通过$L_2(0,1]$中的多分辨率重构量子信息
Quantum information recast via multiresolution in $L_2(0,1]$
AI总结：
本研究结合Borel同构与Haar基构建多分辨率方法，建立$L_2(0,1]$与无穷量子比特阵列希尔伯特空间的对应，实现用量子几何算符表示量子操作与可观测量，还发现反导数是该体系下的自然算符。
摘要：
我们提出了一种用于量子信息理论的多分辨率方法。该方法源于为分析无穷量子比特阵列（即可被解释为量子超材料的结构）开发系统性数学方法的尝试。我们方法的基础是两个具有经典根源的数学构造：Borel同构与Haar基。在此，我们将这两个构造相结合，建立$L_2(0,1]$与无穷量子比特阵列的希尔伯特空间之间的对应关系，从而能够分析作用于（有限或无穷）量子比特阵列的算符。这两个概念的融合使我们能够通过几何算符表示量子操作与可观测量。一个意外的结果是，我们发现微积分的基本概念内蕴于无穷量子比特阵列中；实际上，反导数在该语境下是一种自然且不可或缺的算符。
英文摘要：
We present a multiresolution approach to the theory of quantum information. It arose from an effort to develop a systematic mathematical approach to the analysis of an infinite array of qubits, i.e., a structure that may be interpreted as a quantum metamaterial. Foundational to our approach are two mathematical constructions with classical roots: the Borel isomorphism and the Haar basis. Here, these constructions are intertwined to establish an identification between $L_2(0,1]$ and the Hilbert space of an infinite array of qubits and to enable analysis of operators that act on arrays of qubits (either finite or infinite). The fusion of these two concepts empowers us to represent quantum operations and observables through geometric operators. As an unexpected upshot, we observe that the fundamental concept of calculus is inherent in an infinite array of qubits; indeed, the antiderivative arises as a natural and indispensable operator in this context.
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