On possible extensions of quantum mechanics

TL;DR

This paper revisits the limits of extending quantum mechanics' predictive power, identifying errors in previous no-go theorems and exploring how relaxed measurement assumptions impact these limits, especially in entangled systems.

Contribution

It corrects a critical error in earlier proofs, clarifies the conditions under which quantum mechanics is maximally predictive, and investigates bounds on predictive improvements for entangled qubits.

Findings

Quantum mechanics is maximally predictive only under complete certainty or uncertainty.

Relaxing measurement assumptions does not weaken the maximal predictive power of quantum mechanics.

A conjectured upper bound exists for predictive improvements over quantum mechanics in local spin measurements.

Abstract

It was argued [1] that there can be no extension of quantum mechanics with improved predictive power on a measurement freely chosen, independently of any event that is not in its future light cone. The assumption of measurement choice was criticized [2] to be too strong to be physically necessary and extensions of quantum mechanics were shown [3] to be possible under a more relaxed measurement assumption. Here I point out an error in the criticism and observe that the actual mistake of the no-go theorem lies in an unwarranted assumption implicitly made in the proof of [1]. Hence, quantum mechanics is guaranteed to have the maximal predictive power only in situations of complete certainty and complete uncertainty about measurement outcomes. I then show that the measurement assumption can be further relaxed without affecting the conclusion on the predictive power of quantum mechanics versus alternative theories. I further study the optimal predicative improvement over quantum mechanics of local spin measurements on a pair of entangled qubits by any alternative theory and conjecture a strict upper bound.