图上两粒子格罗弗游走中的纠缠熵
Entanglement entropy in two-particle Grover walks on graphs
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中文总结 AI 辅助
研究图上相同粒子的两粒子格罗弗游走，通过关注克罗内克积对称性确保粒子交换不变性，进而研究其演化量子态的纠缠熵，针对完全二分图\(K_{n,n}\)确定了量子态熵达上限时\(n\)的值为\(1\)和\(2\)。
中文摘要 AI 辅助
我们通过在克罗内克积\(G \otimes G\)上的单粒子格罗弗游走，在图\(G\)上定义了相同粒子的两粒子量子游走，并称之为两粒子格罗弗游走。在相同粒子系统中，量子力学要求量子态在粒子交换下具有一定不变性。我们关注克罗内克积\(G \otimes G\)作为图的对称性，表明此游走的时间演化算符与交换算符对易，确保了该要求得到满足。此外，我们研究了此游走演化的量子态的纠缠熵。对于完全二分图\(K_{n,n}\)，我们完全确定了从特定初始态演化而来的量子态在某些时刻达到熵上限时\(n\)的值，证明其恰好为\(1\)和\(2\)。
英文摘要
We define a two-particle quantum walk of identical particles on a graph $G$ via the one-particle Grover walk on the Kronecker product $G \otimes G$, and call it the two-particle Grover walk. In systems of identical particles, quantum mechanics requires that quantum states have a certain invariance with respect to the exchange of particles. Focusing on the symmetry of the Kronecker product $G \otimes G$ as a graph, we show that the time evolution operator of this walk commutes with the swap operator, which ensures that this requirement is satisfied. Furthermore, we study the entanglement entropy of quantum states evolved by this walk. For the complete bipartite graph $K_{n,n}$, we completely determine the values of $n$ for which the quantum states evolved from specific initial states attain the upper bound of the entropy at some time, and prove that they are exactly $1$ and $2$.
