We study the outcomes in a general measurement with postselection, and derive upper bounds for the pointer readings in weak measurement. Using the idea of weak measurement, we study Hardy's gedanken experiment and show how the negative probabilities emerge in weak measurement. By calculating the information gain of the measuring device about which path the particles pass through, we show that the negative probabilities only emerge for cases when the information gain is little due to very weak coupling between the measuring device and the particles. When the coupling strength increases, we can unambiguously determine whether a particle passes through a given path every time, hence the average shifts always represent true probabilities, and the strange negatives probabilities disappear.
We study the outcomes in a general measurement with postselection, and derive upper bounds for the pointer readings in weak measurement. Using the idea of weak measurement, we study Hardy's gedanken experiment and show how the negative probabilities emerge in weak measurement. By calculating the information gain of the measuring device about which path the particles pass through, we show that the negative probabilities only emerge for cases when the information gain is little due to very weak coupling between the measuring device and the particles. When the coupling strength increases, we can unambiguously determine whether a particle passes through a given path every time, hence the average shifts always represent true probabilities, and the strange negatives probabilities disappear.
A measurement consists in coupling a system to a probe and reading the output of the probe to gather information about the system. The weaker the coupling, the smaller the back-action on the system, but also the less information conveyed. If the system undergoes a second measurement, the statistics of the first output can be conditioned on the value of the second one. This procedure is known as postselection. A postselected weak measurement of an observable can give a large average output of the probe when the postselected state is nearly orthogonal to the initial state of the system. This large value is an interference effect in the readout of the probe, which is initially in a coherent superposition of readout states (also known as pointer states). Usually, the weak interaction between system and probe is considered instantaneous, so that the dynamics of the probe can be neglected. However, for a weak measurement in solid-state devices, an interaction of finite duration is likely needed. Here, we show how this finite duration generates a contribution of the dynamical phase to the readout statistics. Furthermore, we derive interpolation formulas that are able to describe the statistics of the weak measurement for the whole range of pre- and postselected states. Phase-space averages appear in the expansion, suggesting an interpretation in terms of non-positive probabilities. Decoherence in the probe is also accounted for and it is pointed out the existence of a regime of intermediate coupling strength in which coherent oscillations can be observed in the probability of the readout.
The Uncertainty Principle forbids one to determine which of the two paths a quantum system has travelled, unless interference between the alternatives had been destroyed by a measuring device, e.g., by a pointer. One can try to weaken the coupling between the device and the system in order to avoid the veto. We demonstrate, however, that a weak pointer is at the same time an inaccurate one, and the information about the path taken by the system in each individual trial is inevitably lost. We show also that a similar problem occurs if a classical system is monitored by an inaccurate quantum meter. In both cases, one can still determine some characteristic of the corresponding statistical ensemble, a relation between path probabilities in the classical case, and a relation between the probability amplitudes if a quantum system is involved.
A quantum transition can be seen as a result of interference between various pathways (e.g., Feynman paths), which can be labeled by a variable f. An attempt to determine the value of f without destroying the coherence between the pathways produces a weak value of f. We show f to be an average obtained with an amplitude distribution which can, in general, take negative values, which, in accordance with the uncertainty principle, need not contain information about the actual range of f which contributes to the transition. It is also demonstrated that the moments of such alternating distributions have a number of unusual properties which may lead to a misinterpretation of the weak-measurement results. We provide a detailed analysis of weak measurements with and without post-selection. Examples include the double-slit diffraction experiment, weak von Neumann and von Neumann--like measurements, traversal time for an elastic collision, phase time, and local angular momentum.
Suppose we make a series of measurements on a chosen quantum system. The outcomes of the measurements form a sequence of random events, which occur in a particular order. The system, together with a meter or meters, can be seen as following the paths of a stochastic network connecting all possible outcomes. The paths are shaped from the virtual paths of the system, and the corresponding probabilities are determined by the measuring devices employed. If the measurements are highly accurate, the virtual paths become “real”, and the mean values of a quantity (a functional) are directly related to the frequencies with which the paths are traveled. If the measurements are highly inaccurate, the mean (weak) values are expressed in terms of the relative probabilities’ amplitudes. For pre- and post-selected systems they are bound to take arbitrary values, depending on the chosen transition. This is a direct consequence of the uncertainty principle, which forbids one from distinguishing between interfering alternatives, while leaving the interference between them intact.
Weak values and measurements have been proposed as a means to achieve dramatic enhancements in metrology based on the greatly increased range of possible measurement outcomes. Unfortunately, the very large values of measurement outcomes occur with highly suppressed probabilities. This raises three vital questions in weak-measurement-based metrology. Namely, (Q1) Does postselection enhance the measurement precision? (Q2) Does weak measurement offer better precision than strong measurement? (Q3) Is it possible to beat the standard quantum limit or to achieve the Heisenberg limit with weak measurement using only classical resources? We analyze these questions for two prototypical, and generic, measurement protocols and show that while the answers to the first two questions are negative for both protocols, the answer to the last is affirmative for measurements with phase-space interactions, and negative for configuration space interactions. Our results, particularly the ability of weak measurements to perform at par with strong measurements in some cases, are instructive for the design of weak-measurement-based protocols for quantum metrology.
We study quantum measurement with preselection and postselection, and derive the precise expressions of the measurement results without any restriction on the coupling strength between the system and the measuring device. For a qubit system, we derive the maximum pointer shifts by choosing appropriate initial and finial states. A significant amplification effect is obtained when the interaction between the system and the measuring device is very weak, and typical ideal quantum measurement results are obtained when the interaction is strong. The improvement of the signal-to-noise ratio (SNR) and the enhancement of the measurement sensitivity (MS) by weak measurements are studied. Without considering the probability decrease due to postselection, the SNR and the MS can be both significantly improved by weak measurements; however, neither SNR nor MS can be effectively improved when the probability decrease is considered.