递归量子卷积编码器是灾难性的：一个简单的证明
Recursive quantum convolutional encoders are catastrophic: A simple proof
摘要：
Poulin、Tillich和Ollivier发现了卷积编码的经典理论和量子理论之间的一个重要区别，证明了量子卷积编码器不能同时是非灾难性的和递归的。非灾变性是期望的，使得迭代解码算法在解码其组成是量子卷积码的量子turbo码时收敛，并且递归性也是期望的，使得量子turbo码具有分别随着码的长度几乎线性增长的最小距离。他们对上述定理的证明是公认的“相当复杂”，因此，自从他们的结果以来，人们一直希望找到一个更简单的证明。在本文中，我们提供了一个证明，可以说是更简单。我们的方法是群论的-我们证明了量子卷积编码器的零物理重量循环的一部分的记忆状态的子组等价于其“有限记忆”子组的中心化器（记忆状态的子组，其最终通过信息量子比特的身份运算符输入和辅助量子比特的身份或Pauli-Z运算符输入到达身份记忆状态）。在证明这种对称性适用于任何量子卷积编码器之后，如果编码器是非灾难性的，那么很容易得出编码器是非递归的。我们的证明也阐明了为什么这个不可行定理不适用于纠缠辅助的量子卷积编码器-引入共享纠缠作为资源允许打破上述对称性。
英文摘要：
Poulin, Tillich, and Ollivier discovered an important separation between the classical and quantum theories of convolutional coding, by proving that a quantum convolutional encoder cannot be both non-catastrophic and recursive. Non-catastrophicity is desirable so that an iterative decoding algorithm converges when decoding a quantum turbo code whose constituents are quantum convolutional codes, and recursiveness is as well so that a quantum turbo code has a minimum distance growing nearly linearly with the length of the code, respectively. Their proof of the aforementioned theorem was admittedly "rather involved," and as such, it has been desirable since their result to find a simpler proof. In this paper, we furnish a proof that is arguably simpler. Our approach is group-theoretic---we show that the subgroup of memory states that are part of a zero physical-weight cycle of a quantum convolutional encoder is equivalent to the centralizer of its "finite-memory" subgroup (the subgroup of memory states which eventually reach the identity memory state by identity operator inputs for the information qubits and identity or Pauli-Z operator inputs for the ancilla qubits). After proving that this symmetry holds for any quantum convolutional encoder, it easily follows that an encoder is non-recursive if it is non-catastrophic. Our proof also illuminates why this no-go theorem does not apply to entanglement-assisted quantum convolutional encoders---the introduction of shared entanglement as a resource allows the above symmetry to be broken.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
