Nonlinear Schr\"odinger equations and generalized Heisenberg uncertainty principle violating the principle of estimation independence

TL;DR

This paper explores nonlinear Schrödinger equations and generalized uncertainty principles arising from estimation errors within a generalized quantum framework, highlighting how violations of estimation independence lead to potential extensions of quantum mechanics.

Contribution

It introduces a class of nonlinear Schrödinger equations and uncertainty principles derived from estimation errors, constrained by the principle of estimation independence.

Findings

Nonlinear Schrödinger equations can be derived from estimation errors.

Deviations from Heisenberg uncertainty are linked to estimation error generalizations.

Violations of estimation independence constrain possible quantum extensions.

Abstract

One of the advantages of a reconstruction of quantum mechanics based on transparent physical axioms is that it may offer insight to naturally generalize quantum mechanics by relaxing the axioms. Here, we discuss possible extensions of quantum mechanics within a general epistemic framework based on an operational scheme of estimation of momentum given positions under epistemic restriction. The epistemic restriction is parameterized by a global-nonseparable random variable on the order of Planck constant, an ontic extension to the separable classical phase space variables. Within the estimation scheme, the canonical quantum laws is reconstructed for a specific estimator and estimation error. In the present work, keeping the Born's quadratic law intact, we construct a class of nonlinear variants of Schr\"odinger equation and generalized Heisenberg uncertainty principle within the estimation scheme by assuming a more general class of estimation errors. The nonlinearity of the Schr\"odinger equation and the deviation from the Heisenberg uncertainty principle thus have a common transparent operational origin in terms of generalizations of estimation errors. We then argue that a broad class of nonlinearities and deviations from Heisenberg uncertainty principle arise from estimation errors violating a plausible inferential-causality principle of estimation independence which is respected by the standard quantum mechanics. This result therefore constrains possible extensions of quantum mechanics, and suggests directions to generalize quantum mechanics which comply with the principle of estimation independence.