Environment-Assisted Invariance Does Not Necessitate Born's Rule for Quantum Measurement

The argument of environment-assisted invariance (known as envariance) implying Born's rule is widely used in models for quantum measurement to reason that they must yield the correct statistics, specifically for linear models. However, it has recently been shown that linear collapse models can never give rise to Born's rule. Here, we address this apparent contradiction and point out an inconsistency in the assumptions underlying the arguments based on envariance. We use a construction in which the role of the measurement machine is made explicit and shows that the presence of envariance does not imply that every measurement will behave according to Born's rule. Rather, it implies that every quantum state allows a measurement machine to be constructed, which yields Born's rule when measuring that particular state. This resolves the paradox and is in agreement with the recent result of objective collapse models necessarily being nonlinear.

Automated summary

The argument of environment-assisted invariance (envariance) is commonly used in quantum measurement models to justify their ability to produce accurate statistics, particularly linear models. However, recent research has shown that linear collapse models cannot reproduce Born's rule. In this study, we address this contradiction and identify an inconsistency in the assumptions underlying envariance-based arguments. By explicitly considering the construction of a measurement machine, we demonstrate that envariance does not guarantee that every measurement will follow Born's rule. Instead, it implies that a measurement machine can be constructed for every quantum state, which yields Born's rule when measuring that specific state. This resolution aligns with the recent finding that objective collapse models must be nonlinear.