Any modification of the Born rule leads to a violation of the purification and local tomography principles

TL;DR

Modifying the Born rule in quantum theory leads to violations of the purification and local tomography principles, revealing fundamental differences and constraints in alternative theories compared to standard quantum mechanics.

Contribution

The paper classifies all measurement postulate modifications and proves they violate key quantum principles, highlighting the uniqueness of quantum measurement structure.

Findings

All alternative theories violate the purification principle.

State tomography with local measurements becomes impossible in these theories.

A toy model demonstrates modifications to the Born rule without violating no-signalling.

Abstract

Using the existing classification of all alternatives to the measurement postulates of quantum theory we study the properties of bi-partite systems in these alternative theories. We prove that in all these theories the purification principle is violated, meaning that some mixed states are not the reduction of a pure state in a larger system. This allows us to derive the measurement postulates of quantum theory from the structure of pure states and reversible dynamics, and the requirement that the purification principle holds. The violation of the purification principle implies that there is some irreducible classicality in these theories, which appears like an important clue for the problem of deriving the Born rule within the many-worlds interpretation. We also prove that in all such modifications the task of state tomography with local measurements is impossible, and present a simple toy theory displaying all these exotic non-quantum phenomena. This toy model shows that, contrarily to previous claims, it is possible to modify the Born rule without violating the no-signalling principle. Finally, we argue that the quantum measurement postulates are the most non-classical amongst all alternatives.