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Heikon jälkivalitun kvanttimekaanisen mittauksen simuloiminen kvanttitilapolkuja käyttäen

Antti P. Vepsäläinen

Published Jan 1, 2014• 0 citations

Abstract

The concept of weak value of a quantum mechanical observable is based on a weak measurement followed by a strong measurement of a complementary observable. By applying post-selection, the weak value can, under certain circumstances, have values much higher than would be possible without post-selection, even above the highest eigenvalue of the measurement operator. Weak values have a wide range of applications, starting from the amplification of the resolution in high sensitivity metrology. The weak values have been especially successful in the area of quantum optics, which provides a suitable set of tools for performing the weak measurements with post-selection. The topic of this thesis is to lay the ground for the adaptation of weak values also in other fields of physics, the target being to harness the weak values in cavity quantum electrodynamics (cQED), and furthermore in circuit quantum electrodynamics. The main complication in the adaptation in these fields is the de-coherence, which is almost nonexistent in quantum optics. In this thesis we use quantum trajectory simulation to model the weak values under experimentally realistic conditions. We show how the weak values stem from the stochastic description of the quantum mechanical measurement, and that the second order approximation of the weak values is not sufficient in describing the effects of the measurement strength on the weak values. Using the simulation we demonstrate that the measurement resolution can be increased by applying post-selection even in the presence of de-coherence. Finally, we study the correlations between two entangled qubit, and show that post-selection appears to strengthen it.

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The ability to post-select the outcomes of an experiment is a useful theoretical concept and experimental tool. In the context of weak measurements, post-selection can lead to surprising results such as complex weak values outside the range of eigenvalues. Usually post-selection is realized by a projective measurement, which is hard to implement in ensemble systems such as NMR. We demonstrate the first experiment of a weak measurement with post-selection on an NMR quantum information processor. Our setup is used for measuring complex weak values and weak values outside the range of eigenvalues. The scheme for overcoming the problem of post-selection in an ensemble quantum computer is general and can be applied to any circuit-based implementation. This experiment paves the way for studying and exploiting post-selection and weak measurements in systems where projective measurements are hard to realize experimentally.

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Yakir Aharonov, Lev Vaidman A

A description of quantum systems at the time interval between two successive measurements is presented. Two wave functions, the first pre-selected by the initial measurement and the second post-selected by the final measurement describe quantum systems at a single time. It is shown how this approach leads to a new concept: a weak value of an observable. Weak values represent novel characteristics of quantum systems between two measurements. They are outcomes of a standard measuring procedure that fulfills certain requirements of “weakness”. We call it weak measurement . Physical meaning and underlying mathematical structure of weak measurements are explored.

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精密測定への弱値増幅の応用可能性について

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The novel notion of a physical quantity called 'weak value', which was proposed by Aharonov et al. in 1988, has attracted much attention in the fields of quantum foundations and quantum measurement.In particular, the fact that through proper state selections the weak value can be made arbitrarily large, exceeding the conventional bound set by the maximum eigenvalue of the physical observable, has been widely regarded as a promising source for precision measurement.Indeed, in recent years we have seen increasingly many reports of successful applications of the weak value in photonic systems with various instances of observables amplified effectively.However, it has also been recognized that there are two major obstacles in realizing this weak value amplification (WVA).First, it is unclear under what conditions the WVA becomes effective due to the conflict between the ambiguity caused by the state selections and the amplification effect.Second, the required state selections may not be available depending on the systems one is dealing with.In order to expand the applicability of the WVA, one must solve these problems properly.In this dissertation, we first review the basics of precision measurement and the WVA and then present our method of evaluation for the WVA considering the measurement uncertainty including the ambiguity of the WVA.We applied our method of evaluation for the two celebrated experiments to confirm the basic validity of the results of the two experiments.In addition, as a new application of the WVA, we consider the time evolution of unstable particles to find that the decay time distribution is generally characterized by the weak value of the Hamiltonian and that the state selections can indeed be performed by measuring the particles produced during the decay.Our result indicates that the WVA may also be applicable to the system of unstable particles.

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A general description of weak measurements is presented. The “weak value” of an operator in pre- and post-selected systems is described both mathematically and using physical arguments. An optical experiment in which a measurement of a weak value has been realized is described. The possible use of pre- and post-selection as a means to amplify and detect weak effects is also discussed.

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By analysing the concept of contextuality (Bell–Kochen–Specker) in terms of pre- and post-selection, it is possible to assign definite values to observables in a new and surprising way. Physical reasons are presented for restrictions on these assignments. When measurements are performed which do not disturb the pre- and post-selection (i.e. weak measurements), then novel experimental aspects of contextuality can be demonstrated. We also prove that every PPS-paradox with definite predictions directly implies 'quantum contextuality' which is introduced as the analogue of contextuality at the level of quantum mechanics rather than at the level of hidden variable theories. Finally, we argue that certain results of these measurements (e.g. eccentric weak values outside the eigenvalue spectrum) cannot be explained by a 'classical-like' hidden variable theory.

arXiv:quant-ph/0602226•Journal of Physics A Mathematical and Theoretical•Jul 12, 2007•63 citations

The concept of weak value of a quantum mechanical observable is based on a weak measurement followed by a strong measurement of a complementary observable. By applying post-selection, the weak value can, under certain circumstances, have values much higher than would be possible without post-selection, even above the highest eigenvalue of the measurement operator. Weak values have a wide range of applications, starting from the amplification of the resolution in high sensitivity metrology. The weak values have been especially successful in the area of quantum optics, which provides a suitable set of tools for performing the weak measurements with post-selection. The topic of this thesis is to lay the ground for the adaptation of weak values also in other fields of physics, the target being to harness the weak values in cavity quantum electrodynamics (cQED), and furthermore in circuit quantum electrodynamics. The main complication in the adaptation in these fields is the de-coherence, which is almost nonexistent in quantum optics. In this thesis we use quantum trajectory simulation to model the weak values under experimentally realistic conditions. We show how the weak values stem from the stochastic description of the quantum mechanical measurement, and that the second order approximation of the weak values is not sufficient in describing the effects of the measurement strength on the weak values. Using the simulation we demonstrate that the measurement resolution can be increased by applying post-selection even in the presence of de-coherence. Finally, we study the correlations between two entangled qubit, and show that post-selection appears to strengthen it.

Abstract Hamiltonian estimation is a cardinal task in applications of quantum metrology, quantum computation, quantum simulation, etc. When the evolution time is short in order to suppress noise and enhance sensitivity, the measurement and postprocessing may incur high cost. In this paper, we propose an adaptive weak measurement scheme for Hamiltonian estimation, which adjusts the pre- and post-selection according to the difference of the mean shift between two orthogonal post-selected states. According to the angle of the post-selected state, the proposed scheme can change from weak value amplification (WVA) to joint weak measurement (JWM). A study of the quantum Fisher information shows that the JWM situation is always optimal. Meanwhile, the WVA approach loses a little information but it can inherit the WVA’s technical advantages, which are suitable for short evolution times. The widely applicable scope of the proposed scheme makes us believe that it offers a new path for the understanding of unknown Hamiltonians.

The ability to post-select the outcomes of an experiment is a useful theoretical concept and experimental tool. In the context of weak measurements, post-selection can lead to surprising results such as complex weak values outside the range of eigenvalues. Usually post-selection is realized by a projective measurement, which is hard to implement in ensemble systems such as NMR. We demonstrate the first experiment of a weak measurement with post-selection on an NMR quantum information processor. Our setup is used for measuring complex weak values and weak values outside the range of eigenvalues. The scheme for overcoming the problem of post-selection in an ensemble quantum computer is general and can be applied to any circuit-based implementation. This experiment paves the way for studying and exploiting post-selection and weak measurements in systems where projective measurements are hard to realize experimentally.

A description of quantum systems at the time interval between two successive measurements is presented. Two wave functions, the first pre-selected by the initial measurement and the second post-selected by the final measurement describe quantum systems at a single time. It is shown how this approach leads to a new concept: a weak value of an observable. Weak values represent novel characteristics of quantum systems between two measurements. They are outcomes of a standard measuring procedure that fulfills certain requirements of “weakness”. We call it weak measurement . Physical meaning and underlying mathematical structure of weak measurements are explored.

The novel notion of a physical quantity called 'weak value', which was proposed by Aharonov et al. in 1988, has attracted much attention in the fields of quantum foundations and quantum measurement.In particular, the fact that through proper state selections the weak value can be made arbitrarily large, exceeding the conventional bound set by the maximum eigenvalue of the physical observable, has been widely regarded as a promising source for precision measurement.Indeed, in recent years we have seen increasingly many reports of successful applications of the weak value in photonic systems with various instances of observables amplified effectively.However, it has also been recognized that there are two major obstacles in realizing this weak value amplification (WVA).First, it is unclear under what conditions the WVA becomes effective due to the conflict between the ambiguity caused by the state selections and the amplification effect.Second, the required state selections may not be available depending on the systems one is dealing with.In order to expand the applicability of the WVA, one must solve these problems properly.In this dissertation, we first review the basics of precision measurement and the WVA and then present our method of evaluation for the WVA considering the measurement uncertainty including the ambiguity of the WVA.We applied our method of evaluation for the two celebrated experiments to confirm the basic validity of the results of the two experiments.In addition, as a new application of the WVA, we consider the time evolution of unstable particles to find that the decay time distribution is generally characterized by the weak value of the Hamiltonian and that the state selections can indeed be performed by measuring the particles produced during the decay.Our result indicates that the WVA may also be applicable to the system of unstable particles.

By analysing the concept of contextuality (Bell–Kochen–Specker) in terms of pre- and post-selection, it is possible to assign definite values to observables in a new and surprising way. Physical reasons are presented for restrictions on these assignments. When measurements are performed which do not disturb the pre- and post-selection (i.e. weak measurements), then novel experimental aspects of contextuality can be demonstrated. We also prove that every PPS-paradox with definite predictions directly implies 'quantum contextuality' which is introduced as the analogue of contextuality at the level of quantum mechanics rather than at the level of hidden variable theories. Finally, we argue that certain results of these measurements (e.g. eccentric weak values outside the eigenvalue spectrum) cannot be explained by a 'classical-like' hidden variable theory.