标题： 布洛赫球上的拓扑序 显示英文标题

标题： Topological order on the Bloch sphere

摘要： 泡利球是任意二维希尔伯特空间的几何表示。 由[M. Boyer, R. Liss, T. Mor, PRA 95, 032308 (2017)]提出了泡利球上纯态和混合态的纠缠和可分离性的可能类别。 在这里，我们构建了一个泡利球，该球由基沃夫环面码模型的一个基态和一个最近的乘积态所张成的希尔伯特空间。 我们证明这个球只包含一个可分离态，因此属于该论文提出的第四类。 我们进一步研究了其表面上纯态的拓扑序，并得出结论，根据传统的定义，只有一个态（环面码基态）似乎表现出非平凡的拓扑序。 我们猜想，这个泡利球上的大多数态既不是“平凡”态（即它们不能通过一个平凡电路从乘积态生成），也不是拓扑有序的。 此外，我们表明整个设置可以用带有规范对称性的格罗弗旋转来理解，类似于量子搜索算法。 显示英文摘要

摘要： A Bloch sphere is the geometrical representation of an arbitrary two-dimensional Hilbert space. Possible classes of entanglement and separability for the pure and mixed states on the Bloch sphere were suggested by [M. Boyer, R. Liss, T. Mor, PRA 95, 032308 (2017)]. Here we construct a Bloch sphere for the Hilbert space spanned by one of the ground states of Kitaev's toric code model and one of its closest product states. We prove that this sphere contains only one separable state, thus belonging to the fourth class suggested by the said paper. We furthermore study the topological order of the pure states on its surface and conclude that, according to conventional definitions, only one state (the toric code ground state) seems to present non-trivial topological order. We conjecture that most of the states on this Bloch sphere are neither ``trivial'' states (namely, they cannot be generated from a product state using a trivial circuit) nor topologically ordered. In addition, we show that the whole setting can be understood in terms of Grover rotations with gauge symmetry, akin to the quantum search algorithm.