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Using the convex optimization method and Helstrom family of ensemblesintroduced in Ref. [1], we have discussed optimal ambiguous discrimination inqubit systems. We have analyzed the problem of the optimal discrimination of Nknown quantum states and have obtained maximum success probability and optimalmeasurement for N known quantum states with equiprobable prior probabilitiesand equidistant from center of the Bloch ball, not all of which are on the onehalf of the Bloch ball and all of the conjugate states are pure. An exactsolution has also been given for arbitrary three known quantum states. Thegiven examples which use our method include: 1. Diagonal N mixed states; 2. Nequiprobable states and equidistant from center of the Bloch ball which theircorresponding Bloch vectors are inclined at the equal angle from z axis; 3.Three mirror-symmetric states; 4. States that have been prepared with equalprior probabilities on vertices of a Platonic solid.  Keywords: minimum-error discrimination, success probability, measurement,POVM elements, Helstrom family of ensembles, convex optimization, conjugatestates PACS Nos: 03.67.Hk, 03.65.Ta

The minimum-error discrimination via Helstrom family of ensembles and Convex Optimization