AI 中文总结
本文证明非局域稳定子Rényi熵的猜想，建立其谱几何，给出可计算上下界，并揭示其随纠缠熵的对数增长特性。
AI 中文摘要
魔法（magic），或称非稳定子性（nonstabilizerness），是将稳定子操作提升为通用量子计算所需的资源。对于二分纯态，其内在关联于子系统间相关性的分量，即非局域魔法，通过对局域幺正操作最小化某种魔法度量而获得。对于稳定子Rényi熵（SRE），由此得到的非局域SRE被猜想可由计算基态达到，该态的Schmidt向量是按Schmidt系数递减排序的计算基态。在本工作中，我们证明了这一猜想对两个态族在任意系统尺寸下成立：具有二进阶梯Schmidt谱的态以及Schmidt秩至多为六的态。在不限制谱的情况下，我们证明非局域SRE等于计算基态的SRE加上一个有界常数，这确定了非局域SRE任何发散标度的主导项。我们进一步证明非局域SRE随纠缠熵至多对数增长，因此对于面积律态它保持有限，而对于临界态它随系统尺寸至多双重对数增长。我们以横场Ising链为例说明这些结果，在其中我们确定了非局域SRE在能隙相和临界点处的值。我们的结果建立了非局域SRE的定量谱几何，并将其在多体态中的值限定在可计算的上下界之间。
英文摘要
Magic, or nonstabilizerness, is the resource that promotes stabilizer operations to universal quantum computation. For bipartite pure states, its component intrinsic to the correlations between the subsystems, the nonlocal magic, is obtained by minimizing a magic measure over local unitaries. For the stabilizer Rényi entropy (SRE), the minimum is conjectured to be attained by the computational-basis (CB) representative, the state obtained by assigning the Schmidt coefficients in decreasing order to matching computational-basis labels. We prove this conjecture for two families of states at every system size and bipartition: states with dyadic-staircase Schmidt spectra and states of Schmidt rank at most six. For arbitrary bipartite pure states, we show that the CB value exceeds the nonlocal SRE by at most $4$, fixing the leading term of any divergent scaling of nonlocal SRE. We further show that the nonlocal SRE grows at most logarithmically with the entanglement entropy. Consequently, one-dimensional area-law states have bounded nonlocal SRE, while critical states with logarithmic entanglement entropy permit at most doubly logarithmic growth with system size. We apply these results to the transverse-field Ising chain, where we tightly bound the nonlocal SRE in the gapped phases and identify its double-logarithmic growth at criticality.
Commentsseveral minor presentation improvements, 18+24 pages, comments welcome!
