Nonlinear Schrödinger equations and generalized Heisenberg uncertainty principle violating the principle of estimation independence

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 Schrödinger equation and generalized Heisenberg uncertainty principle within the estimation scheme by assuming a more general class of estimation errors. The nonlinearity of the Schrödinger 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.

References in corpus (17)

• A Spin Entanglement Witness for Quantum Gravity
• Gravitationally-induced entanglement between two massive particles is sufficient evidence of quantum effects in gravity
• The uncertainty principle determines the non-locality of quantum mechanics
• Complex weak values in quantum measurement
• A derivation of quantum theory from physical requirements
• A generalized no-broadcasting theorem
• Quantum Optomechanics - throwing a glance
• Towards Quantum Gravity: A Framework for Probabilistic Theories with Non-Fixed Causal Structure
• Grounding Bohmian Mechanics in Weak Values and Bayesianism
• Non-locality distillation and post-quantum theories with trivial communication complexity
• On two recent proposals for witnessing nonclassical gravity
• Implications of Superstrong Nonlocality for Cryptography
• Quantum mechanics as classical statistical mechanics with an ontic extension and an epistemic restriction
• Epistemically restricted phase space representation, weak momentum value, and reconstruction of quantum wave function
• Quantum mechanics is a calculus for estimation under epistemic restriction
• The chain rule implies Tsirelson's bound: an approach from generalized mutual information
• Estimation independence as an axiom for quantum uncertainty

Cited by in corpus (3)

• Quantum uncertainty as classical uncertainty of real-deterministic variables constructed from complex weak values and a global random variable
• Efficient classical computation of expectation values in a class of quantum circuits with an epistemically restricted phase space representation
• Epistemic uncertainty from an averaged Hamilton-Jacobi formalism