On infinite-dimensional state spaces
On infinite-dimensional state spaces
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关于无限维状态空间
DOI：
10.1063/1.4807079
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发表时间：
2012
影响因子：
1.3
通讯作者：
T. Fritz
中科院分区：
文献类型：
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作者：
T. Fritz
众所周知，正则对易关系[x，p] = i只能在无限维希尔伯特空间上实现。虽然任何有限的实验数据集也可以通过近似对易关系在有限维希尔伯特空间中进行解释，但奥卡姆剃刀更喜欢无限维模型，其中[x，p] = i在鼻子上。在任何试图探测无限维的方法中，都必须进行这种推理。为此目的使用规范对易关系的一个缺点是它具有不明确的操作意义。在这里，我们确定了一个操作上定义良好的上下文，从中可以得出类似的结论：如果量子系统上的两个幺正变换U，V满足关系V− 1 U2 V = U3，那么有限维必然包含关系UV− 1 UV = V− 1 UVU;这个含义在某些无限维实现中强烈失败。这是一个结果，从组合群论，我们给出了一个新的p…
It is well known that the canonical commutation relation [x, p] = i can be realized only on an infinite-dimensional Hilbert space. While any finite set of experimental data can also be explained in terms of a finite-dimensional Hilbert space by approximating the commutation relation, Occam's razor prefers the infinite-dimensional model in which [x, p] = i holds on the nose. This reasoning one will necessarily have to make in any approach which tries to detect the infinite-dimensionality. One drawback of using the canonical commutation relation for this purpose is that it has unclear operational meaning. Here, we identify an operationally well-defined context from which an analogous conclusion can be drawn: if two unitary transformations U, V on a quantum system satisfy the relation V−1U2V = U3, then finite-dimensionality entails the relation UV−1UV = V−1UVU; this implication strongly fails in some infinite-dimensional realizations. This is a result from combinatorial group theory for which we give a new p...