Process estimation in qubit systems: a quantum decision theory approach

We address quantum decision theory as a convenient framework to analyse process discrimination and estimation in qubit systems. In particular, we discuss the following problems: (1) how to discriminate whether or not a given unitary perturbation has been applied to a qubit system; (2) how to determine the amplitude of the minimum detectable perturbation. In order to solve the first problem, we exploit the so-called Bayes strategy and look for the optimal measurement to discriminate, with minimum error probability, whether or not the unitary transformation has been applied to a given signal. Concerning the second problem, the strategy of Neyman and Pearson is used to determine the ultimate bound posed by quantum mechanics to the minimum detectable amplitude of the qubit transformation. We consider both pure and mixed initial preparations of the qubit and solve the corresponding binary decision problems. We also analyse the use of entangled qubits in the estimation protocol and found that entanglement, in general, improves stability rather than precision. Finally, we take into account the possible occurrence of different kinds of background noise and evaluate the corresponding effects on the discrimination strategies.