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Imaginary weak values and the Hubble diagram

João Magueijo, Michael Fine

Published Oct 14, 2026• 0 citations

Abstract

Imaginary weak values are a peculiarity of standard quantum mechanics subject to post-selection and weak measurements. We examine their possible role and physical meaning in teleological or post-selected quantum cosmology, in the context of the Hubble diagram. We only consider the linear Hubble diagram, controlled by the Hubble parameter and the redshift effect. We set up a variety of von Neumann models entangling the geometry with different pointers, either upstream, via the redshift itself, or downstream, via the instruments used to measure the arriving frequency of calibrated lines. We find that the AAV (Aharonov, Albert and Vaidman) formula for weak values remains valid in these models, even though the assumptions are wildly different. Hence the imaginary part of the weak value of the Hubble parameter translates into a shift in the conjugate variable of the pointer, and we evaluate the very diverse expressions this takes for the various meters considered. Furthermore, in situations where meters are coupled, the effect may only appear in the first meter explicitly sensitive to the geometry and, unlike real weak values, need not propagate down the chain. The imaginary weak value is therefore distinctly in the ``eye of the beholder'', as a result of quantum measurement theory. This could distinguish teleological quantum explanations of cosmic acceleration from dark-energy alternatives that reproduce the same real Hubble diagram.

DOI: 10.5281/zenodo.23371235

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Imaginary weak values are a peculiarity of standard quantum mechanics subject to post-selection and weak measurements. We examine their possible role and physical meaning in teleological or post-selected quantum cosmology, in the context of the Hubble diagram. We only consider the linear Hubble diagram, controlled by the Hubble parameter and the redshift effect. We set up a variety of von Neumann models entangling the geometry with different pointers, either upstream, via the redshift itself, or downstream, via the instruments used to measure the arriving frequency of calibrated lines. We find that the AAV (Aharonov, Albert and Vaidman) formula for weak values remains valid in these models, even though the assumptions are wildly different. Hence the imaginary part of the weak value of the Hubble parameter translates into a shift in the conjugate variable of the pointer, and we evaluate the very diverse expressions this takes for the various meters considered. Furthermore, in situations where meters are coupled, the effect may only appear in the first meter explicitly sensitive to the geometry and, unlike real weak values, need not propagate down the chain. The imaginary weak value is therefore distinctly in the ``eye of the beholder'', as a result of quantum measurement theory. This could distinguish teleological quantum explanations of cosmic acceleration from dark-energy alternatives that reproduce the same real Hubble diagram.

The physical meaning of the quantum weak value still remains a matter of debate. Originally introduced by Aharonov et al. [Phys. Rev. Lett. 60, 1351 (1988).] as another counterintuitive feature of quantum mechanics, it eventually evolved into a practical tool that is widely used. The theoretical framework in which weak values were introduced was given by von Neumann's model of quantum measurements. In this model, a system is submitted to measurement by coupling it to a ``pointer,'' or meter. In the weak-coupling regime, one may perform a low-order Taylor expansion for the evolution operator of the system and meter. This standard approach ties weak values with weak couplings and weak measurements. We report closed-form expressions that can be used to untie weak values from weak measurements. The regime of strong measurements thereby becomes accessible to weak values, without leaving the framework in which the latter were originally introduced. The reported results should also help us to better understand the physical meaning of weak values.

The contribution of the detector dynamics to the weak measurement is analyzed. According to the usual theory [Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988)] the outcome of a weak measurement with preselection and postselection can be expressed as the real part of a complex number: the weak value. By accounting for the Hamiltonian evolution of the detector, here we find that there is a contribution proportional to the imaginary part of the weak value to the outcome of the weak measurement. This is due to the coherence of the probe being essential for the concept of complex weak value to be meaningful. As a particular example, we consider the measurement of a spin component and find that the contribution of the imaginary part of the weak value is sizable.

Weak measurement of a quantum system followed by postselection based on a subsequent strong measurement gives rise to a quantity called the weak value : a complex number for which the interpretation has long been debated. We analyse the procedure of weak measurement and postselection, and the interpretation of the associated weak value, using a theory of classical mechanics supplemented by an epistemic restriction that is known to be operationally equivalent to a subtheory of quantum mechanics. Both the real and imaginary components of the weak value appear as phase space displacements in the postselected expectation values of the measurement device's position and momentum distributions, and we recover the same displacements as in the quantum case by studying the corresponding evolution in our theory of classical mechanics with an epistemic restriction. By using this epistemically restricted theory, we gain insight into the appearance of the weak value as a result of the statistical effects of post selection, and this provides us with an operational interpretation of the weak value, both its real and imaginary parts. We find that the imaginary part of the weak value is a measure of how much postselection biases the mean phase space distribution for a given amount of measurement disturbance. All such biases proportional to the imaginary part of the weak value vanish in the limit where disturbance due to measurement goes to zero. Our analysis also offers intuitive insight into how measurement disturbance can be minimized and the limits of weak measurement.

In the weak value formalism of Aharonov et al., the weak value

{A}_{w}

of any observable

A

is generally a complex number. We derive a physical interpretation of its value in terms of the shift in the measurement pointer's mean position and mean momentum. In particular, we show that the mean position shift contains a term jointly proportional to the imaginary part of the weak value and the rate at which the pointer is spreading in space as it enters the measurement interaction.

For continuous weak measurement of qubits, we obtain exact expressions for weak values (WVs) from the post-selection restricted average of measurement outputs, by using both the quantumtrajectory- equation (QTE) and quantum Bayesian approach. The former is applicable to short-time weak measurement, while the latter can relax the measurement strength to finite. We find that even in the "very" weak limit the result can be essentially different from the one originally proposed by Aharonov, Albert and Vaidman (AAV), in a sense that our result incorporates non-perturbative correction which could be important when the AAV's WV is large. Within the Bayesian framework, we obtain also elegant expressions for finite measurement strength and find that the amplifier's noise in quantum measurement has no effect on the WVs. In particular, we obtain very useful result for homodyne measurement in circuit-QED system, which allows for measuring the real and imaginary parts of the AAV's WV by simply tuning the phase of the local oscillator. This advantage can be exploited as efficient state-tomography technique.

In a weak measurement the real and imaginary parts of a weak value participate in the shifts of the complementary variables of a pointer. While the real part represents the value of an observable in the limit of zero measurement strength, the imaginary one is regarded as the back-action due to the measurement with a post-selection, which has an influence on the post-selection probability. In this paper I give a case in which a real part could also appear as such a back-action in a post-selection probability on an equal footing with an imaginary one. It is also shown that both of the real and imaginary parts can be inferred by observing the probability in practice, which has an advantage that an additional system of a pointer is not needed.