RSV-free formulation of quantum mondemolition theory
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Volume 92A, number 1 PHYSICS LET1’ERS 18 October 1982 RSV-FREE FORMULATION OF QUANTUM NONDEMOLITION ThEORY Robert LYNCH Physics Department, University of Petroleum and Minerals, Dhahran, Saudi Arabia Received 25 May 1982 The entire validity of the “quantum nondemolition” (QND) concept has been called into question because of its deep reli~nce on “reduction of the state vector” (RSV) in the detailed development of the theory. In this letter QND theory is reformulated without use of RSV, except as found in the overall interpretation of the wave function. The recently developed “quantum nondemoli- The problem is to make a set of measurements on the tion” (QND) theory [1] is probably one of the more system to determine the (extremely weak) force F(t). exciting advances in quantum measurement theory in One introduces two new variables recent years. A theory of great generality and power, ~ cos wt — (a/mw) sin wt, (2) resting firmly on the axioms of quantum theory, it is nonetheless marred by a heavy use of “reduction of ~2 ~ sin wt + (p/mw) cos wt. (3) the state vector” (RSV) in its detailed elaboration. In fact, Vager [2] has recently published a critique of In general initially the system is in a superposition the theory which concludes that use of RSV in the state * of ~ 1 (in the Schrodinger picture) QND theory causes “... a serious inconsistency which makes its conclusions invalid” [1]. i~’(t0)) = ~ c11 ~ (t0)>, (4) The author of the present paper has published [3] a rebuttal to Vager’s critique based on a point-to- where ~1(t0) is one of the eigenvalues of X1. (We point examination of Vager’s arguments in the frame- choose to work with X1 we could just as easily use work of general quantum theory. However, this rebut- tal might not convince a disinterested observer, since ~2) An initial ideal measurement of ~ is made, and the resulting RSV projects the system into an eigen- she may view the disagreement on the issue to be due state of X~, say, I~m(to)~ Which state will result can- to a lack of universal agreement on the interpretation not be predicted with certainty, all that can be said is of quantum theory, particularly quantum measure- ‘2 that this state will appear with probability cm I ment theory. After this initial measurement n further measure- It is the purpose of this letter to reformulate QND ments ofX1 are made, at times t1, t2, ..., t~. As is theory without use of RSV (except as found in the shown in ref. [1] the system remains in an eigenstate overall interpretation of the wave function). Hence use of RSV in the QND theory is shown to be a ques- of~1, with eigenvalue tion of usefulness rather than necessity. Before this is done let us briefly outline the ele- *1 Since~1 has a continuous spectrum eq. (4) should be ments of the QND theory as it pertains to this paper. written with an integral rather than a summation, proba- The system of interest is a forced linear harmonic biities become densities, etc. However, for ease of nota- oscillator with ham iltonian tion the theory will be developed as though 15 has a dis- crete spectrum, as this makes no difference to the argu- ft=p 2/2m+~mw2.e2—.~tF(t). (1) ment. 0 031-9l63/82/0000—0000/$02.75 © 1982 North-Holland 9
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Transcript of RSV-free formulation of quantum mondemolition theory
Volume92A, number1 PHYSICSLET1’ERS 18 October1982
RSV-FREE FORMULATION OF QUANTUM NONDEMOLITION ThEORY
RobertLYNCHPhysicsDepartment,University ofPetroleumandMinerals, Dhahran,SaudiArabia
Received25 May 1982
The entirevalidity of the “quantumnondemolition”(QND) concepthasbeencalled into questionbecauseof itsdeepreli~nceon “reduction of thestatevector” (RSV)in the detaileddevelopmentof thetheory. In this letter QND theoryisreformulatedwithout useof RSV, exceptasfoundin theoverall interpretationof thewavefunction.
The recentlydeveloped“quantum nondemoli- Theproblemis to makea set of measurementson thetion” (QND) theory [1] is probablyone of themore systemto determinethe(extremelyweak) forceF(t).excitingadvancesin quantummeasurementtheory in Oneintroducestwo newvariablesrecentyears.A theoryof greatgeneralityand power,
~ coswt — (a/mw) sinwt, (2)restingfirmly on the axiomsof quantumtheory, it isnonethelessmarredby a heavyuseof “reductionof ~2 ~ sin wt + (p/mw) coswt. (3)the statevector” (RSV) in its detailedelaboration.Infact,Vager [2] hasrecentlypublisheda critique of In generalinitially the systemis in a superpositionthe theorywhichconcludesthatuseof RSV in the state* of ~1 (in the Schrodingerpicture)QND theorycauses“... a seriousinconsistencywhichmakesits conclusionsinvalid” [1]. i~’(t0))= �~c11 ~ (t0)>, (4)
Theauthorof thepresentpaperhaspublished[3]a rebuttalto Vager’scritiquebasedon apoint-to- where~1(t0) is oneof theeigenvaluesof X1. (Wepointexaminationof Vager’sargumentsin the frame-
choosetowork withX1 we could just aseasily usework of generalquantumtheory. However,this rebut-tal might notconvincea disinterestedobserver,since ~2) An initial idealmeasurementof ~ is made,and
theresulting RSV projectsthesysteminto aneigen-shemay viewthe disagreementon the issueto be due
stateof X~,say,I~m(to)~Which statewill result can-to alackof universalagreementon theinterpretation
notbe predictedwith certainty,all thatcanbe saidisof quantumtheory, particularlyquantummeasure- ‘2thatthis statewill appearwith probability cm I
ment theory. After this initial measurementn furthermeasure-It is the purposeof this letterto reformulateQND mentsofX1 aremade,at timest1, t2, ..., t~.As is
theorywithout useof RSV (exceptasfoundin the shownin ref. [1] the systemremainsin an eigenstateoverallinterpretationof thewave function).Henceuseof RSV in theQND theory is shownto be a ques- of~1,with eigenvaluetion of usefulnessratherthannecessity.
Beforethis is donelet usbriefly outlinetheele- *1 Since~1hasa continuousspectrumeq.(4) shouldbe
mentsof theQND theory asit pertainsto this paper. written with anintegralratherthana summation,proba-
Thesystemof interestis a forcedlinearharmonic biities becomedensities,etc.However, for easeof nota-oscillatorwith hamiltonian tion thetheorywill bedevelopedas though15 hasa dis-
cretespectrum,asthismakesno differenceto theargu-ft=p
2/2m+~mw2.e2—.~tF(t). (1) ment.
0 031-9l63/82/0000—0000/$02.75© 1982North-Holland 9
Volume92A, number1 PHYSICSLETTERS 18 October1982
tk with thenext memoryapparatuswith initial wave
~m(tk) = ~m(to) f [F(t’)/mw] sin(wt’) dt’ (5) function V1)1) we have
for themeasurementat tk, allowing determination I’P(t1))= ~cj I~(t1)>~ ~ {~~}). (9)of F(t).
Eachof thesemeasurementsinvolvesan RSV, but Continuing in this way, at theendof theprogramthis is consideredto be of no consequence,since the statevectorhasbecometheseare“measurementsof the first kind” [4] anddo not “demolish” thestate.The RSVsmentioned I’I’(t~))= ~ I~.(t~))I(I~){~-}>1){~.~>...above,particularly the initial one,are at the coreof / / “ / / /
the QND theory, and also are thesource of Vager’s (10)objections.
Having seenthe role of RSV in theusualQND Note that in the aboveargumentthe timedevelop-theorywe now proceedto reformulatethetheory ment of the systemin the interval from t0 to t~,is en-with thecentralityof RSV removed.As beforethe tirely causal, i.e., in accordancewith Schrodinger’sinitial stateof thesystemis givenby the superposi- equation.No RSV hasbeenusedto achievetheresult.tion in eq.(4). We areusingEverett’stheoryonly, for the conve-
Now,however,to carry outour measurementpro- nience of thetechniqueand notation,sowe do notgram we haveour systeminteractat times t0, t1 interpret(10) in termsof “splits” ~. Instead,at timet0 with (n + 1) Everett-type[51memoryapparatuses, t,~or laterwe read the contentsof thememorybanks,eachone designedto measure~. andin termsof orthodoxquantumtheory, we see
As a resultof the interactionof the zerothmem- that with probability cm 2 thememorybanksrevealory apparatuswith the systemat time t0, the state thevalues~m(t0), ~m(t1) ~m(ta) for thevariablevectorcanbewritten as [5, p. 1571: X1, asa result of theoverall RSV of’!’>. This is the
sameresultwhich is foundwhenRSV is takenafter
= UIiJi(t0)> I4~>= ~ ~I~(t0)> I4~f~}), (6) eachmeasurement./ In spiteof the economyof Everett’snotation,
where (1)0 {~1F>is theso-called“relative state”of the equationssuchas(10) are somewhatunwieldy.Sincezerothmem~ryapparatus this paperhasdemonstratedthat RSV is not central
to QND theory,it cannotbe a causeof lackof valid-
~ ~L)= r + >~A ~ ity of the theory. Hence, in theeverydayelaboration~ ~ —J IA g~1(t0) o( ) . ~ of QND theory it is permissible(ashasbeendonein
ref. [11)to takeRSVsat convenientpoints in orderHereA is the memoryvariableof the apparatusin to keepthenotationundercontrol and increasetheinitial state I4~~),couplingto the systemwith cou- clarity of thetheory.pling strengthg,andc1)0~A)= (AI’
1)c~>.The statevec-tor at t = t
1 is *2 It is amusing to notethat a nice featureof the QND theo-ry is that it lessensthe “schizophrenia”[5, P. 1781 of theuniverse(s).After theinitial “split” theuniverseis kept in-
= T(t1, t0)I’P(t0)> = ~ c11 ~1(t1)> ~ {~j}) tact,rather thanundergoingn further “splits”!
/ (8)[1] See,for example,C.M. Caves,K.S. Thorne, R.W.I’.
Drever,V.D. Sandbergand M. Zimmermann,Rev. Mod.smce the time-developmentoperatorT(t1,t0) oper- Phys.52 (1980) No. 2, part 1.ating on l~1(t0)>givesI~1(t1)).After the interaction [21Z. Vager,Phys.Lett. 84A (1981)163.
[3] R. Lynch, Phys.Lett. 87A (1982) 277.*2 We areassumingthat the apparatusesstayin stationary [4] W. Pauli,Encyclopediaof physics,Vol. V, part 1, p. 73.
statesbetweenmeasurements,andare also suppressing [5] B.S. DeWitt andN. Graham,eds.,Themany-worldsinter-unimportantphaseswhich appearfrom thetime evolution pretationof quantummechanics(PrincetonUniversityof thesestationarystates. Press,1973).
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