Physics Today
Consistent treatments of quantum mechanics
Robert B. Griffiths
Published Oct 1, 2011• 1 citation
Abstract
The Letters reply by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen gives a misleading impression of the consistent histories interpretation of quantum mechanics. As I pointed out in the same issue, consistent histories is an alternative to their time-symmetric formulation. It resolves the “collapse problem” of quantum foundations, which Aharonov and coauthors admit they have failed to do. However, that is not its only or even its principal virtue. It also solves what I call the second quantum measurement problem: how to infer from the measurement outcome the state of the measured system at an earlier time before the measurement took place. Thus the consistent histories approach, without any fuzzy references to “ensembles,” provides a proper probabilistic framework for inferring that Sx for a spin-1⁄2 particle actually was −1⁄2 before the measurement took place. This approach evades the criticisms expressed by Michael Nauenberg and Art Hobson in their letters in the same issue.Furthermore, consistent histories, unlike certain aspects of the approach of Aharonov and coauthors, is fully consistent with the Hilbert-space structure of quantum mechanics introduced by John von Neumann and now taught to all students of the subject. Indeed, with a consistent histories analysis one can locate fairly precisely11. R. E. Kastner, Stud. Hist. Phil. Mod. Phys. 35, 57 (2004), http://arxiv.org/abs/quant-ph/0207182. https://doi.org/10.1016/j.shpsb.2003.02.001 the error in reasoning that leads to claims such as that the measured spin of a spin-1⁄2 particle can equal 100.22. Y. Aharonov, D. Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988). https://doi.org/10.1103/PhysRevLett.60.1351 Claims about what weak measurements actually measure, when made by those who have not yet solved either the collapse problem or the second measurement problem, should be treated with great caution.The claim that the consistent histories approach is controversial was true back in the 1990s, and the controversy led to a clearer formulation of the interpretation. However, no significant problems have come to light in the decade since the publication of my book.33. R. B. Griffiths, Consistent Quantum Theory Cambridge U. Press, New York (2002), http://quantum.phys.cmu.edu/CQT/. The fact that something has been ignored does not imply it is controversial. Admirers of Richard Feynman, some of whose writings were cited by Aharonov and coauthors in their reply, may be interested to know that shortly before his death he was quite sympathetic to an early form of the consistent histories or decoherent histories approach to understanding quantum mechanics (see PHYSICS TODAY, February 1999, page 11)—an open-mindedness worth imitating.REFERENCESSection:ChooseTop of pageREFERENCES <<1. R. E. Kastner, Stud. Hist. Phil. Mod. Phys. 35, 57 (2004), http://arxiv.org/abs/quant-ph/0207182. https://doi.org/10.1016/j.shpsb.2003.02.001 , Google ScholarCrossref, ISI2. Y. Aharonov, D. Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988). https://doi.org/10.1103/PhysRevLett.60.1351 , Google ScholarCrossref, ISI3. R. B. Griffiths, Consistent Quantum Theory Cambridge U. Press, New York (2002), http://quantum.phys.cmu.edu/CQT/. Google Scholar© 2011 American Institute of Physics.
- DOI
- 10.1063/pt.3.1272
Related papers
Time-symmetric quantum mechanics questioned and defended
Robert B. Griffiths
Readers of the article by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen might be interested in an alternative time-symmetric formulation of quantum mechanics, known as “consistent histories,” that was developed over roughly the same time period as Aharonov’s work (see the article by Robert Griffiths and Roland Omnès, PHYSICS TODAY, August 1999, page 26). Closely related is the “decoherent histories” approach of Murray Gell-Mann and James Hartle,11. M. Gell-Mann, J. B. Hartle, in Complexity, Entropy and the Physics of InformationW. H. Zurek,ed.,Addison-Wesley, Redwood City, CA (1990), p. 425. but as that is not usually formulated in a way that is transparently time-symmetric, the following remarks refer to the consistent histories approach; see reference 22. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press, New York (2002); available online at http://quantum.phys.cmu.edu/CQT. for an up-to-date formulation.Both the consistent histories approach and that of Aharonov and coworkers pay attention to events at several different times, are formulated in a way that is time-symmetrical, and address a number of quantum paradoxes. Both are consistent with the calculational procedures taught to students in a typical quantum mechanics course, so they are “standard quantum mechanics,” without the additional variables of de Broglie–Bohm or the additional collapses of Ghirardi-Rimini-Weber. And both approaches do not accept the “shut up and calculate” mentality that alas continues to dominate much classroom instruction. So far as I can tell, all the results mentioned by Aharonov and coauthors and in the earlier work they cite are fairly readily translated into the language of consistent histories, though the reverse is not true (see below); therefore, the consistent histories view is more general.In the treatment by Aharonov and coauthors, measurement, as in textbook quantum theory, remains a black box: It collapses the wavefunction, but nothing more can be said. And for good reason: The textbook approach of introducing probabilities by reference to measurement yields what appear to be insoluble difficulties if one attempts to apply quantum theory to the measurement process itself—that is, to actual apparatus constructed out of entities that are quantum mechanical. In the consistent histories approach, that difficulty does not arise, because it treats quantum dynamics as fundamentally probabilistic, not deterministic, and the same rules apply to measurements as to all other physical processes. Speaking metaphorically, the probabilistic approach used in consistent histories allows one to open the black measurement box and watch the quantum gears turn.The other major difference between the two approaches is their treatment of quantum paradoxes. We owe many of the most striking and delightful paradoxes of quantum theory to Aharonov and his coworkers, and he and Daniel Rohrlich have written a book on the topic.33. Y. Aharonov, D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed, Wiley, Weinheim, Germany (2005) . But he leaves the paradoxes largely unresolved; the reader is encouraged to study but not unravel them. The consistent histories approach is exactly opposite: Paradoxes should be—and a large number of them have been—resolved by the correct application of well-formulated and fully consistent quantum principles (see reference 22. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press, New York (2002); available online at http://quantum.phys.cmu.edu/CQT. , chapters 19–25).Students new to quantum theory are often confused and deserve reasoned responses to their queries. Although paradoxes are valuable illustrations of how the quantum world differs from our everyday experience, I prefer to provide students with the conceptual tools needed to resolve and make sense of them. In particular, students benefit from learning a fully consistent approach to probabilities in the quantum domain, one not based on measurements but on general quantum principles. A colleague and I have just finished using that approach in teaching the first term of our introductory graduate quantum mechanics course. Although it requires extra time and effort to learn how to think about quantum processes rather than just do calculations, the reward comes in a deeper understanding of how the real (quantum) world works.REFERENCESSection:ChooseTop of pageREFERENCES <<1. M. Gell-Mann, J. B. Hartle, in Complexity, Entropy and the Physics of InformationW. H. Zurek,ed.,Addison-Wesley, Redwood City, CA (1990), p. 425. Google Scholar2. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press, New York (2002); available online at http://quantum.phys.cmu.edu/CQT. Google Scholar3. Y. Aharonov, D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed, Wiley, Weinheim, Germany (2005) . Google ScholarCrossref© 2011 American Institute of Physics.
Physics Today•May 1, 2011•1 citation
Consistent treatments of quantum mechanics
N. David Mermin
Letters to the editor from Michael Nauenberg and Art Hobson (PHYSICS TODAY, May 2011, page 8) both maintain that Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen make elementary mistakes in quantum mechanics in their November 2010 article (page 27). The authors reply that they have made no mistakes but do not otherwise answer the questions Nauenberg and Hobson raise. The authors are correct. Nauenberg and Hobson have misunderstood them. But I believe the misunderstandings were induced by the rather peculiar language used in the article.The experiment at issue is elementary. At an initial time t0 an ensemble of spin-1⁄2 particles is prepared, all with spin up along the z-axis. At a later intermediate time t all spins are measured along a single direction, either z or x—we do not know which. At a still later final time t1 the spin of each particle is measured along x and a subensemble is formed of those particles with spin up along x. All statements that follow apply to the particles in that “post- selected” subensemble.No magnetic fields act on the spins between measurements, so no spin states change except as the result of measurements. It follows that if the intermediate measurements at t had been along z, then their results would have been up, because the initial state was up along z. But if the intermediate measurements had been along x, then their results would also have been up, because if any had been down, the result of the final measurement at t1 along x would also have been down, and such a spin would not be in the postselected subensemble. (If you still doubt that nothing is needed but plain ordinary quantum mechanics [and Bayes’s theorem for conditional probabilities] to determine the probabilities of intermediate measurements on the postselected ensemble, see the homework assignment in the box at right.)I have no doubt Nauenberg and Hobson would both have agreed with that description of the results at the intermediate time t, had they not been distracted by the strange way the authors prefer to talk about it. The language led them astray, as it led me astray the first time I encountered these entertaining games by Aharonov and his collaborators 16 years ago. I think I finally got that aspect straight in “Limits to quantum mechanics as a source of magic tricks.”11. N. D. Mermin, Phys. Rev. Lett. 74, 831 (1995). https://doi.org/10.1103/PhysRevLett.74.831 For example, Aharonov and coauthors say, “The results at t depend . . . on what happens later at t1.” Nauenberg says, and it sounds right, that this contradicts standard quantum mechanics. He surely would not have objected if the authors had instead said, “What we can learn about the results at t depends . . . on what we learn later at t1.” But the way they do state it suggests causality acting backwards in time, which would indeed contradict quantum orthodoxy. Similarly, they say, “If at t we measure the spin along x, we must also find it up, because otherwise the measurement at t1 wouldn’t find it up.” Nauenberg says that this is incorrect, and he sounds right again. But surely he would not have objected if Aharonov and coauthors had instead said, “If the spin measurement at t were along x and not along z, then it would have given spin up because if it had given spin down, then the measurement at t1 would also have given spin down, and that particle would not have been in the postselected ensemble.” By putting everything in the historical present tense and refusing to use the subjunctive mood to indicate counterfactual statements, Aharonov and coauthors encourage such misunderstandings.In the same vein, the authors say “The idea that they [Sz and Sx] are both well defined stems from the fact that measuring either one yields +1⁄2 with certainty,” and Hobson objects. I doubt he would have objected if the authors had said, “The idea that they are both well defined in the postselected ensemble stems from the fact that no matter which one was actually measured at the intermediate time, the result of that measurement would have been +1⁄2.” Here again, the choice of tense and mood seem designed to mislead.Why has the Aharonov group been confusing some of us in this way for two decades? There is a clue in the final section of the authors’ PHYSICS TODAY article, where they talk about the flow of time. There they suggest taking seriously the idea that time “propagates forward from the past boundary condition and backward from the future boundary condition.” If you believe that, then conventional use of tense is indeed obsolete and distracting, and using the subjunctive mood in counterfactual statements becomes problematic.Aharonov, Popescu, Tollaksen, and their collaborators have uncovered some intriguingly provocative features of plain old ordinary quantum mechanics. Those features suggest to them a radical new interpretation. If they want to get the prerevolutionary masses to pay due attention, they might stop addressing us only in postrevolutionary language, or at least accompany their Newspeak with translations into the Oldspeak that most of us still use.Homework assignmentAt an initial time a spin-1⁄2 particle is prepared with spin up along direction a. At a later intermediate time the spin is measured along direction b. At a still later final time the spin is measured along direction c. Show that if the final measurement gives spin up, then the probability that the intermediate measurement gave spin up is Homework assignmentAt an initial time a spin-1⁄2 particle is prepared with spin up along direction a. At a later intermediate time the spin is measured along direction b. At a still later final time the spin is measured along direction c. Show that if the final measurement gives spin up, then the probability that the intermediate measurement gave spin up is REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. N. D. Mermin, Phys. Rev. Lett. 74, 831 (1995). https://doi.org/10.1103/PhysRevLett.74.831 , Google ScholarCrossref, ISI© 2011 American Institute of Physics.
Physics Today•Oct 1, 2011•5 citations
Time-symmetric quantum mechanics questioned and defended
Shaul Mukamel
The article by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen is stimulating and raises some interesting and profound issues regarding the foundations of quantum mechanics and quantum measurements. One point the authors make is that a measurement of spin component √2‾/2 of a spin-1⁄2 particle is unphysical and can be attributed only to errors in weak measurements performed on a collection of N spins. I offer a more mundane interpretation that does not require the introduction of any error or postselection concepts. The physical, textbook picture of spin-1⁄2 is a vector of length √S(S + 1) = √3‾/2 with some distribution of orientations (that is, polar coordinates θ and φ). A conventional single measurement of Sz can yield only one of the eigenvalues ±1⁄2. This may be interpreted in the classical vector model by envisioning that the spin lies on a cone with θ defined by cosθ = √3‾/3. The expectation value of Sx then vanishes due to the uniform distribution of φ. Having some control over φ can yield a finite value of Sx . Thus I would argue that only an expectation value greater than √3‾/2 is unphysical; a value of √2‾/2 is quite physical and can even be interpreted classically in terms of some distribution of φ and θ.I would phrase the important observation of Aharonov and coauthors in a different way. While spin-1⁄2 has a magnitude of √3‾/2, a fundamental limitation of conventional single-point quantum measurements is that they can yield only the values +1⁄2 and −1⁄2 and any expectation value must therefore lie between those two extremes. Classically, therefore, the spin may not be fully aligned along the z-axis with θ = 0. Such alignment is impossible since it would yield well-defined values of the three noncommuting variables Sz = √3‾/2 and Sx = Sy = 0. However, once multiple time measurements are performed, one can define a reduced distribution of some of the measurements that is conditional on the outcome in the other measurements; such a distribution can be interpreted in terms of Sz greater than 1⁄2 but less than or equal to √3‾/2. That interpretation does not violate any of quantum mechanics' fundamental rules, which do not usually consider such quantities.Rather than a new, time-symmetric formulation of quantum mechanics that involves pre- and postselection, one can simply treat the scheme of the authors' figure 1b as a three-point correlation function, whereas figures 2a and 2b show a four-point correlation function, each panel having its own set of conditional probabilities. Error, time reversal, and postselection need not be invoked. Instead, one may think in multiple dimensions and develop the right language for the interpretation of the observables.The combination of pre- and postselection is an attempt to create an artificial ensemble that reproduces the results of multiple-point measurements in a single one-point measurement. As shown by Aharonov and coauthors, that is possible by introducing errors. An alternative physical picture is obtained by retaining the multipoint analysis. Multidimensional thinking is well developed in coherent nonlinear spectroscopy, where a system of spins or optical chromophores are subjected to sequences of short impulsive pulses.11. R. R. Ernst G. Bodenhausen A. Wokaun Principles of Nuclear Magnetic Resonance in One and Two Dimensions Clarendon Press, New York (1978) . , 22. S. Mukamel Annual Review of Physical Chemistry 51 1 691 (2000). https://doi.org/10.1146/annurev.physchem.51.1.691 ; Physical Review A 61 2 21804 (2000). https://doi.org/10.1103/PhysRevA.61.021804 ; Principles of Nonlinear Optical Spectroscopy Oxford U. Press, New York (1995) . Similar ideas may be applied for the interpretation of multiple measurements. One can think of various types of n-point observables obtained by combining n − m perturbations and m measurements. The nonlinear n-point response functions in spectroscopy represent n − 1 impulsive perturbations followed by a single measurement. The objects Aharonov and colleagues considered correspond to n = m. Proper multiple-distribution functions could then be naturally used for the interpretation of such generalized measurements.REFERENCESSection:ChooseTop of pageREFERENCES <<1. R. R. Ernst G. Bodenhausen A. Wokaun Principles of Nuclear Magnetic Resonance in One and Two Dimensions Clarendon Press, New York (1978) . Google Scholar2. S. Mukamel Annual Review of Physical Chemistry 51 1 691 (2000). https://doi.org/10.1146/annurev.physchem.51.1.691 ; Google ScholarCrossref, ISI Physical Review A 61 2 21804 (2000). https://doi.org/10.1103/PhysRevA.61.021804 ; , Google ScholarCrossref, ISI Principles of Nonlinear Optical Spectroscopy Oxford U. Press, New York (1995) . , Google Scholar© 2011 American Institute of Physics.
Physics Today•May 1, 2011•3 citations
Consistent treatments of quantum mechanics
Yakir Aharonov, Sandu Popescu, Jeff Tollaksen
Aharonov, Popescu, and Tollaksen reply: We are grateful to David Mermin for stating loud and clear, in his unmistakable style, that “the authors are correct.” As for the rest of his letter, a lesson on clear writing from a great master of the subject is always welcome. We did strive for clarity in our article, but there is always space for improvement, and Freudian slips of Newspeak often creep in—that’s their very nature.However, as Mermin certainly knows, analyzing sentences in isolation without the many clues found in context doesn’t do them justice. The initial setup is indeed elementary, and we put in all the details, so it is hard to see how it could be misinterpreted. The situation becomes far more interesting and surprising, though, when weak measurements are taken into account; to ignore them, as Michael Nauenberg and Art Hobson did, is to completely miss the point. We believe that no true understanding of quantum mechanics can be obtained without taking weak measurements seriously.The letter by Robert Griffiths is mainly concerned with the merits of the consistent histories approach to quantum mechanics. It is an interesting subject but one that has very little to do with our article. We presented a number of quantum effects that arise in the context of pre- and postselection, first and foremost the effects that appear when weak measurements are involved. And no, there is no error in reasoning when we claim that in a situation like that described in our article, spin measurements may, in a consistent way, indicate that spin-1⁄2 particles have value 100; that is a statement of fact about results of measurements, which incidentally have been confirmed experimentally. One is free to disagree with the meaning we attach to those facts but cannot deny the facts themselves. Of course, since our effects are purely quantum mechanical, they can be analyzed using any interpretation of quantum mechanics, including consistent histories. However, before having such a discussion, it helps if the effects are first discovered! Their existence is our main message.© 2011 American Institute of Physics.
Physics Today•Oct 1, 2011•6 citations
Time-symmetric quantum mechanics questioned and defended
Art Hobson
“A Time-Symmetric Formulation of Quantum Mechanics” by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen is riddled with errors. That is not surprising, given that their starting point is an erroneous “postselection” process. Postselection ignores the measurement postulate of standard quantum physics, according to which a system’s quantum state abruptly switches, upon measurement, into an eigenstate of the measured observable.With that postulate abandoned, it’s to be expected that Aharonov and coauthors obtain results that persistently violate the standard quantum uncertainty principle. For example, in their discussion on page 28 of a spin-1⁄2 system, they incorrectly state that “the measurement of Sπ /4 [the spin component along the diagonal between the x-axis and z-axis] is effectively a simultaneous measurement of Sx and Sz .” If that were true, it would violate the uncertainty principle. But the statement is not true. A measurement of Sπ /4 could be made by positioning a Stern–Gerlach apparatus with its inhomogeneous magnetic field in the π/4 direction, without measuring either Sz or Sx . In fact, the measurement would put the system into an eigenstate of Sπ /4, leaving both Sz and Sx indeterminate, in agreement with the uncertainty principle.Aharonov and coauthors continue: “The idea that they [Sz and Sx ] are both well defined stems from the fact that measuring either one yields +1⁄2 with certainty.” But that supposed violation of the uncertainty principle is wrong. Using an obvious notation, in their example the system is claimed to be in the eigenstate |+Sz 〉 at time t. This eigenstate can also be written as (|+Sx 〉 + |−Sx 〉)/√2‾, showing that the spin component Sx is indeterminate whenever the system is in the state |+Sz 〉, in agreement with the uncertainty principle.One sentence later, the authors state, “If we first measure Sz and then Sx , . . . then, given the pre- and postselection, both measurements yield +1⁄2 with certainty.” That supposed violation of the uncertainty principle is wrong. If we first measure Sz , we’ll get +1⁄2 because the system was previously prepared in that state. But if we then measure Sx , we’ll find the result +1⁄2 with 0.5 probability and −1⁄2 with 0.5 probability, consistent with the uncertainty principle. To use the misleading “ensemble” language of Aharonov and coauthors, every member of the ensemble is in the state |+Sz 〉 = (|+Sx 〉 + |−Sx 〉)/√2‾. Contrary to the authors’ postselection process, it’s not true that 50% of the ensemble is in the state |+Sx 〉 (which would violate the uncertainty principle), and 50% is in |−Sx 〉. To misinterpret quantum superpositions such as (|+Sx 〉 + |−Sx 〉)/√2‾ in this manner is an elementary misconception. It also directly contradicts the experimental facts about measurements of the Stern–Gerlach type.Contrary to the assertion of Aharonov and coauthors on page 32 that they “have not modified quantum mechanics by one iota,” their postselection process would change the foundations of quantum mechanics. The fallacy in that process was pointed out by Asher Peres 16 years ago.11. A. Peres, Physics Letters A 203 2-3 150(1995). https://doi.org/10.1016/0375-9601(95)00395-J REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. A. Peres, Physics Letters A 203 2-3 150(1995). https://doi.org/10.1016/0375-9601(95)00395-J , Google ScholarCrossref, ISI© 2011 American Institute of Physics.
Physics Today•May 1, 2011•2 citations
Time-symmetric quantum mechanics questioned and defended
Yakir Aharonov, Sandu Popescu, Jeff Tollaksen
Aharonov, Popescu, and Tollaksen reply: We thank the letter writers for their interest and for the opportunity to better clarify our ideas.Michael Nauenberg and Art Hobson make essentially the same point—namely, that our ideas are completely wrong. To put their criticism in the right context, we point out that the outcome of our research program is twofold. First, we have discovered an entirely new class of quantum effects; second, we present a new way of thinking about quantum mechanics.The fact that quantum mechanics predicts the effects we discovered is just that, a fact. The effects are computed using standard quantum mechanics, without additions or modifications. As such, their prediction by quantum mechanics is beyond doubt (unless one suspects algebraic mistakes). Furthermore, many of our effects have been verified experimentally; in particular, different versions of our amplification method have been used as novel technological tools. Both Nauenberg and Hobson completely ignore our effects. But one should not ignore them. They are novel and they are strange. Even more, they don’t appear in isolation, but they form a well-structured pattern. Surely there is a lesson here that quantum mechanics wants to teach us; one ignores it at one’s peril.On the other hand, our way of looking at quantum mechanics is certainly unconventional; it introduces new concepts, and it approaches old concepts in a new way. That is essentially what the two letter writers point out, Hobson most emphatically when he writes that our article “is riddled with errors.” We are criticized for thinking in a different way and for asking new questions. But our way of thinking leads to the same predictions as the conventional way, so as far as experiments are concerned they are completely equivalent. As Richard Feynman says in his book The Character of Physical Law (Modern Library, 1994), suppose we have “two theories” that “have all the consequences . . . exactly the same. . . . How are we going to decide which one is right? There is no way by science, because they both agree with the experiment to the same extent.” So the criticism is baseless.At the same time, if our approach is completely equivalent to the standard one, why bother? Again Feynman gives the best answer: “For psychological reasons . . . , these two things may be far from equivalent, because one gives a man different ideas from the other. . . . There will be something, for instance, in theory A that talks about something, . . . but to find out what the corresponding thing is . . . in [theory] B may be very complicated—it may not be a very simple idea at all.” As a consequence, a new way of thinking allows one to ask new questions that, although they could be asked in the old theory as well, would have been very difficult to even envisage. That is precisely what we did. First, we raised the issue of the physics in pre- and postselected ensembles. And to reply to Hobson, no, there is nothing “erroneous” in the process of postselection. Postselection is a question about results of experiments, and every question about the results of actual measurements is legitimate. Subsequently, we discovered the concept of weak measurements, which in turn led us to discover the various effects we presented. Since the power of any new approach is given by its ability to predict new effects, one should conclude that ours is strong indeed.Furthermore, as many physicists agree, an intuitive understanding of quantum mechanics is still missing. That is why quantum physicists are surprised over and over again by the discovery of strange and unexpected fundamental effects. We hope that our new way of thinking is a step toward the long-sought intuition. Even more important, the new way of thinking may give us new ideas about what to change, if experiments ever turn out to contradict quantum mechanics and therefore require its modification. In particular, since we tinker with the idea of time— one of the most important concepts in physics—starting the change from there may be a very potent method.Shaul Mukamel refers to our experiment in which the component along some given axis of a spin-1⁄2 particle is found, by a weak measurement, to have the value √2‾/2 which is √2‾ times larger than the largest eigenvalue. He suggests an alternative explanation based on a classical vector model of spin. According to his explanation, values up to √3‾/2 should be possible. However, we presented the experiment showing √2‾/2 only because it was mathematically simple; by choosing a different postselection, we could have obtained, as results of weak measurements, values as large as we wanted. Hence the above simple classical-vector view doesn’t work.Robert Griffiths points out that there are two other time-symmetric formulations of quantum mechanics besides ours—“consistent histories” and “decoherent histories.” In particular, Griffiths is right when he emphasizes a major difference in spirit between our theory and consistent histories. The main goal of consistent histories is to find an explanation for the (apparent?) collapse of the wavefunction during a quantum measurement; although that solution is hotly disputed, as are all other solutions to the collapse problem, it is certainly a very ingenious one. However, an answer to the collapse problem is not our primary interest (though we are starting to see glimmers of an alternative answer to it using our approach).The letter by Griffiths, however, runs the risk of being misread as implying that solving the collapse problem will by itself clarify most or all of the counterintuitive aspects of quantum mechanics. That conclusion would be wrong. Quantum mechanics is strange and unusual and defies intuition in many ways; solving the collapse problem is by no means its only interesting fundamental issue. Nor can the solution of that one problem lead to a complete understanding of quantum phenomena. That particles tunnel in the first place is surprising by itself; even more so is the fact that, as we showed, perfectly good measuring devices, working with arbitrarily high precision, indicate consistently that the tunneling particles have negative kinetic energy. Equally surprising is that we can arrange a situation in which perfectly good measurements, made with as high a precision as we want, indicate that spin-1⁄2 particles have arbitrarily large spin. And these are only two examples. As far as we are aware, none of the many proposed solutions to the collapse problem make these effects seem less surprising, let alone predict them.© 2011 American Institute of Physics.
Physics Today•May 1, 2011•7 citations
The Letters reply by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen gives a misleading impression of the consistent histories interpretation of quantum mechanics. As I pointed out in the same issue, consistent histories is an alternative to their time-symmetric formulation. It resolves the “collapse problem” of quantum foundations, which Aharonov and coauthors admit they have failed to do. However, that is not its only or even its principal virtue. It also solves what I call the second quantum measurement problem: how to infer from the measurement outcome the state of the measured system at an earlier time before the measurement took place. Thus the consistent histories approach, without any fuzzy references to “ensembles,” provides a proper probabilistic framework for inferring that Sx for a spin-1⁄2 particle actually was −1⁄2 before the measurement took place. This approach evades the criticisms expressed by Michael Nauenberg and Art Hobson in their letters in the same issue.Furthermore, consistent histories, unlike certain aspects of the approach of Aharonov and coauthors, is fully consistent with the Hilbert-space structure of quantum mechanics introduced by John von Neumann and now taught to all students of the subject. Indeed, with a consistent histories analysis one can locate fairly precisely11. R. E. Kastner, Stud. Hist. Phil. Mod. Phys. 35, 57 (2004),
http://arxiv.org/abs/quant-ph/0207182.
https://doi.org/10.1016/j.shpsb.2003.02.001 the error in reasoning that leads to claims such as that the measured spin of a spin-1⁄2 particle can equal 100.22. Y. Aharonov, D. Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).
https://doi.org/10.1103/PhysRevLett.60.1351 Claims about what weak measurements actually measure, when made by those who have not yet solved either the collapse problem or the second measurement problem, should be treated with great caution.The claim that the consistent histories approach is controversial was true back in the 1990s, and the controversy led to a clearer formulation of the interpretation. However, no significant problems have come to light in the decade since the publication of my book.33. R. B. Griffiths, Consistent Quantum Theory Cambridge U. Press, New York (2002),
http://quantum.phys.cmu.edu/CQT/. The fact that something has been ignored does not imply it is controversial. Admirers of Richard Feynman, some of whose writings were cited by Aharonov and coauthors in their reply, may be interested to know that shortly before his death he was quite sympathetic to an early form of the consistent histories or decoherent histories approach to understanding quantum mechanics (see PHYSICS TODAY, February 1999, page 11)—an open-mindedness worth imitating.REFERENCESSection:ChooseTop of pageREFERENCES <<1. R. E. Kastner, Stud. Hist. Phil. Mod. Phys. 35, 57 (2004),
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http://quantum.phys.cmu.edu/CQT/. Google Scholar© 2011 American Institute of Physics.
Readers of the article by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen might be interested in an alternative time-symmetric formulation of quantum mechanics, known as “consistent histories,” that was developed over roughly the same time period as Aharonov’s work (see the article by Robert Griffiths and Roland Omnès, PHYSICS TODAY, August 1999, page 26). Closely related is the “decoherent histories” approach of Murray Gell-Mann and James Hartle,11. M. Gell-Mann, J. B. Hartle, in Complexity, Entropy and the Physics of InformationW. H. Zurek,ed.,Addison-Wesley, Redwood City, CA (1990), p. 425. but as that is not usually formulated in a way that is transparently time-symmetric, the following remarks refer to the consistent histories approach; see reference 22. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press, New York (2002); available online at http://quantum.phys.cmu.edu/CQT. for an up-to-date formulation.Both the consistent histories approach and that of Aharonov and coworkers pay attention to events at several different times, are formulated in a way that is time-symmetrical, and address a number of quantum paradoxes. Both are consistent with the calculational procedures taught to students in a typical quantum mechanics course, so they are “standard quantum mechanics,” without the additional variables of de Broglie–Bohm or the additional collapses of Ghirardi-Rimini-Weber. And both approaches do not accept the “shut up and calculate” mentality that alas continues to dominate much classroom instruction. So far as I can tell, all the results mentioned by Aharonov and coauthors and in the earlier work they cite are fairly readily translated into the language of consistent histories, though the reverse is not true (see below); therefore, the consistent histories view is more general.In the treatment by Aharonov and coauthors, measurement, as in textbook quantum theory, remains a black box: It collapses the wavefunction, but nothing more can be said. And for good reason: The textbook approach of introducing probabilities by reference to measurement yields what appear to be insoluble difficulties if one attempts to apply quantum theory to the measurement process itself—that is, to actual apparatus constructed out of entities that are quantum mechanical. In the consistent histories approach, that difficulty does not arise, because it treats quantum dynamics as fundamentally probabilistic, not deterministic, and the same rules apply to measurements as to all other physical processes. Speaking metaphorically, the probabilistic approach used in consistent histories allows one to open the black measurement box and watch the quantum gears turn.The other major difference between the two approaches is their treatment of quantum paradoxes. We owe many of the most striking and delightful paradoxes of quantum theory to Aharonov and his coworkers, and he and Daniel Rohrlich have written a book on the topic.33. Y. Aharonov, D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed, Wiley, Weinheim, Germany (2005) . But he leaves the paradoxes largely unresolved; the reader is encouraged to study but not unravel them. The consistent histories approach is exactly opposite: Paradoxes should be—and a large number of them have been—resolved by the correct application of well-formulated and fully consistent quantum principles (see reference 22. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press, New York (2002); available online at http://quantum.phys.cmu.edu/CQT. , chapters 19–25).Students new to quantum theory are often confused and deserve reasoned responses to their queries. Although paradoxes are valuable illustrations of how the quantum world differs from our everyday experience, I prefer to provide students with the conceptual tools needed to resolve and make sense of them. In particular, students benefit from learning a fully consistent approach to probabilities in the quantum domain, one not based on measurements but on general quantum principles. A colleague and I have just finished using that approach in teaching the first term of our introductory graduate quantum mechanics course. Although it requires extra time and effort to learn how to think about quantum processes rather than just do calculations, the reward comes in a deeper understanding of how the real (quantum) world works.REFERENCESSection:ChooseTop of pageREFERENCES <<1. M. Gell-Mann, J. B. Hartle, in Complexity, Entropy and the Physics of InformationW. H. Zurek,ed.,Addison-Wesley, Redwood City, CA (1990), p. 425. Google Scholar2. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press, New York (2002); available online at http://quantum.phys.cmu.edu/CQT. Google Scholar3. Y. Aharonov, D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed, Wiley, Weinheim, Germany (2005) . Google ScholarCrossref© 2011 American Institute of Physics.
Letters to the editor from Michael Nauenberg and Art Hobson (PHYSICS TODAY, May 2011, page 8) both maintain that Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen make elementary mistakes in quantum mechanics in their November 2010 article (page 27). The authors reply that they have made no mistakes but do not otherwise answer the questions Nauenberg and Hobson raise. The authors are correct. Nauenberg and Hobson have misunderstood them. But I believe the misunderstandings were induced by the rather peculiar language used in the article.The experiment at issue is elementary. At an initial time t0 an ensemble of spin-1⁄2 particles is prepared, all with spin up along the z-axis. At a later intermediate time t all spins are measured along a single direction, either z or x—we do not know which. At a still later final time t1 the spin of each particle is measured along x and a subensemble is formed of those particles with spin up along x. All statements that follow apply to the particles in that “post- selected” subensemble.No magnetic fields act on the spins between measurements, so no spin states change except as the result of measurements. It follows that if the intermediate measurements at t had been along z, then their results would have been up, because the initial state was up along z. But if the intermediate measurements had been along x, then their results would also have been up, because if any had been down, the result of the final measurement at t1 along x would also have been down, and such a spin would not be in the postselected subensemble. (If you still doubt that nothing is needed but plain ordinary quantum mechanics [and Bayes’s theorem for conditional probabilities] to determine the probabilities of intermediate measurements on the postselected ensemble, see the homework assignment in the box at right.)I have no doubt Nauenberg and Hobson would both have agreed with that description of the results at the intermediate time t, had they not been distracted by the strange way the authors prefer to talk about it. The language led them astray, as it led me astray the first time I encountered these entertaining games by Aharonov and his collaborators 16 years ago. I think I finally got that aspect straight in “Limits to quantum mechanics as a source of magic tricks.”11. N. D. Mermin, Phys. Rev. Lett. 74, 831 (1995). https://doi.org/10.1103/PhysRevLett.74.831 For example, Aharonov and coauthors say, “The results at t depend . . . on what happens later at t1.” Nauenberg says, and it sounds right, that this contradicts standard quantum mechanics. He surely would not have objected if the authors had instead said, “What we can learn about the results at t depends . . . on what we learn later at t1.” But the way they do state it suggests causality acting backwards in time, which would indeed contradict quantum orthodoxy. Similarly, they say, “If at t we measure the spin along x, we must also find it up, because otherwise the measurement at t1 wouldn’t find it up.” Nauenberg says that this is incorrect, and he sounds right again. But surely he would not have objected if Aharonov and coauthors had instead said, “If the spin measurement at t were along x and not along z, then it would have given spin up because if it had given spin down, then the measurement at t1 would also have given spin down, and that particle would not have been in the postselected ensemble.” By putting everything in the historical present tense and refusing to use the subjunctive mood to indicate counterfactual statements, Aharonov and coauthors encourage such misunderstandings.In the same vein, the authors say “The idea that they [Sz and Sx] are both well defined stems from the fact that measuring either one yields +1⁄2 with certainty,” and Hobson objects. I doubt he would have objected if the authors had said, “The idea that they are both well defined in the postselected ensemble stems from the fact that no matter which one was actually measured at the intermediate time, the result of that measurement would have been +1⁄2.” Here again, the choice of tense and mood seem designed to mislead.Why has the Aharonov group been confusing some of us in this way for two decades? There is a clue in the final section of the authors’ PHYSICS TODAY article, where they talk about the flow of time. There they suggest taking seriously the idea that time “propagates forward from the past boundary condition and backward from the future boundary condition.” If you believe that, then conventional use of tense is indeed obsolete and distracting, and using the subjunctive mood in counterfactual statements becomes problematic.Aharonov, Popescu, Tollaksen, and their collaborators have uncovered some intriguingly provocative features of plain old ordinary quantum mechanics. Those features suggest to them a radical new interpretation. If they want to get the prerevolutionary masses to pay due attention, they might stop addressing us only in postrevolutionary language, or at least accompany their Newspeak with translations into the Oldspeak that most of us still use.Homework assignmentAt an initial time a spin-1⁄2 particle is prepared with spin up along direction a. At a later intermediate time the spin is measured along direction b. At a still later final time the spin is measured along direction c. Show that if the final measurement gives spin up, then the probability that the intermediate measurement gave spin up is Homework assignmentAt an initial time a spin-1⁄2 particle is prepared with spin up along direction a. At a later intermediate time the spin is measured along direction b. At a still later final time the spin is measured along direction c. Show that if the final measurement gives spin up, then the probability that the intermediate measurement gave spin up is REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. N. D. Mermin, Phys. Rev. Lett. 74, 831 (1995). https://doi.org/10.1103/PhysRevLett.74.831 , Google ScholarCrossref, ISI© 2011 American Institute of Physics.
The article by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen is stimulating and raises some interesting and profound issues regarding the foundations of quantum mechanics and quantum measurements. One point the authors make is that a measurement of spin component √2‾/2 of a spin-1⁄2 particle is unphysical and can be attributed only to errors in weak measurements performed on a collection of N spins. I offer a more mundane interpretation that does not require the introduction of any error or postselection concepts. The physical, textbook picture of spin-1⁄2 is a vector of length √S(S + 1) = √3‾/2 with some distribution of orientations (that is, polar coordinates θ and φ). A conventional single measurement of Sz can yield only one of the eigenvalues ±1⁄2. This may be interpreted in the classical vector model by envisioning that the spin lies on a cone with θ defined by cosθ = √3‾/3. The expectation value of Sx then vanishes due to the uniform distribution of φ. Having some control over φ can yield a finite value of Sx . Thus I would argue that only an expectation value greater than √3‾/2 is unphysical; a value of √2‾/2 is quite physical and can even be interpreted classically in terms of some distribution of φ and θ.I would phrase the important observation of Aharonov and coauthors in a different way. While spin-1⁄2 has a magnitude of √3‾/2, a fundamental limitation of conventional single-point quantum measurements is that they can yield only the values +1⁄2 and −1⁄2 and any expectation value must therefore lie between those two extremes. Classically, therefore, the spin may not be fully aligned along the z-axis with θ = 0. Such alignment is impossible since it would yield well-defined values of the three noncommuting variables Sz = √3‾/2 and Sx = Sy = 0. However, once multiple time measurements are performed, one can define a reduced distribution of some of the measurements that is conditional on the outcome in the other measurements; such a distribution can be interpreted in terms of Sz greater than 1⁄2 but less than or equal to √3‾/2. That interpretation does not violate any of quantum mechanics' fundamental rules, which do not usually consider such quantities.Rather than a new, time-symmetric formulation of quantum mechanics that involves pre- and postselection, one can simply treat the scheme of the authors' figure 1b as a three-point correlation function, whereas figures 2a and 2b show a four-point correlation function, each panel having its own set of conditional probabilities. Error, time reversal, and postselection need not be invoked. Instead, one may think in multiple dimensions and develop the right language for the interpretation of the observables.The combination of pre- and postselection is an attempt to create an artificial ensemble that reproduces the results of multiple-point measurements in a single one-point measurement. As shown by Aharonov and coauthors, that is possible by introducing errors. An alternative physical picture is obtained by retaining the multipoint analysis. Multidimensional thinking is well developed in coherent nonlinear spectroscopy, where a system of spins or optical chromophores are subjected to sequences of short impulsive pulses.11. R. R. Ernst G. Bodenhausen A. Wokaun Principles of Nuclear Magnetic Resonance in One and Two Dimensions Clarendon Press, New York (1978) . , 22. S. Mukamel Annual Review of Physical Chemistry 51 1 691 (2000). https://doi.org/10.1146/annurev.physchem.51.1.691 ; Physical Review A 61 2 21804 (2000). https://doi.org/10.1103/PhysRevA.61.021804 ; Principles of Nonlinear Optical Spectroscopy Oxford U. Press, New York (1995) . Similar ideas may be applied for the interpretation of multiple measurements. One can think of various types of n-point observables obtained by combining n − m perturbations and m measurements. The nonlinear n-point response functions in spectroscopy represent n − 1 impulsive perturbations followed by a single measurement. The objects Aharonov and colleagues considered correspond to n = m. Proper multiple-distribution functions could then be naturally used for the interpretation of such generalized measurements.REFERENCESSection:ChooseTop of pageREFERENCES <<1. R. R. Ernst G. Bodenhausen A. Wokaun Principles of Nuclear Magnetic Resonance in One and Two Dimensions Clarendon Press, New York (1978) . Google Scholar2. S. Mukamel Annual Review of Physical Chemistry 51 1 691 (2000). https://doi.org/10.1146/annurev.physchem.51.1.691 ; Google ScholarCrossref, ISI Physical Review A 61 2 21804 (2000). https://doi.org/10.1103/PhysRevA.61.021804 ; , Google ScholarCrossref, ISI Principles of Nonlinear Optical Spectroscopy Oxford U. Press, New York (1995) . , Google Scholar© 2011 American Institute of Physics.
Aharonov, Popescu, and Tollaksen reply: We are grateful to David Mermin for stating loud and clear, in his unmistakable style, that “the authors are correct.” As for the rest of his letter, a lesson on clear writing from a great master of the subject is always welcome. We did strive for clarity in our article, but there is always space for improvement, and Freudian slips of Newspeak often creep in—that’s their very nature.However, as Mermin certainly knows, analyzing sentences in isolation without the many clues found in context doesn’t do them justice. The initial setup is indeed elementary, and we put in all the details, so it is hard to see how it could be misinterpreted. The situation becomes far more interesting and surprising, though, when weak measurements are taken into account; to ignore them, as Michael Nauenberg and Art Hobson did, is to completely miss the point. We believe that no true understanding of quantum mechanics can be obtained without taking weak measurements seriously.The letter by Robert Griffiths is mainly concerned with the merits of the consistent histories approach to quantum mechanics. It is an interesting subject but one that has very little to do with our article. We presented a number of quantum effects that arise in the context of pre- and postselection, first and foremost the effects that appear when weak measurements are involved. And no, there is no error in reasoning when we claim that in a situation like that described in our article, spin measurements may, in a consistent way, indicate that spin-1⁄2 particles have value 100; that is a statement of fact about results of measurements, which incidentally have been confirmed experimentally. One is free to disagree with the meaning we attach to those facts but cannot deny the facts themselves. Of course, since our effects are purely quantum mechanical, they can be analyzed using any interpretation of quantum mechanics, including consistent histories. However, before having such a discussion, it helps if the effects are first discovered! Their existence is our main message.© 2011 American Institute of Physics.
“A Time-Symmetric Formulation of Quantum Mechanics” by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen is riddled with errors. That is not surprising, given that their starting point is an erroneous “postselection” process. Postselection ignores the measurement postulate of standard quantum physics, according to which a system’s quantum state abruptly switches, upon measurement, into an eigenstate of the measured observable.With that postulate abandoned, it’s to be expected that Aharonov and coauthors obtain results that persistently violate the standard quantum uncertainty principle. For example, in their discussion on page 28 of a spin-1⁄2 system, they incorrectly state that “the measurement of Sπ /4 [the spin component along the diagonal between the x-axis and z-axis] is effectively a simultaneous measurement of Sx and Sz .” If that were true, it would violate the uncertainty principle. But the statement is not true. A measurement of Sπ /4 could be made by positioning a Stern–Gerlach apparatus with its inhomogeneous magnetic field in the π/4 direction, without measuring either Sz or Sx . In fact, the measurement would put the system into an eigenstate of Sπ /4, leaving both Sz and Sx indeterminate, in agreement with the uncertainty principle.Aharonov and coauthors continue: “The idea that they [Sz and Sx ] are both well defined stems from the fact that measuring either one yields +1⁄2 with certainty.” But that supposed violation of the uncertainty principle is wrong. Using an obvious notation, in their example the system is claimed to be in the eigenstate |+Sz 〉 at time t. This eigenstate can also be written as (|+Sx 〉 + |−Sx 〉)/√2‾, showing that the spin component Sx is indeterminate whenever the system is in the state |+Sz 〉, in agreement with the uncertainty principle.One sentence later, the authors state, “If we first measure Sz and then Sx , . . . then, given the pre- and postselection, both measurements yield +1⁄2 with certainty.” That supposed violation of the uncertainty principle is wrong. If we first measure Sz , we’ll get +1⁄2 because the system was previously prepared in that state. But if we then measure Sx , we’ll find the result +1⁄2 with 0.5 probability and −1⁄2 with 0.5 probability, consistent with the uncertainty principle. To use the misleading “ensemble” language of Aharonov and coauthors, every member of the ensemble is in the state |+Sz 〉 = (|+Sx 〉 + |−Sx 〉)/√2‾. Contrary to the authors’ postselection process, it’s not true that 50% of the ensemble is in the state |+Sx 〉 (which would violate the uncertainty principle), and 50% is in |−Sx 〉. To misinterpret quantum superpositions such as (|+Sx 〉 + |−Sx 〉)/√2‾ in this manner is an elementary misconception. It also directly contradicts the experimental facts about measurements of the Stern–Gerlach type.Contrary to the assertion of Aharonov and coauthors on page 32 that they “have not modified quantum mechanics by one iota,” their postselection process would change the foundations of quantum mechanics. The fallacy in that process was pointed out by Asher Peres 16 years ago.11. A. Peres, Physics Letters A 203 2-3 150(1995). https://doi.org/10.1016/0375-9601(95)00395-J REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. A. Peres, Physics Letters A 203 2-3 150(1995). https://doi.org/10.1016/0375-9601(95)00395-J , Google ScholarCrossref, ISI© 2011 American Institute of Physics.
Aharonov, Popescu, and Tollaksen reply: We thank the letter writers for their interest and for the opportunity to better clarify our ideas.Michael Nauenberg and Art Hobson make essentially the same point—namely, that our ideas are completely wrong. To put their criticism in the right context, we point out that the outcome of our research program is twofold. First, we have discovered an entirely new class of quantum effects; second, we present a new way of thinking about quantum mechanics.The fact that quantum mechanics predicts the effects we discovered is just that, a fact. The effects are computed using standard quantum mechanics, without additions or modifications. As such, their prediction by quantum mechanics is beyond doubt (unless one suspects algebraic mistakes). Furthermore, many of our effects have been verified experimentally; in particular, different versions of our amplification method have been used as novel technological tools. Both Nauenberg and Hobson completely ignore our effects. But one should not ignore them. They are novel and they are strange. Even more, they don’t appear in isolation, but they form a well-structured pattern. Surely there is a lesson here that quantum mechanics wants to teach us; one ignores it at one’s peril.On the other hand, our way of looking at quantum mechanics is certainly unconventional; it introduces new concepts, and it approaches old concepts in a new way. That is essentially what the two letter writers point out, Hobson most emphatically when he writes that our article “is riddled with errors.” We are criticized for thinking in a different way and for asking new questions. But our way of thinking leads to the same predictions as the conventional way, so as far as experiments are concerned they are completely equivalent. As Richard Feynman says in his book The Character of Physical Law (Modern Library, 1994), suppose we have “two theories” that “have all the consequences . . . exactly the same. . . . How are we going to decide which one is right? There is no way by science, because they both agree with the experiment to the same extent.” So the criticism is baseless.At the same time, if our approach is completely equivalent to the standard one, why bother? Again Feynman gives the best answer: “For psychological reasons . . . , these two things may be far from equivalent, because one gives a man different ideas from the other. . . . There will be something, for instance, in theory A that talks about something, . . . but to find out what the corresponding thing is . . . in [theory] B may be very complicated—it may not be a very simple idea at all.” As a consequence, a new way of thinking allows one to ask new questions that, although they could be asked in the old theory as well, would have been very difficult to even envisage. That is precisely what we did. First, we raised the issue of the physics in pre- and postselected ensembles. And to reply to Hobson, no, there is nothing “erroneous” in the process of postselection. Postselection is a question about results of experiments, and every question about the results of actual measurements is legitimate. Subsequently, we discovered the concept of weak measurements, which in turn led us to discover the various effects we presented. Since the power of any new approach is given by its ability to predict new effects, one should conclude that ours is strong indeed.Furthermore, as many physicists agree, an intuitive understanding of quantum mechanics is still missing. That is why quantum physicists are surprised over and over again by the discovery of strange and unexpected fundamental effects. We hope that our new way of thinking is a step toward the long-sought intuition. Even more important, the new way of thinking may give us new ideas about what to change, if experiments ever turn out to contradict quantum mechanics and therefore require its modification. In particular, since we tinker with the idea of time— one of the most important concepts in physics—starting the change from there may be a very potent method.Shaul Mukamel refers to our experiment in which the component along some given axis of a spin-1⁄2 particle is found, by a weak measurement, to have the value √2‾/2 which is √2‾ times larger than the largest eigenvalue. He suggests an alternative explanation based on a classical vector model of spin. According to his explanation, values up to √3‾/2 should be possible. However, we presented the experiment showing √2‾/2 only because it was mathematically simple; by choosing a different postselection, we could have obtained, as results of weak measurements, values as large as we wanted. Hence the above simple classical-vector view doesn’t work.Robert Griffiths points out that there are two other time-symmetric formulations of quantum mechanics besides ours—“consistent histories” and “decoherent histories.” In particular, Griffiths is right when he emphasizes a major difference in spirit between our theory and consistent histories. The main goal of consistent histories is to find an explanation for the (apparent?) collapse of the wavefunction during a quantum measurement; although that solution is hotly disputed, as are all other solutions to the collapse problem, it is certainly a very ingenious one. However, an answer to the collapse problem is not our primary interest (though we are starting to see glimmers of an alternative answer to it using our approach).The letter by Griffiths, however, runs the risk of being misread as implying that solving the collapse problem will by itself clarify most or all of the counterintuitive aspects of quantum mechanics. That conclusion would be wrong. Quantum mechanics is strange and unusual and defies intuition in many ways; solving the collapse problem is by no means its only interesting fundamental issue. Nor can the solution of that one problem lead to a complete understanding of quantum phenomena. That particles tunnel in the first place is surprising by itself; even more so is the fact that, as we showed, perfectly good measuring devices, working with arbitrarily high precision, indicate consistently that the tunneling particles have negative kinetic energy. Equally surprising is that we can arrange a situation in which perfectly good measurements, made with as high a precision as we want, indicate that spin-1⁄2 particles have arbitrarily large spin. And these are only two examples. As far as we are aware, none of the many proposed solutions to the collapse problem make these effects seem less surprising, let alone predict them.© 2011 American Institute of Physics.