[論文レビュー] Inaccurate (weak) measurements classical and quantum

D. Sokolovski, D. Alonso+1 more|arXiv (Cornell University)|Mar 13, 2026

Quantum Mechanics and Applications被引用数 0

ひとこと要約

The paper compares highly inaccurate (weak) measurements in classical and quantum systems, showing that individual-trial information is lost while ensemble statistics and quantum quasi-probabilities can be extracted, and that post-selection reshapes distributions without providing evidence of anomalous quantum values.

ABSTRACT

We consider highly inaccurate measurements made on classical stochastic and quantum systems. In the quantum case such a \e{weak} measurement preserves coherence between the system's alternatives. We demonstrate that in both cases the information about the scenario realised in each individual trial is lost. However, ensemble parameters such as classical path probabilities, and quantum quasi-probabilities can be extracted from the obtained statistics. In both cases causality ensures that additional post-selection only redistributes individual outcomes between the system's final states. Quantum quasi-probabilities may change sign, which allows for anomalously large meter's (pointer's) reading for some final states. These, we show, result from mere \e{reshaping} of a broad distribution obtained earlier, and provide no \e{experimental evidence} of quantum variables taking, on rare occasions, exceptionally large values.

研究の動機と目的

• Motivate understanding of highly inaccurate measurements and their effect on extracting information about intermediate conditions in classical and quantum transitions.
• Analyze whether weak-pointer shifts correspond to a unique value of the measured quantity in each trial.
• Examine how post-selection redistributes outcomes and affects the interpretation of measurement results in both frameworks.
• Introduce quasi-probabilities for virtual paths and discuss their properties and implications for quantum measurements.

提案手法

• Develop a classical analogue with an uncertain pointer measuring a variable B along multiple paths from initial to final states.
• Derive asymptotic results for broad pointer distributions showing shifts equal to ensemble averages or path-related quantities.
• Extend the classical model to quantum systems using von Neumann pointers and unitary evolution to obtain analogous distributions.
• Introduce and compute quantum quasi-probabilities for virtual paths and relate them to weak values via post-selected ensembles.
• Provide simple examples, including a double-slit-like setup, to illustrate negative quasi-probabilities and their interpretation.

Figure 1: Two broad Gaussians ( 5 ) in the l.h.s. of Eq.( 1 ), $\Delta x=30$ , $A_{1}=1$ , $B_{1}=0$ , and $A_{2}=-0.8$ , $B_{2}=-1$ , add up to a smaller Gaussian shifted to the right by approximately 4 units. The dot-dashed line shows the same Gaussian centred at $x=0$ . In the inset the added das

実験結果

リサーチクエスチョン

• RQ1 Can the shift of an inaccurate pointer be interpreted as a unique value of a measured quantity for each transition in both classical and quantum cases?
• RQ2 Do anomalously large pointer readings in quantum weak measurements indicate actual large values of observables or are they reshaped artefacts of post-selection?
• RQ3 How do post-selection and interference affect the extraction of path probabilities and ensemble statistics in weak measurements?
• RQ4 What is the role and interpretation of quantum quasi-probabilities in describing virtual paths and weak values?
• RQ5 Can classical-like causality and restructuring arguments account for peculiar quantum-reading outcomes without implying new quantum variables taking extreme values?

主な発見

• In both classical and quantum setups, highly inaccurate measurements erase single-trial path information but preserve ensemble statistics such as path probabilities (classical) or weak-value–like averages (quantum).
• Post-selection merely reshapes the aggregated readings among final states, without creating new information about individual trials in the sense of revealing hidden values.
• Quantum quasi-probabilities can take negative values and, unlike true probabilities, do not represent observable frequencies; they provide a coherent accounting for interference between paths and can yield large apparent shifts via reshaping of broad distributions.
• The observed shifts in highly inaccurate quantum measurements correspond to weak values when expressed for pre- and post-selected states, yet these do not imply that the observable assumes anomalously large values on rare events; rather, they reflect redistribution within the measurement statistics.
• A simple double-slit–like example demonstrates that some quasi-probabilities are negative, illustrating non-classical correlations, while causality and normalization are preserved in the overall distributions.
• The reshaping mechanism explains anomalously large pointer readings as results of post-selection and distribution structure, not as evidence of exceptional quantum values in single trials.

Figure 2: a) Alice’s classical setup ( $N=3$ ). The particle passes through a state $|i{\rangle}$ on the way to its final state $j$ . b) Each of the nine paths available to the particle is equipped with the the probability in Eq.( 6 ). c) initial uncertainty of an accurate pointer’s position, $\Delt