Random qubit-states and how best to measure them

The state of a quantum system is not an observable. Given a large number of copies of the system of interest we can obtain a good estimation of the state [1], but it is not possible to determine the state given only a single copy.1 Fundamentally, we can understand this limitation as a manifestation of complementarity, that is, the existence of incompatible observables. It is this limitation, moreover, that underlies the security of quantum key distribution [3–6] and gives quantum communications theory its distinctive character [3,6–8]. In a quantum communication system, we typically only have a single copy of each state and the receiving party is faced with the task of determining, as well as possible, the state originally prepared. This difficult task is usually made simpler by prior knowledge in the form of a set of possible states and associated probabilities for each of them. Quantum communications is, of course, is an important motivation for the study of quantum state discrimination [6, 9, 10]. The quality of the measurement strategy may be measured by reference to a variety of figures of merit. Amongst those most commonly employed are state discrimination with minimum probability of error or minimum Bayes cost [11, 12], unambiguous state discrimination [13–15], and maximizing the accessible information [16–18]. We may also be interested in the measurement that allows us to prepare a state most likely to pass as the original, which leads us to maximize the fidelity [19], or to maximize our confidence in the state identified by our measurement [20]. A number of these optimal detection strategies have been realized in experiments using optical polarization [21–26]. In this paper we consider the problem of measuring a single qubit, or a string of such qubits, about which we have only a bare minimum of information. We consider, in particular, how best to measure a qubit prepared in a pure state randomly selected either from all possible states or from all the real states. These sets of states correspond, respectively, to a uniform probability distribution of states over the whole surface of the Bloch sphere, and a uniform probability distribution on a single great circle on the Bloch sphere. The minimum-error figure of merit is not applicable in this case; the set of states is continuous and therefore the error probability is always unity. Similarly, the maximum confidence, that is the greatest probability for correctly identifying the state, is necessarily zero, and there is no strategy for unambiguous discrimination between the states. It is possible, however, to maximize the fidelity and the accessible information. In each case, we find that any sensible measurement strategy is optimal.