Convert quantum state with thermal operation

• are two states are connected by a Gibbs-stochastic matrix?

  • “Use thermomajorization; Analogously to how Majorization decides state convertibility under LOCC in the resource theory of entanglement.

  • It works like this: One associates to a density matrix ρ a curve fρ(x) (called thermomajorization curve).

  • For two density matrices ρ and σ: ∀x, fρ(x) ≥ fσ(x) → ρ thermomajorizes σ.

import Thermal_operation

theorem (ρ σ) [quasiclassical_state ρ σ] :
  ∃ thermal_op T, T ρ = σ ↔ ρ thermomajorizes σ := sorry

/-- Diagonal in the energy basis -/
def quasiclassical_state [quantum_state ρ] :=
sorry 

lemma quasi_classical_iff_eq_twirl :
  quasi_classical ρ ↔ twirl ρ = ρ := sorry

variables
[quasiclassical_state ρ] [quasiclassical_state σ]

notation `GSₙ` := monoid of n × n stochastic matrices that preserve the diagonal of the thermal state.

/-- iff ∃ Gibbs-stochastic matrix from the diagonal p of ρ to the diagonal q of σ: -/
theorem Gibbs_stoch_iff_reachable_by_thermal_op :
  ∃ thermal_op T, T ρ = σ
  ↔ ∃ G ∈ GSₙ, q = G p :=
sorry

/- thermal operations on quasiclassical states:
Map the population vector of an initial state
to the population vector of a final state. -/

References

• Spaventa2021;