Reconstruction of Quantum States of Spin Systems: From Quantum Bayesian Inference to Quantum Tomography

We study in detail the reconstruction of spin-1/2 states and analyze the connection between (1) quantum Bayesian inference, (2) reconstruction via the Jaynes principle of maximum entropy, and (3) complete reconstruction schemes such asdiscrete quantum tomography. We derive an expression for a density operator estimated via Bayesian quantum inference in the limit of an infinite number of measurements. This expression is derived under the assumption that the reconstructed system is in a pure state. In this case the estimation corresponds to averaging over a microcanonical ensemble of pure states satisfying a set of constraints imposed by the measured mean values of the observables under consideration. We show that via a "purification" ansatz, statistical mixtures can also be consistently reconstructed via the quantum Bayesian inference scheme. In this case the estimation corresponds to averaging over the generalized grand canonical ensemble of states satisfying the given constraints, and in the limit of large number of measurements this density operator is equal to the generalized canonical density operator, which can be obtained with the help of the Jaynes principle of the maximum entropy. We also discuss inseparability of reconstructed density operators of two spins-1/2.