AI 中文总结
该研究基于Miller的反例分析，证明任意维度$d\uff1e3$时不存在固定成本算子使量子最优输运的最优成本或其平方根为度量，信道诱导耦合还会导致$d\uff1e2$时无法实现点分离。
AI 中文摘要
基于耦合的量子最优输运通过将输运计划表示为具有指定边际的 bipartite 态，并将其成本评估为固定厄米特算子的期望，推广了经典最优输运。Friedland等人[Phys. Rev. Lett. 129, 110402 (2022)]推测，SWAP投影算子对应的最优成本的平方根在任意维度下都是度量，且该性质在量子成本矩阵临近时仍成立。Miller [arXiv:2607.07764]通过构造明确的对角qutrit反例否定了这两个推测。基于他的分析，我们证明了标准耦合的统一不可行性定理：在任意维度$d\uff1e3$时，不存在固定成本算子使得最优成本或其平方根成为度量，且违反情况已出现在对易态中；该阻碍在SWAP成本稳定时仍存在。对于信道诱导的耦合，全局非负性和自成本为零的要求迫使成本算子为零，导致$d\uff1e2$时无法实现点分离。综上，这些不可行性结果表明，固定成本耦合公式无法在完整量子态空间上生成度量。
英文摘要
Coupling-based quantum optimal transport generalizes classical optimal transport by representing transport plans as bipartite states with prescribed marginals and evaluating their cost as the expectation of a fixed Hermitian operator. Friedland et al. [Phys. Rev. Lett. 129, 110402 (2022)] conjectured that the square root of the optimal cost associated with the SWAP projector is a metric in every dimension and that this property persists for nearby quantum cost matrices. Miller [arXiv:2607.07764] disproved both conjectures by constructing explicit diagonal qutrit counterexamples. Building on his analysis, we prove a uniform no-go theorem for standard couplings. In every dimension $d\geq3$, no fixed cost operator makes either the optimal cost or its square root a metric, with violations occurring already among commuting states. The obstruction persists under stabilization of the SWAP cost. For channel-induced couplings, global nonnegativity and vanishing self-cost force the cost operator to be zero, precluding point separation when $d\geq2$. Taken together, these no-go results show that fixed-cost coupling formulations do not lead to metrics on the full quantum state space.
Comments17 pages
