Symmetry of entangled states under a swap of outcomes (“envariance”) implies their equiprobability and leads to Born’s rule $p_k=|{\psi }_k{|}^2$. Here I show the converse: I demonstrate that the amplitude of a state given by a superposition of sequences of events that share the same total count (e.g., $n$ detections of 0 and $m$ of 1 in a spin-$\frac{1}{2}$ measurement) is proportional to the square root of the fraction—square root of the relative frequency—of all the equiprobable sequences of 0’s and 1’s with that $n$ and $m$.

• Received 26 March 2011

DOI:https://doi.org/10.1103/PhysRevLett.106.250402