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

Anurag Anshu

doi:10.1088/1367-2630/18/8/083011

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 σ 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 . This bound holds whenever . 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 . This bound holds whenever .

Highlights

A question of primary interest for local hamiltonian spin systems is to determine the energy distribution of natural class of states with respect to a given local hamiltonian
Given a product state on a collection of spins which has average energy e with respect to a local hamiltonian, we show that the overlap of this state with ei√genspace of energy f is at most exp(−(e − f )2/nm2)
For quantum states with finite correlation length on finite sized lattice, a quantum version of Berry-Esseen theorem was recently shown to hold in [BC15],[BCG15]. These results give a strong indication that states satisfying exponential decay of correlation behave similar to product states, even when their energy distributions are measured with respect to the eigenspectrum of a non-commuting hamiltonian

Summary

Introduction

A question of primary interest for local hamiltonian spin systems is to determine the energy distribution of natural class of states with respect to a given local hamiltonian. For quantum states with finite correlation length (which includes product states) on finite sized lattice, a quantum version of Berry-Esseen theorem was recently shown to hold in [BC15],[BCG15] These results give a strong indication that states satisfying exponential decay of correlation behave similar to product states, even when their energy distributions are measured with respect to the eigenspectrum of a non-commuting (but local) hamiltonian. A concentration result has been noted in [KAAV17] (Section 5 in this reference) for ground states of gapped local hamiltonians on finite dimensional quantum lattice systems, which exhibit exponential decay of correlation ([Has04]). The probability distribution has been shown to be concentrated about the me√dian of the distribution with the weight of the distribution above energy ε decaying as e−|ε−f|/O(1) nσ (f being the median of the distribution, n being the number of local terms in the local hamiltonian and σ being the correlation length of the ground state).

Physical set-up and basic facts

A combinatorial lemma

Energy distribution of a product state

Conclusion