圆稳定子态与禁止任意顶点子式态中的渐近纠缠
Asymptotic entanglement in circle stabilizer states and states forbidding arbitrary vertex-minors
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中文总结 AI 辅助
本文基于Geelen的顶点子式弱结构猜想，建立稳定子态渐近纠缠与禁止顶点子式的关联，证明不包含固定稳定子态为顶点子式的稳定子态的纠缠受渐近约束，为量子器件纠缠研究提供支撑。
中文摘要 AI 辅助
稳定子态在量子信息论中占据核心地位，对其纠缠性质的研究催生了大量工作。一个被广泛研究的问题是：何时稳定子态$|\boldsymbol{\theta}\rangle$仅通过单量子比特克利福德操作和泡利测量即可转化为另一个稳定子态$|\boldsymbol{\theta}\rangle$，若能实现，则称$|\boldsymbol{\theta}\rangle$是$|\boldsymbol{\theta}\rangle$的一个顶点子式。基于Geelen关于顶点子式的弱结构猜想，本文建立了如下一般性结论：对任意固定的稳定子态$|\boldsymbol{\theta}\rangle$，不将$|\boldsymbol{\theta}\rangle$作为顶点子式的稳定子态$|\boldsymbol{\theta}\rangle$的纠缠受到渐近约束。更具体地，本文证明：任何足够秩连通且不将$|\boldsymbol{\theta}\rangle$作为顶点子式的$|\boldsymbol{\theta}\rangle$的距离增长为$O(\boldsymbol{\theta} n)$，并对所谓局部可达信息（该量刻画了通过单量子比特泡利测量可获取的信息量）证明了类似结果。本文的结果依赖于三项工作：（i）将上述两种纠缠测度与多拟阵的秩函数关联；（ii）将圆稳定子态的秩函数与4-正则多重图上的秩函数关联，该关联渐近约束了圆稳定子态的纠缠；（iii）利用Geelen关于顶点子式的弱结构猜想，将前述结果“提升”至稳定子态的真顶点子式闭族中足够连通的态。本文的结果建立了渐近稳定子纠缠与禁止顶点子式之间的关联，对量子器件中可产生的纠缠具有直接意义。
英文摘要
Stabilizer states play a central role in quantum information theory, and understanding their entanglement has motivated a large body of work. A well-studied question in particular is when a stabilizer state $|ψ\rangle$ can be transformed into another stabilizer state $|ϕ\rangle$ using only single-qubit Clifford operations and Pauli measurements. If this is possible, we say that $|ϕ\rangle$ is a vertex-minor of $|ψ\rangle$. Assuming Geelen's weak structural conjecture on vertex-minors, we establish the following general statement. For any fixed stabilizer state $|ϕ\rangle$, the entanglement in stabilizer states $|ψ\rangle$ that do not contain $|ϕ\rangle$ as a vertex-minor is asymptotically constrained. More concretely, we show that the distance of any sufficiently rank-connected $|ψ\rangle$ not containing $|ϕ\rangle$ as a vertex-minor grows as $O(\log n)$, and prove similar results for the so-called locally accessible information, a quantity that captures the amount of information that can be learned through single-qubit Pauli measurements. Our results rely on (i) connecting the above two entanglement measures to rank functions of multimatroids, (ii) connecting the rank functions of circle stabilizer states to rank functions on $4$-regular multigraphs, which asymptotically constrains the entanglement of circle stabilizer states, and (iii) using Geelen's weak structural conjecture on vertex-minors to `lift' the previous result to sufficiently connected states in proper vertex-minor-closed families of stabilizer states. Our results establish a connection between asymptotic stabilizer entanglement and forbidden vertex-minors, with direct implications for the entanglement that can be generated in quantum devices.
