多量子比特正交乘积基
Multiqubit orthogonal product bases
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中文总结 AI 辅助
该研究通过形式矩阵与边着色多重图的关联，将多量子比特正交乘积基的分类简化为图同构，推导了其等价类数量的上下界及渐近行为，并提出了等价判定的两阶段指数复杂度算法。
中文摘要 AI 辅助
我们通过多量子比特正交乘积基（OPB）的形式矩阵形式主义研究n量子比特系统的完备正交乘积基（OPB）。我们为每个OPB形式矩阵关联一个边着色完全多重图，并证明两个形式矩阵等价当且仅当它们的多重图同构，从而将OPB的分类简化为图同构问题。通过子立方体划分和塔尔斯基（Tarsi）引理，我们将变量数量的上界限定为2ⁿ - 1。我们还研究n量子比特OPB等价类数量aₙ的上下界，证明了组合数不等式：C(a_{n-1}+1,2) ≤ aₙ ≤ B_{2^{n-1}}ⁿ，其中Bₘ表示m元集合的划分数。这些界给出了渐近行为aₙ = 2^(2ⁿ⁺ᵒ⁽ⁿ⁾)。最后，我们得到一个两阶段算法，用于判定两个OPB形式矩阵是否等价，并给出其正确性证明和复杂度分析，该算法在最坏情况下的复杂度为指数级。
英文摘要
We use edge-colored complete multigraphs to study complete orthogonal product bases (OPBs) in $n$-qubit systems. We prove that two OPBs are equivalent if and only if their associated multigraphs are isomorphic, thereby reducing OPB classification to graph isomorphism. Within this framework, we establish the upper bound $v\le 2^n-1$ on the number of variables of an $n$-qubit OPB. We also derive $\binom{a_{n-1}+1}{2}\le a_n\le B_{2^{n-1}}^n$ for the number $a_n$ of equivalence classes of $n$-qubit OPBs, where $B_m$ denotes the number of partitions of an $m$-element set. These bounds imply the asymptotic behavior $a_n=2^{2^{n+o(n)}}$. For every OPB, the connectivity pattern of its color layers characterizes local irreducibility, which in turn implies indistinguishability by finite-round local operations and classical communication (LOCC); the existence of a complete color-splitting tree characterizes perfect distinguishability by finite-round LOCC. Finally, we give an algorithm for testing OPB equivalence and a recursive graph algorithm that constructs a finite-round LOCC protocol whenever such perfect discrimination is possible.
发表机构
- School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）
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