Neural Network Learning of One-Bit Protocols for Qubit Measurement Simulation

Communication complexity provides a natural framework for quantifying the classical resources required to reproduce quantum statistics. In the qubit prepare-and-measure scenario, two classical bits have been shown to be necessary and sufficient to simulate arbitrary qubit states and arbi- trary quantum measurements exactly. However, this result does not exclude the possibility that restricted families of measurements may admit accurate 1-bit classical approximations. We use a neural network procedure to demonstrate that a single bit can achieve high average accuracy for specific measurement families. A performance analysis of our neural network reveals that symmet- ric measurements with uniformly weighted elements, such as those forming regular polyhedra, are particularly amenable to this restricted communication. By analyzing the patterns learned by the neural network, we derive an analytical protocol that is extremely accurate for finite information- ally complete symmetric configurations and becomes exact in the limit of a continuous isotropic measurement.