Analytical approach to optimal discrimination with unambiguous measurements

A fundamental aspect of quantum information theory is that nonorthogonal quantum states cannot be perfectly distinguished. Therefore, a central problem in quantum mechanics is to design measurements optimized to distinguish between a collection of nonorthogonal quantum states. One possible discrimination strategy is the so-called Unambiguous State Discrimination (USD) where the states are successfully identified with non-unit probability, but without error. The reason why until recently the area has shown relatively slow progress within the rapidly evolving field of quantum information is that it poses quite formidable mathematical challenges. Except for a handful of very special cases, no general exact solution has been available involving more than two arbitrary states and mostly numerical algorithms are proposed for finding optimal measurements for quantum-state discrimination, where the theory of the semi-definite programming provides a simple check of the optimality of the numerically obtained results.

In this work we present an exact analytic solution to an optimum measurement problem involving an arbitrary number of pure linearly independent quantum states. To this end we have reduced the theory of the semi-definite programming to a linear programming one with a feasible region of polygon type which can be solved via simplex method. The strength of the method is illustrated on some explicit examples. Also using the close connection between the Lewenstein-Sanpera decomposition and semi-definite programming, we have been able to obtain the optimal positive operator valued measure for some of the well known examples via Lewenstein-Sanpera decomposition method.