Athermality

• system S. In state ρS. Evolving under Hamiltonian HS.

• Athermality: ρS’s deviation from thermal equilibrium.

  • Thermodynamics

• Distill it?

• Use it to do work?

  • Quantum heat engine;

• When can we convert one athermal state to another?

  • Convert quantum state with catalyst”

import Quantum_relative_entropy

/-- Athermality of state (ρ,γ) -/
def athermality [quantum_state ρ] γ : ℝ :=
quantum_relative_entropy ρ γ

lemma athermality_rw :
athermality ρ γ = β * (free_energy ρ - free_energy γ) :=
begin
  sorry,
end

def energy_preserving [unitary U] :=
[U, HS + HB] = 0

• Axiom: If a unitary on S+B is energy preserving then we can perform it.

• Free states:

  • Thermal state

• Free operations:

  • Thermal operations (TO),

  • closed thermal operations (CTO),

  • Gibbs preserving covariant operations (GPC).”

/- "an energy shift of the form HA → HA + (c : ℝ)*1" -/
def thermal_state_invariant_under_energy_shifts :=
invariant_under_energy_shifts thermal_state

---

“golden unit of Athermality”

References

• Gour2022;