Anomalous Weak Values Without Post-Selection

Alastair A. Abbott1, Ralph Silva2,3, Julian Wechs1, Nicolas Brunner2, and Cyril Branciard1

1Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France
2Département de Physique Appliquée, Université de Genève, 1211 Genève, Switzerland
3Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland

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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.

A hallmark of quantum mechanics is the “no information without disturbance” principle: to learn about a quantum system one must measure it in some way, but doing so generally perturbs it irreversibly. As a result, it is often useful to perform “weak measurements”, in which one only disturbs the system of interest very weakly but, in turn, only learns a small amount of information about it. Such weak measurements can lead to surprising phenomena, such as “anomalous weak values”: when a weak-measurement is followed by a “post-selection” on certain final conditions, the average result of the measurement can be far outside the range of expected values (for example, one may find that the average score of a “quantum dice” is -100). This paradoxical effect has even found applications in quantum metrology and beyond.

The need to post-select on certain outcomes, thereby discarding the unselected data, is traditionally seen as an essential prerequisite to obtaining these anomalous weak-values. In this paper we show that, in some situations, this popular belief is in fact incorrect. In situations in which one performs several weak measurements sequentially on a system, one can obtain anomalous outcomes without any post-selection. We analyse this unexpected phenomenon in detail, showing how the final measurement can act as an “effective post-selection” but, crucially, without the need to discard any data. These results and insights shed new light on the phenomenon of anomalous weak-values, and raise questions about the relation between “weak values” – the formal property used to calculate the measurement outcomes in such scenarios – and the actual reading shown by the measurement device.

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[2] Surya Narayan Sahoo, Sanchari Chakraborti, Arun K. Pati, and Urbasi Sinha, "Quantum State Interferography", Physical Review Letters 125 12, 123601 (2020).

[3] Jianning Li, Haodi Liu, Zhihai Wang, and X. X. Yi, "Enhanced parameter estimation by measurement of non-Hermitian operators", AAPPS Bulletin 33 1, 22 (2023).

[4] Le Bin Ho, "Quantum backaction effects in sequential measurements", Annals of Physics 453, 169310 (2023).

[5] Chetan Waghela and Shubhrangshu Dasgupta, "Simulation of exceptional-point systems on quantum computers for quantum sensing", AVS Quantum Science 6 1, 014403 (2024).

[6] Eliahu Cohen, "Quantum measurements - yet another surprise", Quantum Views 3, 27 (2019).

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