z . Phys. 53 (1929) 840.

Why is it necessary to use a macroscopic device to [9] W.H. Zurek, in: Quantum optics, experimental gravitation

measure a microscopic system? A macroscopic sys- and measurement theory, eds. P. Meystre and MO. Scully

tern has a large number of microscopic constituents ( P l e n u m , New Y o r k , 1 9 8 3 ) .

and a large interaction cross section with its envi- [10] M.H. Partovi, Phys. Rev. Lett. 50 (1983)1883;R. Blankenbecler and M.H. Partovi, Phys. Rev. Lett. 54

ronment. This renders many ofits possible states ex- (1985) 373.

tremely improbable, hence unstable. Its stable states, [11] S. Fl{igge, ed., Handbuch der Physik, Vol. V (Springer,on the other hand, provide the pointer basis ~ for B e r l i n , 1 9 5 8 ) p . 7 3 .

~ Z u r e k h as a d v a n c e d a n d e m p h a s i z e d t h i s p o i n t o f v i e w , [5,9].

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Volume 151, number 8 PHYSICS LETTERS A 24 December 1990

Entropic formulation ofchaos for quantum dynamics

M. Hossein Partovi

DepartmentofPhysicsand Astronomy, California State University, Sacramento, CA 95819, USA

Received 20 August 1990; revised manuscript received 10 October 1990; accepted for publication 19 October 1990

Communicated by A.R. Bishop

A general formulation ofquantum chaos based on theasymptotic growth rate ofmeasurement entropy is presented and shown

to be closely related to theclassical formulation in terms ofLyapunov characteristic exponents and the KolmogorovSinai invar-

iant. Examples areused to illustrate theformalism.

Despite extensive activity in recent years on the behavior is sensitivity to initial conditions, a prop-existence or meaning ofchaos for quantum systems erty that for bounded systems gives rise to a strong

and a large number ofuseful results ~, there isas yet mixing behavior. Accordingly, for both quantum andno general formulation of the meaning of chaos for classical systems, one should expect chaoticbehavior

quantum dynamics, nor any indication ofits exis- to be associated with a steady increasein the degreetence in an actual physical system. Indeed there are ofindeterminacy inthe measuredvaluesofthe phys-good reasons to believe that the most common quan- ical observables. We shall see below that this is in-

tum dynamical models are essentially nonchaotic deed the case, and that in general the asymptotic rate

[2]. Nevertheless, the objective of formulating a at which measurement entropy increases is directlyuniversal definition of quantum chaos similar to relatedto the characteristic exponents associated withthose available for classical dynamical systems is an the dynamical system. In particular, we shall find that

important problem deserving of serious attention, in the case of classical dynamics, the present ap-Thepurpose ofthis Letter is to present a formulation proach parallels the ergodic formulation ofchaos in

ofquantum chaos using the concept ofmeasurement terms of Lyapunov characteristic exponents. Forentropy, a notion which is equally applicable to clas- quantum dynamics, we shall definechaotic behaviorsical and quantum dynamical systems. to signify a positive value for the above asymptotic

The physical idea underlying the present approach rate, and quantify the degree of stochasticity (or

originates in the realization that there is a natural chaoticity) for such systems by means ofan indexand powerful measure ofinformation in quantum which is the quantum version ofthe Kolmogorovmechanics which embodiesvirtually every statistical Sinai (KS) invariant. Illustrative toy models will then

aspect of the theory *2 This is the above-mentioned be used to demonstrate various aspects of the for-measurement entropy, a quantity which provides an malism, including the calculation of characteristicinformation-theoretic measure ofthe degree ofscat- indices, and to gain insight into the question ofcha-

ter or indeterminacy in the measured values of a otic behavior for an actual quantum system. A gen-physical observable. Now for classical dynamical eral quantumclassical correspondence showing the

systems, the most characteristic feature of chaotic (formal) reduction, in the classical limit, of thequantum indices to the corresponding classical ones

will be presented to demonstratethe correct classical~ Numerous papers in this andotherjournals have appeared in .

therecent past; for useful sources to the literaturesee ref. [1]. hmtt of the formalism.~ The development ofthe concept and its applications can be To start, we shall consider the general case ofa

found in ref. [3]. measurement carried out on a quantum system, and

0375-9601/90/s 03.50 1990 Elsevier Science Publishers B.V. (North-Holland) 389


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Volume 151. number 8 PHYSICS LETTERS A 24 December 1990

define the corresponding measurement entropy. In gressively refined by reducing the size of each bingeneral, the state ofa quantum system is described toward zero. Assuming the existence of the limit, weby a density matrix, denoted by /5 , and the results of arrive at ,~,a quantity that depends on the complex

the measurement of some observable A (a self-ad- {H, / 5 , A} but not on the particular device used tojoint operator) by a set of probabilities, each cor- measure A. Note the obvious but significant fact thatresponding to a group of possible values of

A .Let the if A, or / 5 , commutes with I - ? , then A~vanishes (bar-

experimental arrangement used to measure A be re- ring an explicit time dependence in A).ferred to as the measuring device and be denoted by Having carefully definedA~by means of a limitingD. Then in general D

4 entails a partitioning of the procedure, we now revert to a more compact (but

spectrum ofA into a collection of(non-overlapping) less rigorous) notation in terms ofinfinite partitionssubsets ~ct/} , which we shall call bins, and a corre- and continuous distributions to facilitate the presen-sponding orthogonal decomposition of the under- tation. Thus in place of ~ we introduce the prob-lying Hubert space given by {~} , where ~ repre- ability density ~ (a) which is related to / 5 by meanssents the projection operator corresponding to the of ~ ~ (a) da=tr [ftA (da)~3(t)], where ft4(da) is thebin a~.Clearly, we have u = spectrum ofA projection operator corresponding to the bin ct~ (a.~,,it,1 and ~=i. The results ofthe mea- a+da]. In case the spectrum of A is sufficientlysurement process are summarized in a set ofprob- smooth, one can also write ~ (a) =tr(aA)/5(t),

abilities {.~},where ~ is the probability that the and in place ofeq. (2) obtain

outcome of the measurement turned up in bin a~.According to the rules of quantum mechanics, ,~= ~-~-J da ~3,4(a)In ~~a) . (2)~A =tr ,4 I5 . where tr denotes the trace operation.

Among the various measures ofinformation, thequantity It is important to realize that the passage to the con-

- tinuous form (2) will in general lead to a divergentS(p ID4) = ~ ;?~;~In ~ Y , (1) result ifcarried out for the entropy S itself. We can

now define quantum chaos:A boundedquantum sys-

named measurement entropy, plays a fundamental tern ~H,/ 5 } is chaotic ifitpossesses an observable withrole [3]. I t i s in terms of this quantity that we shall a positive index.

definethe characteristic indices. Note that S is a pos- To establish the basis for this definition and showitive number, vanishing only if all but one of the the afore-mentioned relation to the characteristic cx-~A

} vanish, andsaturating

at the value ln(number ponents for classical mechanics, we consider a gen-ofbins) if all ~ are equal. Consider next a series eral dynamical system whose evolution is given byofmeasurements carried out by DA at times 1,,, with a set ofMfirst-order differential equations ~ =F~(~measurement results ~ (t,,), and define the index 1), where the vector ~= (~)represents the dynami-

cal variables. Let the trajectories ofthis system be= lim

7-S(p~ lD~), (2) given by ~1(t)=R~(1I~0), corresponding to the initialconditions ~,(t0) =~,. We will assume the R 1 to be

where/ 5 , , =/5(t,,) represents the state ofthe system at differentiableand invertible, so that the equality ~=~j

time t,,. As usual, / 5(t)= U ( I) / 5U~(t), where the evo- at time I implies ~= ~ at the earlier time 1 0 (a validlutionoperator Uisdefinedby (i/t)U(t)_fiU(t); assumption for most systems of interest). Next wei? is the Hamiltonian operator for the system (11=1 consider the probability density function Q(~,t)throughout). representing a distribution in the phase space ofthe

Before proceeding to the interpretation ofthe in- system. The evolution of Q is given bydex, let us note that the effect ofrefining the parti-tion {a~}is, in general, to increase S(/5j~),hence Q(~,t) J d~0 M)( R(tI~))Q(~0, t0),to increase the value of the index in eq. (2). It istherefore advantageous to considerthe supremum of an obvious relation. From the full distribution func-the limit in eq. (2) as the partition {a~} is pro- tion Q, one can derive probability distributions for

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various dynamical variables. For example, the prob- therefore expect that in the limit t~oo,the contri-ability density function ~?1~(a) for the dynamical bution ofthe logarithm to A~is equal to ~~~Q.maxt,variable A(~,t) is given by so that A~ AQmax. In other words, the asymptotic

S dM~Q(~t) growth rate ofthe measurement entropy ofa dynam-(aA(~1)) . ical variable is in general equal to the largestchar-acteristic exponent associated with the phase-space

The index ~ is now defined from ~ (a) ac- region that supports Q (under certain conditions,

cording to eq. (2), and can be expressed in terms characteristic exponents can be associated with in-of the trajectories by using the evolution equation decomposable regions in phase space; see ref. [4]).

for Q. After a few straightforward steps, one arrives Higher-dimensional indices are similarly defined

at in terms of the joint measurement entropy of a set

= urn -~-5 ~ Q(i~,t~)ln( ~ dM ofdynamical variables. Thus associated with n in-,~oQ(~,t~) dependent variablesA,B,...,Z, n~


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Volume 151. number 8 PHYSICS LETTERS A 24 December 1990

must await detailed information on the existence and statement that Q(q, p, I)=Q(q0, P o 0) togetherwith

nature of the spectrum of characteristic exponents the condition that the phase point (q0, Po) at timefor quantum dynamical systems, we are in a position zero evolves to (q, p) at time t. This statement, byto define a global measure of chaos for quantum sys- way of Liouvilles theorem, is equivalent to time

terns that is the analog of the KS invariant of clas- evolution according to classical dynamics.

sical dynamics: For a bounded quantum system, the In recent years much has been learned about thestochasticity index is equal to the supremum ofthe set spectral characteristics ofquantum systems that are

ofindices {)~A.B Z}~A system is chaoticifiihas apos- known to be chaotic classically, and examples haveitive stochasticity index, been constructed of open systems that obey non-

Before proceeding to examples, we will outline the Harniltonian evolution and exhibit chaotic excur-

derivation ofa quantumclassical correspondence in sions in some dynamical variable [11. However, asthe h~0limit between the quantum andclassical in- mentioned at the outset, there is as yet no known cx-dices defined above. It is important to realize that ample of a realistic quantum system that exhibits

this can only be aformal correspondence, relying as chaos in an unambiguous way. Consequently, the ex-it does on the interchangeability ofthe t~cx limit in amples considered below are only illustrative models,eq. (2) on the one hand, and the classical limit 1 5 -~0 although they are Hamiltonian systems and they doon the other. Although there are good reasons to be- illustrate the workings of the formalism.

lieve that the two limits in general fail to commute, The first example will test whether the formalismit is still important to establish the correspondence works for linear systems, and whether it correctly

for those cases where they do commute. Now what produces a positive characteristic index at a point of

we wish to show is the reduction ,~~ in the clas- linear instability. A convenient example is the har-sical limit. This requires (a) a physically meaningful monic oscillator, defined by ii=15

2/2m+k.*2/2. For

way ofrelating/5 and Q for a given system, and (b) / 5 , we choose the pure state ~i> 0, which corre-latter in fact reduces to the classical law ofevolution. sponds tox=0 being a stable equilibrium point (k=0The main quantum mechanical object in these equa- describes a free particle, also with a vanishing in-

tions is the evolution operator U introduced earlier. dex). The more interesting case, however, is that ofThe essential ingredient in the derivation is the re- k


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Volume 151, number 8 PHYSICSLETTERS A 24 December 1990

To construct it , we start with the differential equa- ...). Once again one can use eq. (2) to calculatetion 0=wsin 0 it~ ~ ~ it, where the endpointsit 2~for some choice of ~3;the result is w for both in-are identified and w is a positive constant. The so- dices,just as in the classical case. Ifthis were an ac-lution is ceptable quantum system, we would identify w as the

stochasticity index. However, this system is un-O(t)=2arctan[tan(

0/2)exp(wt)] ,bounded in momentum, both classically and quan-

where O~=(0).It has two fixed points, the first, at turn mechanically, and cannot properly be consid-

0=0, is unstable, and the second, at 0=it, is sta- ered a chaotic system. Nonetheless, it is an exampleble. The stable point has a basin ofattraction which ofa spatially bounded quantum system that exhibitsconsists ofthe entire 0-space minus 0=0. permanent diffusion in momentum space, albeit a

Next we develop a prescription for quantizing the diffusion that is accompanied by unbounded growth.

M-dimensional classical dynamical system ~=F(~, Note also that the spectrum ofI? consists of the en-t) considered earlier. To each ~, we assign a con- tire set of real numbers, continuous and unbounded

jugate momentum y, and define the Hamiltonian by from below, a fact that explains the apparent con-H( ~, y, t)=~,LL1F,.The equations ofmotion for ~ re- tradictions with the quasiperiodicity results found

produce the original dynamical system, while those by Hogg and Huberman [2].

for y, constitute an additional set ofMfirst-order The equality of the classical and quantum indices

equations not present originally. The resulting Ham- in the above models isthe exception rather than theiltonian system can now be quantized canonically rule, and can be traced to the fact that the semiclas-

(subject to the well-known hazards ofthe ordering sical approximation to the corresponding path in-problem). tegrals is actually an exact result for these cases. In-

Following the procedurejust outlined, we first ar- deed as emphasized above, the formal equality in the

rive at the classical Hamiltonian H(, p)=Wpsin, h~0limit will fail in general, since the various lim-where p is the momentum conjugate to 0. Using the iting processes involved may not commute. Thus aconstancy of H, we can then verify that p ( t)= classically chaotic system may well produce a zero

p0sin00/sin 0(t). The new fixed points are (0, 0) stochasticity index when quantized.and (it, 0), both unstable; the rest of the phase The formulation presented in this paper relies onspace is attracted to (it,cx), so that for almost the structuralsimilarity ofclassical andquantum dy-

all initial conditions, p(t) eventually tends to cx. namics, and on the key role measurement entropy

At this point one can calculate the indices ~~P; the plays in all statistical aspects ofdynamical measure-result is that for any Q(0, P o , 0) which is well-be- ments. It provides a general and rigorous criterion

haved at 0~=0,both indices are equal to w. Thus w for quantum chaos, one that applies equally natu-is the largest characteristic exponent, a fact that can rally to classical and quantum dynamics. The en-be verified by means ofthe matrix ofpartial deny- tropic formulation thus produces a faithful quantum

atives 8(0, p)/0(Oo, Po) mentioned earlier, implementation ofthe standard classical definition.

The quantized version ofthe above system is an The reason for this strict adherence to the standard

unusual rotor defined by the Hamiltonian definition is that the concepts of randomness and~w(p sin ~+ sin ~j3), with [0, jS ] =i (j S is the unpredictability that underlie the notion of chaos

usual angular momentum operator). The evolution concern the measured values ofthe observables ofa

operator U= exp ( i/it) can be calculated explicitly system, not the particular dynamics that producesfor this example; it is given by = them. Whether this standard notion ofchaos is ofC~~[O()],where C=(coshwt+cossinhWt)12 any use or relevance when dealing with quantum

and 0(0) is the angle 2 arctan [tan (0/2) exp ( Wt)]. phenomena is a separate and as yet unansweredThese equations show that the 0-space wave func- question. As already noted, most common quantum

tion evolves by steadily piling up at 0=it while dynamical models are quasiperiodic and thereforethinning everywhere else, and correspondingly the nonchaotic, and it appears likely that a continuous

momentum wavefunction continually spreads out energy spectrum is a necessary condition for a non-overthe spectrum ofj S (which consists of0,1 ,2, zero stochasticity index. Time-dependent Hamilto-

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nians and unorthodox models such as the one intro- G . boss, R.H.G. Helleman and R. Stora, eds., Chaoticduced in this paper appear to be fertile searching behavior in deterministic systems (North-Holland,

grounds for chaotic behavior, although it is already Amsterdam, 1983);B. Eckhardt, Phys. Rep. 16 3 (1988) 205.

clear that chaos is not the omnipresent phenomenon [2] T. Hogg and B.A. Huberman, Phys. Rev. A 28 (1983) 22;

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