Partially Unitary Learning

The problem of an optimal mapping between Hilbert spaces IN of left|psirightrangle and OUT of left|φrightrangle based on a set of wavefunction measurements (within a phase) psil → φl, l=1dots M, is formulated as an optimization problem maximizing the total fidelity suml=1M ω(l) left|langleφl|U|psilrangleright|2 subject to probability preservation constraints on U (partial unitarity). The constructed operator U can be considered as an IN to OUT quantum channel; it is a partially unitary rectangular matrix (an isometry) of dimension dim(OUT) × dim(IN) transforming operators as AOUT=U AIN Udagger. An iterative algorithm for finding the global maximum of this optimization problem is developed, and its application to a number of problems is demonstrated. A software product implementing the algorithm is available from the authors.