Self-consistent state and measurement tomography with fewer measurements

We describe a technique for self-consistently characterizing both the quantum state of a single-qubit system, and the positive-operator-valued measure (POVM) that describes measurements on the system. The method works with only 10 measurements. We assume that a series of unitary transformations performed on the quantum state are fully known, while making minimal assumptions about both the density operator of the state and the POVM. The technique returns maximum-likely estimates of both the density operator and the POVM. To experimentally demonstrate the method, we perform reconstructions of over 300 state-measurement pairs, and we compare them to their expected density operators and POVMs. We find that 95% of the reconstructed POVMs have fidelities of 0.98 or greater, and 92% of the density operators have fidelities that are 0.98 or greater.