Non-testability of instrument validity under continuous treatments

This note presents a proof of the conjecture in Pearl (1995) about testing the validity of an instrumental variable in hidden variable models. It implies that instrument validity cannot be tested in the case where the endogenous treatment is continuously distributed. This stands in contrast to the classical testability results for instrument validity when the treatment is discrete. However, imposing weak structural assumptions on the model, such as continuity between the observable variables, can re-establish theoretical testability in the continuous setting.

Using changes in host demographic rates to reveal the effects of infection with hidden variable models

The impacts of disease on host vital rates can be clearly demonstrated using longitudinal studies, but these studies can be expensive and logistically challenging. We examined the utility of hidden variable models to infer the individual effects of disease, caused by infection, from population-level measurements of survival and fecundity when longitudinal studies are not possible. Our approach seeks to explain temporal changes in population-level vital rates by coupling observed changes in the infection status of individuals to an epidemiological model. We tested the approach using both single and coinfection viral challenge experiments on populations of fruit flies (Drosophila melanogaster). Specifically, we determined whether our approach yielded reliable estimates of disease prevalence and of the effects of disease on survival and fecundity rates for treatments of single infections and coinfection. We found two conditions are necessary for reliable estimation. First, diseases must drive detectable changes in vital rates, and second, there must be substantial variation in the degree of prevalence over time. This approach could prove useful for detecting epidemics from public health data in regions where standard surveillance techniques are not available, and in the study of epidemics in wildlife populations, where longitudinal studies can be especially difficult to implement.

Analysis of single-particle nonlocality through the prism of weak measurements

Although regarded today as an important resource in quantum information, nonlocality has yielded over the years many conceptual conundrums. Among the latter are nonlocal aspects of single particles which have been of major interest. In this paper, the nonlocality of single quanta is studied in a square nested Mach–Zehnder interferometer with spatially separated detectors using a delayed choice modification of quantum measurement outcomes that depend on the complex-valued weak values. We show that if spacelike separated Bob and Alice are allowed to freely control their quantum devices, the geometry of the setup constrains the local hidden variable models. In particular, hidden signaling and a list of contextual instructions are required to split a quantum state characterized by a positive Wigner function into two quantum states with nonpositive Wigner functions. This implies that local hidden variable models could rely neither on only two hidden variables for position and momentum, nor on simultaneous factorizability of both the hidden probability densities and weights of splitting to reproduce the correct quantum distributions. While our analysis does not fully exclude the existence of nonfactorizable local hidden variable models, it demonstrates that the recently proposed weak values of quantum histories necessitate contextual splitting of prior commitments to measurement outcomes, due to functional dependence on the total Feynman sum that yields the complex-valued quantum probability amplitude for the studied quantum transition. This analysis also highlights the quantum nature of weak measurements.

Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constantKG(3)

We consider the problem of reproducing the correlations obtained by arbitrary local projective measurements on the two-qubit Werner stateρ=v|ψ−⟩⟨ψ−|+(1−v)14via a local hidden variable (LHV) model, where|ψ−⟩denotes the singlet state. We show analytically that these correlations are local forv=999×689×10−6cos2⁡(π/50)≃0.6829. In turn, as this problem is closely related to a purely mathematical one formulated by Grothendieck, our result implies a new bound on the Grothendieck constantKG(3)≤1/v≃1.4644. We also present a LHV model for reproducing the statistics of arbitrary POVMs on the Werner state forv≃0.4553. The techniques we develop can be adapted to construct LHV models for other entangled states, as well as bounding other Grothendieck constants.