We address the issue of how to properly treat, and in a more general setting, the concept of a weak value of a weak measurement in quantum mechanics. We show that for this purpose, one must take in account the effects of the measuring process on the entire phase space of the measuring system. By using coherent states, we go a step further than Jozsa in a recent paper and we present an example where the result of the measurement is symmetrical in the position and momentum observables and seems to be much better suited for quantum optical implementation.
We address the issue of how to properly treat, and in a more general setting, the concept of a weak value of a weak measurement in quantum mechanics. We show that for this purpose, one must take in account the effects of the measuring process on the entire phase space of the measuring system. By using coherent states, we go a step further than Jozsa in a recent paper and we present an example where the result of the measurement is symmetrical in the position and momentum observables and seems to be much better suited for quantum optical implementation.
Summary form only given. Measurements are the very basis of Physics, especially in Quantum Mechanics, where they assume even a more fundamental role because of wave function collapse after a “strong” (projective) measurement. Furthermore, measuring a quantum-mechanical observable completely erases the information on its conjugate one (e.g. measurement of position erases information on momentum, and vice-versa). Nevertheless, in Quantum Mechanics other kinds of measurement are possible. For example, wave function collapse can be overcome through Weak Measurements, i.e. measurements performed with an interaction sufficiently weak not to collapse the original state, featuring several interesting properties. An example of these are weak values [1-3], realised for the first time in [4-6], that have been used for addressing fundamental questions [7-12] as Contextuality and are also a tool for Quantum Metrology [13-19]. One of the most intriguing properties of Weak Measurements is that, since they are not affected by wave function collapse, they can allow gathering simultaneous information on non-commuting observables [20], impossible with the standard (projective) measurement protocols. A second example is offered by Protective Measurements [21], a new technique able to extract information on the expectation value of an observable even measuring a single particle. In this talk, after a general introduction to WMs, we present the first realisation of sequential weak value measurements [22], i.e. a joint measurement of the weak value of (incompatible) polarizations on a single photon. Then, we present an experiment addressed to explore the connection between anomalous weak values and Contextuality [23], showing a clear violation of the inequality proposed in [12] while satisfying all the additional theoretical requests, unequivocally demonstrating the contextual nature of weak values. Finally, we present and discuss the first implementation of Protective Measurements, showing unprecedented measurement capability and demonstrating how the expectation value of an observable can be obtained with a single measurement on a single particle.
We generalize the concept of the weak value of a quantum observable to cover arbitrary real positive operator measures. We show that the definition is operationally meaningful in the sense that it can be understood within the quantum theory of sequential measurements. In particular, we show that the weak value can be obtained from a single measurement scheme. We then present a detailed analysis of the recent experiment [J. S. Lundeen et al., Nature (London) 474, 188 (2011)] concerning the reconstruction of the state of a photon using weak measurements. We compare their method with the reconstruction method through informationally complete phase space measurements. In particular, we show that unlike with phase space measurements, the reconstruction of a completely unknown state is not always possible using the method of weak measurements.
A weak measurement on a system is made by coupling a pointer weakly to the system and then measuring the position of the pointer. If the initial wave function for the pointer is real, the mean displacement of the pointer is proportional to the so-called weak value of the observable being measured. This gives an intuitively direct way of understanding weak measurement. However, if the initial pointer wave function takes complex values, the relationship between pointer displacement and weak value is not quite so simple, as pointed out recently by Jozsa [R. Jozsa, Phys. Rev. A 76, 044103 (2007)]. This is even more striking in the case of sequential weak measurements [G. Mitchison, R. Jozsa, and S. Popescu, Phys. Rev. A 76, 062105 (2007)]. These are carried out by coupling several pointers at different stages of the evolution of the system, and the relationship between the products of the measured pointer positions and the sequential weak values can become extremely complicated for an arbitrary initial pointer wave function. Surprisingly, all this complication vanishes when one calculates the cumulants of pointer positions. These are directly proportional to the cumulants of sequential weak values. This suggests that cumulants have a fundamental physical significance for weak measurement.
The main goal of this thesis is to introdude to readers the concept of weak measure- ments in quantum mechanics. Firstly a model of indirect measuremnts is established as an alternative to Von Neumann measurement postulate. This model leads to a possible generalization in a form of weak measurements. Then the concept of postselection is introduced and involved into the issue of weak measurements which leads to the weak values of physical quantities. All these theoretical concepts are presented using simple model examples. Finally, a discussion in provided on the possible interpretation and of weak values and their experimental verification.
In Chapter 4, we take a limit where the coupling of the measurement apparatus to the quantum system is very small, and in this limit, discuss weak measurements and weak values. The later involved a sequence of a weak and a strong measurement. Generalizations of these effects are discussed using the concepts in Chapter 3, and we discuss generalized eigenvalues of quantum observables that can exceed the eigenvalue range and reincorporate the concept of observables in generalized measurement theory.
Three recent results on weak measurements are presented. They are: i) repeated measurements on a single copy can not provide any information on it and further, that in the limit of very large such measurements, weak measurements have exactly the same characterstics as strong measurements, ii) the apparent non-invasiveness of weak measurements is \emph{illusory} and they are no more advantageous than strong measurements even in the specific context of establishing Leggett-Garg inequalities, when errors are properly taken into account, and, finally, iii) weak value measurements are optimal, in the precise sense of Wootters and Fields, when the post-selected states are mutually unbiased with respect to the eigenstates of the observable whose weak values are being measured. Notion of weak value coordinates for state spaces are introduced and elaborated.