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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Can we undo quantum measurements?

Asher Peres

Published Aug 15, 1980• 59 citations
Publisher

Abstract

The Schr\"odinger equation cannot convert a pure state into a mixture (just as Newton's equations cannot display irreversibility). However, to observe phase relationships between macroscopically distinguishable states, one has to measure very peculiar operators. An example, constructed explicitly, shows that the classical analog of such an operator cannot be measured, because to do so would violate classical irreversibility. This result justifies von Neumann's measurement theory, without any hypothesis on the role of the observer.

DOI
10.1103/physrevd.22.879

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Observability of relative phases of macroscopic quantum states

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After a measurement, to observe the relative phases of macroscopically distinguishable states we have to ``undo'' a quantum measurement. We generalise an earlier model of Peres from two state to N-state quantum system undergoing measurement process and discuss the issue of observing relative phases of different branches. We derive an inequality which is satisfied by the relative phases of macroscopically distinguishable states and consequently any desired relative phases can not be observed in interference setups. The principle of macroscopic complementarity is invoked that might be at ease with the macroscopic world. We illustrate the idea of limit on phase observability in Stern-Gerlach measurements and the implications are discussed.

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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
The Schr"odinger equation cannot convert a pure state into a mixture (just as Newton's equations cannot display irreversibility). However, to observe phase relationships between macroscopically distinguishable states, one has to measure very peculiar operators. An example, constructed explicitly, shows that the classical analog of such an operator cannot be measured, because to do so would violate classical irreversibility. This result justifies von Neumann's measurement theory, without any hypothesis on the role of the observer.
Traditionally, quantum measurements have been associated with irreversible transformations of pure states into mixtures. In this paper we show that this old hypothesis is redundant. A simple model is constructed in which the measuring process is described by a continuous unitary transformation of the state vector of the observed system and the apparatus. The final state is pure, but appears to be a mixture if we ignore the degrees of freedom of the apparatus. This property is most strikingly illustrated by reformulating the problem in the Heisenberg picture: The state vector is fixed, and the dynamical variables are changed by the measurement.
This is the second, mathematically more detailed part of a paper consisting of two articles, the first having appeared in the immediately preceding issue of this Journal∗. It shows that a measurement converts a pure case into a mixture with reducible probabilities. The measurement as such permits no inference whatever as to the state of the physical system subjected to measurement after the measurement has been performed. But because the probabilities after the act are classical and therefore reducible, it is often possible to adjust them so that Von Neumann's projection postulate is true. Among the more specific features dealt with in Part II is the occurrence of negative joint probabilities for the measurement of non-commuting operators in certain (not all!) quantum states. The general conclusions reached are stated at the end of the article.
Quantum mechanics is widely recognised as being incomplete. It is not consistent with the second law of thermodynamics and does not provide a scientifically credible physical account of the measurement process, the means by which coherence is broken and classically observable states are recorded. This has led to many ad hoc assumptions being used to account for various properties of quantum systems, among which is the coherence time of quantum devices that determines their ability to perform computations. Here, we show that all these properties can be accommodated naturally and consistently in the context of mixing quantum systems which exhibit continuous spectra, as arises in the thermodynamic limit in quantum statistical mechanics and quantum gravity. In particular, for isolated systems we show that the time-reversal symmetry associated with unitary time evolution of the quantum state gives rise to time-symmetry breaking and a semi-group evolution which attains thermodynamic equilibrium at long times. Moreover, the emergence of this non-unitary time-asymmetry leads to microcanonical equilibrium states in which all quantum coherence is lost and is accompanied by the transformation of pure states into mixtures, leading in turn to an increase in entropy. Inclusion of a macroscopic measurement apparatus shows how the outcome of a measurement corresponds to the von Neumann projection postulate, arising with probabilities in conformance with the Born rule. The mathematical structure of the theory which applies to quantum systems with continuous spectra is closely analogous to the classical ergodic theory of dynamical systems and the conditions under which they attain equilibrium states.
After a measurement, to observe the relative phases of macroscopically distinguishable states we have to ``undo'' a quantum measurement. We generalise an earlier model of Peres from two state to N-state quantum system undergoing measurement process and discuss the issue of observing relative phases of different branches. We derive an inequality which is satisfied by the relative phases of macroscopically distinguishable states and consequently any desired relative phases can not be observed in interference setups. The principle of macroscopic complementarity is invoked that might be at ease with the macroscopic world. We illustrate the idea of limit on phase observability in Stern-Gerlach measurements and the implications are discussed.
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.