Description
The Quantum Prepare-and-Measure scenario is a fundamental framework for studying the boundaries of Classical/Quantum communication, and studying the grounds for quantum advantage from a communication complexity perspective. One way of quantifying the quantum advantage would be by measuring classical communication, that should be added to a Local-Hidden-Variable model, so that it can reproduce the statistics from the quantum counterpart. Renner, Tavakoli and Quintino have shown in Classical Cost of Transmitting a Qubit that to reproduce any general quantum measurement on a general prepared qubit, the statistics will be simulated exactly, by a classical communication protocol using 2 bits of information, added to two uniformly distributed 3d real vectors, as shared randomness, accessible to the parties. For higher dimensions though, the problem turns out to be very challenging, and the protocol remains not clearly generalizable. Our previous work Prepare-and-measure and entanglement simulation beyond qubits was to generalize the exact case of $d=2$ analytically, from a new perspective, which led to a novel protocol, which turns out to approximate the statistics pretty accurately in $d=3$ and $d=4$. However, an exact protocol remains open. Motivated by this, one possibility is the intricacy of parameterization, and number of free parameters, blowing up as the dimension increases. This fact, makes the scope of the problem a very well-suited context of applying Machine Learning techniques. As a result, we design a new neural network, respecting the structure of a classical communication strategy, and show how it finds the known protocol of $d=2$ as the test bed, and show that it enables the automated discovery of novel protocols with different shared randomness structures for this case. The design provided is promising in its flexibility in prospective application to higher dimensions, different quantum scenarios (Bell-scenarios), or assuming the communication under certain restrictions (1-bit restricted protocols).