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Convergence Rates for the Quantum Central Limit Theorem.

Simon Becker1, Nilanjana Datta1, Ludovico Lami2,3

  • 1Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences University of Cambridge, Cambridge, CB3 0WA UK.

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This study refines the quantum central limit theorem for quantum optics, establishing convergence rates for quantum states in n-splitters. The research provides new bounds for quantum channels, crucial for understanding quantum communication systems.

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Area of Science:

  • Quantum Information Theory
  • Quantum Optics
  • Mathematical Physics

Background:

  • Quantum analogues of the central limit theorem are crucial for quantum optics.
  • The Cushen-Hudson quantum central limit theorem describes state convergence in n-splitters.
  • Previous analyses lacked refined convergence rate details.

Purpose of the Study:

  • To analyze the rate of convergence in the Cushen-Hudson quantum central limit theorem using phase space formalism.
  • To extend these techniques to non-independent and identically distributed (i.i.d.) settings for lossy optical fiber analysis.
  • To establish bounds on classical and quantum capacities of cascade channels.

Main Methods:

  • Exploitation of the phase space formalism for refined convergence rate analysis.
  • Derivation of Hilbert-Schmidt norm convergence rates dependent on finite third moments.
  • Extension of proof techniques to non-i.i.d. settings for analyzing cascaded beam splitters.

Main Results:

  • Convergence rate of O(1/sqrt(N)) in Hilbert-Schmidt norm for n-splitters with finite third moments.
  • Convergence of effective channel in a lossy optical fiber model to a thermal attenuator at a rate O(1/sqrt(n)).
  • Derivation of uniform bounds for quantum characteristic functions.

Conclusions:

  • The refined analysis provides tight bounds on convergence rates for quantum states in optical systems.
  • The study offers new insights into the behavior of quantum channels, particularly in cascaded systems.
  • Established bounds on channel capacities have implications for quantum communication and information processing.