Maximum Entropy Methods for Quantum State Compatibility Problems

Inferring a quantum system from incomplete information is a common problem in many aspects of quantum information science and applications, where the principle of maximum entropy (MaxEnt) plays an important role. The quantum state compatibility problem asks whether there exists a density matrix $\rho$ compatible with some given measurement results. Such a compatibility problem can be naturally formulated as a semidefinite programming (SDP), which searches directly for the existence of a $\rho$. However, for large system dimensions, it is hard to represent $\rho$ directly, since it requires too many parameters. In this work, MaxEnt is applied to solve various quantum state compatibility problems, including the quantum marginal problem. An immediate advantage of the MaxEnt method is that it only needs to represent $\rho$ via a relatively small number of parameters, which is exactly the number of the operators measured. Furthermore, in case of incompatible measurement results, the method will further return a witness that is a supporting hyperplane of the compatible set. The method has a clear geometric meaning and can be computed effectively with hybrid quantum-classical algorithms.