We consider unambiguous discrimination of three pure quantum states. Necessary and sufficient conditions to decide which states should be detected for optimal measurement of unambiguous discrimination are provided in terms of inner products and the geometric phase Phi. We get the optimal measurement and the optimal failure probability when the optimal unambiguous discrimination does not require the detection of every given quantum state. When at least two quantum states are orthogonal to each other, we supply the optimal measurement and optimal failure probability in analytic form. When all three quantum states are not orthogonal to each other and Phi not equal 0, we find an analytic condition to determine the zero and nonzero elements for an optimal positive operator valued measure. We explain how to determine the solution in a geometric manner. Using the known solution of a case where the mutual inner products are real, we check the necessary and sufficient conditions when Phi = pi and analyze the property of the singular point when Phi = 0 and the relation between the optimal points.
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