Towards Optimal Convergence Rates for the Quantum Central Limit Theorem

2023/10/15 by Salman Beigi, Beigi, Salman, Hami Mehrabi +1 · 2 citations
Computer Science · Mathematics · #FOS: Physical sciences #Mathematical functions and polynomials #Quantum Information and Cryptography #Quantum Physics (quant-ph) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2310.09812

openalex publication_date 2023/10/15 · openalex created_date 2023/10/18 · openalex updated_date 2026/07/28

Abstract

The quantum central limit theorem for bosonic quantum systems states that the sequence of states ρ^{\boxplus n} obtained from the n-fold convolution of a centered quantum state ρ converges to a quantum Gaussian state ρ_G that has the same first and second moments as ρ. In this paper, we contribute to the problem of finding the optimal rate of convergence for this quantum central limit theorem. We first show that if an m-mode quantum state has a finite moment of order max\3, 2m\, then we have ‖ρ^{\boxplus n} - ρ_G‖₁=\mathcal O(n^-1/2). We also introduce a notion of Poincaré inequality for quantum states and show that if ρ satisfies this Poincaré inequality, then D(ρ^{\boxplus n}‖ ρ_G)= \mathcal O(n^-1). By giving an explicit example, we verify that both these convergence rates are optimal.

Cited by

• The Central Limit Theorem for random exponents on a Hilbert space in the Weak Operator Topology
• Efficient Measurement of Bosonic Non-Gaussianity

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