Entropy-energy inequalities for qudit states

TL;DR

This paper introduces a method to approximate the energy spectra of finite qudit Hamiltonians using extremal density matrices and establishes new entropy-energy inequalities applicable to these systems.

Contribution

It presents a procedure to find extremal density matrices for finite qudit Hamiltonians and derives new inequalities linking energy and entropy.

Findings

Extremal density matrices approximate energy spectra of qudit Hamiltonians.

New entropy-energy inequalities are established for qudit states.

The approach recovers known spectra for qubits and qutrits, and conjectures applicability to arbitrary qudits.

Abstract

We establish a procedure to find the extremal density matrices for any finite Hamiltonian of a qudit system. These extremal density matrices provide an approximate description of the energy spectra of the Hamiltonian. In the case of restricting the extremal density matrices by pure states, we show that the energy spectra of the Hamiltonian is recovered for $d=2$ and $3$. We conjecture that by means of this approach the energy spectra can be recovered for the Hamiltonian of an arbitrary finite qudit system. For a given qudit system Hamiltonian, we find new inequalities connecting the mean value of the Hamiltonian and the entropy of an arbitrary state. We demonstrate that these inequalities take place for both the considered extremal density matrices and generic ones.