Energy upper bound for structurally stable N-passive states

TL;DR

This paper establishes an upper energy bound for large N-passive, structurally stable quantum states, linking their energy content to the spectral properties of the Hamiltonian and approaching thermal Gibbs states.

Contribution

It provides a quantitative upper bound on the energy of N-passive, structurally stable states based on Hamiltonian spectra, extending to non-structurally stable states for large N.

Findings

Upper bound depends only on spectral properties of the Hamiltonian.

Bound is tight and can be saturated under certain conditions.

Extension of the bound applies to non-structurally stable states for large N.

Abstract

Passive states are special configurations of a quantum system which exhibit no energy decrement at the end of an arbitrary cyclic driving of the model Hamiltonian. When applied to an increasing number of copies of the initial density matrix, the requirement of passivity induces a hierarchical ordering which, in the asymptotic limit of infinitely many elements, pinpoints ground states and thermal Gibbs states. In particular, for large values of N the energy content of a N-passive state which is also structurally stable (i.e. capable to maintain its passivity status under small perturbations of the model Hamiltonian), is expected to be close to the corresponding value of the thermal Gibbs state which has the same entropy. In the present paper we provide a quantitative assessment of this fact, by producing an upper bound for the energy of an arbitrary N-passive, structurally stable state which only depends on the spectral properties of the Hamiltonian of the system. We also show the condition under which our inequality can be saturated. A generalization of the bound is finally presented that, for sufficiently large N, applies to states which are N-passive, but not necessarily structurally stable.