Convergence rates for the quantum central limit theorem

Understand

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.