Weak Values and Quantum Properties


We investigate in this work the meaning of weak values through the prism of property ascription in quantum systems. Indeed, the weak measurements framework contains only ingredients of the standard quantum formalism, and as such weak measurements are from a technical point of view uncontroversial. However attempting to describe properties of quantum systems through weak values—the output of weak measurements—goes beyond the usual interpretation of quantum mechanics, that relies on eigenvalues. We first recall the usual form of property ascription, based on the eigenstate-eigenvalue link and the existence of “elements of reality”. We then describe against this backdrop the different meanings that have been given to weak values. We finally argue that weak values can be related to a specific form of property ascription, weaker than the eigenvalues case but still relevant to a partial description of a quantum system.

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  1. 1.

    Note that this reasoning assumes the pointer states \(\left| \varphi _{x_{0} -ga_{k}}\right\rangle \) of Eq. (7) are orthogonal. Otherwise a pointer state does not correlate with a single eigenstate and no element of reality can be defined.

  2. 2.

    This implementation of the three-box paradox [2] has been described in details elsewhere [23, 24].

  3. 3.

    The estimate minimizes a specific distance d in Hilbert space, namely \(d=\mathrm {Tr}\left[ \left| \psi (t_{i} )\right\rangle \left\langle \psi (t_{i})\right| \left( A-A_{\mathrm {est} }\right) ^{2}\right] ,\) and the resulting best estimate is [40] \(A_{\mathrm {est} }=\sum _{f}{\text {Re}}A_{f}^{w}\left| b_{f}\right\rangle \left\langle b_{f}\right| {\text {Re}}A_{f}^{w}\) where \(A_{f}^{w}\) is the weak value (19).

  4. 4.

    We have already noticed above that even in the absence of postselection, no element of reality can be defined if the pointer states are not orthogonal.

  5. 5.

    Unsurprisingly, the current density appears in the numerator of the following weak value of the momentum, \(\mathrm {Re}\frac{\left\langle x\right| P\left| \psi (t)\right\rangle }{\left\langle x\right| \left. \psi (t)\right\rangle }=\frac{mj(x,t)}{\rho (x,t)}\).

  6. 6.

    The corresponding classical expression is


    where \(\rho (q)\) is the configuration space classical distribution. The integral is taken over \(\mathcal {B}_{f}\) which is the set of all q’s taken at \(t_{w}\) such that at the final time \(t_{f}\) we have \(B(q,t_{f})=b_{f}\). In a classical setting, the filtering needs to be done before the weak interaction takes place. Note that the denominator is simply the normalization constant for the density due to the filtering (see [46] for details).


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Dipankar Home (Bose Institute, Kolkata) and Urbasi Sinha (Raman Research Institute, Bangalore) are thanked for useful discussions on an earlier version of the manuscript. Partial support from the Templeton Foundation (Project 57758) is gratefully acknowledged.

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Matzkin, A. Weak Values and Quantum Properties. Found Phys 49, 298–316 (2019). https://doi.org/10.1007/s10701-019-00245-3

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  • Measurement in quantum mechanics
  • Properties of quantum systems
  • Weak measurements
  • Post-selected quantum systems