On 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⁾ → ⁽ˡ⁾, l=1 M, is formulated as an optimization problem maximizing the total fidelity F=∑ₗ₌₁M ω⁽ˡ⁾ F(⁽ˡ⁾,∑ₛ Bₛ ρ⁽ˡ⁾ B^ₛ) subject to probability preservation constraints on Kraus operators Bₛ. For F(,σ) in the form that total fidelity can be represented as a quadratic form with superoperator F=∑ₛ Bₛ|S| Bₛ (either exactly or as an approximation) an iterative algorithm is developed to find the global maximum. The result comprises in Nₛ operators Bₛ that collectively form an $IN$ to $OUT$ quantum channel AOUT=∑ₛ Bₛ AIN Bₛ^. 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 general quantum channel. This marks a crucial advancement from the commonly studied unitary mapping of pure states φₗ=U ψₗ to a general quantum channel, what allows us to distinguish probabilistic mixture of states and their superposition. An application of the approach is demonstrated on unitary learning of density matrix mapping ⁽ˡ⁾=U ρ⁽ˡ⁾ U^, in this case a quadratic on U fidelity can be constructed by considering √ρ⁽ˡ⁾ → √⁽ˡ⁾ mapping, and on a general quantum channel of Kraus rank Nₛ, where quadratic on Bₛ fidelity is an approximation -- a quantum channel is then built as a hierarchy of unitary mappings. The approach can be applied to study decoherence effects, spontaneous coherence, synchronizing, etc.