Abstract
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 n-splitter, which is fed with n copies of a centred state rho with finite second moments, converges to the Gaussian state with the same first and second moments as rho . 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 mathcal {O}left( n^{-1/2}right) in the Hilbert–Schmidt norm whenever the third moments of rho 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 m-mode state enters a cascade of n beam splitters of equal transmissivities lambda ^{1/n} 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 mathcal {O}Big (n^{-frac{1}{2(m+1)}}Big ). 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 chi _rho is uniformly bounded by some eta _rho <1 outside of any neighbourhood of the origin; also, eta _rho can be made to depend only on the energy of the state rho .
Highlights
The Central Limit Theorem (CLT) is one of the cornerstones of probability theory
We state our results on rates of convergence in the Cushen–Hudson quantum central limit theorem
Our first theorem provides convergence rates O n−1/2 in the quantum central limit theorem under a fourth-order moment condition
Summary
The Central Limit Theorem (CLT) is one of the cornerstones of probability theory. This theorem and its various extensions have found numerous applications in diverse fields including mathematics, physics, information theory, economics and psychology. . .}, with each pair acting on a distinct copy of the Hilbert space H1 They showed that sequences that are stochastically independent and identically distributed, and have finite covariance matrix and zero mean with respect to a quantum state ρ (given by a density operator on H1), are such that their scaled sums converge in distribution to a normal limit distribution [2, Theorem 1]. Their result admits a physical interpretation in terms of a passive quantum optical element known as the n-splitter. The paper contains a technical appendix (Appendix A) that makes the connection between moments and the regularity of the quantum characteristic function and shows that our definition of moments induces a canonical family of interpolation spaces
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