2026/07/22 by Alberto Acevedo, Antonio Falcó
#quant-ph
We deform the Carlen--Maas--Wirth framework for noncommutative dynamical optimal transport by an Arnold--Nielsen type complexity operator. A positive state-independent operator G compatible with the Hilbert bimodule structure of a noncommutative differential calculus ∂\colon\M→\Hcal can be absorbed into the calculus itself, ∂G:=G1/2∂. The corresponding complexity-weighted transport problem is exactly the unweighted transport problem generated by ∂G, whenever the deformed quadratic form remains Dirichlet. In finite dimensions we prove existence of minimizers for density-dependent Petz-class metrics and for fixed physical complexity weights, the latter without commutation between G and the state-dependent mobility. On unitary orbits we identify the induced distance with a quotient metric coming from a right-invariant complexity geometry. This yields an exact Bell-state preparation result via Clairaut's relation and an exactly computed restricted-path upper bound for GHZ preparation; the Lindblad detailed-balance case is included only as entropy-gradient-flow background.