通过量子信道传输代数
Transmitting algebras through quantum channels
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中文总结 AI 辅助
本研究通过量子信道精确传输有限维$C^*$-代数，发现支配代数在低维即失效，引入混合容量并给出支配的有限刻画，证明联合编码可产生新代数类型且支配可能失败，构造出增长规模最大的信道。
中文摘要 AI 辅助
我们发展了一种通过量子信道精确传输有限维$C^*$-代数的理论。这些代数描述了混合经典-量子信息，允许量子系统的维度依赖于经典消息。允许任意编码和解码产生了一个按嵌入排序的代数类型传输集，该集合将零误差信息论与算子代数纠错统一起来。我们询问该集合是否承认一个支配代数，使得每个可传输代数都能嵌入其中。我们的核心发现是，即使在低维情况下，支配也可能失败。在其缺失的情况下，几个不可比较的极大代数可以描述同一信道的不同最优使用方式，迫使使用者根据操作任务和要保留的信息类型选择一个。我们引入了混合容量，它重构了唯一可能的支配代数类型，并使用最小禁止代数类型给出了支配的完整有限刻画。我们识别了其算子系统与$*$-代数图同构的信道（包括高度可除信道）以及具有零一次性零误差量子容量的信道的支配代数。在张量积下，我们证明了联合编码可以产生新的代数类型，给出了香农理论中超可加性的代数类比。即使两个信道分别承认支配代数，它们的联合使用也可能导致支配失败。最后，我们构造了一个信道，其$n$重张量幂具有双重指数多的极大代数类型，达到了有限维信道可能的最大增长规模。因此，对于该信道，新的可传输代数类型在任意大的块长度下出现，因此其完整传输结构不能由任何有限的有界块长度码集合生成。
英文摘要
We develop a theory of exact transmission of finite-dimensional $C^*$-algebras through quantum channels. These algebras describe hybrid classical-quantum information, allowing the dimension of the quantum system to depend on the classical message. Allowing arbitrary encodings and decodings yields a transmission set of algebra types, ordered by embedding, that unifies zero-error information theory with operator-algebraic error correction. We ask if this set admits a dominating algebra into which every transmittable algebra embeds. Our central finding is that domination can fail even in small dimensions. In its absence, several incomparable maximal algebras can describe different optimal uses of the same channel, forcing the user to select one depending on the operational task and the type of information to be preserved. We introduce hybrid capacities that reconstruct the only possible dominating algebra type and present a complete finite characterization of domination using minimal forbidden algebra types. We identify dominating algebras for channels whose operator systems are graph-isomorphic to $*$-algebras, including highly divisible channels, and for channels with zero one-shot zero-error quantum capacity. Under tensor products, we prove that joint coding can produce new algebra types, giving an algebraic analogue of superadditivity from Shannon theory. Domination can fail for the joint use of two channels, even when both channels separately admit dominating algebras. Finally, we construct a channel whose $n$-fold tensor powers have doubly exponentially many maximal algebra types, attaining the largest possible growth scaling for finite-dimensional channels. Consequently, new transmittable algebra types appear at arbitrarily large block-lengths for this channel, so its full transmission structure cannot be generated from any finite collection of bounded-block-length codes.
发表机构
- RWTH Aachen University（亚琛工业大学）
- Inria, ENS Lyon, UCBL, LIP（法国国家信息与自动化研究所、里昂高等师范学院、里昂第一大学、里昂信息处理实验室）
- Technical University of Munich（慕尼黑工业大学）
- Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与工程中心）
机构由 AI 辅助整理，请以论文原文为准。
