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arXiv · 2301.05921

Quantum Geometry of Expectation Values

Chaoming Song
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Abstract

We propose a novel 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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Chaoming Song. 2023-01-14. Quantum Geometry of Expectation Values. https://doi.org/10.1103/physreva.107.062207

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