Title: Quantum information recast via multiresolution in $L_2(0,1]$ Show Chinese title

Title: 量子信息通过多分辨率在$L_2(0,1]$重新表述

Abstract: 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. Show Chinese abstract

Abstract: 我们提出了一种多分辨率方法来探讨量子信息理论。这一方法源自于试图系统地发展一种数学方法来分析无限数量的量子位（qubits）数组，即可以被解读为量子超材料的一种结构。我们方法的基础包括两个具有经典根源的数学构造：博雷尔同构和哈尔基。在这里，这些构造相互交织，以建立一个从 $L_2(0,1]$ 到无限数量量子位的希尔伯特空间之间的识别，并且能够分析作用于量子位数组（无论是有限还是无限）上的算子。这两个概念的融合使我们能够通过几何算子来表示量子操作和可观测量。作为一项意外的成果，我们观察到微积分的基本概念存在于无限数量的量子位数组中；实际上，在这种情况下，原函数作为一个自然且不可或缺的操作符出现。