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Physical Review A

Reply to “Comment on ‘Protective measurement of the wave function of a single squeezed harmonic-oscillator state”’

Orly Alter, Y. Yamamoto

Published Jul 1, 1997• 11 citations
Publisher

Abstract

The adiabatic ``protective measurement'' suggested by Aharonov, Anandan, and Vaidman [Phys. Lett. A 178, 38 (1993)] is a measurement of a discrete energy eigenstate, in which the entanglement between the measured state and the quantum probe is avoided approximately when they interact adiabatically, and the measured state is left approximately unchanged (up to a phase factor) for all measured observables. This measurement scheme, however, does not overcome the reduction of the measured state, because this state is only approximately, and not exactly, disentangled from the quantum probe. Our recent analysis of a measurement scheme of a squeezed harmonic-oscilllator state [Phys. Rev. A 53, R2911 (1996)], in which the entanglement between the measured state and the probe is avoided exactly, and the measured state is left exactly unchanged (up to a phase factor) for all measured observables, suggests that in order to measure the quantum state of a single system without changing this state, full a priori information about this state is required.

DOI
10.1103/physreva.56.1057

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Protective measurement of the wave function of a single squeezed harmonic-oscillator state

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A scheme for the "protective measurement" [Phys. Rev. A 47, 4616 (1993)] of the wave function of a squeezed harmonic-oscillator state is described. This protective measurement is shown to be equivalent to a measurement of an ensemble of states. The protective measurement, therefore, allows for a definition of the quantum wave function on a single system. Yet, this equivalency also suggests that both measurement schemes account for the epistemological meaning of the wave function only. The protective measurement requires a full a priori knowledge of the measured state. The intermediate cases, in which only partial a priori information is given, are also discussed.

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Comment on “Protective measurement of the wave function of a single squeezed harmonic-oscillator state”

Yakir Aharonov, Lev Vaidman A

Alter and Yamamoto [Phys. Rev. A 53, R2911 (1996)] claimed to consider ``protective measurements'' that we have recently introduced. We show that the measurements discussed by Alter and Yamamoto are not the protective measurements we proposed. Therefore, their results are irrelevant to the nature of protective measurements.

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A new type of procedures, called protective measurements, was proposed by Aharonov, Anandan, and Vaidman [Phys. Rev. A 97, 4616 (1993); Found. Phys. 26, 117 (1996)]. These authors argued that a protective measurement allows the determination of arbitrary observables of a single quantum system, and claimed that this favors a realistic interpretation of the quantum state. This paper proves that only observables that commute with the system's Hamiltonian can be measured protectively. It is argued that this restriction saves the coherence of alternative interpretations.

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We examine the entanglement and state disturbance arising in a protective measurement and argue that these inescapable effects doom the claim that protective measurement establishes the reality of the wave function. An additional challenge to this claim results from the exponential number of protective measurements required to reconstruct multi-qubit states. We suggest that the failure of protective measurement to settle the question of the meaning of the wave function is entirely expected, for protective measurement is but an application of the standard quantum formalism, and none of the hard foundational questions can ever be settled in this way. Introduction From the start, the technical result of protective measurement has been suggested to have implications for the interpretation of quantum mechanics. Consider how Aharonov and Vaidman [2] chose to begin their original paper introducing the idea of protective measurement: We show that it is possible to measure the Schrödinger wave of a single quantum system. This provides a strong argument for associating physical reality with the quantum state of a single system.… Since then, the pioneers of protective measurement seem to have taken a more moderate stance. Vaidman [42], in a recent synopsis of protective measurement, concedes that the protective measurement procedure is not a proof that we should adopt one interpretation instead of the other, but it is a good testbed which shows advantages and disadvantages of various interpretations. Notwithstanding this more subtle perspective and a number of critical studies of the technical and foundational aspects of protective measurement, Gao [21] has maintained, if not amplified, the force of Aharonov and Vaidman's original argument: An immediate implication is that the result of a protective measurement, namely the expectation value of the measured observable in the measured state, reflects the actual physical property of the measured system, as the system is not disturbed after this result has been obtained. […]

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Protective measurement: an introduction

Shan Geo

Protective measurement, in the language of standard quantum mechanics, is a method to measure the expectation value of an observable on a single quantum system (Aharonov and Vaidman, 1993; Aharonov, Anandan and Vaidman, 1993). For a conventional impulsive measurement, the state of the measured system is strongly entangled with the state of the measuring device during the measurement, and the measurement result is one of the eigenvalues of the measured observable. By contrast, during a protective measurement, the measured state is protected by an appropriate procedure so that it neither changes nor becomes entangled with the state of the measuring device appreciably. In this way, such protective measurements can measure the expectation values of observables on a single quantum system, and in particular, the wave function of the system can also be measured as expectation values of certain observables. It is expected that protective measurements can be performed in the near future with the rapid development of weak measurement technologies (e.g. Kocsis et al., 2011; Lundeen et al., 2011). In this chapter, we will give a clear introduction to protective measurement in quantum mechanics. Standard quantum mechanics and impulsive measurement The standard formulation of quantum mechanics, which was first developed by Dirac (1930) and von Neumann (1955), is based on the following basic principles. 1 Physical states The state of a physical system is represented by a normalized wave function or unit vector | ψ ( t )〉 in a Hilbert space. The Hilbert space is complete in the sense that every possible physical state can be represented by a state vector in the space. 2 Physical properties Every measurable property or observable of a physical system is represented by a Hermitian operator on the Hilbert space associated with the system. A physical system has a determinate value for an observable if and only if it is in an eigenstate of the observable (this is often called the eigenvalue–eigenstate link).

Cambridge University Press eBooks•Nov 30, 2014•2 citations
The adiabatic ``protective measurement'' suggested by Aharonov, Anandan, and Vaidman [Phys. Lett. A 178, 38 (1993)] is a measurement of a discrete energy eigenstate, in which the entanglement between the measured state and the quantum probe is avoided approximately when they interact adiabatically, and the measured state is left approximately unchanged (up to a phase factor) for all measured observables. This measurement scheme, however, does not overcome the reduction of the measured state, because this state is only approximately, and not exactly, disentangled from the quantum probe. Our recent analysis of a measurement scheme of a squeezed harmonic-oscilllator state [Phys. Rev. A 53, R2911 (1996)], in which the entanglement between the measured state and the probe is avoided exactly, and the measured state is left exactly unchanged (up to a phase factor) for all measured observables, suggests that in order to measure the quantum state of a single system without changing this state, full a priori information about this state is required.
A scheme for the "protective measurement" [Phys. Rev. A 47, 4616 (1993)] of the wave function of a squeezed harmonic-oscillator state is described. This protective measurement is shown to be equivalent to a measurement of an ensemble of states. The protective measurement, therefore, allows for a definition of the quantum wave function on a single system. Yet, this equivalency also suggests that both measurement schemes account for the epistemological meaning of the wave function only. The protective measurement requires a full a priori knowledge of the measured state. The intermediate cases, in which only partial a priori information is given, are also discussed.
A new type of procedures, called protective measurements, was proposed by Aharonov, Anandan, and Vaidman [Phys. Rev. A 97, 4616 (1993); Found. Phys. 26, 117 (1996)]. These authors argued that a protective measurement allows the determination of arbitrary observables of a single quantum system, and claimed that this favors a realistic interpretation of the quantum state. This paper proves that only observables that commute with the system's Hamiltonian can be measured protectively. It is argued that this restriction saves the coherence of alternative interpretations.
An experiment that would measure noncommuting quantum-mechanical observables without collapsing the wave function has been recently proposed by Y. Aharonov, J. Anandan, and L. Vaidman [Phys. Rev. A 47, 4616 (1993)]. These authors argue that this ``protected measurement'' may give an indication on ``the reality of the wave function.'' We show that the proposed experiment admits a much simpler interpretation: it is an experiment in which the quantum system is (assumed to be) in an eigenstate of the observable being measured. As such, the fact that no collapse occurs is trivial and can be understood without the need of the concept of ``protective'' devices. We argue that the experiment does not provide a way for measuring noncommuting observables without a collapse, does not bear on the issue of the ``reality of the wave function,'' and does not add any particular insight into our understanding (or nonunderstanding) of quantum mechanics.
We examine the entanglement and state disturbance arising in a protective measurement and argue that these inescapable effects doom the claim that protective measurement establishes the reality of the wave function. An additional challenge to this claim results from the exponential number of protective measurements required to reconstruct multi-qubit states. We suggest that the failure of protective measurement to settle the question of the meaning of the wave function is entirely expected, for protective measurement is but an application of the standard quantum formalism, and none of the hard foundational questions can ever be settled in this way. Introduction From the start, the technical result of protective measurement has been suggested to have implications for the interpretation of quantum mechanics. Consider how Aharonov and Vaidman [2] chose to begin their original paper introducing the idea of protective measurement: We show that it is possible to measure the Schrödinger wave of a single quantum system. This provides a strong argument for associating physical reality with the quantum state of a single system.… Since then, the pioneers of protective measurement seem to have taken a more moderate stance. Vaidman [42], in a recent synopsis of protective measurement, concedes that the protective measurement procedure is not a proof that we should adopt one interpretation instead of the other, but it is a good testbed which shows advantages and disadvantages of various interpretations. Notwithstanding this more subtle perspective and a number of critical studies of the technical and foundational aspects of protective measurement, Gao [21] has maintained, if not amplified, the force of Aharonov and Vaidman's original argument: An immediate implication is that the result of a protective measurement, namely the expectation value of the measured observable in the measured state, reflects the actual physical property of the measured system, as the system is not disturbed after this result has been obtained. […]
Protective measurement, in the language of standard quantum mechanics, is a method to measure the expectation value of an observable on a single quantum system (Aharonov and Vaidman, 1993; Aharonov, Anandan and Vaidman, 1993). For a conventional impulsive measurement, the state of the measured system is strongly entangled with the state of the measuring device during the measurement, and the measurement result is one of the eigenvalues of the measured observable. By contrast, during a protective measurement, the measured state is protected by an appropriate procedure so that it neither changes nor becomes entangled with the state of the measuring device appreciably. In this way, such protective measurements can measure the expectation values of observables on a single quantum system, and in particular, the wave function of the system can also be measured as expectation values of certain observables. It is expected that protective measurements can be performed in the near future with the rapid development of weak measurement technologies (e.g. Kocsis et al., 2011; Lundeen et al., 2011). In this chapter, we will give a clear introduction to protective measurement in quantum mechanics. Standard quantum mechanics and impulsive measurement The standard formulation of quantum mechanics, which was first developed by Dirac (1930) and von Neumann (1955), is based on the following basic principles. 1 Physical states The state of a physical system is represented by a normalized wave function or unit vector | ψ ( t )〉 in a Hilbert space. The Hilbert space is complete in the sense that every possible physical state can be represented by a state vector in the space. 2 Physical properties Every measurable property or observable of a physical system is represented by a Hermitian operator on the Hilbert space associated with the system. A physical system has a determinate value for an observable if and only if it is in an eigenstate of the observable (this is often called the eigenvalue–eigenstate link).