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.
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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