A general framework for unambiguous detection of quantum states

Full Text:The problem of detecting information stored in the state of a quantum system is a fundamental problem in quantum information theory. Several approaches have emerged to distinguishing between a collection of non-orthogonal quantum states. We consider the problem of unambiguous detection where we seek a measurement that with a certain probability returns an inconclusive result, but such that if the measurement returns an answer, then the answer is correct with probability 1. We begin by considering unambiguous discrimination between a set of linearly independent pure quantum states. We show that the design of the optimal measurement that minimizes the probability of an inconclusive result can be formulated as a semidefinite programming problem. Based on this formulation, we develop a set of necessary and sufficient conditions for an optimal quantum measurement. We show that the optimal measurement can be computed very efficiently in polynomial time by exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum. Using the general conditions for optimality, we derive necessary and sufficient conditions so that the measurement that results in an equal probability of an inconclusive result for each one of the quantum states is optimal. We refer to this measurement as the equal-probability measurement (EPM). We then show that for any state set, the prior probabilities of the states can be chosen such that the EPM is optimal. Finally, we consider state sets with strong symmetry properties and equal prior probabilities for which the EPM is optimal. We next develop a general framework for unambiguous state discrimination between a collection of mixed quantum states, which can be applied to any number of states with arbitrary prior probabilities. In particular, we derive a set of necessary and sufficient conditions for an optimal measurement that minimizes the probability of an inconclusive result, by exploiting principles of duality theory in vector space optimization