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A short walk in quantum probability.

Robin Hudson1

  • 1Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK r.hudson@lboro.ac.uk.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|March 21, 2018
PubMed
Summary
This summary is machine-generated.

This study explores quantum probability, introducing new joint distributions and quantum Brownian motions that modify classical concepts. These findings offer a novel perspective on quantum mechanics and probability theory.

Keywords:
quantum central limit theoremquantum planar Brownian motionquantum probability

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Area of Science:

  • Quantum Probability
  • Mathematical Physics
  • Quantum Mechanics

Background:

  • The Wigner distribution for canonical pairs often lacks non-negativity, posing challenges for joint probability interpretations.
  • Classical planar Brownian motion serves as a benchmark for quantum analogues.

Purpose of the Study:

  • To develop a more satisfactory quantum notion of joint distributions for canonical pairs.
  • To investigate a central limit theorem for quantum canonical pairs.
  • To introduce quantum planar Brownian motions and quantum stochastic areas.

Main Methods:

  • Utilizing the non-negativity failure of the Wigner distribution.
  • Applying a central limit theorem to quantum canonical pairs.
  • Developing deformations of classical planar Brownian motion.

Main Results:

  • A novel family of quantum joint distributions is proposed.
  • A central limit theorem for quantum canonical pairs is established.
  • Quantum planar Brownian motions and stochastic areas are derived, deforming classical counterparts.

Conclusions:

  • The study provides a new framework for understanding joint distributions in quantum probability.
  • The introduced quantum Brownian motions offer insights into quantum stochastic processes.
  • This work contributes to the broader context of Hilbert's sixth problem.