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we’ll find the result +1⁄2 with 0.5 proba-bility and −1⁄2 with 0.5 probability, con-sistent with the uncertainty principle.To use the misleading “ensemble” lan-guage of Aharonov and coauthors,every member of the ensemble is in thestate |+Sz⟩ = (|+Sx⟩ + |−Sx⟩)/√2‾. Contraryto the authors’ postselection process, it’snot true that 50% of the ensemble is inthe state |+Sx⟩ (which would violate theuncertainty principle), and 50% is in|−Sx⟩. To misinterpret quantum super-positions such as (|+Sx⟩ + |−Sx⟩)/√2‾ inthis manner is an elementary miscon-ception. It also directly contradicts theexperimental facts about measure-ments of the Stern–Gerlach type.

Contrary to the assertion ofAharonov and coauthors on page 32that they “have not modified quantummechanics by one iota,” their postselec-tion process would change the founda-tions of quantum mechanics. The fal-lacy in that process was pointed out byAsher Peres 16 years ago.1

Reference1. A. Peres, Phys. Lett. A 203, 150 (1995).

Art Hobson(ahobson@uark.edu)

University of ArkansasFayetteville

The article by Yakir Aharonov,Sandu Popescu, and Jeff Tollaksen isstimulating and raises some interestingand profound issues regarding thefoundations of quantum mechanics andquantum measurements. One point theauthors make is that a measurement ofspin component √2‾/2 of a spin-1⁄2 particleis unphysical and can be attributed onlyto errors in weak measurements per-formed on a collection of N spins. I offera more mundane interpretation thatdoes not require the introduction of anyerror or postselection concepts. Thephysical, textbook picture of spin-1⁄2 is a vector of length √S(S + 1) = √3‾/2 withsome distribution of orientations (thatis, polar coordinates θ and φ). A con-ventional single measurement of Sz canyield only one of the eigenvalues ±1⁄2.This may be interpreted in the classicalvector model by envisioning that thespin lies on a cone with θ defined bycosθ = √3‾/3. The expectation value of Sxthen vanishes due to the uniform distri-bution of φ. Having some control overφ can yield a finite value of Sx. Thus Iwould argue that only an expectationvalue greater than √3‾/2 is unphysical; avalue of √2‾/2 is quite physical and caneven be interpreted classically in termsof some distribution of φ and θ.

I would phrase the important obser-vation of Aharonov and coauthors in a

different way. While spin-1⁄2 has a mag-nitude of √3‾/2, a fundamental limitationof conventional single-point quantummeasurements is that they can yieldonly the values +1⁄2 and −1⁄2 and any ex-pectation value must therefore lie be-tween those two extremes. Classically,therefore, the spin may not be fullyaligned along the z-axis with θ = 0. Suchalignment is impossible since it wouldyield well-defined values of the threenoncommuting variables Sz = √3‾/2 andSx = Sy = 0. However, once multiple timemeasurements are performed, one candefine a reduced distribution of some ofthe measurements that is conditional onthe outcome in the other measure-ments; such a distribution can be inter-preted in terms of Sz greater than 1⁄2 butless than or equal to √3‾/2. That interpre-tation does not violate any of quantummechanics’ fundamental rules, whichdo not usually consider such quantities.

Rather than a new, time-symmetricformulation of quantum mechanicsthat involves pre- and postselection,one can simply treat the scheme of theauthors’ figure 1b as a three-point cor-relation function, whereas figures 2aand 2b show a four-point correlationfunction, each panel having its own setof conditional probabilities. Error, timereversal, and postselection need not beinvoked. Instead, one may think inmultiple dimensions and develop theright language for the interpretation ofthe observables.

The combination of pre- and post -selection is an attempt to create an arti-ficial ensemble that reproduces the re-sults of multiple-point measurementsin a single one-point measurement. Asshown by Aharonov and coauthors,that is possible by introducing errors.An alternative physical picture is ob-tained by retaining the multipointanalysis. Multidimensional thinking iswell developed in coherent nonlinearspectroscopy, where a system of spinsor optical chromophores are subjectedto sequences of short impulsivepulses.1,2 Similar ideas may be appliedfor the interpretation of multiple meas-urements. One can think of varioustypes of n-point observables obtainedby combining n − m perturbations andm measurements. The nonlinear n-point response functions in spec-troscopy represent n − 1 impulsive per-turbations followed by a singlemeasurement. The objects Aharonovand colleagues considered correspondto n = m. Proper multiple-distributionfunctions could then be naturally usedfor the interpretation of such general-ized measurements.

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References1. R. R. Ernst, G. Bodenhausen, A. Wokaun,

Principles of Nuclear Magnetic Resonance inOne and Two Dimensions, ClarendonPress, New York (1978).

2. S. Mukamel, Annu. Rev. Phys. Chem. 51,691 (2000); Phys. Rev. A. 61, 21804 (2000);Principles of Nonlinear Optical Spec-troscopy, Oxford U. Press, New York(1995).

Shaul Mukamel(smukamel@uci.edu)

University of California, Irvine

Readers of the article by YakirAharonov, Sandu Popescu, and Jeff Tol-laksen might be interested in an alterna-tive time-symmetric formulation ofquantum mechanics, known as “consis-tent histories,” that was developed overroughly the same time period asAharonov’s work (see the article byRobert Griffiths and Roland Omnès,PHYSICS TODAY, August 1999, page 26).Closely related is the “decoherent histo-ries” approach of Murray Gell-Mannand James Hartle,1 but as that is not usu-ally formulated in a way that is trans-parently time-symmetric, the followingremarks refer to the consistent historiesapproach; see reference 2 for an up-to-date formulation.

Both the consistent histories ap-proach and that of Aharonov andcoworkers pay attention to events atseveral different times, are formulatedin a way that is time-symmetrical, andaddress a number of quantum para-doxes. Both are consistent with the cal-culational procedures taught to stu-dents in a typical quantum mechanicscourse, so they are “standard quantummechanics,” without the additionalvariables of de Broglie–Bohm or the ad-ditional collapses of Ghirardi-Rimini-Weber. And both approaches do not ac-cept the “shut up and calculate”mentality that alas continues to domi-nate much classroom instruction. So faras I can tell, all the results mentioned byAharonov and coauthors and in the ear-lier work they cite are fairly readilytranslated into the language of consis-tent histories, though the reverse is nottrue (see below); therefore, the consis-tent histories view is more general.

In the treatment by Aharonov andcoauthors, measurement, as in textbookquantum theory, remains a black box: Itcollapses the wavefunction, but nothingmore can be said. And for good reason:The textbook approach of introducing

probabilities by reference to measure-ment yields what appear to be insolubledifficulties if one attempts to applyquantum theory to the measurementprocess itself—that is, to actual appara-tus constructed out of entities that arequantum mechanical. In the consistenthistories approach, that difficulty doesnot arise, because it treats quantum dy-namics as fundamentally probabilistic,not deterministic, and the same rulesapply to measurements as to all otherphysical processes. Speaking metaphor-ically, the probabilistic approach used inconsistent histories allows one to openthe black measurement box and watchthe quantum gears turn.

The other major difference betweenthe two approaches is their treatment ofquantum paradoxes. We owe many ofthe most striking and delightful para-doxes of quantum theory to Aharonovand his coworkers, and he and DanielRohrlich have written a book on thetopic.3 But he leaves the paradoxeslargely unresolved; the reader is en-couraged to study but not unravelthem. The consistent histories approachis exactly opposite: Paradoxes shouldbe—and a large number of them havebeen—resolved by the correct applica-tion of well-formulated and fully con-sistent quantum principles (see refer-ence 2, chapters 19–25).

Students new to quantum theory areoften confused and deserve reasonedresponses to their queries. Althoughparadoxes are valuable illustrations ofhow the quantum world differs fromour everyday experience, I prefer toprovide students with the conceptualtools needed to resolve and make senseof them. In particular, students benefitfrom learning a fully consistent ap-proach to probabilities in the quantumdomain, one not based on measure-ments but on general quantum princi-ples. A colleague and I have just fin-ished using that approach in teachingthe first term of our introductory grad-uate quantum mechanics course. Al-though it requires extra time and effortto learn how to think about quantumprocesses rather than just do calcula-tions, the reward comes in a deeper un-derstanding of how the real (quantum)world works.

References1. M. Gell-Mann, J. B. Hartle, in Complexity,

Entropy and the Physics of Information, W. H. Zurek, ed., Addison-Wesley, Red-wood City, CA (1990), p. 425.

2. R. B. Griffiths, Consistent Quantum Theory,Cambridge U. Press, New York (2002);available online at http://quantum.phys.cmu.edu/CQT.

3. Y. Aharonov, D. Rohrlich, Quantum Para-

doxes: Quantum Theory for the Perplexed,Wiley, Weinheim, Germany (2005).

Robert B. Griffiths(rgrif@cmu.edu)

Carnegie Mellon UniversityPittsburgh, Pennsylvania

Aharonov, Popescu, and Tollaksenreply: We thank the letter writers fortheir interest and for the opportunity tobetter clarify our ideas.

Michael Nauenberg and Art Hobsonmake essentially the same point—namely, that our ideas are completelywrong. To put their criticism in the rightcontext, we point out that the outcomeof our research program is twofold.First, we have discovered an entirelynew class of quantum effects; second, wepresent a new way of thinking aboutquantum mechanics.

The fact that quantum mechanicspredicts the effects we discovered is justthat, a fact. The effects are computedusing standard quantum mechanics,without additions or modifications. Assuch, their prediction by quantum me-chanics is beyond doubt (unless onesuspects algebraic mistakes). Further-more, many of our effects have beenverified experimentally; in particular,different versions of our amplificationmethod have been used as novel tech-nological tools. Both Nauenberg andHobson completely ignore our effects.But one should not ignore them. Theyare novel and they are strange. Evenmore, they don’t appear in isolation, butthey form a well-structured pattern.Surely there is a lesson here that quan-tum mechanics wants to teach us; oneignores it at one’s peril.

On the other hand, our way of look-ing at quantum mechanics is certainlyunconventional; it introduces new con-cepts, and it approaches old concepts ina new way. That is essentially what thetwo letter writers point out, Hobsonmost emphatically when he writes thatour article “is riddled with errors.” Weare criticized for thinking in a differentway and for asking new questions. Butour way of thinking leads to the samepredictions as the conventional way, soas far as experiments are concerned theyare completely equivalent. As RichardFeynman says in his book The Characterof Physical Law (Modern Library, 1994),suppose we have “two theories” that“have all the consequences . . . exactlythe same. . . . How are we going to de-cide which one is right? There is no wayby science, because they both agree withthe experiment to the same extent.” Sothe criticism is baseless.

At the same time, if our approach iscompletely equivalent to the standard

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