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Journal of Physics Conference Series

Weak Values, the Reconstruction Problem, and the Uncertainty Principle

Charlyne de Gosson, Maurice A. de Gosson

Published Mar 1, 2016• 2 citations
Publisher

Abstract

Closely associated with the notion of weak value is the problem of reconstructing the post-selected state: this is the so-called reconstruction problem. We show that the reconstruction problem can be solved by inversion of the cross-Wigner transform, using an ancillary state. We thereafter show, using the multidimensional Hardy uncertainty principle, that maximally concentrated cross-Wigner transforms corresponds to the case where a weak measurement reduces to an ordinary von Neumann measurement.

DOI
10.1088/1742-6596/701/1/012011

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Closely associated with the notion of weak value is the problem of reconstructing the post-selected state: this is the so-called reconstruction problem. We show that the reconstruction problem can be solved by inversion of the cross-Wigner transform, using an ancillary state. We thereafter show, using the multidimensional Hardy uncertainty principle, that maximally concentrated cross-Wigner transforms corresponds to the case where a weak measurement reduces to an ordinary von Neumann measurement.
A so called “weak value” of an observable in quantum mechanics (QM) may be obtained in a weak measurement + post-selection procedure on the QM system under study. Applied to number operators, it has been invoked in revisiting some QM paradoxes ( e.g. , the so called Three-Box Paradox and Hardy’s Paradox). This requires the weak value to be interpreted as a bona fide property of the system considered, a par with entities like operator mean values and eigenvalues. I question such an interpretation; it has no support in the basic axioms of quantum mechanics and it leads to unreasonable results in concrete situations.
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 measurements allow a probe system to encode the transitionary weak value of a desired observable on a signal system that is both pre- and post-selected. Such interactions alter the state and measurement statistics of the probe system. We establish that by encoding the imaginary part of a weak value onto an initially entangled probe, we can modify its entanglement content, provided the probe observable can distinguish between system states of different von Neumann entropies. Apart from fundamental interest, this result illustrates the utility of the imaginary weak value as a calculational tool in certain entanglement concentration protocols.
The outcomes of a series of measurements, made on a quantum system, 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) is directly related to the frequencies with which the paths are travelled. 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 to distinguish between interfering alternatives, while leaving the interference between them intact.
A weak measurement performed on a pre- and post-selected quantum system can result in an average value that lies outside of the observable's spectrum. This effect, usually referred to as an ``anomalous weak value'', is generally believed to be possible only when a non-trivial post-selection is performed, i.e., when only a particular subset of the data is considered. Here we show, however, that this is not the case in general: in scenarios in which several weak measurements are sequentially performed, an anomalous weak value can be obtained without post-selection, i.e., without discarding any data. We discuss several questions that this raises about the subtle relation between weak values and pointer positions for sequential weak measurements. Finally, we consider some implications of our results for the problem of distinguishing different causal structures.