AI 中文总结
研究量子信息论中正算子均分问题，利用等变上同调，证明了鲁斯凯-奥德纳尔特猜想的强形式在多类信道成立，弱形式在所有三量子比特信道成立，并得出量子信道凸分解的更多结果。
AI 中文摘要
量子信息论中的几个开放问题可表述为正算子的均分问题，即在统一约束下分解为有界秩正部。SIC-POVM和MUB的存在问题以及鲁斯凯-奥德纳尔特猜想都属于此类。首先表明可利用等变上同调解决某些此类问题，接着给出鲁斯凯-奥德纳尔特猜想的新结果。其弱形式断言每个量子信道可凸分解为最少数量的广义极点；强形式要求权重相等。证明了在所有维度上，对于一组非零测度的信道，包括所有cq和qc信道以及所有具有量子比特输入的信道，强猜想成立，还证明了所有三量子比特信道的弱猜想以及量子信道凸分解的进一步结果。
英文摘要
Several open problems in quantum information theory can be formulated as equipartition problems for positive operators, asking for a decomposition into bounded-rank positive parts under uniform constraints. The existence problems for SIC-POVMs and MUBs are of this type, as is the Ruskai-Audenaert conjecture. We first show that certain problems of this form can be attacked using equivariant cohomology, and then present new results on the Ruskai-Audenaert conjecture. In its weak form, this conjecture asserts that every quantum channel admits a convex decomposition into a minimal number of generalized extreme points; in its strong form, one with equal weights. We prove the strong conjecture in all dimensions for a set of channels of nonzero measure, including all cq- and qc-channels, as well as for all channels with qubit inputs. We also prove the weak conjecture for all qutrit channels, along with further results on convex decompositions of quantum channels.
Comments21 pages
