  Nonlinear modifications of quantum mechanics have a troubled history. They
were initially studied for many promising reasons: resolving the measurement
problem, testing the limits of standard quantum mechanics, and reconciling it
with gravity. Two results substantially undermined the credibility of
non-linear theories. Some have been experimentally refuted, and more
importantly, all deterministic non-linear theories can be used for superluminal
communication. However, these results are unconvincing because they overlook
the fact that the distribution of measurement results predicted by non-linear
quantum mechanics depends on the interpretation of quantum mechanics that one
uses. For instance, although the Everett and Copenhagen interpretations agree
on the expression of Born's rule for the outcomes of multiple measurements in
linear quantum mechanics, they disagree in non-linear quantum mechanics. We
present the range of expressions of Born's rule that can be obtained by
applying different formulations of quantum mechanics to a class of non-linear
quantum theories. We then determine that many do not allow for superluminal
communication but only two seem to have a reasonable justification. The first
is the Everett interpretation, and the second, which we name
causal-conditional, states that a measurement broadcasts its outcome to degrees
of freedom in its future light-cone, who update the wavefunction that their
non-linear Hamiltonian depends on according to this new information.
