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Quantum geometry of expectation values

Chaoming Song*

  • Department of Physics, University of Miami, Coral Gables, Florida 33146, USA

  • *c.song@miami.edu
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Phys. Rev. A 107, 062207 – Published 9 June, 2023

DOI: https://doi.org/10.1103/PhysRevA.107.062207

Abstract

We propose a framework for the quantum geometry of expectation values over arbitrary sets of operators and establish a link between this geometry and the eigenstates of Hamiltonian families generated by these operators. We show that the boundary of expectation value space corresponds to the ground state, which presents a natural bound that generalizes Heisenberg's uncertainty principle. To demonstrate the versatility of our framework, we present several practical applications, including providing a stronger nonlinear quantum bound that violates the Bell inequality and an explicit construction of the density functional. Our approach provides an alternative time-independent quantum formulation that transforms the linear problem in a high-dimensional Hilbert space into a nonlinear algebro-geometric problem in a low dimension, enabling us to gain new insights into quantum systems.

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Phys. Rev. A 107, 062207– Published 9 June, 2023
  • Received 17 January 2023
  • Revised 20 April 2023
  • Accepted 30 May 2023
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