Physics Today
More on the quantum measurement problem
Art Hobson
Published Nov 1, 2022• 1 citation
Abstract
In his June 2022 Quick Study (page 62), David Mermin argues that “the quantum state of a system expresses only the belief of the particular physicist who assigns it to the system.” Applying that to quantum measurements, he finds that “the acquisition of further information by that physicist … can lead to an abrupt change in those probabilities and thus to an updating of the quantum state that the physicist uses to represent them. There is no quantum measurement problem.”But wavefunctions have been collapsing ever since the Big Bang, with no assistance from physicists. An apparatus’s display of a measurement outcome occurs even if the experimenters happen to be out of the room. When a cosmic-ray proton strikes a sand grain on Mars and moves the grain, a quantum measurement occurs and the proton’s wavefunction collapses regardless of the absence of humans.Roger Carpenter and Andrew Anderson of the University of Cambridge performed a “Schrödinger’s cat” experiment that demolishes Mermin’s interpretation. Instead of connecting a Geiger counter to a cat-killing device, they mercifully connected it to a hammer that would fall without harm. Their strategy was to split information about the experimental result between two observers in such a way that neither observer can know the outcome. The observers learn the outcome later by sharing their information. The question is then, Did the hammer fall at the time of the experiment or later, when the observers became conscious of the outcome? The result: The hammer fell when the nucleus decayed, not later when the observers became conscious of the outcome. I think nearly all physicists would have predicted that. My hat is off to Carpenter and Anderson, who reported it with a straight face.11. R. H. S. Carpenter, A. J. Anderson, Ann. Fond. Louis de Broglie 31, 45 (2006).Humans and their consciousness have nothing to do with quantum physics. Photons, electrons, and the like, as well as their states, are real configurations of fields that have existed throughout the universe since the Big Bang.22. A. Hobson, Am. J. Phys. 81, 211 (2013). https://doi.org/10.1119/1.4789885Nevertheless, I agree with Mermin’s title: There is no quantum measurement problem, because quantum physics, with no special interpretation and without a collapse postulate, logically implies that superpositions collapse nonlocally to a single definite outcome.33. A. Hobson, Quantum Eng. 2022, 1 (2022). https://doi.org/10.1155/2022/5889159ReferencesSection:ChooseTop of pageReferences <<CITING ARTICLES1. R. H. S. Carpenter, A. J. Anderson, Ann. Fond. Louis de Broglie 31, 45 (2006). Google Scholar2. A. Hobson, Am. J. Phys. 81, 211 (2013). https://doi.org/10.1119/1.4789885, Google ScholarCrossref, ISI3. A. Hobson, Quantum Eng. 2022, 1 (2022). https://doi.org/10.1155/2022/5889159, Google ScholarCrossref© 2022 American Institute of Physics.
- DOI
- 10.1063/pt.3.5112
Related papers
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
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
Measured responses to quantum Bayesianism
Art Hobson
Because it’s important that physicists discuss the problems of quantum foundations, I was delighted to see David Mermin’s commentary on “fixing the shifty split.” But I must disagree with my friend’s view.There are multiple ways of interpreting probabilities.11. A. Hobson, Concepts in Statistical Mechanics, Gordon and Breach, New York (1971), p. 27. In quantum physics, the Bayesian “degree of belief” interpretation can be appropriate for mixed states, but only the frequentist or “ensemble” interpretation applies to the pure states that Mermin discusses.A recent paper by Matthew Pusey and coauthors22. M. F. Pusey, J. Barrett, T. Rudolph, http://arxiv.org/abs/1111.3328v1. concludes that if two different pure quantum states are assigned to a single physical situation, one of the states is objectively wrong. The proof’s main assumption is that any quantum system has some set of real physical properties, labeled λ. Pusey and coauthors show that two different pure states cannot represent the same λ. So, although their paper implies but does not assume that quantum states are physically real, it does assume that each specific physical situation has behind it some kind of physical reality. Chris Fuchs, quoted favorably by Mermin, might dispute the notion that some kind of physical reality actually exists, but most physicists are probably sufficiently realist to grant that notion. Without such a notion, science lies somewhere between solipsism and superstition.Whenever I hear subjective interpretations of quantum physics, I wonder about such questions as the one Mermin quotes from Albert Einstein: Can a mouse collapse a wavefunction? Mermin’s Bayesian response, that a mouse cannot but a physics student can, doesn’t reassure me. Were wavefunctions not collapsing before there were physicists?The “Wigner’s friend” paradox, as described in Mermin’s commentary, doesn’t need a subjective interpretation to find a solution. In fact, Eugene Wigner abandoned his interpretation around 1970 once he became aware of the ideas underlying quantum decoherence.33. M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition, Springer, Berlin (2007), p. 364. Suppose Wigner’s friend makes a measurement, the outcome of which is known to the friend but not to Wigner. Regardless of whether the outcome is known, and in fact regardless of the presence of the friend, the measurement device must decohere the quantum system that’s being measured, putting the system into a mixed state of the either/or form, with no interferences, just like a coin that’s been flipped. A definite outcome has thus occurred, and Wigner and his friend will both agree on this. Neither the friend nor Wigner affects the outcome. She has simply “looked” and now knows the outcome, while Wigner doesn’t know. That is not paradoxical and does not require a revision of our understanding of pure quantum states.There’s no reason to give up on a realist interpretation of quantum physics, much less a realist interpretation of nature itself.REFERENCESSection:ChooseTop of pageREFERENCES <<1. A. Hobson, Concepts in Statistical Mechanics, Gordon and Breach, New York (1971), p. 27. Google Scholar2. M. F. Pusey, J. Barrett, T. Rudolph, http://arxiv.org/abs/1111.3328v1. Google Scholar3. M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition, Springer, Berlin (2007), p. 364. Google Scholar© 2012 American Institute of Physics.
Physics Today•Nov 30, 2012•1 citation
On measurement and quantum nondemolition
H. D. Zeh
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.
Physics Today•Jul 1, 2011•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
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
In his June 2022 Quick Study (page 62), David Mermin argues that “the quantum state of a system expresses only the belief of the particular physicist who assigns it to the system.” Applying that to quantum measurements, he finds that “the acquisition of further information by that physicist … can lead to an abrupt change in those probabilities and thus to an updating of the quantum state that the physicist uses to represent them. There is no quantum measurement problem.”But wavefunctions have been collapsing ever since the Big Bang, with no assistance from physicists. An apparatus’s display of a measurement outcome occurs even if the experimenters happen to be out of the room. When a cosmic-ray proton strikes a sand grain on Mars and moves the grain, a quantum measurement occurs and the proton’s wavefunction collapses regardless of the absence of humans.Roger Carpenter and Andrew Anderson of the University of Cambridge performed a “Schrödinger’s cat” experiment that demolishes Mermin’s interpretation. Instead of connecting a Geiger counter to a cat-killing device, they mercifully connected it to a hammer that would fall without harm. Their strategy was to split information about the experimental result between two observers in such a way that neither observer can know the outcome. The observers learn the outcome later by sharing their information. The question is then, Did the hammer fall at the time of the experiment or later, when the observers became conscious of the outcome? The result: The hammer fell when the nucleus decayed, not later when the observers became conscious of the outcome. I think nearly all physicists would have predicted that. My hat is off to Carpenter and Anderson, who reported it with a straight face.11. R. H. S. Carpenter, A. J. Anderson, Ann. Fond. Louis de Broglie 31, 45 (2006).Humans and their consciousness have nothing to do with quantum physics. Photons, electrons, and the like, as well as their states, are real configurations of fields that have existed throughout the universe since the Big Bang.22. A. Hobson, Am. J. Phys. 81, 211 (2013).
https://doi.org/10.1119/1.4789885Nevertheless, I agree with Mermin’s title: There is no quantum measurement problem, because quantum physics, with no special interpretation and without a collapse postulate, logically implies that superpositions collapse nonlocally to a single definite outcome.33. A. Hobson, Quantum Eng. 2022, 1 (2022).
https://doi.org/10.1155/2022/5889159ReferencesSection:ChooseTop of pageReferences <<CITING ARTICLES1. R. H. S. Carpenter, A. J. Anderson, Ann. Fond. Louis de Broglie 31, 45 (2006). Google Scholar2. A. Hobson, Am. J. Phys. 81, 211 (2013).
https://doi.org/10.1119/1.4789885, Google ScholarCrossref, ISI3. A. Hobson, Quantum Eng. 2022, 1 (2022).
https://doi.org/10.1155/2022/5889159, Google ScholarCrossref© 2022 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.
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.
Because it’s important that physicists discuss the problems of quantum foundations, I was delighted to see David Mermin’s commentary on “fixing the shifty split.” But I must disagree with my friend’s view.There are multiple ways of interpreting probabilities.11. A. Hobson, Concepts in Statistical Mechanics, Gordon and Breach, New York (1971), p. 27. In quantum physics, the Bayesian “degree of belief” interpretation can be appropriate for mixed states, but only the frequentist or “ensemble” interpretation applies to the pure states that Mermin discusses.A recent paper by Matthew Pusey and coauthors22. M. F. Pusey, J. Barrett, T. Rudolph, http://arxiv.org/abs/1111.3328v1. concludes that if two different pure quantum states are assigned to a single physical situation, one of the states is objectively wrong. The proof’s main assumption is that any quantum system has some set of real physical properties, labeled λ. Pusey and coauthors show that two different pure states cannot represent the same λ. So, although their paper implies but does not assume that quantum states are physically real, it does assume that each specific physical situation has behind it some kind of physical reality. Chris Fuchs, quoted favorably by Mermin, might dispute the notion that some kind of physical reality actually exists, but most physicists are probably sufficiently realist to grant that notion. Without such a notion, science lies somewhere between solipsism and superstition.Whenever I hear subjective interpretations of quantum physics, I wonder about such questions as the one Mermin quotes from Albert Einstein: Can a mouse collapse a wavefunction? Mermin’s Bayesian response, that a mouse cannot but a physics student can, doesn’t reassure me. Were wavefunctions not collapsing before there were physicists?The “Wigner’s friend” paradox, as described in Mermin’s commentary, doesn’t need a subjective interpretation to find a solution. In fact, Eugene Wigner abandoned his interpretation around 1970 once he became aware of the ideas underlying quantum decoherence.33. M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition, Springer, Berlin (2007), p. 364. Suppose Wigner’s friend makes a measurement, the outcome of which is known to the friend but not to Wigner. Regardless of whether the outcome is known, and in fact regardless of the presence of the friend, the measurement device must decohere the quantum system that’s being measured, putting the system into a mixed state of the either/or form, with no interferences, just like a coin that’s been flipped. A definite outcome has thus occurred, and Wigner and his friend will both agree on this. Neither the friend nor Wigner affects the outcome. She has simply “looked” and now knows the outcome, while Wigner doesn’t know. That is not paradoxical and does not require a revision of our understanding of pure quantum states.There’s no reason to give up on a realist interpretation of quantum physics, much less a realist interpretation of nature itself.REFERENCESSection:ChooseTop of pageREFERENCES <<1. A. Hobson, Concepts in Statistical Mechanics, Gordon and Breach, New York (1971), p. 27. Google Scholar2. M. F. Pusey, J. Barrett, T. Rudolph, http://arxiv.org/abs/1111.3328v1. Google Scholar3. M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition, Springer, Berlin (2007), p. 364. Google Scholar© 2012 American Institute of Physics.
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.
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.
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.