A two-step measurement protocol of a quantum system, known as the weak value (WV), was introduced more than two decades ago by Aharonov et al (1988 Phys. Rev. Lett. 60 1351) and has since then been studied in various contexts. In this paper, we discuss another two-step measurement protocol that we dub the null weak value (NWV). The protocol consists of a partial-collapse measurement followed by quantum manipulation on the system and finally a strong measurement. The first step is a strong measurement which takes place with a small probability. The second strong measurement is used as a postselection on the outcome of the earlier step. Not being measured in the partial-collapse stage (null outcome) leads to a nontrivial correlation between the two measurements. The NVW protocol, first defined for a two-level system (Zilberberg et al 2012 arXiv:1205.3877), is then generalized to a multi-level system and compared with the standard-WV protocol.
A two-step measurement protocol of a quantum system, known as the weak value (WV), was introduced more than two decades ago by Aharonov et al (1988 Phys. Rev. Lett. 60 1351) and has since then been studied in various contexts. In this paper, we discuss another two-step measurement protocol that we dub the null weak value (NWV). The protocol consists of a partial-collapse measurement followed by quantum manipulation on the system and finally a strong measurement. The first step is a strong measurement which takes place with a small probability. The second strong measurement is used as a postselection on the outcome of the earlier step. Not being measured in the partial-collapse stage (null outcome) leads to a nontrivial correlation between the two measurements. The NVW protocol, first defined for a two-level system (Zilberberg et al 2012 arXiv:1205.3877), is then generalized to a multi-level system and compared with the standard-WV protocol.
A two-step measurement protocol of a quantum system, known as weak value (WV), has been introduced more than two decades ago by Aharonov et al. [1], and has since been studied in various contexts. Here we discuss another two-step measurement protocol which we dub null weak value (NWV). The protocol consists of a partial-collapse measurement followed by quantum manipulation on the system and finally a strong measurement. The first step is a strong measurement which takes place with small probability. The second strong measurement is used as postselection on the outcome of the earlier step. Not being measured in the partial-collapse stage (null outcome) leads to a non-trivial correlation between the two measurements. The NVW protocol, first defined for a two-level system [2], is generalized to a multi-level system, and compared to the standard-WV protocol.
The one-time and two-time measurements are studied by employing the idea of Aharonov et al. that the strength of a measurement depends on the state of the measuring device in the context of von Neumann measurement theory. A measurement is classified as either a strong or a weak measurement depending on whether the interaction between the system and the measuring device is infinitely large or negligibly small. A single strong measurement yields one of the eigenvalues of the operator while a single weak measurement, in addition to the eigenvalues, has infinite other possible outcomes. Nevertheless, the expectation values of the strong and the weak one-time measurements are the same. However, the expectation values of the two-time measurements depend on the strength of the first measurement. For the strong case the result is consistent with the conventional result without having to introduce the state collapsing. For the weak case the result is the real part of the expectation value of the time-ordered two-time operator. The fact that this expectation value cannot be factorized indicates that the entanglement between the system and the measuring device introduced by a weak measurement cannot be neglected.
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.
The physical meaning of the quantum weak value still remains a matter of debate. Originally introduced by Aharonov et al. [Phys. Rev. Lett. 60, 1351 (1988).] as another counterintuitive feature of quantum mechanics, it eventually evolved into a practical tool that is widely used. The theoretical framework in which weak values were introduced was given by von Neumann's model of quantum measurements. In this model, a system is submitted to measurement by coupling it to a ``pointer,'' or meter. In the weak-coupling regime, one may perform a low-order Taylor expansion for the evolution operator of the system and meter. This standard approach ties weak values with weak couplings and weak measurements. We report closed-form expressions that can be used to untie weak values from weak measurements. The regime of strong measurements thereby becomes accessible to weak values, without leaving the framework in which the latter were originally introduced. The reported results should also help us to better understand the physical meaning of weak values.
Publisher’s Note: This paper, originally published on 10/24/2016, was replaced with a corrected/revised version on 11/8/2016. If you downloaded the original PDF but are unable to access the revision, please contact SPIE Digital Library Customer Service for assistance. In the late 1980s, Aharonov and colleagues developed the notion of a weak measurement of a quantum observable that does not appreciably disturb the system.1, 2 The measurement results are conditioned on both the pre-selected and post-selected state of the quantum system. While any one measurement reveals very little information, by making the same measurement on a large ensemble of identically prepared pre- and post-selected (PPS) states and averaging the results, one may obtain what is known as the weak value of the observable with respect to that PPS ensemble. Recently, weak measurements have been proposed as a method of assessing the security of QKD in the well-known BB84 protocol.3 This weak value augmented QKD protocol (WV-QKD) works by additionally requiring the receiver, Bob, to make a weak measurement of a particular observable prior to his strong measurement. For the subset of measurement results in which Alice and Bob's measurement bases do not agree, the weak measurement results can be used to detect any attempt by an eavesdropper, Eve, to correlate her measurement results with Bob's. Furthermore, the well-known detector blinding attacks, which are known to perfectly correlate Eve's results with Bob's without being caught by conventional BB84 implementations, actually make the eavesdropper more visible in the new WV-QKD protocol. In this paper, we will introduce the WV-QKD protocol and discuss its generalization to the 6-state single qubit protocol. We will discuss the types of weak measurements that are optimal for this protocol, and compare the predicted performance of the 6- and 4-state WV-QKD protocols.