Joint Quantum State and Measurement Tomography with Incomplete Measurements
Adam C. Keith, Charles H. Baldwin, Scott C. Glancy, Emanuel H. Knill
Estimation of quantum states and measurements is crucial for the implementation of quantum information protocols. The standard method for each is quantum tomography (QT). However, QT suffers from systematic errors caused by imperfect knowledge of the system. We present a procedure to simultaneously characterize quantum states and measurements that mitigates systematic errors by use of a single high-fidelity state preparation and a limited set of high- fidelity unitary operations. Such states and operations are typical of many state-of-the-art systems. We design a set of experiments for the situation in question and use alternating maximum likelihood iterations to estimate measurement operators and unknown quantum states from the data. In some cases, the procedure does not enable unique estimation of the states. For these cases, we show how one may identify a set of density matrices compatible with the measurements and use a semi-definite program to place bounds on the state's expectation values. We demonstrate the procedure on data from a simulated experiment with two trapped ions.