A quantum code is a subspace of a Hilbert space of a physical system chosen to be correctable against a given class of errors, where information can be encoded. Ideally, the quantum code lies within the ground space of the physical system. When the physical model is the Heisenberg ferromagnet in the absence of an external magnetic field, the corresponding ground-space contains all permutation-invariant states. We use techniques from combinatorics and operator theory to construct families of permutation-invariant quantum codes. These codes have length proportional to $t^2$; one family of codes perfectly corrects arbitrary weight $t$ errors, while the other family of codes approximately correct $t$ spontaneous decay errors. The analysis of our codes' performance with respect to spontaneous decay errors utilizes elementary matrix analysis, where we revisit and extend the quantum error correction criterion of Knill and Laflamme, and Leung, Chuang, Nielsen and Yamamoto.
Permutation symmetry in large N Matrix Quantum Mechanics and Partition Algebras
我们描述了一般大小的矩阵$N$的量子力学系统的状态空间和动力学的置换对称性的影响。我们求解了一般的11参数置换不变量子矩阵谐振子哈密顿量，计算了正则配分函数。对于一般的哈密顿算子，希尔伯特空间的置换不变扇区可以用划分代数图来描述，划分代数图形成了划分代数塔$P_k(N)$的基底。整数$k$在量子力学中被解释为矩阵振子多项式的次数。家庭的相互作用的哈密顿被对角化的表示理论的基础上，我们构造的置换不变的子空间$ N \ge 2k $。这些包括低能量状态是置换不变的哈密顿量，可以引起大的基态退化有关的分区代数的尺寸。文献中讨论的量子多体伤疤的一种基于量子力学的机制可以在这些具有置换对称性的矩阵系统中实现。的矩阵索引值的晶格位置的映射允许在修改后的玻色-哈伯德模型的上下文中的机制的实现。极端的adS/CFT研究类似于那些遵守选择规则的基础上Clebsch-Gordan多重性（克罗内克系数）的对称群。
