Quantum Measurement

11 Key excerpts on "Quantum Measurement"

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  The Undivided Universe

  An Ontological Interpretation of Quantum Theory

  • David Bohm, Basil J. Hiley(Authors)
  • 2006(Publication Date)
  • Routledge

    (Publisher)

  Chapter 6 Measurement as a special case of quantum process

  As pointed out in chapter 2 , in the conventional interpretation of the quantum theory (which is basically epistemological in nature) measurement plays a key role in the sense that without it the mathematical equations would have no physical meaning. In our ontological interpretation, however, we have started with a treatment of the individual, actual process, e.g. a particle penetrating a barrier, undergoing transition between stationary states etc. Evidently it is necessary to deal with the measurement process in essentially the same way; i.e. as a special case of quantum processes in general. What is particularly significant in a measurement process is that from a large scale result that is observable in a piece of measuring apparatus, one can infer the state of the observed system, or at least the state in which it has been left after the measurement process is over.

  In this chapter we shall treat the measurement process in some detail and show how it is to be understood ontologically. We shall first discuss the general principles of this process. We shall show that the essential new feature of Quantum Measurement is that there is mutual and irreducible participation of the measuring instrument and the observed object in each other. As a result, any attempt to discuss this process as measuring ‘a property of the observed object alone’ will not be consistent with our interpretation. Rather we say that the result of a measurement is a potentiality of the combined system and can be determined only in terms of the properties of the particles, along with the wave function of the combined system as a whole. On the basis of this notion, we shall go into several examples that help illustrate the principles involved and clarify some of the puzzles to which these examples give rise in the conventional treatment.

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  Universal Measurements: How To Free Three Birds In One Move

  How to Free Three Birds in One Move

  • Massimiliano Sassoli De Bianchi, Diederik Aerts(Authors)
  • 2017(Publication Date)
  • World Scientific

    (Publisher)

  The Measurement Problem 21 measurement being precisely the process that was going to actualize one (and only one) among the different potential properties , a priori possible. In other words, if in classical physics a measurement process was considered only to be a process of discovery , after the advent of quan-tum mechanics it was also (and especially) understood as a process of creation . If we compare the apple to a microscopic entity, it was as if the apple did not have an a priori circumference, and that it did not become real until the moment of its being measured. This strangeness of the Quantum Measurements gave rise to sev-eral tentative explanations, associated with different possible inter-pretations of the theory. Basically, we can say that all these inter-pretations tried to do one and the same thing: explain the origin of the “ observer effect ” in the Quantum Measurements. Clearly, there are many other “strange” aspects of the theory which are considered to be poorly understood, and which also require an explanation, but the measurement problem unanimously remains a central one, and it is unthinkable to be able to explain the other peculiarities of the theory if we are not able to explain what it means, in the first place, to observe a microscopic system. As we shall see, the notion of universal average will prove very important in proposing a solution to the measurement problem, in the sense that Quantum Measurements can be understood as uni-versal measurements , i.e., as measurements that produce a universal average . But to understand what this means, we must proceed in stages. We will start by explaining how a Quantum Measurement occurs in practice. To do so, we will use the example of one of the experi-ments that had the greatest impact on modern physics: the Stern– Gerlach experiment .

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  Quantum Mechanical Irreversibility And Measurement

  • Paolo Grigolini(Author)
  • 1993(Publication Date)
  • World Scientific

    (Publisher)

  Chapter 1 The Conventional Theory of Measurement We devote the first Chapter of this book to an elementary exposition of the conceptual problems posed by the theory of measurement in quantum mechanics. The reader is presumed to have at least an introductory knowledge of quantum mechanics at the level of textbooks such as Messiah (Messiah, 1958) or Cohen-Tannoudji, Diu and Laloe (1977). The interested reader can find a deeper discussion of the basic concepts of quantum mechanics in the book by d'Espagnat (1989). We shall remind the reader that for quantum mechanics to have dynamical significance, the linear superpositions of states with different eigenvalues must exist. Due to the linear nature of quantum mechanics this seemingly innocent property cannot be confined to the microscopic world and has the tendency to spread over the macroscopic world, where it is perceived as paradoxical. This is the essence of the famous paradox of Schrodinger (1935). The theory of measurement developed by von Neumann (1955) is the elegant formulation of the process of interaction between a microscopic system to be measured and the macroscopic apparatus to carry out the measurement. This theory shows that Schrodinger's cat is the ineluctable consequence of the measurement act. 1.1 Schrodinger's Cat One of the most devastating features of quantum mechanics is the fact that it rests on the linear superposition of distinct states. This aspect of quantum mechanics turns out to be 1 2 indispensable for the picture of microscopic dynamical processes and cannot be rejected without rejecting quantum mechanics itself. However, this fundamental aspect of quantum mechanics is also the root of the lack of realism of this fascinating theoretical picture of the physical world. Indeed, as we go from microscopic to macroscopic physics, while retaining this property, we are unavoidably led to predict the existence of unrealistic effects such as the existence of Schrodinger's cat.

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  Quantum Theory and Measurement

  • John Archibald Wheeler, Wojciech Hubert Zurek, John Archibald Wheeler, Wojciech Hubert Zurek, John Wheeler, Wojciech Zurek(Authors)
  • 2014(Publication Date)
  • Princeton University Press

    (Publisher)

  II.4 PROBLEM OF MEASUREMENT 329 It is well known that statistical correlations of the nature just described play a most important role in the structure of quantum me- chanics. One of the earliest observations in this direction is Mott's explanation of the straight track left by the spherical wave of outgoing a particles. 8 In fact, the principal conceptual difference between quan- tum mechanics and the earlier Bohr-Kramers-Slater theory is that the former, by its use of configuration space rather than ordinary space for its waves, allows for such statistical correlations. Returning to the problem of measurement, we see that we have not arrived either at a conflict between the theory of measurement and the equations of motion, nor have we obtained an explanation of that theory in terms of the equations of motion. The equations of motion permit the description of the process whereby the state of the object is mirrored by the state of an apparatus. The problem of a measurement on the object is thereby transformed into the problem of an observation on the apparatus. Clearly, further transfers can be made by introducing a sec- ond apparatus to ascertain the state of the first, and so on. However, the fundamental point remains unchanged and a full description of an observation must remain impossible since the quantum-mechanical equations of motion are causal and contain no statistical element, where- as the measurement does. It should be admitted that when the quantum theorist discusses mea- surements, he makes many idealizations. He assumes, for instance, that the measuring apparatus will yield some result, no matter what the initial state of the object was. This is clearly unrealistic since the object may move away from the apparatus and never come into contact with it. More importantly, he has appropriated the word "measurement" and used it to characterize a special type of interaction by means of which information can be obtained on the state of a definite object.

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  Quantum Mechanics

  • P. J. E. Peebles, P. James E. Peebles(Authors)
  • 2020(Publication Date)
  • Princeton University Press

    (Publisher)

  CHAPTER 4 MEASUREMENT THEORY We come now to the last element of quantum mechanics, the prescrip-tion for how the mathematical theory is to be related to the results of measurements. This is required for any physical theory, not just quan-tum mechanics. For example, the formalism of general relativity theory is quite misleading if one does not understand that coordinate depen-dent objects like the four-velocity of a particle must be distinguished from the scalars that are in principle measurable. The measurement prescription in quantum mechanics can be stated in a few lines (as has already been done in sections 14 and 21 on the principles of quantum mechanics, and is done in a little more detail in the first section of this chapter). There is no controversy over this or the other elements of quantum physics. The theory has found an enormous range of applications, in all of which it has proved to be con-sistent with logic and experimental tests. Why then is this chapter so long? It is because the implications seem so bizarre that such thoughtful people as Einstein and Wigner have argued that the theory cannot be physically complete as it stands. A review of these bizarre implications, and the ways in which the reservations about the physical complete-ness of quantum physics have been rationalized or otherwise laid to rest to the satisfaction of many (but certainly not all) physicists gives us an excellent framework for a discussion of some of the physics of this subject. The first section in this chapter reviews measurement theory in quantum mechanics, and extends the prescription to the case where the state vector is not known. Some of the paradoxes of quantum mechan-ics (none of which prove to be paradoxical) are discussed in section 27. Section 28 presents Bell's theorem, that shows there cannot be a local underlying deterministic theory for which quantum mechanics plays the

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  Handbook of Measurements

  Benchmarks for Systems Accuracy and Precision

  • Adedeji B. Badiru, LeeAnn Racz, Adedeji B. Badiru, LeeAnn Racz(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  Since the position operator and momen-tum operator do not commute, a measurement of position collapses the wave function into an eigenfunction of the position operator and precludes any knowledge of momentum. If an attempt to measure momentum is performed, the system will collapse into an eigen-function of the momentum operator and eliminate any knowledge of the position. The postulates of quantum mechanics establish a link between measurement and the collapse of the wave function. However, the mechanism by which this collapse occurs is not predicted by Schrödinger’s equation. The lack of a mechanism for the collapse of the wave function represents an important area of inquiry regarding the foundations of quantum theory (Greenstein and Zajonc 2006). Another foundational question is how the probabilistic quantum description of atoms and molecules on the microscopic scale relates to the deterministic classical description that works so well for macroscopic assemblies of these atoms and molecules (Bokulich 2008). Why for example, when I observe that my car is under a bridge, do I not become concerned that the wave function of the car will collapse making the momentum of my car completely uncertain? Perhaps, the most remarkable difference between classical measurement and Quantum Measurement is our picture of what we are measuring before the measurement occurs. 201 Chapter nine: Measurement and quantum mechanics Classically, an object has a preexisting position independent of our knowledge. The clas-sical measurement of an object’s position is just our determination of the object’s preexist-ing condition. In quantum mechanics, an object does not have a preexisting position prior to a measurement. There is only a probability that it has a particular position. The act of making a Quantum Measurement creates the reality of an object’s position. In this sense, a quantum mechanical measurement gives the observer an active role in generating reality.

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  John S Bell On The Foundations Of Quantum Mechanics

  • Mary Bell, Kurt Gottfried, Martinus J G Veltman(Authors)
  • 2001(Publication Date)
  • World Scientific

    (Publisher)

  It must have a section to itself. Against ‘measurement’ When I say that the word ‘measurement’ is even worse than the others, I do not have in mind the use of the word in phrases like ‘measure the mass and width of the Z boson’. I do have in mind its use in the fundamental interpretive rules of quantum mechanics. For example, here they are as given by Dirac (Quantum Mechanics Oxford University Press 1930): ‘. . . any result of a measurement of a real dynamical variable is one of its eigenvalues . . .’ ‘. . . if the measurement of the observable . . . is made a large number of times the average of all the results obtained will be . . .’ ‘. . . a measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured . . .’ It would seem that the theory is exclusively concerned about ‘results of measurement’, and has nothing to say about anything else. What exactly qualifies some physical systems to play the role of ‘measurer’? Was the wavefunc- tion of the world waiting to jump for thousands of millions of years until a single-celled living creature appeared? Or did it have to wait a little longer, for some better qualified system . . . with a PhD? If the theory is to apply to anything but highly idealised laboratory operations, are we not obliged to admit that more or less ‘measurement-like’ processes are going on more or less all the time, more or less everywhere? Do we not have jumping then all the time? The first charge against ‘measurement’, in the fundamen- tal axioms of quantum mechanics, is that it anchors there the shifty split of the world into ‘system’ and ‘apparatus’. A second charge is that the word comes loaded with meaning from everyday life, meaning which is entirely inappropriate in the quantum context. When it is said that something is ‘measured’ it is difficult not to think of the result as referring to some pre-existing property of the object in question.

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  Measurements and Time Reversal in Objective Quantum Theory

  International Series in Natural Philosophy

  • F. J. Belinfante, D. Ter Haar(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  C H A P T E R 1 MEASUREMENTS IN OBJECTIVE QUANTUM THEORY 1.1. Introduction The basic elements of quantum theory consist of two parts. On the one hand, it describes pure states by state vectors ψ (often called wavefunctions) of which the time dependence is given by a Schrödinger equation ihd^/dt = ^ ψ . On the other hand, there is a set of rules which tells us how these ψ physically are to be interpreted. Without the one or the other, the theory is incomplete. If the one or the other is fundamentally and drastically altered, we may still have a theory, but it would no longer be quantum theory. Whether the new theory would be better or would be worse than conventional quantum theory would have to be decided by experiment. If the new theory would not be experimentally distinguishable from quantum theory, the simpler of the two theories would be preferred by most people. (Occam's razor.) The rules for interpreting ψ are well known. Each ψ predicts a probability distribution over possible outcomes of successful measure-ments of arbitrary observables A, The (theoretically) possible out-comes are the eigenvalues A„ of the operators corresponding to these observables. If ψ is expanded in terms of the corresponding eigenfunc-tions φ„ of such an operator, the absolute squares cj^ of the expan-sion coefficients are interpreted as probabilities for these outcomes of successful measurements of A. I

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  Selected Topics in Applications of Quantum Mechanics

  • Mohammad Reza Pahlavani(Author)
  • 2015(Publication Date)
  • IntechOpen

    (Publisher)

  a measurement, it requires the application of [a] process [of the first kind].” M. Jammer points out that “this argument for the indispensability of processes of the first kind also seems to suggest that these processes do not occur in the observed portions of the world, however deeply in the observer’s body the boundary is drawn. They can thus occur only in his consciousness. A complete measurement, according to von Neumann’s theory, involves therefore the consciousness of the observer. ” [2; emphases added] 6. Entangled states: The Schrödinger cat and the EPR paradox The state ψ given by (6) is an entangled state where each term in the sum is the product of a possible state σ n of a microsystem, the corresponding final state α n of the apparatus, and the number c n . So, as long as the total system+apparatus remains isolated, we have a linear superposition of different states of the apparatus, the coefficients being c n σ n . Entangled states do not have a classical equivalent and are an unavoidable consequence of the superpo‐ sition principle , considered by some physicists the fundamental principle of quantum mechan‐ ics. Schrödinger showed how strange some entangled states are with his well-known example of the cat: Imagine that the microsystem is a radioactive element with two possible states: σ 1 (atom non-decayed), and σ 2 (atom decayed). If the atom decays, a mechanism is activated and The Measurement Problem in Quantum Mechanics Revisited http://dx.doi.org/10.5772/59209 145 kills the cat, a macrosystem with the possible states α 1 (cat alive, if the atom has not decayed) and α 2 (cat dead, if the atom has decayed). Then, if at a given instant the probability the radioactive element has of being decayed is ½, the coefficients take on the value c 2 = c 1 = (½) and ( ) ( ) y + = 1 1 2 2 ½ ½ σ α σ α (7) This entangled state is a superposition in which the two states “cat alive” and “cat dead” are mixed or smeared together by equal amounts.

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  Protective Measurement and Quantum Reality

  Towards a New Understanding of Quantum Mechanics

  • Shan Gao(Author)
  • 2015(Publication Date)
  • Cambridge University Press

    (Publisher)

  It follows that whatever form the “measurement of the wave function” takes in protective measurement, it must be weaker than the condition we stated in the previous paragraph: “If we could reliably measure the unknown quantum state of a single quantum system without changing that state . . . . ” The italicized words indicate possibilities for relaxing this condition. We might be content with mea- surements that are not 100% reliable and may change the state, as long as the disturbance can be made arbitrarily small (or unlikely). Or we might be able to show that the measurement is possible only for certain quantum states, or under certain conditions, or both. Indeed, all of these concessions must be made in the case of protective mea- surement [3, 13; see also 38, 40]. Of these, the fact that protective measurement only works under carefully designed conditions and for special quantum states – specifically, the system must be in a non-degenerate eigenstate of its Hamiltonian – may well be of least concern. After all, if protective measurement allowed us to operationally establish the reality of an unknown quantum state in certain situa- tions, perhaps it would not be so far-fetched to extend this interpretation to the rest of the states. The more serious issue, however, arises from the inevitable system– apparatus entanglement in protective measurement. This entanglement introduces an irreducible randomness into the readout; there is a non-zero probability for the system to end up in a state different from the initial state. While this issue has been pointed out before [6, 13], here we will take it up in more detail, by describing the creation of entanglement in protective measurement (Section 13.2) and discussing 1 See, for example, Schwinger [35], Rovelli [28], Samuel and Nityananda [29], Unruh [41], Dass and Qureshi [13], Alter and Yamamoto [6], Uffink [38, 40].

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  Decoherence And Quantum Measurements

  • Mikio Namiki, Saverio Pascazio, Hiromichi Nakazato(Authors)
  • 1998(Publication Date)
  • World Scientific

    (Publisher)

  Von Neumann and Wigner strongly asserted that quantum mechanics, in particular the superposition principle, should be strictly applied to the mea-surement process. With the help of (3.6) and (3.7), therefore, they obtained |*J ot >H|^Vl*?> = I>l*>®l*?> -»■ 5>l*>®l*F> = l*F 0t > (3-8) i i for the whole measurement process. This is often called the von Neumann measurement process. However, the phase correlation among different Q eigenstates is still pres-ent in 1*1?'), as we easily understand by writing the corresponding density matrix pr = i*r>{«ri = Ei^i 3 «?®i*?><*?i+EE^i»> i. (3.9) where £j~ — |xfc)(Xfcl- Obviously, the above is still a projection operator repre-senting a pure state with non-vanishing off-diagonal components (the last term in the r.h.s.). Thus, we are led to the conclusion that the von Neumann-Wigner approach can never realize the wave function collapse as a physical process. As is well known and was already mentioned in Chapter 1, the von Neu-mann-Wigner theory required that the measuring process is not completed by the interaction of Q with D, which is instead followed by the so-called von Neumann's chain of measurements, connecting Q to the observer via many steps. Eventually the theory brings in an Abstraktes Ich or consciousness at the end of the chain, which should be responsible for the wave function col-lapse by measurement. On the basis of these arguments, Wigner claimed that quantum mechanics is incomplete. This theory provoked the famous paradoxes of Schrodinger's cat and Wigner's friend, roughly sketched in Section 2.1 of Chapter 1. The whole state in the cat paradox is considered to be I*) = -/= [|alive) cat ® |+) q + |dead) cat ® | -) Q ] , (3.10) von Neumann- Wigner 57 where the subscripts cat and Q stand for the cat and Q states, and ± for the excited and ground states of Q, respectively.

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