Becker Simon, Datta Nilanjana, Lami Ludovico, Rouzé Cambyse
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences University of Cambridge, Cambridge, CB3 0WA UK.
School of Mathematical Sciences and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, University Park, Nottingham, NG7 2RD UK.
Commun Math Phys. 2021;383(1):223-279. doi: 10.1007/s00220-021-03988-1. Epub 2021 Feb 15.
Various quantum analogues of the central limit theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It implies that the state in any single output arm of an -splitter, which is fed with copies of a centred state with finite second moments, converges to the Gaussian state with the same first and second moments as . Here we exploit the phase space formalism to carry out a refined analysis of the rate of convergence in this quantum central limit theorem. For instance, we prove that the convergence takes place at a rate in the Hilbert-Schmidt norm whenever the third moments of are finite. Trace norm or relative entropy bounds can be obtained by leveraging the energy boundedness of the state. Via analytical and numerical examples we show that our results are tight in many respects. An extension of our proof techniques to the non-i.i.d. setting is used to analyse a new model of a lossy optical fibre, where a given -mode state enters a cascade of beam splitters of equal transmissivities fed with an arbitrary (but fixed) environment state. Assuming that the latter has finite third moments, and ignoring unitaries, we show that the effective channel converges in diamond norm to a simple thermal attenuator, with a rate . This allows us to establish bounds on the classical and quantum capacities of the cascade channel. Along the way, we derive several results that may be of independent interest. For example, we prove that any quantum characteristic function is uniformly bounded by some outside of any neighbourhood of the origin; also, can be made to depend only on the energy of the state .
概率论的基石之一——中心极限定理有各种各样的量子类似物，这在文献中是已知的。其中一种类似物，由库申（Cushen）和哈德森（Hudson）提出，对量子光学具有特别的相关性。它意味着在一个分束器的任何单个输出臂中的状态，若输入的是具有有限二阶矩的中心态的多个副本，则该状态会收敛到具有与输入态相同的一阶和二阶矩的高斯态。在这里，我们利用相空间形式理论对这个量子中心极限定理中的收敛速率进行精细分析。例如，我们证明只要输入态的三阶矩有限，那么在希尔伯特 - 施密特范数下收敛速率为 。通过利用态的能量有界性可以得到迹范数或相对熵界。通过解析和数值例子，我们表明我们的结果在许多方面都是紧密的。我们将证明技术扩展到非独立同分布的情形，用于分析一种有损光纤的新模型，在该模型中，给定的多模态进入由具有相等透射率 的多个分束器组成的级联结构，并输入任意（但固定）的环境态。假设后者具有有限的三阶矩，并且忽略酉算子，我们表明有效信道在钻石范数下收敛到一个简单的热衰减器，收敛速率为 。这使我们能够建立级联信道的经典容量和量子容量的界。在此过程中，我们推导出了几个可能具有独立研究价值的结果。例如，我们证明任何量子特征函数 在原点的任何邻域之外都由某个 一致有界；而且， 可以使得仅依赖于态 的能量。
