Beyond the Projection Postulate and Back: Quantum Theories with Generalised State-Update Rules

Abstract: Are there consistent and physically reasonable alternatives to the projection postulate? Does it have unique properties compared to acceptable alternatives? We answer these questions by systematically investigating hypothetical state-update rules for quantum systems that nature could have chosen over the L\"uders rule. Among other basic properties, any prospective rule must define unique post-measurement states and not allow for superluminal signalling. Particular attention will be paid to consistently defining post-measurement states when performing local measurements in composite systems. Explicit examples of valid unconventional update rules are presented, each of them resulting in a distinct, well-defined foil of quantum theory. This framework of state-update rules allows us to identify operational properties that distinguish the projective update rule from all others and to put earlier derivations of the projection postulate into perspective.