Concentration bounds for quantum states with finite correlation length on quantum spin lattice systems

TL;DR

This paper derives concentration bounds for quantum states with finite correlation length and product states, showing exponential and Gaussian decay of energy distribution tails in quantum spin lattice systems.

Contribution

It provides new bounds on energy distribution tails for states with finite correlation length and for product states, extending understanding of quantum state energy localization.

Findings

Energy distribution tail decays exponentially in 1D systems.

Product states exhibit Gaussian decay in energy distribution.

Bounds depend on correlation length, local terms, and overlap structure.

Abstract

We consider the problem of determining the energy distribution of quantum states that satisfy exponential decay of correlation and product states, with respect to a quantum local hamiltonian on a spin lattice. For a quantum state on a $D$-dimensional lattice that has correlation length $\sigma$ and has average energy $e$ with respect to a given local hamiltonian (with $n$ local terms, each of which has norm at most $1$), we show that the overlap of this state with eigenspace of energy $f$ is at most $exp(-((e-f)^2\sigma)^{\frac{1}{D+1}}/n^{\frac{1}{D+1}}D\sigma)$. This bound holds whenever $|e-f|>2^{D}\sqrt{n\sigma}$. Thus, on a one dimensional lattice, the tail of the energy distribution decays exponentially with the energy. For product states, we improve above result to obtain a Gaussian decay in energy, even for quantum spin systems without an underlying lattice structure. Given a product state on a collection of spins which has average energy $e$ with respect to a local hamiltonian (with $n$ local terms and each local term overlapping with at most $m$ other local terms), we show that the overlap of this state with eigenspace of energy $f$ is at most $exp(-(e-f)^2/nm^2)$. This bound holds whenever $|e-f|>m\sqrt{n}$.