Quantum Geometry of Expectation Values

TL;DR

This paper introduces a new geometric framework for quantum expectation values that links boundary states to ground states, leading to stronger bounds and novel insights into quantum systems.

Contribution

It develops a novel quantum geometric framework connecting expectation values, eigenstates, and bounds, with applications to Bell inequalities and density functional theory.

Findings

Boundary of expectation value space corresponds to ground states

Provides a stronger nonlinear quantum bound violating Bell inequality

Constructs an explicit density functional

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.