AI 中文总结
研究通过阿诺德 - 尼尔森型算子变形非交换动力学最优输运框架，证明有限维下相关极小值存在性，在酉轨道上建立诱导距离与商度量关系，得到贝尔态制备结果及GHZ制备上界。
AI 中文摘要
我们通过阿诺德 - 尼尔森型复杂度算子对非交换动力学最优输运的卡伦 - 马斯 - 维尔特框架进行变形。一个与非交换微分演算\(\partial\colon\M\to\Hcal\)的希尔伯特双模结构兼容的正定与态无关算子\(G\)可被吸收到演算本身，即\(\partial_G := G^{1/2}\partial\)。当变形二次型仍为狄利克雷型时，相应的复杂度加权输运问题恰好是由\(\partial_G\)产生的未加权输运问题。在有限维中，我们证明了对于密度依赖的佩茨类度量和固定物理复杂度权重存在极小值，后者在\(G\)与态依赖迁移率之间无交换的情况下成立。在酉轨道上，我们将诱导距离与来自右不变复杂度几何的商度量等同起来。这通过克莱罗关系产生了精确的贝尔态制备结果，并为GHZ制备给出了精确计算的受限路径上界；林德布拉德详细平衡情形仅作为熵梯度流背景包含在内。
英文摘要
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 $\partial\colon\M\to\Hcal$ can be absorbed into the calculus itself, \[ \partial_G:=G^{1/2}\partial. \] The corresponding complexity-weighted transport problem is exactly the unweighted transport problem generated by $\partial_G$, 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.
