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arXiv:2609.04812

Transition between weak and strong measurements in the presence of post-selection

Shogo Hanashiro, Holger F. Hofmann

quant-phSubmitted Sep 4, 2026• Announced Sep 7, 2026• 0 citations
v1 · Sep 4, 2026v2 · Oct 21, 2026
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Abstract

In the weak measurement regime, post-selection can result in the observation of anomalous weak values, seemingly contradicting the eigenvalue statistics observed when the measurement interaction is strong. Here, we investigate the dependence of meter statistics on measurement strength in a post-selected measurement. We find that the meter statistics in the intermediate regime between weak and strong measurements is nearly independent of measurement strength and show that, in this regime, the system performs a measurement of momentum on the meter. The transition between weak and strong measurements is explained by a reversal of the roles of the system and the meter, where the post-selection acts as a readout of information about the meter.

Comments
11 pages, 9 figures

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In the weak measurement regime, post-selection can result in the observation of anomalous weak values, seemingly contradicting the eigenvalue statistics observed when the measurement interaction is strong. Here, we investigate the dependence of meter statistics on measurement strength in a post-selected measurement. We find that the meter statistics in the intermediate regime between weak and strong measurements is nearly independent of measurement strength and show that, in this regime, the system performs a measurement of momentum on the meter. The transition between weak and strong measurements is explained by a reversal of the roles of the system and the meter, where the post-selection acts as a readout of information about the meter.
It is widely assumed that the quantum fluctuations of the meter readout in a weak measurement make it impossible to identify the contributions originating from the individual values of the physical property observed in the measurement. Here, we show that a careful analysis of the quantum dynamics of the meter system allows us to identify a universal relation between quantum interference in the post-selection probability, and quantum interference in the statistics of the meter readout. The analysis reveals that quantum interference modifies the readout distribution of the meter in two ways, a diffusion term that identifies the appropriate operator ordering in the post-selected variance of the observed system property, and a wavefunction-dependent update of the initial meter statistics based on the effect of back action on the post-selection probability. Our results show that the statistical patterns described by meter interference provide important details about the physics of post-selection in weak measurements, allowing us to exclude the possibility of statistical artefacts in the experimental observation of weak values.
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.
Measurements map the value of a target observable onto a meter shift, resulting in a meter readout that combines the initial statistics of the meter state with the quantum statistics of the target observable. Even in the limit of weak measurement interactions, some information about the fluctuations of the target observable can be extracted from the change in the readout fluctuations caused by the measurement interaction. Here, we apply the Heisenberg picture to analyze the changes in the meter readout statistics caused by sufficiently weak measurement interactions, including the effects of nonlinearities in the meter response. When additional information is obtained in a subsequent measurement of the system, the meter fluctuations are modified based on the postselected statistics of the target observable. In addition, our analysis reveals a direct modification of the meter fluctuations due to the dependence of the postselection probability on the dynamics induced by the meter in the measurement interaction. We point out that the quantum formalism makes it difficult to distinguish this dynamic term from the physical fluctuations of the target observable and stress the importance of distinguishing between genuine conditional fluctuations of the target observable and the dynamic pseudovariance associated with the measurement back-action.
Quantum measurement is physically realized through a finite dynamical interaction between a system and a measuring apparatus, giving rise to a continuous transition from weak to strong regimes. While this crossover is well understood under ideal conditions, the combined role of thermal instabilities and pre- and post-selection open dynamics has not been systematically addressed. Here, we develop a generalized open-system framework to analyze the weak-to-strong measurement transition in the simultaneous presence of environmental decoherence and thermal noise. We model the probe as a thermal Gaussian state, explicitly incorporating temperature-dependent fluctuations in the measuring device, and include open-system evolution of the measured system prior to post-selection. By deriving the apparatus's final state, we show that the measurement statistics are modified in a nontrivial, highly sensitive manner by the temperature regime of the system's thermal instabilities, the probe's thermal properties, and the particular choice of pre- and post-selection. This approach allows us to characterize how thermal effects reshape the weak-value condition, the anomalous amplification, and the resulting metrological sensitivity, demonstrating the protocol's practical utility for precision measurements across the full measurement crossover.
We show using statistically rigorous arguments that the technique of weak value amplification (WVA) does not perform better than standard statistical techniques for the tasks of single parameter estimation and signal detection. Specifically we prove that post-selection, a necessary ingredient for WVA, decreases estimation accuracy and, moreover, arranging for anomalously large weak values is a suboptimal strategy. In doing so, we explicitly provide the optimal estimator, which in turn allows us to identify the optimal experimental arrangement to be the one in which all outcomes have equal weak values (all as small as possible) and the initial state of the meter is the maximal eigenvalue of the square of the system observable. Finally, we give precise quantitative conditions for when weak measurement (measurements without post-selection or anomalously large weak values) can mitigate the effect of uncharacterized technical noise in estimation.
Information about the internal properties of a system can only be obtained through interactions of the system with an external meter. However, such interactions generally result in entanglement between the system and the meter, making it difficult to trace the measurement result back to a specific value of the physical property in the system. It is therefore possible that the outcomes of quantum measurements depend in a nontrivial manner on the dynamics of the measurement interaction, possibly providing a physical explanation for the role of measurement contexts in quantum mechanics. Here we show that the effects of the measurement interaction on the meter can be described entirely in terms of the quantum coherent system dynamics associated with the back-action on the system. For sufficiently small back-action uncertainties, the physical property of the system is described by a weak value obtained from the Hamilton-Jacobi equation of the back-action dynamics. At higher measurement resolutions, the observed values are determined by quantum interferences between different amounts of back-action. Eigenvalues emerge when the quantum interferences between different back-actions correspond to a Fourier transform in the back-action parameter. We conclude that the values of physical properties obtained in quantum measurements originate from the quantum coherent properties of the back-action dynamics generated by that physical property during an interaction. Measurement outcomes represent elements of the dynamics and cannot be explained by measurement independent elements of reality.