Arnold--Nielsen Geometry for Complexity-Deformed Noncommutative Transport

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 $\mathrm{\partial }\text{\hspace{0.17em}⁣}:\text{M}\to \text{Hcal}$ can be absorbed into the calculus itself, $${\mathrm{\partial }}_G:=G^{1\mathrm{/}2}\mathrm{\partial }\mathrm{.}$$ The corresponding complexity-weighted transport problem is exactly the unweighted transport problem generated by ${\mathrm{\partial }}_G$, 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.