Nash states versus eigenstates for many-body quantum systems

TL;DR

This paper introduces 'Nash states' as a new concept generalizing eigenstates for multiple observables in many-body quantum systems, exploring their mathematical properties and potential applications.

Contribution

It proposes Nash states and Nash varieties as a novel framework for understanding many-body quantum systems beyond traditional eigenstates.

Findings

Existence of Nash states demonstrated analytically and numerically

Nash varieties characterized geometrically

Connections established to quantum game theory and Hamiltonian minimization

Abstract

Eigenstates of observables such as the Hamiltonian play a central role in quantum mechanics. Inspired by the pure Nash equilibria that arise in classical game theory, we propose ''Nash states'' of multiple observables as a generalization of eigenstates of single observables. This generalization is mathematically natural for many-body quantum systems, which possess an intrinsic tensor product structure. Every set of observables gives rise to algebraic varieties of Nash state vectors that we call ''Nash varieties''. We present analytical and numerical results on the existence of Nash states and on the geometry of Nash varieties. We relate these ideas to earlier, pioneering work on the Nash equilibria of few-body quantum games and discuss connections to the variational minimization of local Hamiltonians.