Universal operations in resource theories and local thermodynamics - PowerPoint PPT Presentation

Universal operations in resource theories and local thermodynamics Henrik Wilming , Rodrigo Gallego, Jens Eisert @ Freie Universität Berlin January 16th, 2015

Thermodynamics

Charge battery Unitary dynamics of the form charges the battery by an amount of work Tr Thermalising map Goal: Charge the battery as much as possible Quantum Thermodynamics p e ∆ 1 − p e

Charge battery Unitary dynamics of the form charges the battery by an amount of work Tr Thermalising map Goal: Charge the battery as much as possible Quantum Thermodynamics p e ∆ 1 − p e

Charge battery Unitary dynamics of the form charges the battery by an amount of work Tr Thermalising map Goal: Charge the battery as much as possible Quantum Thermodynamics p e ∆ 1 − p e

Charge battery Unitary dynamics of the form charges the battery by an amount of work Tr Thermalising map Goal: Charge the battery as much as possible Quantum Thermodynamics p ′ e ∆ 1 − p ′ e

Charge battery Unitary dynamics of the form charges the battery by an amount of work Tr Goal: Charge the battery as much as possible Quantum Thermodynamics p ′ e ∆ 1 − p ′ e Thermalising map

Charge battery Unitary dynamics of the form charges the battery by an amount of work Tr Thermalising map Goal: Charge the battery as much as possible Quantum Thermodynamics p ′ e ∆ 1 − p ′ e

Charge battery Unitary dynamics of the form charges the battery by an amount of work Tr Thermalising map Goal: Charge the battery as much as possible Quantum Thermodynamics p ′ e ∆ 1 − p ′ e

Unitary dynamics of the form charges the battery by an amount of work Tr Thermalising map Charge battery Goal: Charge the battery as much as possible Quantum Thermodynamics p ′ e ∆ ′ 1 − p ′ e

Unitary dynamics of the form charges the battery by an amount of work Tr Thermalising map Goal: Charge the battery as much as possible Quantum Thermodynamics p ′ e ∆ ′ 1 − p ′ e Charge battery

Charge battery Thermalising map Goal: Charge the battery as much as possible Quantum Thermodynamics Unitary dynamics of the form ( ρ 0 , H 0 ) �→ ( U t ρ 0 U † t , H t ) charges the battery by an amount of work ( ) ⟨ W ⟩ = Tr ρ 0 H 0 − U t ρ 0 U † t H t .

Charge battery Unitary dynamics of the form charges the battery by an amount of work Tr Thermalising map Quantum Thermodynamics p ′ e ∆ ′ 1 − p ′ e Goal: Charge the battery as much as possible

Def.: Thermal Operations (TO) [7] Def.: Gibbs-perserving map (GP-map) Def.: Weak thermal contact (WTC) Let be a Hamiltonian (on a bath). A thermal operation A Gibbs-preserving map is a transformation on pairs that is of the form cannot bring the system out of thermal equilibrium: WTC puts the system into thermal equilibrium: Tr e GP with a quantum channel . where is any unitary such that 1 1 . WTC GP Thermodynamical Operations p e p e ∆ 1 − p e ∆

Def.: Thermal Operations (TO) [7] Def.: Gibbs-perserving map (GP-map) Def.: Weak thermal contact (WTC) Let be a Hamiltonian (on a bath). A thermal operation A Gibbs-preserving map is a transformation on pairs that is of the form cannot bring the system out of thermal equilibrium: WTC puts the system into thermal equilibrium: Tr e GP with a quantum channel . where is any unitary such that 1 1 . WTC GP Thermodynamical Operations p e p e p e ∆ 1 − p e ∆ ∆

Def.: Thermal Operations (TO) [7] Def.: Gibbs-perserving map (GP-map) Def.: Weak thermal contact (WTC) Let be a Hamiltonian (on a bath). A thermal operation A Gibbs-preserving map is a transformation on pairs that is of the form cannot bring the system out of thermal equilibrium: WTC puts the system into thermal equilibrium: Tr e GP with a quantum channel . where is any unitary such that 1 1 . WTC GP Thermodynamical Operations p e p e ∆ 1 − p e ∆

Def.: Thermal Operations (TO) [7] Def.: Gibbs-perserving map (GP-map) Def.: Weak thermal contact (WTC) Let be a Hamiltonian (on a bath). A thermal operation A Gibbs-preserving map is a transformation on pairs that is of the form cannot bring the system out of thermal equilibrium: WTC puts the system into thermal equilibrium: Tr e GP with a quantum channel . where is any unitary such that 1 1 . WTC GP Thermodynamical Operations p e p e p e ∆ 1 − p e ∆ ∆

Def.: Thermal Operations (TO) [7] Def.: Gibbs-perserving map (GP-map) Def.: Weak thermal contact (WTC) Let be a Hamiltonian (on a bath). A thermal operation A Gibbs-preserving map is a transformation on pairs that is of the form cannot bring the system out of thermal equilibrium: WTC puts the system into thermal equilibrium: Tr e GP with a quantum channel . where is any unitary such that 1 1 . WTC GP Thermodynamical Operations p e p e ∆ 1 − p e ∆

Def.: Thermal Operations (TO) [7] Def.: Gibbs-perserving map (GP-map) Let be a Hamiltonian (on a bath). A thermal operation A Gibbs-preserving map is a transformation on pairs that is of the form cannot bring the system out of thermal equilibrium: Tr GP with a quantum channel . where is any unitary such that 1 1 . WTC GP Thermodynamical Operations p e p e ∆ 1 − p e Def.: Weak thermal contact (WTC) WTC puts the system into thermal equilibrium: ω H := e − βH ( ρ, H ) �→ ( ω H , H ) , . Z H ∆

Def.: Thermal Operations (TO) [7] Def.: Gibbs-perserving map (GP-map) Def.: Weak thermal contact (WTC) Let be a Hamiltonian (on a bath). A thermal operation A Gibbs-preserving map is a transformation on pairs that is of the form cannot bring the system out of thermal equilibrium: WTC puts the system into thermal equilibrium: Tr e GP with a quantum channel . where is any unitary such that 1 1 . WTC GP Thermodynamical Operations p e p e ∆ 1 − p e ∆

Def.: Thermal Operations (TO) [7] Def.: Gibbs-perserving map (GP-map) Def.: Weak thermal contact (WTC) Let be a Hamiltonian (on a bath). A thermal operation A Gibbs-preserving map is a transformation on pairs that is of the form cannot bring the system out of thermal equilibrium: WTC puts the system into thermal equilibrium: Tr e GP with a quantum channel . where is any unitary such that 1 1 . GP Thermodynamical Operations p e p e ∆ 1 − p e WTC ∆

Def.: Thermal Operations (TO) [7] Def.: Gibbs-perserving map (GP-map) Def.: Weak thermal contact (WTC) Let be a Hamiltonian (on a bath). A thermal operation A Gibbs-preserving map is a transformation on pairs that is of the form cannot bring the system out of thermal equilibrium: WTC puts the system into thermal equilibrium: Tr e GP with a quantum channel . where is any unitary such that 1 1 . WTC GP Thermodynamical Operations p e p e ∆ 1 − p e ∆

Def.: Gibbs-perserving map (GP-map) Def.: Weak thermal contact (WTC) A Gibbs-preserving map is a transformation on pairs that cannot bring the system out of thermal equilibrium: WTC puts the system into thermal equilibrium: e GP with a quantum channel . WTC GP Thermodynamical Operations p e p e ∆ 1 − p e Def.: Thermal Operations (TO) [7] Let H B be a Hamiltonian (on a bath). A thermal operation is of the form ( ( Uρ ⊗ ω H B U † ) ) Tr B ( ρ, H ) �→ , H , where U is any unitary such that [ U, H ⊗ 1 + 1 ⊗ H B ] = 0 . ∆

Def.: Thermal Operations (TO) [7] Def.: Weak thermal contact (WTC) Let be a Hamiltonian (on a bath). A thermal operation is of the form WTC puts the system into thermal equilibrium: Tr e GP where is any unitary such that 1 1 . WTC GP Thermodynamical Operations p e p e ∆ 1 − p e Def.: Gibbs-perserving map (GP-map) A Gibbs-preserving map is a transformation on pairs that cannot bring the system out of thermal equilibrium: ( ρ, H ) �→ ( G H ( ρ ) , H ) , G H ( ω H ) = ω H with a quantum channel G H . ∆

Def.: Thermal Operations (TO) [7] Def.: Gibbs-perserving map (GP-map) Def.: Weak thermal contact (WTC) Let be a Hamiltonian (on a bath). A thermal operation A Gibbs-preserving map is a transformation on pairs that is of the form cannot bring the system out of thermal equilibrium: WTC puts the system into thermal equilibrium: Tr e GP with a quantum channel . where is any unitary such that 1 1 . WTC GP Thermodynamical Operations p e p e ∆ 1 − p e ∆

Def.: Thermal Operations (TO) [7] Def.: Gibbs-perserving map (GP-map) Def.: Weak thermal contact (WTC) Let be a Hamiltonian (on a bath). A thermal operation A Gibbs-preserving map is a transformation on pairs that is of the form cannot bring the system out of thermal equilibrium: WTC puts the system into thermal equilibrium: Tr e GP with a quantum channel . where is any unitary such that 1 1 . WTC Thermodynamical Operations p e p e ∆ 1 − p e GP ∆

Def.: Thermal Operations (TO) [7] Def.: Gibbs-perserving map (GP-map) Def.: Weak thermal contact (WTC) Let be a Hamiltonian (on a bath). A thermal operation A Gibbs-preserving map is a transformation on pairs that is of the form cannot bring the system out of thermal equilibrium: WTC puts the system into thermal equilibrium: Tr e with a quantum channel . where is any unitary such that 1 1 . WTC GP Thermodynamical Operations p e p e ∆ 1 − p e GP ∆