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

Quantum Thermodynamics

pe ∆ 1 − pe

Thermalising map Charge battery

Goal: Charge the battery as much as possible Unitary dynamics of the form charges the battery by an amount of work Tr

Quantum Thermodynamics

pe ∆ 1 − pe

Thermalising map Charge battery

Goal: Charge the battery as much as possible Unitary dynamics of the form charges the battery by an amount of work Tr

Quantum Thermodynamics

pe ∆ 1 − pe

Thermalising map Charge battery

Goal: Charge the battery as much as possible Unitary dynamics of the form charges the battery by an amount of work Tr

Quantum Thermodynamics

p′

e

∆ 1 − p′

e

Thermalising map Charge battery

Goal: Charge the battery as much as possible Unitary dynamics of the form charges the battery by an amount of work Tr

Quantum Thermodynamics

p′

e

∆ 1 − p′

e

Thermalising map Charge battery

Goal: Charge the battery as much as possible Unitary dynamics of the form charges the battery by an amount of work Tr

Quantum Thermodynamics

Thermalising map

p′

e

∆ 1 − p′

e

Charge battery

Goal: Charge the battery as much as possible Unitary dynamics of the form charges the battery by an amount of work Tr

Quantum Thermodynamics

Thermalising map

p′

e

∆ 1 − p′

e

Charge battery

Goal: Charge the battery as much as possible Unitary dynamics of the form charges the battery by an amount of work Tr

Quantum Thermodynamics

Thermalising map

p′

e

∆′ 1 − p′

e

Charge battery

Goal: Charge the battery as much as possible Unitary dynamics of the form charges the battery by an amount of work Tr

Quantum Thermodynamics

Thermalising map

p′

e

∆′ 1 − p′

e

Charge battery

Goal: Charge the battery as much as possible Unitary dynamics of the form charges the battery by an amount of work Tr

Quantum Thermodynamics

Thermalising map Charge battery

Goal: Charge the battery as much as possible Unitary dynamics of the form (ρ0, H0) → (Utρ0U †

t , Ht)

charges the battery by an amount of work ⟨W⟩ = Tr ( ρ0H0 − Utρ0U †

t Ht

) .

Quantum Thermodynamics

Thermalising map Charge battery

p′

e

∆′ 1 − p′

e

Goal: Charge the battery as much as possible Unitary dynamics of the form charges the battery by an amount of work Tr

Thermodynamical Operations

pe ∆ 1 − pe ∆ pe

Def.: Weak thermal contact (WTC)

WTC puts the system into thermal equilibrium: e WTC

Def.: Thermal Operations (TO) [7]

Let be a Hamiltonian (on a bath). A thermal operation is of the form Tr where is any unitary such that 1 1 .

Def.: Gibbs-perserving map (GP-map)

A Gibbs-preserving map is a transformation on pairs that cannot bring the system out of thermal equilibrium: with a quantum channel . GP GP

Thermodynamical Operations

pe ∆ 1 − pe ∆ pe ∆ pe

Def.: Weak thermal contact (WTC)

WTC puts the system into thermal equilibrium: e WTC

Def.: Thermal Operations (TO) [7]

Let be a Hamiltonian (on a bath). A thermal operation is of the form Tr where is any unitary such that 1 1 .

Def.: Gibbs-perserving map (GP-map)

A Gibbs-preserving map is a transformation on pairs that cannot bring the system out of thermal equilibrium: with a quantum channel . GP GP

Thermodynamical Operations

pe ∆ 1 − pe ∆ pe

Def.: Weak thermal contact (WTC)

WTC puts the system into thermal equilibrium: e WTC

Def.: Thermal Operations (TO) [7]

Let be a Hamiltonian (on a bath). A thermal operation is of the form Tr where is any unitary such that 1 1 .

Def.: Gibbs-perserving map (GP-map)

A Gibbs-preserving map is a transformation on pairs that cannot bring the system out of thermal equilibrium: with a quantum channel . GP GP

Thermodynamical Operations

pe ∆ 1 − pe ∆ pe ∆ pe

Def.: Weak thermal contact (WTC)

WTC puts the system into thermal equilibrium: e WTC

Def.: Thermal Operations (TO) [7]

Let be a Hamiltonian (on a bath). A thermal operation is of the form Tr where is any unitary such that 1 1 .

Def.: Gibbs-perserving map (GP-map)

A Gibbs-preserving map is a transformation on pairs that cannot bring the system out of thermal equilibrium: with a quantum channel . GP GP

Thermodynamical Operations

pe ∆ 1 − pe ∆ pe

Def.: Weak thermal contact (WTC)

WTC puts the system into thermal equilibrium: e WTC

Def.: Thermal Operations (TO) [7]

Let be a Hamiltonian (on a bath). A thermal operation is of the form Tr where is any unitary such that 1 1 .

Def.: Gibbs-perserving map (GP-map)

A Gibbs-preserving map is a transformation on pairs that cannot bring the system out of thermal equilibrium: with a quantum channel . GP GP

Thermodynamical Operations

pe ∆ 1 − pe ∆ pe

Def.: Weak thermal contact (WTC)

WTC puts the system into thermal equilibrium: (ρ, H) → (ωH, H) , ωH := e−βH ZH . WTC

Def.: Thermal Operations (TO) [7]

Let be a Hamiltonian (on a bath). A thermal operation is of the form Tr where is any unitary such that 1 1 .

Def.: Gibbs-perserving map (GP-map)

A Gibbs-preserving map is a transformation on pairs that cannot bring the system out of thermal equilibrium: with a quantum channel . GP GP

Thermodynamical Operations

pe ∆ 1 − pe ∆ pe

Def.: Weak thermal contact (WTC)

WTC puts the system into thermal equilibrium: e WTC

Def.: Thermal Operations (TO) [7]

Let be a Hamiltonian (on a bath). A thermal operation is of the form Tr where is any unitary such that 1 1 .

Def.: Gibbs-perserving map (GP-map)

A Gibbs-preserving map is a transformation on pairs that cannot bring the system out of thermal equilibrium: with a quantum channel . GP GP

Thermodynamical Operations

pe ∆ 1 − pe ∆ pe

Def.: Weak thermal contact (WTC)

WTC puts the system into thermal equilibrium: e WTC

Def.: Thermal Operations (TO) [7]

Let be a Hamiltonian (on a bath). A thermal operation is of the form Tr where is any unitary such that 1 1 .

Def.: Gibbs-perserving map (GP-map)

A Gibbs-preserving map is a transformation on pairs that cannot bring the system out of thermal equilibrium: with a quantum channel . GP GP

Thermodynamical Operations

pe ∆ 1 − pe ∆ pe

Def.: Weak thermal contact (WTC)

WTC puts the system into thermal equilibrium: e WTC

Def.: Thermal Operations (TO) [7]

Let be a Hamiltonian (on a bath). A thermal operation is of the form Tr where is any unitary such that 1 1 .

Def.: Gibbs-perserving map (GP-map)

A Gibbs-preserving map is a transformation on pairs that cannot bring the system out of thermal equilibrium: with a quantum channel . GP GP

Thermodynamical Operations

pe ∆ 1 − pe ∆ pe

Def.: Weak thermal contact (WTC)

WTC puts the system into thermal equilibrium: e WTC

Def.: Thermal Operations (TO) [7]

Let HB be a Hamiltonian (on a bath). A thermal operation is of the form (ρ, H) → ( TrB ( Uρ ⊗ ωHBU †) , H ) , where U is any unitary such that [U, H ⊗ 1 + 1 ⊗ HB] = 0.

Def.: Gibbs-perserving map (GP-map)

A Gibbs-preserving map is a transformation on pairs that cannot bring the system out of thermal equilibrium: with a quantum channel . GP GP

Thermodynamical Operations

pe ∆ 1 − pe ∆ pe

Def.: Weak thermal contact (WTC)

WTC puts the system into thermal equilibrium: e WTC

Def.: Thermal Operations (TO) [7]

Let be a Hamiltonian (on a bath). A thermal operation is of the form Tr where is any unitary such that 1 1 .

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) → (GH(ρ), H) , GH(ωH) = ωH with a quantum channel GH. GP GP

Thermodynamical Operations

pe ∆ 1 − pe ∆ pe

Def.: Weak thermal contact (WTC)

WTC puts the system into thermal equilibrium: e WTC

Def.: Thermal Operations (TO) [7]

Let be a Hamiltonian (on a bath). A thermal operation is of the form Tr where is any unitary such that 1 1 .

Def.: Gibbs-perserving map (GP-map)

A Gibbs-preserving map is a transformation on pairs that cannot bring the system out of thermal equilibrium: with a quantum channel . GP GP

Thermodynamical Operations

pe ∆ 1 − pe ∆ pe

Def.: Weak thermal contact (WTC)

WTC puts the system into thermal equilibrium: e WTC

Def.: Thermal Operations (TO) [7]

Let be a Hamiltonian (on a bath). A thermal operation is of the form Tr where is any unitary such that 1 1 .

Def.: Gibbs-perserving map (GP-map)

A Gibbs-preserving map is a transformation on pairs that cannot bring the system out of thermal equilibrium: with a quantum channel . GP GP

Thermodynamical Operations

pe ∆ 1 − pe ∆ pe

Def.: Weak thermal contact (WTC)

WTC puts the system into thermal equilibrium: e WTC

Def.: Thermal Operations (TO) [7]

Let be a Hamiltonian (on a bath). A thermal operation is of the form Tr where is any unitary such that 1 1 .

Def.: Gibbs-perserving map (GP-map)

A Gibbs-preserving map is a transformation on pairs that cannot bring the system out of thermal equilibrium: with a quantum channel . GP GP

Thermalising maps

GP TO WTC Weak thermal contact (WTC): (ρ, H) → (ωH, H). Thermal operations (TO): (ρ, H) → ( TrB ( Uρ ⊗ ωHBU †) , H ) . Gibbs-Preserving maps (GP): (ρ, H) → ((GH(ρ), H) , GH(ωH) = ωH.

The game of work-extraction

You:

Unitary GP Continue? Extract

GP

times

Me:

Unitary WTC Continue? Extract

WTC

times

I win if

GP WTC.

If you begin, I always win. (And vice-versa.)

The game of work-extraction

You:

(ρ0, H0) Unitary GP Continue? Extract

GP

times

Me:

Unitary WTC Continue? Extract

WTC

times

I win if

GP WTC.

If you begin, I always win. (And vice-versa.)

The game of work-extraction

You:

(ρ0, H0) Unitary GP Continue? Extract

GP

times

Me:

Unitary WTC Continue? Extract

WTC

times

I win if

GP WTC.

If you begin, I always win. (And vice-versa.)

The game of work-extraction

You:

(ρ0, H0) Unitary GP Continue? Extract

GP

times

Me:

Unitary WTC Continue? Extract

WTC

times

I win if

GP WTC.

If you begin, I always win. (And vice-versa.)

The game of work-extraction

You:

(ρ0, H0) Unitary GP Continue? Extract

GP

n times

Me:

Unitary WTC Continue? Extract

WTC

times

I win if

GP WTC.

If you begin, I always win. (And vice-versa.)

The game of work-extraction

You:

(ρ0, H0) Unitary GP Continue? (ρn, H0) Extract

GP

n times

Me:

Unitary WTC Continue? Extract

WTC

times

I win if

GP WTC.

If you begin, I always win. (And vice-versa.)

The game of work-extraction

You:

(ρ0, H0) Unitary GP Continue? (ρn, H0) Extract ⟨W⟩GP n times

Me:

Unitary WTC Continue? Extract

WTC

times

I win if

GP WTC.

If you begin, I always win. (And vice-versa.)

The game of work-extraction

You:

(ρ0, H0) Unitary GP Continue? (ρn, H0) Extract ⟨W⟩GP n times

Me:

(ρ0, H0) Unitary WTC Continue? (ρm, H0) Extract ⟨W⟩WTC m times

I win if

GP WTC.

If you begin, I always win. (And vice-versa.)

The game of work-extraction

You:

(ρ0, H0) Unitary GP Continue? (ρn, H0) Extract ⟨W⟩GP n times

Me:

(ρ0, H0) Unitary WTC Continue? (ρm, H0) Extract ⟨W⟩WTC m times

I win if ⟨W⟩GP ≤ ⟨W⟩WTC.

If you begin, I always win. (And vice-versa.)

The game of work-extraction

You:

(ρ0, H0) Unitary GP Continue? (ρn, H0) Extract ⟨W⟩GP n times

Me:

(ρ0, H0) Unitary WTC Continue? (ρm, H0) Extract ⟨W⟩WTC m times

I win if ⟨W⟩GP ≤ ⟨W⟩WTC.

If you begin, I always win. (And vice-versa.)

Universality of WTC

Theorem [2, 4]

The work yield of a cyclic Hamiltonian process using GP-maps as thermalising maps is bounded as ⟨W⟩GP(ρ0, H0) ≤ 1 β S(ρ0||ωH0). The bound can be saturated arbitrarily well already with WTC as thermalising maps.

[2] J. Aberg, Phys. Rev. Lett. 113, 150402 (2014). [4] H. Wilming, R. Gallego, J. Eisert, arXiv:1411.3754 (2014).

Optimal Protocol

pe ∆ 1 − pe ∆ pe

Weak thermal contact is universal for work extraction.

Optimal Protocol

pe ∆ 1 − pe ∆ pe ∆ pe

Weak thermal contact is universal for work extraction.

Optimal Protocol

pe ∆ 1 − pe ∆ pe ∆ pe

Weak thermal contact is universal for work extraction.

Optimal Protocol

pe ∆ 1 − pe ∆ pe ∆ pe

Weak thermal contact is universal for work extraction.

Optimal Protocol

pe ∆ 1 − pe ∆ pe ∆ pe

Weak thermal contact is universal for work extraction.

Optimal Protocol

pe ∆ 1 − pe ∆ pe ∆ pe

Weak thermal contact is universal for work extraction.

Optimal Protocol

pe ∆ 1 − pe ∆ pe ∆ pe

Weak thermal contact is universal for work extraction.

Optimal Protocol

pe ∆ 1 − pe ∆ pe ∆ pe

Weak thermal contact is universal for work extraction.

Optimal Protocol

pe ∆ 1 − pe ∆ pe ∆ pe

Weak thermal contact is universal for work extraction.

Optimal Protocol

pe ∆ 1 − pe ∆ pe

Weak thermal contact is universal for work extraction.

New rules for the game

pe ∆ 1 − pe ∆ pe WTC GP

You can win! WTC is not universal for restricted work extraction.

New rules for the game

pe ∆ 1 − pe ∆ pe WTC GP

You can win! WTC is not universal for restricted work extraction.

New rules for the game

pe ∆ 1 − pe ∆ pe WTC GP

You can win! WTC is not universal for restricted work extraction.

New rules for the game

pe ∆ 1 − pe ∆ pe WTC GP

You can win! WTC is not universal for restricted work extraction.

New rules for the game

pe ∆ 1 − pe ∆ pe WTC GP

You can win! WTC is not universal for restricted work extraction.

New rules for the game

pe ∆ 1 − pe ∆ pe WTC GP ∆ pe

You can win! WTC is not universal for restricted work extraction.

New rules for the game

pe ∆ 1 − pe ∆ pe WTC GP ∆ pe

You can win! WTC is not universal for restricted work extraction.

New rules for the game

pe ∆ 1 − pe ∆ pe WTC GP ∆ pe

You can win! WTC is not universal for restricted work extraction.

New rules for the game

pe ∆ 1 − pe ∆ pe WTC GP

You can win! WTC is not universal for restricted work extraction.

More general restrictions Given an initial state (ρ0, H0) we restrict the possible Hamiltonians to a set H(H0).

WTC under restrictions

Theorem (General work bound)

The work that can be extracted from a pair (ρ0, H0) using WTC is bounded as ⟨W⟩H

WTC(ρ0, H0) ≤ 1

β S(ρ0||ωH0) − inf

Ht∈H(H0) U∈U[H0]

1 β S(Uρ0U †||ωHt), with U[H0] being the unitary group generated by H(H0). The bound can be saturated arbitrarily well.

Restriction on locality

Hloc(H0) := { H0 + ∑

i

Hi | Hi local } Models the situation of local control over an interacting Hamiltonian.

Theorem (Non-universality of WTC)

Under the restriction to

loc, there exist initial states

such that no work can be extracted using WTC but work can be extracted using GP-maps.

Restriction on locality

Hloc(H0) := { H0 + ∑

i

Hi | Hi local } Models the situation of local control over an interacting Hamiltonian.

Theorem (Non-universality of WTC)

Under the restriction to Hloc, there exist initial states (ρ0, H0) such that no work can be extracted using WTC but work can be extracted using GP-maps.

Proof (Example)

Consider two qubits with maximally mixed initial state Ω, Hamiltonian Z ⊗ Z and β = 1. W.l.o.g. assume all Hamiltonians are traceless. General bound implies sup

loc

sup

loc

sup

loc

with the free energy Tr

Peierls-Bogoliubov inequality:

tr Any traceless local Hamiltonian is

• rthogonal to

. Hence . Thermomajorization [5, 6, 7] implies that there exists a thermal operation such that

1

for . Since 1

loc

, we can extract

1

Proof (Example)

Consider two qubits with maximally mixed initial state Ω, Hamiltonian Z ⊗ Z and β = 1. W.l.o.g. assume all Hamiltonians are traceless. General bound implies ⟨W⟩(Ω, Z ⊗ Z) ≤ sup

Ht∈Hloc(H0) U∈U[H0]

( S(Ω||ωZ⊗Z) − S(UΩU †||ωHt) ) , sup

loc

sup

loc

with the free energy Tr

Peierls-Bogoliubov inequality:

tr Any traceless local Hamiltonian is

• rthogonal to

. Hence . Thermomajorization [5, 6, 7] implies that there exists a thermal operation such that

1

for . Since 1

loc

, we can extract

1

Proof (Example)

Consider two qubits with maximally mixed initial state Ω, Hamiltonian Z ⊗ Z and β = 1. W.l.o.g. assume all Hamiltonians are traceless. General bound implies ⟨W⟩(Ω, Z ⊗ Z) ≤ sup

Ht∈Hloc(H0) U∈U[H0]

( S(Ω||ωZ⊗Z) − S(UΩU †||ωHt) ) , = sup

Ht∈Hloc(H0)

(S(Ω ||ωZ⊗Z) − S(Ω ||ωHt)) , sup

loc

with the free energy Tr

Peierls-Bogoliubov inequality:

tr Any traceless local Hamiltonian is

• rthogonal to

. Hence . Thermomajorization [5, 6, 7] implies that there exists a thermal operation such that

1

for . Since 1

loc

, we can extract

1

Proof (Example)

Consider two qubits with maximally mixed initial state Ω, Hamiltonian Z ⊗ Z and β = 1. W.l.o.g. assume all Hamiltonians are traceless. General bound implies ⟨W⟩(Ω, Z ⊗ Z) ≤ sup

Ht∈Hloc(H0) U∈U[H0]

( S(Ω||ωZ⊗Z) − S(UΩU †||ωHt) ) , = sup

Ht∈Hloc(H0)

(S(Ω ||ωZ⊗Z) − S(Ω ||ωHt)) , sup

loc

1 β S(ρ||ωH) = F(ρ, H) − F(ωH, H) with the free energy F(ρ, H) = Tr(ρH) − 1 β S(ρ).

Peierls-Bogoliubov inequality:

tr Any traceless local Hamiltonian is

• rthogonal to

. Hence . Thermomajorization [5, 6, 7] implies that there exists a thermal operation such that

1

for . Since 1

loc

, we can extract

1

Proof (Example)

Consider two qubits with maximally mixed initial state Ω, Hamiltonian Z ⊗ Z and β = 1. W.l.o.g. assume all Hamiltonians are traceless. General bound implies ⟨W⟩(Ω, Z ⊗ Z) ≤ sup

Ht∈Hloc(H0) U∈U[H0]

( S(Ω||ωZ⊗Z) − S(UΩU †||ωHt) ) , = sup

Ht∈Hloc(H0)

(S(Ω ||ωZ⊗Z) − S(Ω ||ωHt)) , = sup

Ht∈Hloc(H0)

(F(ωHt, Ht) − F(ωZ⊗Z, Z ⊗ Z)) . 1 β S(ρ||ωH) = F(ρ, H) − F(ωH, H) with the free energy F(ρ, H) = Tr(ρH) − 1 β S(ρ).

Peierls-Bogoliubov inequality:

tr Any traceless local Hamiltonian is

• rthogonal to

. Hence . Thermomajorization [5, 6, 7] implies that there exists a thermal operation such that

1

for . Since 1

loc

, we can extract

1

Proof (Example)

Consider two qubits with maximally mixed initial state Ω, Hamiltonian Z ⊗ Z and β = 1. W.l.o.g. assume all Hamiltonians are traceless. General bound implies ⟨W⟩(Ω, Z ⊗ Z) ≤ sup

Ht∈Hloc(H0) U∈U[H0]

( S(Ω||ωZ⊗Z) − S(UΩU †||ωHt) ) , = sup

Ht∈Hloc(H0)

(S(Ω ||ωZ⊗Z) − S(Ω ||ωHt)) , = sup

Ht∈Hloc(H0)

(F(ωHt, Ht) − F(ωZ⊗Z, Z ⊗ Z)) . with the free energy Tr

Peierls-Bogoliubov inequality:

F(ωA+B, A + B) ≤ F(ωA, A) + tr(ωAB) Any traceless local Hamiltonian is

• rthogonal to ωZ⊗Z. Hence

F(ωHt, Ht) ≤ F(ωZ⊗Z, Z ⊗ Z). Thermomajorization [5, 6, 7] implies that there exists a thermal operation such that

1

for . Since 1

loc

, we can extract

1

Proof (Example)

Consider two qubits with maximally mixed initial state Ω, Hamiltonian Z ⊗ Z and β = 1. W.l.o.g. assume all Hamiltonians are traceless. General bound implies ⟨W⟩(Ω, Z ⊗ Z) ≤ sup

Ht∈Hloc(H0) U∈U[H0]

( S(Ω||ωZ⊗Z) − S(UΩU †||ωHt) ) , = sup

Ht∈Hloc(H0)

(S(Ω ||ωZ⊗Z) − S(Ω ||ωHt)) , = sup

Ht∈Hloc(H0)

(F(ωHt, Ht) − F(ωZ⊗Z, Z ⊗ Z)) . ≤ (F(ωZ⊗Z, Z ⊗ Z) − F(ωZ⊗Z, Z ⊗ Z)) ≤ 0 with the free energy Tr

Peierls-Bogoliubov inequality:

F(ωA+B, A + B) ≤ F(ωA, A) + tr(ωAB) Any traceless local Hamiltonian is

• rthogonal to ωZ⊗Z. Hence

F(ωHt, Ht) ≤ F(ωZ⊗Z, Z ⊗ Z). Thermomajorization [5, 6, 7] implies that there exists a thermal operation such that

1

for . Since 1

loc

, we can extract

1

Proof (Example)

Consider two qubits with maximally mixed initial state Ω, Hamiltonian Z ⊗ Z and β = 1. W.l.o.g. assume all Hamiltonians are traceless. General bound implies ⟨W⟩(Ω, Z ⊗ Z) ≤ sup

Ht∈Hloc(H0) U∈U[H0]

( S(Ω||ωZ⊗Z) − S(UΩU †||ωHt) ) , = sup

Ht∈Hloc(H0)

(S(Ω ||ωZ⊗Z) − S(Ω ||ωHt)) , = sup

Ht∈Hloc(H0)

(F(ωHt, Ht) − F(ωZ⊗Z, Z ⊗ Z)) . ≤ (F(ωZ⊗Z, Z ⊗ Z) − F(ωZ⊗Z, Z ⊗ Z)) ≤ 0 with the free energy Tr

Peierls-Bogoliubov inequality:

tr Any traceless local Hamiltonian is

• rthogonal to

. Hence . Thermomajorization [5, 6, 7] implies that there exists a thermal operation G such that G(Ω) = ωZ⊗Z+tZ⊗1, for |t| < 0.46. Since Z ⊗ Z + tZ ⊗ 1 ∈ Hloc(Z ⊗ Z), we can extract ⟨W⟩ = S (ωZ⊗Z+tZ⊗1||ωZ⊗Z) > 0.

[5] E. Ruch, R. Schranner, and T. H. Seligman, J. Chem. Phys. 69, 386 (1978). [6] H. D. Janzing, P. Wocjan, R. Zeier, R. Geiss, and T. Beth, Int. J. Th. Phys. 39, 2717 (2000). [7] M. Horodecki and J. Oppenheim, Nature Comm. 4, 2059 (2013).

Summarizing the effect of restrictions Restrictions on the Hamiltonians Passive states for WTC GP-maps can activate passive states WTC ceases to be universal

Outlook

Resource theory (roughly) A class of operations that is closed under composition and contains the identity together with a distinction between free and costly states and operations [9].

[9] B. Coecke, T. Fritz, R. W. Spekkens, arXiv:1409.5531 (2014).

What happens if we “combine” two resource theories?

A B ??

A B

We call a sub-theory B ⊂ A universal for a specific task if the task can already be achieved optimally only with operations from B.

Suppose Aʼ ⊂ A is universal for some task.

A Aʼ A Aʼ B

Aʼ B

Is Aʼ B still universal in B?

Suppose Aʼ ⊂ A is universal for some task.

A Aʼ A Aʼ B

Aʼ B

Is Aʼ B still universal in B?

Suppose Aʼ ⊂ A is universal for some task.

A Aʼ A Aʼ B

Aʼ ∩ B

Is Aʼ ∩ B still universal in B?

GP

WTC is universal for unrestricted work extraction

WTC

GP WTC H

WTC∩H

WTC is not universal for restricted work extraction Open questions Can we find similar situations outside of thermodynamics? Other operationally meaningful types of restrictions? What about single-shot statistical mechanics / different notions of work?

GP

WTC is universal for unrestricted work extraction

WTC

GP WTC H

WTC∩H

WTC is not universal for restricted work extraction Open questions Can we find similar situations outside of thermodynamics? Other operationally meaningful types of restrictions? What about single-shot statistical mechanics / different notions of work?

GP

WTC is universal for unrestricted work extraction

WTC

GP WTC H

WTC∩H

WTC is not universal for restricted work extraction Open questions Can we find similar situations outside of thermodynamics? Other operationally meaningful types of restrictions? What about single-shot statistical mechanics / different notions of work?

GP

WTC is universal for unrestricted work extraction

WTC

GP WTC H

WTC∩H

WTC is not universal for restricted work extraction Open questions Can we find similar situations outside of thermodynamics? Other operationally meaningful types of restrictions? What about single-shot statistical mechanics / different notions of work?

GP

WTC is universal for unrestricted work extraction

WTC

GP WTC H

WTC∩H

WTC is not universal for restricted work extraction Open questions Can we find similar situations outside of thermodynamics? Other operationally meaningful types of restrictions? What about single-shot statistical mechanics / different notions of work?

Thanks for listening! For details see arXiv:1411.3754.

Funding: A.-v.-H., BMBF, EU (RAQUEL, SIQS, COST, AQuS), and ERC (TAQ) References:

[1]

• J. Aberg, Nature Comm. 4, 1925 (2013).

[2]

• J. Aberg, Phys. Rev. Lett. 113, 150402 (2014).

[3]

• R. Gallego, A. Riera, and J. Eisert, arXiv:1310.8349 (2013).

[4]

• H. Wilming, R. Gallego, and J. Eisert, arXiv:1411.3754 (2014).

[5]

• E. Ruch, R. Schranner, and T. H. Seligman, J. Chem. Phys. 69, 386 (1978).

[6]

• H. D. Janzing, P. Wocjan, R. Zeier, R. Geiss, and T. Beth, Int. J. Th. Phys. 39, 2717

(2000). [7]

• M. Horodecki and J. Oppenheim, Nature Comm. 4, 2059 (2013).

[8]

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