The Schrödinger-Newton equation is a proposed model to explain the localisation of macroscopic particles by suppressing quantum dispersion with the particle's own gravitational attraction. On cosmic scales, however, dark energy also acts repulsively, as witnessed by the accelerating rate of universal expansion. Here, we introduce the effects of dark energy in the form of cosmological constant Λ, that drives the late-time acceleration of the universe, into the Schrödinger-Newton approach. We then ask in which regime dark energy dominates both canonical quantum diffusion and gravitational self-attraction. It turns out that this happens for sufficiently delocalised objects with arbitrary mass and that there exists a minimal delocalisation width of about 67 meters. While extremely macroscopic from a quantum perspective, the value is in principle accessible to laboratories on Earth. Hence, we analyse, numerically, how the dynamics of an initially spherical Gaussian wave packet is modified in the presence of Λ > 0. A notable feature is the gravitational collapse of part of the wave packet, in the core region close to the centre of mass, accompanied by the accelerated expansion of the more distant shell surrounding it. The order of magnitude of the distance separating collapse from expansion matches analytical estimates of the classical turnaround radius for a spherically symmetric body in the presence of dark energy. However, the time required to observe these modifications is astronomical. They can potentially be measured only in physical systems simulating a high effective cosmological constant, or, possibly, via their effects on the inflationary universe.

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We show a purely local protocol to upgrade local realistic correlations to genuine non-local correlations if only one local party has access to a negative probabilistic bit (nebit), i.e., a bit taking 0 with probability 1 + ∆ and 1 with probability −∆. The minimal amount of nebit's negativity ∆ required for the upgrade can serve as a measure of non-locality. The upgrade protocol bears a striking resemblance to ordinary local unitary operations in quasi-stochastic formulations of quantum theory, mathematically equivalent to positive stochastic processes controlled by nebits. This suggests that nebits can be interpreted as units of quantum departure from classical physics as well.

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