Discrimination and identification of quantum states

Determining the state of a quantum system is an essential step in quantum information processing. While the case of N = 2 arbitrary states is well known the extension to N > 2 is highly non-trivial.;Unambiguous discrimination among N > 2 pure states is one of the longest standing unsolved problems in quantum information. We develop a complete geometric picture that encompasses all aspects of the problem: linear independence of the states, positivity of the detection operators, and a graphic method for finding and classifying the optimal solutions. We illustrate it on the example of three states and also show that the problem depends on an invariant combination of the phases of the complex inner products, the Berry phase. For arbitrary inner products and prior probabilities only numerical solutions are possible but the features of the solution are universal, they hold for any value of the Berry phase up to o = pi at which point it greatly simplifies. We, therefore, present the complete analytical solution for the case of vanishing Berry phase. The corresponding optimal failure probability exhibits full permutational symmetry for a large range of the parameters. However, when the parameters have very different values, a second-order symmetry-breaking phase transition takes place: at a particular value of the parameters the optimal failure probability becomes bi-valued: a second, less symmetric solution branches away in a continuous way from the symmetric one which is optimal in the new regime for some set of parameters. We also study some special cases where the inner products of two or all three states coincide but the phase is arbitrary as well as the case of weighted equal probability measurement. The optimum measurement is derived and it is a general measurement (POVM). The generalization of our results to the discrimination of more than three states will discussed in the conclusion.;Finally, we address the problem of identifying one probe qudit with one out of N reference qudits. Two strategies, the unambiguous and the minimum error identification, are studied. The reference states are assumed to be pure states and no classical knowledge about them is available. The probe state is guaranteed to match one of the reference states with equal probability. The problem is shown to be equivalent to distinguishing between mixed quantum states. Through the example of three ququartz states the form of the optimal measurement operators is derived for the unambiguous strategy. Using the positivity constraint for the operator of the inconclusive result the optimum success probability is calculated. In the minimum error identification an upper and a lower bound are derived, the latter by using a square-root measurement. Numerical values of the success probability are calculated to which the lower bound compares favorable.