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 ofNthe energy content of aN-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 arbitraryN-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 largeN, applies to states which areN-passive, but not necessarily structurally stable.
Highlights
One of the most striking differences between classical and quantum thermodynamics is that, while the properties of macroscopic system in equilibrium can be described with a small number of degrees of freedom, the unitary evolution prescribed by the laws of quantum mechanics has as many conserved quantities as the dimension of the Hilbert space [1]
There are states of a quantum system which are not in thermal equilibrium, but that are passive states, in the sense that their energy can not decrease under unitary evolution
We give inequalities that apply for N -passive states which are not necessarily structurally stable, in the asymptotic limit of large N
Summary
One of the most striking differences between classical and quantum thermodynamics is that, while the properties of macroscopic system in equilibrium can be described with a small number of degrees of freedom, the unitary evolution prescribed by the laws of quantum mechanics has as many conserved quantities as the dimension of the Hilbert space [1]. Aim of the present work is to investigate how the gap between E(ρ; H) and Eβ(H) reduces as N increases For this purpose we prove an inequality which provides an upper bound for E(ρ; H) in term of Eβ(H), via a multiplicative factor which only depends upon the spectral properties of the Hamiltonian, and which converges asymptotically to 1 as N increases. This allows us to provide a quantitative estimation of the way in which a quantum effect (the gap between ergotropy and free energy) decreases when the size of the system (quantified by the number N of copies) increases, and disappears in the macroscopic limit N → ∞. The manuscript contains few appendixes which provide technical support for the derivation of the main results (in particular in Appendix B we give a new proof of the fact that Gibbs states and ground states are the only density matrices which are completely passive)
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