Quantic Superpositions and the Geometry of Complex Hilbert Spaces
Daniel Lehmann
备注：
29 pages. This 2nd version differs from the first in that superposition of states x and y is now defined as a one-parameter operation: r x + (1-r) y whereas version 1 defined it as a two-parameters operation: r x + (1-r) e^{i ϕ} y
发表信息：
International Journal of Theoretical Physics, 47(5) 2008 pp. 1333-1353
The concept of a superposition is a revolutionary novelty introduced by Quantum Mechanics. If a system may be in any one of two pure states x and y, we must consider that it may also be in any one of many superpositions of x and y. An in-depth analysis of superpositions is proposed, in which states are represented by one-dimensional subspaces, not by unit vectors as in Dirac's notation. Superpositions must be considered when one cannot distinguish between possible paths, i.e., histories, leading to the current state of the system. In such a case the resulting state is some compound of the states that result from each of the possible paths. States can be compounded, i.e., superposed in such a way only if they are not orthogonal. Since different classical states are orthogonal, the claim implies no non-trivial superpositions can be observed in classical systems. The parameter that defines such compounds is a proportion defining the mix of the different states entering the compound. Two quantities, p and theta, both geometrical in nature, relate one-dimensional subspaces in complex Hilbert spaces: the first one is a measure of proximity relating two rays, the second one is an angle relating three rays. The properties of the operation of superposition are very different from those that govern linear combination of vectors.
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RIKEN Center for Quantum Computing, RIKEN（理化学研究所量子计算中心）；Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University（湖南师范大学低维量子结构与量子控制教育部重点实验室、物理系及量子效应与应用协同创新中心）；Quantum Research Institute and Department of Physics, University of Michigan（密歇根大学量子研究所与物理系）
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ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（ICFO-光子科学研究所，巴塞罗那科学与技术研究院）；Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales（布宜诺斯艾利斯大学，理学院）；CONICET - Universidad de Buenos Aires, Instituto de Física de Buenos Aires (IFIBA)（阿根廷国家科学研究委员会-布宜诺斯艾利斯大学，布宜诺斯艾利斯物理研究所 (IFIBA)）
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