In physics, it is crucial to identify operational measurement procedures to give physical meaning to abstract quantities. There has been significant effort to define time operationally using quantum systems, but the same has not been achieved for space. Developing an operational procedure to obtain information about the location of a quantum system is particularly important for a theory combining general relativity and quantum theory, which cannot rest on the classical notion of spacetime. Here, we take a first step towards this goal, and introduce a model to describe an extended material quantum system working as a position measurement device. Such a "quantum ruler" is composed of $N$ harmonically interacting dipoles and serves as a (quantum) reference system for the position of another quantum system. We show that we can define a quantum measurement procedure corresponding to the "superposition of positions", and that by performing this measurement we can distinguish when the quantum system is in a coherent or incoherent superposition in the position basis. The model is fully relational, because the only meaningful variables are the relative positions between the ruler and the system, and the measurement is expressed in terms of an interaction between the measurement device and the measured system.

中文翻译：

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使用材料量子尺进行关系叠加测量

在物理学中，确定可操作的测量程序以赋予抽象量物理意义至关重要。人们在使用量子系统来定义时间方面做出了巨大的努力，但在空间方面还没有实现同样的目标。开发一种操作程序来获取有关量子系统位置的信息对于广义相对论和量子理论相结合的理论尤其重要，因为它不能依赖于经典的时空概念。在这里，我们朝着这个目标迈出了第一步，并引入了一个模型来描述用作位置测量设备的扩展材料量子系统。这样的“量子标尺”由 $N$ 谐波相互作用的偶极子组成，并充当另一个量子系统位置的（量子）参考系统。我们表明，我们可以定义与“位置叠加”相对应的量子测量过程，并且通过执行这种测量，我们可以区分量子系统何时处于位置基础上的相干或非相干叠加。该模型是完全相关的，因为唯一有意义的变量是标尺和系统之间的相对位置，并且测量是用测量设备和被测系统之间的相互作用来表达的。