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arXiv:2608.15209 · Quanta

Anomalous Weak Pointer Shifts as Postselected Interference among Coarse-Grained Histories

Cody Charles Payne, Eliahu Cohen

quant-phSubmitted Aug 15, 2026• Announced Aug 18, 2026• 0 citations
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Abstract

Anomalous weak values and superoscillations allow a pointer to exhibit an effective shift outside the eigenvalue range of the measured observable. We reformulate this effect in a phase-space path-integral language. For each eigenvalue branch $a_j$ of the measured observable, {the interaction couples $a_j$ to the pointer position $\hat{x}_p$ and thereby translates the conjugate pointer momentum by $γa_j$.} After postselection, {the branch-conditioned amplitudes are coherently superposed with coefficients chosen to produce a superoscillatory approximation. When this conditional amplitude acts on an initial pointer momentum wavepacket for which the finite superoscillatory-window condition is satisfied, the resulting momentum wavefunction behaves as if it had been translated by $γA_w$, rather than by any of the eigenvalue-conditioned shifts $γa_j$. This yields an effective path-integral representation of anomalous weak pointer shifts and motivates their interpretation in terms of interference among finite coarse-grained histories.} We discuss how this picture bears on conservation-law questions.

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Anomalous weak values and superoscillations allow a pointer to exhibit an effective shift outside the eigenvalue range of the measured observable. We reformulate this effect in a phase-space path-integral language. For each eigenvalue branch aja_jaj​ of the measured observable, {the interaction couples aja_jaj​ to the pointer position x^p\hat{x}_px^p​ and thereby translates the conjugate pointer momentum by γajγa_jγaj​.} After postselection, {the branch-conditioned amplitudes are coherently superposed with coefficients chosen to produce a superoscillatory approximation. When this conditional amplitude acts on an initial pointer momentum wavepacket for which the finite superoscillatory-window condition is satisfied, the resulting momentum wavefunction behaves as if it had been translated by γAwγA_wγAw​, rather than by any of the eigenvalue-conditioned shifts γajγa_jγaj​. This yields an effective path-integral representation of anomalous weak pointer shifts and motivates their interpretation in terms of interference among finite coarse-grained histories.} We discuss how this picture bears on conservation-law questions.
Superoscillations, namely regions where a band-limited function f (x) varies faster than the fastest of its Fourier components k, generate the illusion that the Fourier content is ?supershifted? so as to lie outside the spectrum of the function. The relation between supershifts and superoscillations, central to the quantum weak measurements scheme, is explored in terms of two different representations of the local Fourier transform in the ?phase space? (x, k). The Wigner function W(x, k), regarded as a function of k for fixed x, inherits the band-limited property of f (x). Neverthless, its local k average can lie outside the spectrum because W, although real, posesses negative values. The local Wigner average of k equals the local wavenumber at x (local weak value of momentum), defined as the phase variation k loc(x)?=?? x arg f (x). By contrast, the Husimi function H(x, k), i.e. the windowed Fourier transform with window width L, corresponding to squeezing of the coherent state associated with (x, k) (and representing the pointer wavefunction after a weak measurement), is positive-definite. But it is not band-limited, and the local Husimi average of k equals k loc if L is small enough. These properties are illustrated numerically with two superoscillatory functions.
The association between large shifts of a pointer in a weak measurement and fast oscillations in an associated function involving the pre- and post-selected states has been clarified in a recent paper (Aharonov et al 2011 J. Phys. A: Math. Theor . 44 365304 ). Here we explore the association further for the case of an observable with N discrete eigenvalues, by calculating and illustrating how the supershift emerges, even for N = 2, as the uncertainty in the pointer position increases. This happens if the initial pointer wavefunction is Gaussian or Lorentzian but not if it is exponential.
In the weak measurement (WM) scenario involving weak interaction and postselection by projective measurement, the empirical significance of weak values is manifested in terms of shifts in the measurement pointer's mean position and mean momentum. In this context, a general quantitative treatment is presented in this paper by taking into account the hitherto unexplored effect of correlations among the pointer degrees of freedom which pertain to an arbitrary multidimensional preselected pointer state. This leads to an extension of the earlier results, showing that, for complex weak values, the correlations among different pointer degrees of freedom can crucially affect the way the imaginary parts of the weak values are related to the observed shifts of the mean pointer position and momentum. The particular relevance of this analysis is discussed in the case of sequential weak interactions followed by a projective measurement enabling postselection (called sequential WM) which, in the special case, reduces to the usual WM scheme (involving a single weak interaction and postseletion) modified by the effect of pointer-state correlations.
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.
Feynman's sum-over-histories formulation of quantum mechanics has been considered a useful calculational tool in which virtual Feynman histories entering into a coherent quantum superposition cannot be individually measured. Here we show that sequential weak values, inferred by consecutive weak measurements of projectors, allow direct experimental probing of individual virtual Feynman histories, thereby revealing the exact nature of quantum interference of coherently superposed histories. Because the total sum of sequential weak values of multitime projection operators for a complete set of orthogonal quantum histories is unity, complete sets of weak values could be interpreted in agreement with the standard quantum mechanical picture. We also elucidate the relationship between sequential weak values of quantum histories with different coarse graining in time and establish the incompatibility of weak values for nonorthogonal quantum histories in history Hilbert space. Bridging theory and experiment, the presented results may enhance our understanding of both weak values and quantum histories.
In Y. Aharonov et al. [Phys. Rev. Lett. 60, 1351 (1988)], it was shown that, in some weak measurements with postselection schemes, it is possible to obtain inferred mean values outside the range of the eigenvalues of the measured observable. In our work, we are going to show that this effect can be associated with the presence of an asymptotically vanishing but nonzero amount of system-detector quantum correlations. These correlations are also related to the presence of a finite (nonzero) amount of coherence of the initial states in the basis of the measured observable.