On Partially Unitary Learning

The problem of an optimal mapping between Hilbert spaces $IN$ of |ψ and $OUT$ of |φ based on a set of wavefunction measurements (within a phase) ψₗ → φₗ, l=1 M, is formulated as an optimization problem maximizing the total fidelity ∑ₗ₌₁M ω⁽ˡ⁾ |φₗ|U|ψₗ|² subject to probability preservation constraints on U (partial unitarity). Constructed operator U can be considered as a $IN$ to $OUT$ quantum channel; it is a partially unitary rectangular matrix of the dimension (OUT) × (IN) transforming operators as AOUT=U AIN U^. An iteration algorithm finding the global maximum of this optimization problem is developed and it's application to a number of problems is demonstrated. A software product implementing the algorithm is available from the authors.