Nonlinear Quantum Mechanics and Contextuality Offer Route to Hidden Variable Models

Research demonstrates that established nonlinear modifications of quantum mechanics—specifically Deutsch’s map, Weinberg’s model, and the Schrödinger–Newton equation—transform contextual quantum states into non-contextual ones, permitting classical hidden variable models. This provides a pathway to experimentally distinguish these modifications or potentially resolve the measurement problem via weak wave function collapse.

The enduring quest to reconcile quantum mechanics with gravity necessitates exploration beyond the standard linear framework. Researchers are investigating whether incorporating nonlinearity into the fundamental equations could resolve inconsistencies and address the measurement problem – the perplexing transition from quantum superposition to definite classical states. A new analysis demonstrates that certain proposed nonlinear modifications to quantum mechanics – including Deutsch’s map, Weinberg’s model, and the Schrödinger-Newton equation – exhibit a surprising property: they can transform inherently quantum, ‘contextual’ systems into ones that behave as if governed by classical, deterministic rules. This allows for the design of experiments to test these theories. Ruben Campos Delgado and Martin Plávala, both from the Institut für Theoretische Physik at Leibniz Universität Hannover, detail these findings in their article, ‘Ruling out nonlinear modifications of quantum theory with contextuality’.

Manipulating Contextuality in Quantum Systems via Nonlinear Transformations

Contextuality, a core tenet of quantum mechanics, describes how the outcome of a measurement depends not only on the system’s state but also on the other measurements performed alongside it. Recent research challenges the assumption that contextuality is an immutable property of quantum states, demonstrating its malleability through specific nonlinear modifications to the underlying quantum formalism. This work has implications for fundamental understandings of quantum mechanics and potentially offers a novel approach to the long-standing measurement problem.

Researchers established that linear transformations are inadequate to eliminate contextuality. Only nonlinear maps successfully transform a contextual set of states into a non-contextual one. They illustrate this principle using a single qubit – the quantum analogue of a classical bit – demonstrating that contextuality isn’t inherent to the states themselves, but a relational property dependent on the measurement context. A set of four or more distinct pure states – states described by a single wavefunction – can exhibit contextuality, which these transformations can remove.

The study analyses Weinberg’s model, a simplified representation of a qubit where the normalisation of the state functions is treated as a variable parameter. This allows researchers to observe how contextuality emerges within the model and how adjusting this parameter manipulates it. The normalisation ensures the total probability of finding the system in some state equals one. Altering this parameter effectively changes the ‘shape’ of the wavefunction, influencing the probabilities of measurement outcomes.

Researchers demonstrate that applying specific nonlinear maps – including Deutsch’s map, Weinberg’s model, and the Schrödinger–Newton equation – alters the linear dependencies within states. This alteration potentially enables a classical hidden variable description where one was previously impossible. Hidden variable theories posit that quantum mechanics is incomplete and that underlying, unobserved variables determine measurement outcomes, restoring determinism.

These findings contribute to the ongoing debate surrounding the foundations of quantum mechanics. They suggest a potential mechanism for weak wavefunction collapse – a gradual reduction in the superposition of states – offering a pathway towards resolving the measurement problem. The measurement problem concerns the transition from a quantum superposition of states to a definite, classical outcome upon measurement. This research suggests that nonlinear transformations may provide a means to model this transition without invoking ad-hoc postulates.