Is Quantum Mechanics An Island In Theoryspace?

TL;DR

This paper explores modifications to quantum mechanics, revealing that changing norms or amplitude restrictions leads to theories with computational or signaling issues, suggesting quantum mechanics is uniquely constrained in theoryspace.

Contribution

It systematically analyzes how altering core aspects of quantum mechanics impacts its mathematical structure and physical implications, highlighting its special status.

Findings

No nontrivial norm-preserving linear maps with p-norms other than 2

Relaxing norm preservation enables solving PP-complete problems and superluminal signaling

Restricting amplitudes to real numbers introduces a simpler fundamental difficulty

Abstract

This recreational paper investigates what happens if we change quantum mechanics in several ways. The main results are as follows. First, if we replace the 2-norm by some other p-norm, then there are no nontrivial norm-preserving linear maps. Second, if we relax the demand that norm be preserved, we end up with a theory that allows rapid solution of PP-complete problems (as well as superluminal signalling). And third, if we restrict amplitudes to be real, we run into a difficulty much simpler than the usual one based on parameter-counting of mixed states.