arXivDaily arXiv每日学术速递 周一至周五更新
arXiv quant-ph/0003063quant-ph
收藏

自然数在量子力学中的表示

The Representation of Natural Numbers in Quantum Mechanics

Paul Benioff

备注：
Much rewrite, including response to comments. To Appear in Phys. Rev. A
发表信息：
Phys. Rev. A63, 032305 (2001)
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

本文提出了一种方法，使明确的一些假设和条件中隐含的广泛表示的数字的复合量子系统。任何非空集合和相关的运算都是一个自然数集合或一个算术模型，如果这个集合和运算满足数论或算术的公理。这项工作仅限于长度为L的k元表示和算术模k^{L}的公理。基于抽象L重张量积Hilbert空间H^{arith}中的状态和算子描述了公理的模型。然后描述该空间到基于物理参数的乘积空间H^{phy}上的酉映射。这些映射中的每一个都在H^{phy}中产生状态，而诱导算子是公理的模型。许多这些地图的存在的后果进行了讨论，随着依赖格罗弗和肖尔的算法对这些地图。的重要性的主要物理要求，即基本的算术运算是有效地实现，进行了讨论。这个条件表明存在物理上可实现的哈密顿量，可以实现基本的算术运算，并且所需的时空和热力学资源是L中的多项式。

英文摘要：

This paper represents one approach to making explicit some of the assumptions and conditions implied in the widespread representation of numbers by composite quantum systems. Any nonempty set and associated operations is a set of natural numbers or a model of arithmetic if the set and operations satisfy the axioms of number theory or arithmetic. This work is limited to k-ary representations of length L and to the axioms for arithmetic modulo k^{L}. A model of the axioms is described based on states in and operators on an abstract L fold tensor product Hilbert space H^{arith}. Unitary maps of this space onto a physical parameter based product space H^{phy} are then described. Each of these maps makes states in H^{phy}, and the induced operators, a model of the axioms. Consequences of the existence of many of these maps are discussed along with the dependence of Grover's and Shor's Algorithms on these maps. The importance of the main physical requirement, that the basic arithmetic operations are efficiently implementable, is discussed. This conditions states that there exist physically realizable Hamiltonians that can implement the basic arithmetic operations and that the space-time and thermodynamic resources required are polynomial in L.

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。

↑