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We address the problem of finding optimal CPTP (completely positive, tracepreserving) maps between a set of binary pure states and another set of binarygeneric mixed state in a two dimensional space. The necessary and sufficientconditions for the existence of such CPTP maps can be discussed within a simplegeometrical picture. We exploit this analysis to show the existence of anoptimal quantum repeater which is superior to the known repeating strategiesfor a set of coherent states sent through a lossy quantum channel. We also showthat the geometrical formulation of the CPTP mapping conditions can be asimpler method to derive a state-dependent quantum (anti) cloning machine thanthe study so far based on the explicit solution of several constraints imposedby unitarity in an extended Hilbert space.Comment: Latex, 13 Pags., 13 Fig

physical review. a, atomic, molecular, and optical physics/physical review, a, atomic, molecular, and optical physics

Geometrical conditions for completely positive trace-preserving maps and their application to a quantum repeater and a state-dependent quantum cloning machine