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Physics Today

On measurement and quantum nondemolition

H. D. Zeh

Published Jul 1, 2011• 2 citations
Publisher

Abstract

In his letter, Christopher Monroe recommends that the term “quantum nondemolition” be abandoned (PHYSICS TODAY, January 2011, page 8). I agree, but for somewhat different reasons. Quantum measurement theory has always used the concept of ideal measurements, or “measurements of the first kind,” in which the state of the measured system does not change—that is, it is not “demolished”—if it is already in an eigenstate of the observable. Although an ideal measurement is often a good approximation when dealing with conserved particles, its realization requires special effort in the case of photons, which are usually absorbed while being measured. The concept of ideal measurements is certainly meaningful, but I do not agree with the suggestion that the repeated preparation of the same photon state should be regarded as being equivalent to its nondemolition during a measurement. The term nondemolition may be quite useful in photon experiments or in the construction of gravitational-wave antennae, where it was first used, as far as I know. But in a general context, the traditional terminology would be more to the point, since nonabsorption of the measured photon does not necessarily define an ideal measurement.Much of the confusion originates in the abridged description of a measurement as a stochastic jump from a superposition a∣0〉 + b∣1〉 of a microscopic system into ∣0〉 or ∣1〉 with probabilities ∣a∣2 or ∣b∣2, respectively. That description is a relic from the times when quantum theory was expected to apply only to isolated atoms, electrons, and so forth. (It does not seem to be a genuine part of the Copenhagen interpretation, which describes measurement outcomes in classical terms and often assumes that the wavefunction loses its meaning thereafter.) In most situations, though, one has to include other interacting systems in the unitary description. In fact, all of the much-discussed weirdness of quantum theory has been derived by using entangled quantum states for all involved systems. Therefore, most of the inflationary confusing vocabulary of quantum weirdness—including spooky action, quantum teleportation, quantum eraser, quantum Darwinism, sudden death, quantum information, and even quantum jumps—could be abandoned if one used instead a consistent description in terms of entangled wavefunctions, at least as far as this entanglement remains relevant.11. See, for example, H. D. Zeh, Foundations of Physics 40 9-10 1476 (2010). https://doi.org/10.1007/s10701-009-9383-9 Usually this is so until irreversible decoherence, which was itself derived from unitary interaction with the environment, has occurred in the chain of interactions leading to observation. However, I think it is definitely wrong to say that decoherence theory allows us to “think of any measurement as a [quantum nondemolition] measurement plus a possible interaction with a reservoir.” Although I personally prefer the assumption that unitary dynamics is universal, I agree that some stochastic dynamics has to be used somewhere along the chain of interactions that ends at the observer. It does not matter whether you describe this observed indeterminism in terms of a collapse or a branching of a universal wavefunction into dynamically autonomous components (unlike the collapse, the branching does not require any new dynamical postulate; it is a consequence of the Schrödinger equation with local interactions). Such a real or apparent collapse is an important part of the effective dynamics of “our quantum world”—our “branch,” if you like—that should be carefully analyzed rather than being used ad hoc in a pragmatic and confusing manner. In neither case would it describe a mere increase of information about an initially incompletely known quantum state of the involved systems; the indeterminism is described by a change of a state, which is known to be different from an ensemble of potential states.REFERENCESSection:ChooseTop of pageREFERENCES <<1. See, for example, H. D. Zeh, Foundations of Physics 40 9-10 1476 (2010). https://doi.org/10.1007/s10701-009-9383-9 , Google ScholarCrossref, ISI© 2011 American Institute of Physics.

DOI
10.1063/pt.3.1143

Related papers

Demolishing quantum nondemolition

Christopher R. Monroe

Most physicists are taught that quantum physics has two parts: coherent wave mechanics, in which quantum states evolve according to the reversible Schrödinger wave equation, and the measurement process, in which quantum states irreversibly collapse into a particular state given by the details of the measurement. So when a quantum superposition state such as a|0〉 + b|1〉 is measured with a device that discriminates perfectly between |0〉 and |1〉, the system evolves to the definite state |0〉 or |1〉 with respective probabilities |a|2 and |b|2. Of course, when such an ideal measurement is repeated on the surviving state, the same definite state will appear again and again. Many of us, starting with Albert Einstein, do not like the apparent disconnect between those two rules and may not even subscribe to that conventional “Copenhagen interpretation” of quantum mechanics. It is nevertheless useful because it correctly predicts the results of all experiments performed to date, even though it gives the appearance that the observer necessarily affects the system.In the past two decades, the advent of quantum information science and the ability of experiments to control quantum systems such as single atoms have provided the happy opportunity to once again ponder the disconnect between the fundamental quantum concepts of coherent evolution and measurement. Quantum systems can now be measured with nearly perfect efficiency, meaning that the observed distribution of a measurement is given almost exactly by the probabilities of the underlying superposition, as prescribed by the conventional Copenhagen rules that we are taught.However, a new term, the quantum nondemolition, or QND, has crept into the literature these past decades. Also known as “quantum back-action-evading measurement,” it’s a needlessly confusing term that seems to highlight the mundane aspect of a perfect quantum measurement: If one already knows that the system is in a particular eigenstate of the measuring device, then, obviously, a measurement on the system will produce that eigenstate and leave the system intact. Zero information is gained from the repeated measurement. On the other hand, when the system is not in an eigenstate of the measuring device, the quantum state can be thought to collapse to one of its eigenstates. In that case, information is gained from the system, and the QND measurement most certainly demolishes the system. The concept of QND measurement adds nothing to the usual rules of quantum measurement, regardless of interpretation. But worse, the term confounds students and newcomers who may even be comfortable with the prescriptions of quantum physics.Not all measurements are perfect, of course, and some people may wish to use the QND nomenclature to distinguish the efficiency of a measuring device. However, in the spirit of quantum decoherence theory (see, for example, Wojciech Zurek’s article in PHYSICS TODAY, October 1991, page 36), we can still think of any measurement as a QND measurement plus a possible interaction with a reservoir that can either mask the information obtained from the system or even uncontrollably change the quantum state. The interaction with the reservoir is what can make quantum problems difficult, and it is there that we confront the conceptual difficulties with quantum mechanics and the quantum–classical divide.As a common example of an imperfect measurement, consider photodetection. When a single photon strikes a photomultiplier tube and generates a pulse of photocurrent, the photon is lost forever. Or is it? The photocurrent may interact with a macroscopic system of a bulk conductor, and we may measure the resulting voltage across the conductor. Sure, the photon has disappeared, but if our detector indicates that we had one photon, we can always create another and get the same answer again and again, exactly like a QND measurement. Even if we begin with a coherent superposition of zero and one photon, which is not an eigenstate of our measurement device, that approach works. After the first measurement collapses the state into |0〉 or |1〉, we know which one came out and we either prepare zero or one photon forever after. A real photodetector is not 100% efficient, of course, meaning that a single photon striking the detector could register nothing, or that even in the absence of photons the noisy detector fires anyway. Again we can think of this as a QND measurement in which we have lost information to the environment. In practice, we trace over the environment and then must describe the system in terms of a density matrix containing the probabilities of various outcomes.In every case, the concept of QND measurement is confusing and unnecessary. It wasn’t needed in the formative years of quantum mechanics; and now with quantum information theory, entanglement, and decoherence theory, we still do not need it. Marvelous advances in many experimental quantum systems, such as individual atoms and superconducting circuits, are allowing quantum measurements to be performed almost exactly like in textbooks, as most of us have learned. Why not demolish the term “QND”?© 2011 American Institute of Physics.

Physics Today•Dec 30, 2010•12 citations

Measured responses to quantum Bayesianism

Michael Nauenberg

Some time ago, John Bell and I wrote a tongue-in-cheek article on the interpretation of quantum mechanics;11. J. S. Bell, M. Nauenberg, in Preludes in Theoretical Physics, A. de Shalit, H. Feshbach, L. van Hove, eds., North Holland, Amsterdam (1966), p. 270;reprinted in J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge U. Press, New York (1987), p. 24. in it we remarked that only a minority of physicists had any interest in the topic and that the typical physicist thought he would understand it “if ever he can spare twenty minutes to think about it.” According to David Mermin, however, that situation has changed dramatically, and “new interpretations [of quantum mechanics] appear every year.”Actually, most textbooks on the subject continue to be based on the well-established Born interpretation of quantum mechanics, which postulates that the absolute square of the Schrödinger wavefunction, projected onto a particular quantum state, gives the probability for the occurrence of that state after an appropriate measurement has been made. Mermin indicated that the problem with the Born interpretation involves a confusion, “prevalent among physicists,” regarding the nature of probability. He presented his preference for a “personalist Bayesian view,” which leads separate “agents,” like Eugene Wigner and his friend, to make different probability assignments based on their beliefs. But as Richard Feynman remarked, “Nature does not know what you are looking at, and she behaves the way she is going to behave whether you bother to take down the data or not.”22. R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. 3, Addison-Wesley, Redwood City, CA (1989), p. 3–7.Indeed, as applied to experiments, probability simply corresponds to the relative frequency for the occurrence of a given final state after identical states are prepared initially and measurements are made by appropriate detectors—devices that, through an irreversible macroscopic process, provide a permanent record of the outcome. Hence, the results of an experiment are independent of any observers, whose only role is to set up appropriate measurement devices, calculate the probabilities from quantum theory, and then write papers based on the recorded results.As Max Born also emphasized, in classical mechanics the situation is analogous, when one takes into account that in practice initial conditions in an experiment are known only imprecisely.33. M. Born, Dan. Mat. Fys. Medd. 30 (2), 1 (1955). Hence the observed evolution of the system is also determined by a probability function that satisfies the Liouville equation. But in that case, after a measurement is performed, no one talks about the “collapse” of this classical function, whereas in quantum mechanics it is common to use that controversial expression for the Schrödinger wavefunction.Mermin discussed a “shifty split” between the classical and the quantum domain, a split that supposedly “haunts quantum mechanics.” He claimed that “regardless of what is split from what, all versions of the shifty split are vague and ambiguous.” However, that is not always the case, as a simple example shows. Consider an initial quantum wavepacket for a particle under an attractive central force that varies inversely with the square of the radial distance—the classic problem of planetary motion from the viewpoint of quantum mechanics, for example, or an excited electron in a Rydberg atom. For a finite time, the wavepacket will follow a classical Keplerian elliptic orbit, but like the Liouville distribution for the analogous classical problem, from the outset it will disperse due to the spread in initial position and momentum.For the classical distribution to be in close correspondence with the quantum wavepacket, its initial spread is assumed to satisfy the uncertainty relation. Eventually, however, the head of the wavepacket, like in the classical distribution, will catch up with its tail. Afterwards, further spreading of the wavepacket leads to interference effects between the two components, and the classical–quantum correspondence ceases to be valid. The time when such interference effects first appear is of order nτ/3, where n is the mean principal quantum number of the wavepacket and τ is the Kepler period.44. M. Nauenberg, Phys. Rev. A 40, 1133 (1989). https://doi.org/10.1103/PhysRevA.40.1133 Experimentally, that transition has been observed to agree with theory in Rydberg atoms55. M. Nauenberg, C. Stroud, J. Yeazell, Sci. Am., June 1994, p. 44. https://doi.org/10.1038/scientificamerican0694-44 for n = 100, τ = 152 picoseconds, and nothing shifty about this classical–quantum correspondence is found. For planetary orbits, however, n is of order 1070, and therefore the answer to Albert Einstein’s often repeated remark “Is the moon there when nobody looks?” (see David Mermin, Physics Today, April 1985, page 38) is yes.REFERENCESSection:ChooseTop of pageREFERENCES <<1. J. S. Bell, M. Nauenberg, in Preludes in Theoretical Physics, A. de Shalit, H. Feshbach, L. van Hove, eds., North Holland, Amsterdam (1966), p. 270; Google Scholarreprinted in J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge U. Press, New York (1987), p. 24. Google Scholar2. R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. 3, Addison-Wesley, Redwood City, CA (1989), p. 3–7. Google Scholar3. M. Born, Dan. Mat. Fys. Medd. 30 (2), 1 (1955). Google Scholar4. M. Nauenberg, Phys. Rev. A 40, 1133 (1989). https://doi.org/10.1103/PhysRevA.40.1133, Google ScholarCrossref, CAS5. M. Nauenberg, C. Stroud, J. Yeazell, Sci. Am., June 1994, p. 44. https://doi.org/10.1038/scientificamerican0694-44, Google ScholarCrossref© 2012 American Institute of Physics.

Physics Today•Nov 30, 2012•0 citations

Quantum Measurement

Bob Coecke, Aleks Kissinger

A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it. – Max Planck, 1936 The only way quantum theory allows us to interact with quantum systems is via quantum processes. Thus, the only way we can extract information about the state of a quantum system is to apply some non-deterministic process and observe which of the branches happened. For that reason, many people refer to the act of applying certain quantum processes as measurement and sometimes even as observation . A consequence of this unfortunate terminology is that one of the most touted ‘strange’ features of quantum theory is often misleadingly described as follows: ‘The mere act of observing a quantum state changes it.’ ‘Misleading’, since the concept of measurement described in the previous paragraph is far from the passive concept of observation familiar from our macroscopic world, but is rather a non-trivial process that will almost always drastically affect the quantum state. So, the mysterious aspect of quantum theory is not that ‘observation’ alters the quantum state but, rather, that it is impossible, even in principle, to ‘observe’ a quantum system in the classical sense. Given this fundamental restriction on how we can extract information from a quantum system, what can we learn about that system by means of a quantum process? The answer is, in fact, very little! First, performing a particular measurement could, for the vast majority of quantum states, yield any outcome i with a non-zero probability. In that case, obtaining an outcome i doesn't tell us much about what the state of the system was. Second, we say ‘was’, because the measurement will moreover irreversibly change the state according to the outcome. So rather than revealing the state of a system, a quantum measurement typically erases that state from history! Nonetheless, these quantum measurements are crucial to quantum theory since they constitute the only interface between us, in our classical world, and the quantum world. The great insight of the quantum computing community is that the non-deterministic changes induced by these quantum measurements are not a nuisance, but rather an extremely useful resource.

Cambridge University Press eBooks•Mar 16, 2017•0 citations

Many Worlds?: Everett, Quantum Theory, and Reality

Robert M. Wald

Few, if any, issues in physics have engendered as much discussion as the measurement problem in quantum mechanics. It is generally agreed that the `normal' dynamical evolution of the state vector in quantum mechanics is given by a unitary map. The linearity of this map implies that the state vector will, in general, be found in a superposition of eigenstates of a given observable (or, similarly, that the density matrix describing a subsystem will not correspond to a definite value of this observable). However, when we make a measurement of an observable, we always obtain a define value—although it is impossible to predict with certainty which value will be obtained. The traditional response to this issue is to postulate that when a measurement is made, the wavefunction `collapses' to an eigenstate of the observable being measured, perhaps due to the inherent classicality of the measuring apparatus (Bohr), or to the consciousness of the observer (Wigner), or possibly to some modification of quantum dynamics that occurs even when observations are not being made. The main motivation for the Everett (`many worlds') interpretation is to avoid introducing any such collapse postulate. This volume commemorates the 50th anniversary of the publication of Everett's paper in 1957 and contains 20 original articles as well as the transcripts of several discussions that took place at meetings devoted to the Everett interpretation at Oxford University and the Perimeter Institute. The attractiveness of the Everett interpretation is very succinctly summarized by a sentence from Vaidman's contribution (p 582): `The collapse, with its randomness, non-locality and the lack of a well-defined moment of occurrence, is such an ugly scar on quantum theory, that I, along with many others, am ready to follow Everett and deny its existence.' But the main drawback of the interpretation is then equally succinctly stated in the next sentence: `The price is the many worlds interpretation, i.e., the existence of numerous parallel worlds.' In other words, if the state vector evolves to have a non-zero amplitude for two different values of an observable that I `measure' (i.e., interact with in an appropriate way), each of these alternatives remains equally `real', i.e., there is a `branch' of the wavefunction in which the observable takes the first value and I believe that it has this first value and a branch of the wavefunction where it takes the second value and I believe that it has this second value. Each of these `me's' sees a definite value for the observable, thus nicely accounting for what I believe I experience without invoking collapse. But then, it seems that I have to believe that there are many other `me's' around—even if they are very difficult for this particular me to access—and no way I can predict which `me' I will be in the future. Indeed, Saunders writes on p 192, `If Alice is a person (of course she is) then we must say, even prior to branching at tj that there are many Alices present, atom for atom duplicates...Alice should be uncertain after all— each Alice should be uncertain—for each as of tj does not (and as a matter of principle, cannot) know which of these branching persons she is.' Thus, if taken seriously, the Everett viewpoint leads directly to questions like `What determines which ``me'' I am now and/or will feel that I am in the future?' The response `You are all of them' somehow doesn't seem satisfying, and further pursuit of these issues would seem to lead to questions as fruitless as questions like `Why am I ``me'' and not ``you''?' I believe that this accounts for why the Everett interpretation does not have universal appeal. But enough of `me' for now (more later); let's discuss the book. Unlike many edited volumes and conference proceedings in physics—which often consist of hastily slapped-together re-hashes of previous papers—this volume meets very high standards in the choice of topics and authors and in the quality of the writing. Although each of the articles is closely tied in with the Everett interpretation, there are many articles of considerable value and interest in their own right, such as the discussion of quasiclassical realms by Hartle and and the discussion of the de Broglie–Bohm interpretation by Valentini. There is also an excellent introduction by Saunders to the quantum measurement problem and the main issues that arise in the Everett interpretation. A central issue in the Everett interpretation is the status of the `Born rule', which asserts that, for state ψ, the probability of obtaining a particular outcome of a measurement is ||Pψ||2, where P is the projection operator onto the eigensubspace associated with the measurement outcome. In traditional interpretations, the Born rule is simply postulated as part of the collapse hypothesis. In the Everett interpretation, it is far from obvious that the Born rule even has any meaning—if all outcomes occur, how can one talk about the probability of a particular outcome? Given the importance of this issue, it is highly appropriate that four chapters of the book (by Saunders, Papineau, Wallace, and Greaves and Myrvold) are devoted to addressing probability and the Born rule from the Everett viewpoint, and three chapters (by Kent, Albert, and Price) are devoted to criticising these views. As indicated by the above quote from Saunders' article, uncertainty and probability can enter the Everett interpretation via each Alice being uncertain as to which Alice she is or will be. However, in view of the difficulties to which this kind of argumentation will lead, it is perhaps not surprising that in the 123 pages of combined discussion by Saunders, Papineau, Wallace, and Greaves and Myrvold, there are only fleeting references to the use of probability along these lines. Rather, the approach taken is reminiscent of that of adept politicians: if you can't answer a question, then answer a different question, and answer it with conviction and detail. Instead of considering probabilities of outcomes—a highly problematical notion when all of the possible outcomes actually occur—consider only decision theory issues. In particular, Wallace gives a precise and detailed mathematical proof that, under certain hypotheses, the only rational strategy that a quantum observer can follow in making decisions is to weight the possible outcomes by Born rule probabilities. To illustrate this more graphically, suppose tomorrow's weather were determined by some quantum mechanical, Stern–Gerlach-like process. I would like to know whether it is likely to rain tomorrow. However, when I ask an Everett follower (who knows the state vector and has done the Born rule calculation), all she will tell me in direct response to my question is that `it will rain and it won't rain.' Nevertheless, she is willing tell me whether or not it is rational for me to take an umbrella. I suppose that if she tells me that it would be highly irrational for me to take an umbrella, I can take the hint and deduce that it is very unlikely to rain. However, there does seem something wrong with the fact that she is not allowed to say this directly. In any case, if the conclusion of a mathematically correct argument is that rational decision strategies require the Born rule, then there must be quite a bit lying in the assumptions. The articles by Kent, Albert, and Price do a good job of fleshing out these assumptions and pointing out the weaknesses and flaws in the probability and decision theory discussions within the Everett framework. I hope it is clear from what I have written above that this is an outstanding volume, which should be of great interest to anyone interested in the foundations of quantum theory. Indeed, I feel quite fortunate to be the particular `me' that happens to exist in the same branch of the wavefunction of the universe in which this book was published.

Classical and Quantum Gravity•Apr 21, 2011•24 citations

More on the quantum measurement problem

Gregory N. Derry

Although I have neither an objection to nor an endorsement of David Mermin’s major premise in his Quick Study, “There is no quantum measurement problem” (Physics Today, June 2022, page 62), I would like to make several contextual points.Mermin’s argument has an extensive history, along with a comparable history of rebuttal. Five decades ago Max Jammer wrote that “measurements in quantum mechanics are no more or less problematic than in classical physics, for the Hilbert space vector is only a purely formal device for relating the statistics associated with these arrangements to the physics of observations in classical physics.” That seems like a condensed version of the argument Mermin presents. On the following page, however, Jammer notes that a “problematic aspect whose serious implications were only gradually understood was the fact that as long as a quantum mechanical one-body or many-body system does not interact with macroscopic objects, as long as its motion is described by the deterministic Schrödinger time-dependent equation, no events could be considered to take place in the system.”11. M. Jammer, The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective, Wiley (1974), pp. 473, 474.Niels Bohr giving a lecture in Copenhagen in April 1929. (Photograph by Samuel Goudsmit, courtesy of the AIP Emilio Segrè Visual Archives, Goudsmit Collection.)PPT|High resolutionRegarding the quotations from Niels Bohr, Mermin’s point is entirely valid and represents Bohr’s position accurately, but Bohr’s thought is always subtle and difficult to capture completely. Another quotation from Bohr, which is consistent with everything else he wrote, is, “The decisive point is that in neither case does the appropriate widening of our conceptual framework imply any appeal to the observing subject, which would hinder unambiguous communication of experience.”22. N. Bohr, Essays 1958–1962 on Atomic Physics and Human Knowledge, Ox Bow Press (1987), p. 7. That quotation seems inconsistent with Mermin’s paragraph that includes the passage “Why should I insist that my interpretation of science, which I use to make sense of the world that I experience, should never make any mention of me?” Bohr always insisted that his framework was a rational “generalization” of classical objectivity.The last and most conceptually important point concerns the ontological status of probability. I don’t necessarily disagree with Mermin and Bruno de Finetti, but it is by no means obvious that they are correct either. David Hawkins, for example, begins 50 pages of rigorous analysis with a brief statement of a major premise.The obvious and natural interpretation of probability in this context is that it introduces, not something that has to do with the nature of the physical system, but something that has to do with the incompleteness of our knowledge….But this opinion turns out to be false. Ignorance of dynamical regularity may be a motive for resorting to probabilities, but it does not explain the probabilities to which we resort.33. D. Hawkins, The Language of Nature: An Essay in the Philosophy of Science, W. H. Freeman (1964), p. 135.Another example can be found in Jammer’s well-known historical account of quantum theory: “For Einstein the notion of probability … was the traditional conception of classical physics, a mathematical objectivization of the human deficiency of complete or exact knowledge but ultimately a creation of the human mind…. For Born probability, as far as it was related to the wave function, was not merely a mathematical fiction but something endowed with physical reality.”44. M. Jammer, The Conceptual Development of Quantum Mechanics, McGraw-Hill (1966), p. 286.Of course, such examples don’t mean that Mermin’s position is incorrect, but I do think that they demand a better argument than a comparison to “Fairies and Witches.” I am confident that Mermin has such an argument, and I look forward to reading it as I continue working toward a coherent position on the unresolved interpretational issues of quantum theory.ReferencesSection:ChooseTop of pageReferences <<CITING ARTICLES1. M. Jammer, The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective, Wiley (1974), pp. 473, 474. Google Scholar2. N. Bohr, Essays 1958–1962 on Atomic Physics and Human Knowledge, Ox Bow Press (1987), p. 7. Google Scholar3. D. Hawkins, The Language of Nature: An Essay in the Philosophy of Science, W. H. Freeman (1964), p. 135. Google Scholar4. M. Jammer, The Conceptual Development of Quantum Mechanics, McGraw-Hill (1966), p. 286. Google Scholar© 2022 American Institute of Physics.

Physics Today•Nov 1, 2022•2 citations

More on the quantum measurement problem

Donald G. Truhlar

Sean Carroll’s July Quick Study, “Addressing the quantum measurement problem” (page 62), brings up the following question: Does the wavefunction still obey the Schrödinger equation when a measurement is made? A system being measured (or interacting with its environment in any other way) is actually a subsystem, and a subsystem is properly described by a reduced density matrix. The density matrix for an entire system corresponds to a wavefunction—that is, to a pure state—but the density matrix for a subsystem does not necessarily correspond to a wavefunction. The reduced density matrix of a subsystem may correspond to an impure state, also called a mixture or an incoherent combination,11. A. Bohm, Quantum Mechanics: Foundations and Applications, 3rd ed., Springer (1993), p. 72. which does not have well-defined pure-state content.22. See, for example, L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific (2015), chap. 2. In the words of Kurt Gottfried and Tung-Mow Yan, “systems in the real world are rarely in pure states.”33. K. Gottfried, T.-M. Yan, Quantum Mechanics: Fundamentals, 2nd ed., Springer (2003), p. 46.The proper way to discuss a measurement is not using a wavefunction but rather a reduced density matrix. The density matrix of a pure state evolves according to the Liouville–von Neumann equation, which is equivalent to the unitary evolution of the wavefunction by the time-dependent Schrödinger equation. For a subsystem (that is, for any system except the entire universe), the reduced density matrix evolves according to the nonunitary Liouville–von Neumann equation, which has an additional contribution causing decoherence and dissipation.44. See, for example, D. A. Micha, Int. J. Quantum Chem. 80, 394 (2000). https://doi.org/10.1002/1097-461X(2000)80:3<394::AID-QUA9>3.0.CO;2-U The nature of the measurement—or, more generally, the nature of the subsystem–environment interaction—selects a preferred basis, called the pointer basis, and the subsystem decoheres into an effectively classical mixture in the pointer basis. (See the article by Wojciech Zurek, Physics Today, October 1991, page 36.)ReferencesSection:ChooseTop of pageReferences <<1. A. Bohm, Quantum Mechanics: Foundations and Applications, 3rd ed., Springer (1993), p. 72. Google Scholar2. See, for example, L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific (2015), chap. 2. Google Scholar3. K. Gottfried, T.-M. Yan, Quantum Mechanics: Fundamentals, 2nd ed., Springer (2003), p. 46. Google ScholarCrossref4. See, for example, D. A. Micha, Int. J. Quantum Chem. 80, 394 (2000). https://doi.org/10.1002/1097-461X(2000)80:3<394::AID-QUA9>3.0.CO;2-U, Google ScholarCrossref© 2022 American Institute of Physics.

Physics Today•Nov 1, 2022•0 citations
In his letter, Christopher Monroe recommends that the term “quantum nondemolition” be abandoned (PHYSICS TODAY, January 2011, page 8). I agree, but for somewhat different reasons. Quantum measurement theory has always used the concept of ideal measurements, or “measurements of the first kind,” in which the state of the measured system does not change—that is, it is not “demolished”—if it is already in an eigenstate of the observable. Although an ideal measurement is often a good approximation when dealing with conserved particles, its realization requires special effort in the case of photons, which are usually absorbed while being measured. The concept of ideal measurements is certainly meaningful, but I do not agree with the suggestion that the repeated preparation of the same photon state should be regarded as being equivalent to its nondemolition during a measurement. The term nondemolition may be quite useful in photon experiments or in the construction of gravitational-wave antennae, where it was first used, as far as I know. But in a general context, the traditional terminology would be more to the point, since nonabsorption of the measured photon does not necessarily define an ideal measurement.Much of the confusion originates in the abridged description of a measurement as a stochastic jump from a superposition a∣0〉 + b∣1〉 of a microscopic system into ∣0〉 or ∣1〉 with probabilities ∣a∣2 or ∣b∣2, respectively. That description is a relic from the times when quantum theory was expected to apply only to isolated atoms, electrons, and so forth. (It does not seem to be a genuine part of the Copenhagen interpretation, which describes measurement outcomes in classical terms and often assumes that the wavefunction loses its meaning thereafter.) In most situations, though, one has to include other interacting systems in the unitary description. In fact, all of the much-discussed weirdness of quantum theory has been derived by using entangled quantum states for all involved systems. Therefore, most of the inflationary confusing vocabulary of quantum weirdness—including spooky action, quantum teleportation, quantum eraser, quantum Darwinism, sudden death, quantum information, and even quantum jumps—could be abandoned if one used instead a consistent description in terms of entangled wavefunctions, at least as far as this entanglement remains relevant.11. See, for example, H. D. Zeh, Foundations of Physics 40 9-10 1476 (2010). https://doi.org/10.1007/s10701-009-9383-9 Usually this is so until irreversible decoherence, which was itself derived from unitary interaction with the environment, has occurred in the chain of interactions leading to observation. However, I think it is definitely wrong to say that decoherence theory allows us to “think of any measurement as a [quantum nondemolition] measurement plus a possible interaction with a reservoir.” Although I personally prefer the assumption that unitary dynamics is universal, I agree that some stochastic dynamics has to be used somewhere along the chain of interactions that ends at the observer. It does not matter whether you describe this observed indeterminism in terms of a collapse or a branching of a universal wavefunction into dynamically autonomous components (unlike the collapse, the branching does not require any new dynamical postulate; it is a consequence of the Schrödinger equation with local interactions). Such a real or apparent collapse is an important part of the effective dynamics of “our quantum world”—our “branch,” if you like—that should be carefully analyzed rather than being used ad hoc in a pragmatic and confusing manner. In neither case would it describe a mere increase of information about an initially incompletely known quantum state of the involved systems; the indeterminism is described by a change of a state, which is known to be different from an ensemble of potential states.REFERENCESSection:ChooseTop of pageREFERENCES <<1. See, for example, H. D. Zeh, Foundations of Physics 40 9-10 1476 (2010). https://doi.org/10.1007/s10701-009-9383-9 , Google ScholarCrossref, ISI© 2011 American Institute of Physics.
Most physicists are taught that quantum physics has two parts: coherent wave mechanics, in which quantum states evolve according to the reversible Schrödinger wave equation, and the measurement process, in which quantum states irreversibly collapse into a particular state given by the details of the measurement. So when a quantum superposition state such as a|0〉 + b|1〉 is measured with a device that discriminates perfectly between |0〉 and |1〉, the system evolves to the definite state |0〉 or |1〉 with respective probabilities |a|2 and |b|2. Of course, when such an ideal measurement is repeated on the surviving state, the same definite state will appear again and again. Many of us, starting with Albert Einstein, do not like the apparent disconnect between those two rules and may not even subscribe to that conventional “Copenhagen interpretation” of quantum mechanics. It is nevertheless useful because it correctly predicts the results of all experiments performed to date, even though it gives the appearance that the observer necessarily affects the system.In the past two decades, the advent of quantum information science and the ability of experiments to control quantum systems such as single atoms have provided the happy opportunity to once again ponder the disconnect between the fundamental quantum concepts of coherent evolution and measurement. Quantum systems can now be measured with nearly perfect efficiency, meaning that the observed distribution of a measurement is given almost exactly by the probabilities of the underlying superposition, as prescribed by the conventional Copenhagen rules that we are taught.However, a new term, the quantum nondemolition, or QND, has crept into the literature these past decades. Also known as “quantum back-action-evading measurement,” it’s a needlessly confusing term that seems to highlight the mundane aspect of a perfect quantum measurement: If one already knows that the system is in a particular eigenstate of the measuring device, then, obviously, a measurement on the system will produce that eigenstate and leave the system intact. Zero information is gained from the repeated measurement. On the other hand, when the system is not in an eigenstate of the measuring device, the quantum state can be thought to collapse to one of its eigenstates. In that case, information is gained from the system, and the QND measurement most certainly demolishes the system. The concept of QND measurement adds nothing to the usual rules of quantum measurement, regardless of interpretation. But worse, the term confounds students and newcomers who may even be comfortable with the prescriptions of quantum physics.Not all measurements are perfect, of course, and some people may wish to use the QND nomenclature to distinguish the efficiency of a measuring device. However, in the spirit of quantum decoherence theory (see, for example, Wojciech Zurek’s article in PHYSICS TODAY, October 1991, page 36), we can still think of any measurement as a QND measurement plus a possible interaction with a reservoir that can either mask the information obtained from the system or even uncontrollably change the quantum state. The interaction with the reservoir is what can make quantum problems difficult, and it is there that we confront the conceptual difficulties with quantum mechanics and the quantum–classical divide.As a common example of an imperfect measurement, consider photodetection. When a single photon strikes a photomultiplier tube and generates a pulse of photocurrent, the photon is lost forever. Or is it? The photocurrent may interact with a macroscopic system of a bulk conductor, and we may measure the resulting voltage across the conductor. Sure, the photon has disappeared, but if our detector indicates that we had one photon, we can always create another and get the same answer again and again, exactly like a QND measurement. Even if we begin with a coherent superposition of zero and one photon, which is not an eigenstate of our measurement device, that approach works. After the first measurement collapses the state into |0〉 or |1〉, we know which one came out and we either prepare zero or one photon forever after. A real photodetector is not 100% efficient, of course, meaning that a single photon striking the detector could register nothing, or that even in the absence of photons the noisy detector fires anyway. Again we can think of this as a QND measurement in which we have lost information to the environment. In practice, we trace over the environment and then must describe the system in terms of a density matrix containing the probabilities of various outcomes.In every case, the concept of QND measurement is confusing and unnecessary. It wasn’t needed in the formative years of quantum mechanics; and now with quantum information theory, entanglement, and decoherence theory, we still do not need it. Marvelous advances in many experimental quantum systems, such as individual atoms and superconducting circuits, are allowing quantum measurements to be performed almost exactly like in textbooks, as most of us have learned. Why not demolish the term “QND”?© 2011 American Institute of Physics.
Some time ago, John Bell and I wrote a tongue-in-cheek article on the interpretation of quantum mechanics;11. J. S. Bell, M. Nauenberg, in Preludes in Theoretical Physics, A. de Shalit, H. Feshbach, L. van Hove, eds., North Holland, Amsterdam (1966), p. 270;reprinted in J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge U. Press, New York (1987), p. 24. in it we remarked that only a minority of physicists had any interest in the topic and that the typical physicist thought he would understand it “if ever he can spare twenty minutes to think about it.” According to David Mermin, however, that situation has changed dramatically, and “new interpretations [of quantum mechanics] appear every year.”Actually, most textbooks on the subject continue to be based on the well-established Born interpretation of quantum mechanics, which postulates that the absolute square of the Schrödinger wavefunction, projected onto a particular quantum state, gives the probability for the occurrence of that state after an appropriate measurement has been made. Mermin indicated that the problem with the Born interpretation involves a confusion, “prevalent among physicists,” regarding the nature of probability. He presented his preference for a “personalist Bayesian view,” which leads separate “agents,” like Eugene Wigner and his friend, to make different probability assignments based on their beliefs. But as Richard Feynman remarked, “Nature does not know what you are looking at, and she behaves the way she is going to behave whether you bother to take down the data or not.”22. R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. 3, Addison-Wesley, Redwood City, CA (1989), p. 3–7.Indeed, as applied to experiments, probability simply corresponds to the relative frequency for the occurrence of a given final state after identical states are prepared initially and measurements are made by appropriate detectors—devices that, through an irreversible macroscopic process, provide a permanent record of the outcome. Hence, the results of an experiment are independent of any observers, whose only role is to set up appropriate measurement devices, calculate the probabilities from quantum theory, and then write papers based on the recorded results.As Max Born also emphasized, in classical mechanics the situation is analogous, when one takes into account that in practice initial conditions in an experiment are known only imprecisely.33. M. Born, Dan. Mat. Fys. Medd. 30 (2), 1 (1955). Hence the observed evolution of the system is also determined by a probability function that satisfies the Liouville equation. But in that case, after a measurement is performed, no one talks about the “collapse” of this classical function, whereas in quantum mechanics it is common to use that controversial expression for the Schrödinger wavefunction.Mermin discussed a “shifty split” between the classical and the quantum domain, a split that supposedly “haunts quantum mechanics.” He claimed that “regardless of what is split from what, all versions of the shifty split are vague and ambiguous.” However, that is not always the case, as a simple example shows. Consider an initial quantum wavepacket for a particle under an attractive central force that varies inversely with the square of the radial distance—the classic problem of planetary motion from the viewpoint of quantum mechanics, for example, or an excited electron in a Rydberg atom. For a finite time, the wavepacket will follow a classical Keplerian elliptic orbit, but like the Liouville distribution for the analogous classical problem, from the outset it will disperse due to the spread in initial position and momentum.For the classical distribution to be in close correspondence with the quantum wavepacket, its initial spread is assumed to satisfy the uncertainty relation. Eventually, however, the head of the wavepacket, like in the classical distribution, will catch up with its tail. Afterwards, further spreading of the wavepacket leads to interference effects between the two components, and the classical–quantum correspondence ceases to be valid. The time when such interference effects first appear is of order nτ/3, where n is the mean principal quantum number of the wavepacket and τ is the Kepler period.44. M. Nauenberg, Phys. Rev. A 40, 1133 (1989). https://doi.org/10.1103/PhysRevA.40.1133 Experimentally, that transition has been observed to agree with theory in Rydberg atoms55. M. Nauenberg, C. Stroud, J. Yeazell, Sci. Am., June 1994, p. 44. https://doi.org/10.1038/scientificamerican0694-44 for n = 100, τ = 152 picoseconds, and nothing shifty about this classical–quantum correspondence is found. For planetary orbits, however, n is of order 1070, and therefore the answer to Albert Einstein’s often repeated remark “Is the moon there when nobody looks?” (see David Mermin, Physics Today, April 1985, page 38) is yes.REFERENCESSection:ChooseTop of pageREFERENCES <<1. J. S. Bell, M. Nauenberg, in Preludes in Theoretical Physics, A. de Shalit, H. Feshbach, L. van Hove, eds., North Holland, Amsterdam (1966), p. 270; Google Scholarreprinted in J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge U. Press, New York (1987), p. 24. Google Scholar2. R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. 3, Addison-Wesley, Redwood City, CA (1989), p. 3–7. Google Scholar3. M. Born, Dan. Mat. Fys. Medd. 30 (2), 1 (1955). Google Scholar4. M. Nauenberg, Phys. Rev. A 40, 1133 (1989). https://doi.org/10.1103/PhysRevA.40.1133, Google ScholarCrossref, CAS5. M. Nauenberg, C. Stroud, J. Yeazell, Sci. Am., June 1994, p. 44. https://doi.org/10.1038/scientificamerican0694-44, Google ScholarCrossref© 2012 American Institute of Physics.
A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it. – Max Planck, 1936 The only way quantum theory allows us to interact with quantum systems is via quantum processes. Thus, the only way we can extract information about the state of a quantum system is to apply some non-deterministic process and observe which of the branches happened. For that reason, many people refer to the act of applying certain quantum processes as measurement and sometimes even as observation . A consequence of this unfortunate terminology is that one of the most touted ‘strange’ features of quantum theory is often misleadingly described as follows: ‘The mere act of observing a quantum state changes it.’ ‘Misleading’, since the concept of measurement described in the previous paragraph is far from the passive concept of observation familiar from our macroscopic world, but is rather a non-trivial process that will almost always drastically affect the quantum state. So, the mysterious aspect of quantum theory is not that ‘observation’ alters the quantum state but, rather, that it is impossible, even in principle, to ‘observe’ a quantum system in the classical sense. Given this fundamental restriction on how we can extract information from a quantum system, what can we learn about that system by means of a quantum process? The answer is, in fact, very little! First, performing a particular measurement could, for the vast majority of quantum states, yield any outcome i with a non-zero probability. In that case, obtaining an outcome i doesn't tell us much about what the state of the system was. Second, we say ‘was’, because the measurement will moreover irreversibly change the state according to the outcome. So rather than revealing the state of a system, a quantum measurement typically erases that state from history! Nonetheless, these quantum measurements are crucial to quantum theory since they constitute the only interface between us, in our classical world, and the quantum world. The great insight of the quantum computing community is that the non-deterministic changes induced by these quantum measurements are not a nuisance, but rather an extremely useful resource.
Few, if any, issues in physics have engendered as much discussion as the measurement problem in quantum mechanics. It is generally agreed that the `normal' dynamical evolution of the state vector in quantum mechanics is given by a unitary map. The linearity of this map implies that the state vector will, in general, be found in a superposition of eigenstates of a given observable (or, similarly, that the density matrix describing a subsystem will not correspond to a definite value of this observable). However, when we make a measurement of an observable, we always obtain a define value—although it is impossible to predict with certainty which value will be obtained. The traditional response to this issue is to postulate that when a measurement is made, the wavefunction `collapses' to an eigenstate of the observable being measured, perhaps due to the inherent classicality of the measuring apparatus (Bohr), or to the consciousness of the observer (Wigner), or possibly to some modification of quantum dynamics that occurs even when observations are not being made. The main motivation for the Everett (`many worlds') interpretation is to avoid introducing any such collapse postulate. This volume commemorates the 50th anniversary of the publication of Everett's paper in 1957 and contains 20 original articles as well as the transcripts of several discussions that took place at meetings devoted to the Everett interpretation at Oxford University and the Perimeter Institute. The attractiveness of the Everett interpretation is very succinctly summarized by a sentence from Vaidman's contribution (p 582): `The collapse, with its randomness, non-locality and the lack of a well-defined moment of occurrence, is such an ugly scar on quantum theory, that I, along with many others, am ready to follow Everett and deny its existence.' But the main drawback of the interpretation is then equally succinctly stated in the next sentence: `The price is the many worlds interpretation, i.e., the existence of numerous parallel worlds.' In other words, if the state vector evolves to have a non-zero amplitude for two different values of an observable that I `measure' (i.e., interact with in an appropriate way), each of these alternatives remains equally `real', i.e., there is a `branch' of the wavefunction in which the observable takes the first value and I believe that it has this first value and a branch of the wavefunction where it takes the second value and I believe that it has this second value. Each of these `me's' sees a definite value for the observable, thus nicely accounting for what I believe I experience without invoking collapse. But then, it seems that I have to believe that there are many other `me's' around—even if they are very difficult for this particular me to access—and no way I can predict which `me' I will be in the future. Indeed, Saunders writes on p 192, `If Alice is a person (of course she is) then we must say, even prior to branching at tj that there are many Alices present, atom for atom duplicates...Alice should be uncertain after all— each Alice should be uncertain—for each as of tj does not (and as a matter of principle, cannot) know which of these branching persons she is.' Thus, if taken seriously, the Everett viewpoint leads directly to questions like `What determines which ``me'' I am now and/or will feel that I am in the future?' The response `You are all of them' somehow doesn't seem satisfying, and further pursuit of these issues would seem to lead to questions as fruitless as questions like `Why am I ``me'' and not ``you''?' I believe that this accounts for why the Everett interpretation does not have universal appeal. But enough of `me' for now (more later); let's discuss the book. Unlike many edited volumes and conference proceedings in physics—which often consist of hastily slapped-together re-hashes of previous papers—this volume meets very high standards in the choice of topics and authors and in the quality of the writing. Although each of the articles is closely tied in with the Everett interpretation, there are many articles of considerable value and interest in their own right, such as the discussion of quasiclassical realms by Hartle and and the discussion of the de Broglie–Bohm interpretation by Valentini. There is also an excellent introduction by Saunders to the quantum measurement problem and the main issues that arise in the Everett interpretation. A central issue in the Everett interpretation is the status of the `Born rule', which asserts that, for state ψ, the probability of obtaining a particular outcome of a measurement is ||Pψ||2, where P is the projection operator onto the eigensubspace associated with the measurement outcome. In traditional interpretations, the Born rule is simply postulated as part of the collapse hypothesis. In the Everett interpretation, it is far from obvious that the Born rule even has any meaning—if all outcomes occur, how can one talk about the probability of a particular outcome? Given the importance of this issue, it is highly appropriate that four chapters of the book (by Saunders, Papineau, Wallace, and Greaves and Myrvold) are devoted to addressing probability and the Born rule from the Everett viewpoint, and three chapters (by Kent, Albert, and Price) are devoted to criticising these views. As indicated by the above quote from Saunders' article, uncertainty and probability can enter the Everett interpretation via each Alice being uncertain as to which Alice she is or will be. However, in view of the difficulties to which this kind of argumentation will lead, it is perhaps not surprising that in the 123 pages of combined discussion by Saunders, Papineau, Wallace, and Greaves and Myrvold, there are only fleeting references to the use of probability along these lines. Rather, the approach taken is reminiscent of that of adept politicians: if you can't answer a question, then answer a different question, and answer it with conviction and detail. Instead of considering probabilities of outcomes—a highly problematical notion when all of the possible outcomes actually occur—consider only decision theory issues. In particular, Wallace gives a precise and detailed mathematical proof that, under certain hypotheses, the only rational strategy that a quantum observer can follow in making decisions is to weight the possible outcomes by Born rule probabilities. To illustrate this more graphically, suppose tomorrow's weather were determined by some quantum mechanical, Stern–Gerlach-like process. I would like to know whether it is likely to rain tomorrow. However, when I ask an Everett follower (who knows the state vector and has done the Born rule calculation), all she will tell me in direct response to my question is that `it will rain and it won't rain.' Nevertheless, she is willing tell me whether or not it is rational for me to take an umbrella. I suppose that if she tells me that it would be highly irrational for me to take an umbrella, I can take the hint and deduce that it is very unlikely to rain. However, there does seem something wrong with the fact that she is not allowed to say this directly. In any case, if the conclusion of a mathematically correct argument is that rational decision strategies require the Born rule, then there must be quite a bit lying in the assumptions. The articles by Kent, Albert, and Price do a good job of fleshing out these assumptions and pointing out the weaknesses and flaws in the probability and decision theory discussions within the Everett framework. I hope it is clear from what I have written above that this is an outstanding volume, which should be of great interest to anyone interested in the foundations of quantum theory. Indeed, I feel quite fortunate to be the particular `me' that happens to exist in the same branch of the wavefunction of the universe in which this book was published.
Although I have neither an objection to nor an endorsement of David Mermin’s major premise in his Quick Study, “There is no quantum measurement problem” (Physics Today, June 2022, page 62), I would like to make several contextual points.Mermin’s argument has an extensive history, along with a comparable history of rebuttal. Five decades ago Max Jammer wrote that “measurements in quantum mechanics are no more or less problematic than in classical physics, for the Hilbert space vector is only a purely formal device for relating the statistics associated with these arrangements to the physics of observations in classical physics.” That seems like a condensed version of the argument Mermin presents. On the following page, however, Jammer notes that a “problematic aspect whose serious implications were only gradually understood was the fact that as long as a quantum mechanical one-body or many-body system does not interact with macroscopic objects, as long as its motion is described by the deterministic Schrödinger time-dependent equation, no events could be considered to take place in the system.”11. M. Jammer, The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective, Wiley (1974), pp. 473, 474.Niels Bohr giving a lecture in Copenhagen in April 1929. (Photograph by Samuel Goudsmit, courtesy of the AIP Emilio Segrè Visual Archives, Goudsmit Collection.)PPT|High resolutionRegarding the quotations from Niels Bohr, Mermin’s point is entirely valid and represents Bohr’s position accurately, but Bohr’s thought is always subtle and difficult to capture completely. Another quotation from Bohr, which is consistent with everything else he wrote, is, “The decisive point is that in neither case does the appropriate widening of our conceptual framework imply any appeal to the observing subject, which would hinder unambiguous communication of experience.”22. N. Bohr, Essays 1958–1962 on Atomic Physics and Human Knowledge, Ox Bow Press (1987), p. 7. That quotation seems inconsistent with Mermin’s paragraph that includes the passage “Why should I insist that my interpretation of science, which I use to make sense of the world that I experience, should never make any mention of me?” Bohr always insisted that his framework was a rational “generalization” of classical objectivity.The last and most conceptually important point concerns the ontological status of probability. I don’t necessarily disagree with Mermin and Bruno de Finetti, but it is by no means obvious that they are correct either. David Hawkins, for example, begins 50 pages of rigorous analysis with a brief statement of a major premise.The obvious and natural interpretation of probability in this context is that it introduces, not something that has to do with the nature of the physical system, but something that has to do with the incompleteness of our knowledge….But this opinion turns out to be false. Ignorance of dynamical regularity may be a motive for resorting to probabilities, but it does not explain the probabilities to which we resort.33. D. Hawkins, The Language of Nature: An Essay in the Philosophy of Science, W. H. Freeman (1964), p. 135.Another example can be found in Jammer’s well-known historical account of quantum theory: “For Einstein the notion of probability … was the traditional conception of classical physics, a mathematical objectivization of the human deficiency of complete or exact knowledge but ultimately a creation of the human mind…. For Born probability, as far as it was related to the wave function, was not merely a mathematical fiction but something endowed with physical reality.”44. M. Jammer, The Conceptual Development of Quantum Mechanics, McGraw-Hill (1966), p. 286.Of course, such examples don’t mean that Mermin’s position is incorrect, but I do think that they demand a better argument than a comparison to “Fairies and Witches.” I am confident that Mermin has such an argument, and I look forward to reading it as I continue working toward a coherent position on the unresolved interpretational issues of quantum theory.ReferencesSection:ChooseTop of pageReferences <<CITING ARTICLES1. M. Jammer, The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective, Wiley (1974), pp. 473, 474. Google Scholar2. N. Bohr, Essays 1958–1962 on Atomic Physics and Human Knowledge, Ox Bow Press (1987), p. 7. Google Scholar3. D. Hawkins, The Language of Nature: An Essay in the Philosophy of Science, W. H. Freeman (1964), p. 135. Google Scholar4. M. Jammer, The Conceptual Development of Quantum Mechanics, McGraw-Hill (1966), p. 286. Google Scholar© 2022 American Institute of Physics.
Sean Carroll’s July Quick Study, “Addressing the quantum measurement problem” (page 62), brings up the following question: Does the wavefunction still obey the Schrödinger equation when a measurement is made? A system being measured (or interacting with its environment in any other way) is actually a subsystem, and a subsystem is properly described by a reduced density matrix. The density matrix for an entire system corresponds to a wavefunction—that is, to a pure state—but the density matrix for a subsystem does not necessarily correspond to a wavefunction. The reduced density matrix of a subsystem may correspond to an impure state, also called a mixture or an incoherent combination,11. A. Bohm, Quantum Mechanics: Foundations and Applications, 3rd ed., Springer (1993), p. 72. which does not have well-defined pure-state content.22. See, for example, L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific (2015), chap. 2. In the words of Kurt Gottfried and Tung-Mow Yan, “systems in the real world are rarely in pure states.”33. K. Gottfried, T.-M. Yan, Quantum Mechanics: Fundamentals, 2nd ed., Springer (2003), p. 46.The proper way to discuss a measurement is not using a wavefunction but rather a reduced density matrix. The density matrix of a pure state evolves according to the Liouville–von Neumann equation, which is equivalent to the unitary evolution of the wavefunction by the time-dependent Schrödinger equation. For a subsystem (that is, for any system except the entire universe), the reduced density matrix evolves according to the nonunitary Liouville–von Neumann equation, which has an additional contribution causing decoherence and dissipation.44. See, for example, D. A. Micha, Int. J. Quantum Chem. 80, 394 (2000). https://doi.org/10.1002/1097-461X(2000)80:3<394::AID-QUA9>3.0.CO;2-U The nature of the measurement—or, more generally, the nature of the subsystem–environment interaction—selects a preferred basis, called the pointer basis, and the subsystem decoheres into an effectively classical mixture in the pointer basis. (See the article by Wojciech Zurek, Physics Today, October 1991, page 36.)ReferencesSection:ChooseTop of pageReferences <<1. A. Bohm, Quantum Mechanics: Foundations and Applications, 3rd ed., Springer (1993), p. 72. Google Scholar2. See, for example, L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific (2015), chap. 2. Google Scholar3. K. Gottfried, T.-M. Yan, Quantum Mechanics: Fundamentals, 2nd ed., Springer (2003), p. 46. Google ScholarCrossref4. See, for example, D. A. Micha, Int. J. Quantum Chem. 80, 394 (2000). https://doi.org/10.1002/1097-461X(2000)80:3<394::AID-QUA9>3.0.CO;2-U, Google ScholarCrossref© 2022 American Institute of Physics.