Quantum channel learning.

The problem of an optimal mapping between Hilbert spaces IN and OUT, based on a series of density matrix mapping measurements ρ^{(l)}→ϱ^{(l)}, l=1⋯M, is formulated as an optimization problem maximizing the total fidelity F=∑_{l=1}^{M}ω^{(l)}F(ϱ^{(l)},∑_{s}B_{s}ρ^{(l)}B_{s}^{†}) subject to probability preservation constraints on Kraus operators B_{s}. For F(ϱ,σ) in the form that total fidelity can be represented as a quadratic form with superoperator F=∑_{s}〈B_{s}|S|B_{s}〉 (either exactly or as an approximation) an iterative algorithm is developed. The work introduces two important generalizations of unitary learning: (1) IN/OUT states are represented as density matrices; (2) the mapping itself is formulated as a mixed unitary quantum channel A^{OUT}=∑_{s}|w_{s}|^{2}U_{s}A^{IN}U_{s}^{†} (no general quantum channel yet).