the thermodynamics of creating correlations · work in collaboration with david e. bruschi...

Optimal creation of total correlationsOptimal creation of entanglement

The Thermodynamics

of Creating Correlations

Limitations and Optimal Protocols

27th SFB FoQus meeting

Innsbruck, October 9th, 2014

Nicolai Friis

Institute for Quantum Optics and Quantum Information - Innsbruck

work in collaboration with David E. Bruschi (Jerusalem), Martı Perarnau-Llobet,Karen V. Hovhannisyan, and Marcus Huber (Barcelona)

Nicolai Friis (IQOQI Innsbruck) The thermodynamics of creating correlations

Motivation — Quantum Thermodynamics

Some approaches to thermodynamics in the quantum domain

• Landauer’s principle: minimum energy for bit erasure (see, e.g., Reeb & Wolf,

arXiv:1306.4352; Esposito & Van den Broeck, Europhys. Lett. 95, 40004 (2011) [arXiv:1104.5165])

• Role of quantum effects in thermal machinesScarani, et al., Phys. Rev. Lett. 88, 097905 (2002) [arXiv:quant-ph/0110088];

Alicki & Fannes, Phys. Rev. E 87, 042123 (2013) [arXiv:1211.1209];

Correa et al., Phys. Rev. E 87, 042131 (2013) [arXiv:1212.4501];

Brunner et al., Phys. Rev. E 89, 032115 (2014) [arXiv:1305.6009];

Gallego, Riera, & Eisert, arXiv:1310.8349;

• Thermodynamic laws in quantum regime (see, e.g., Brandao et al., arXiv:1305.5278)

Connection between resource theories?

thermodynamics ←→ quantum information

energy ←→ correlations

Outline

We ask1 How strongly can one correlate a system at initialtemperature T if the work W is supplied?

Previous work in this direction

• All correlations imply extractable work (Perarnau-Llobet et al.,arXiv:1407.7765)

• Cost of correlating isolated systems of qubits (Huber et al., arXiv:1404.2169)

Outline of our contribution1

Generating total correlations:

• Bound for overall correlations measured by the mutual information

• Optimal protocol saturating this bound

Generating entanglement — bottom-up approach:

• Optimal entanglement generation between two bosonic modes

• Optimal entanglement generation between two fermionic modes

1 D. E. Bruschi, M. Perarnau-Llobet, N. Friis, K. V. Hovhannisyan, and M. Huber, arXiv:1409.4647 [quant-ph].

Outline

Framework

System S & Bath B: HS1⊗HS2

non-interacting HSB = HS1+HS2

thermal state τSB

(β) = Z−1(β) e−βHSB

initial state uncorrelated τSB

= τS1⊗ τ

S2⊗ τ

Cost of moving the system out of thermal equilibrium

Free energy difference: ∆F(τ(β)→ ρ

(ρ)− F

(τ(β)

)≥ 0

where F(ρ)

= E(ρ)− T S

↑ ↖energy entropy

Applying unitaries US or USB to correlate system ⇔ energy cost

Notation: Z . . . partition function, β = 1/T , units: ~ = kB = 1

Framework

thermal state τSB

= τS1⊗ τ

S2⊗ τ

(ρ)− F

(τ(β)

)≥ 0

where F(ρ)

= E(ρ)− T S

Framework

thermal state τSB

= τS1⊗ τ

S2⊗ τ

(ρ)− F

(τ(β)

)≥ 0

where F(ρ)

= E(ρ)− T S

Framework

thermal state τSB

= τS1⊗ τ

S2⊗ τ

(ρ)− F

(τ(β)

)≥ 0

where F(ρ)

= E(ρ)− T S

Framework

thermal state τSB

= τS1⊗ τ

S2⊗ τ

(ρ)− F

(τ(β)

)≥ 0

where F(ρ)

= E(ρ)− T S

Framework

thermal state τSB

= τS1⊗ τ

S2⊗ τ

(ρ)− F

(τ(β)

)≥ 0

where F(ρ)

= E(ρ)− T S

Framework

thermal state τSB

= τS1⊗ τ

S2⊗ τ

(ρ)− F

(τ(β)

)≥ 0

where F(ρ)

= E(ρ)− T S

Framework

thermal state τSB

= τS1⊗ τ

S2⊗ τ

(ρ)− F

(τ(β)

)≥ 0

where F(ρ)

= E(ρ)− T S

Bound on optimal correlationsOptimal protocol for the generation of mutual information

Optimal creation of total correlations

Bound on optimal correlations

Measure of correlation

Mutual information: IS1S2

) = S(ρS1

) + S(ρS2

) − S(ρS

Energy cost for global unitary USB

W = ∆FS

+ ∆FB

+ T ISB

Similarly: ∆FS

= ∆FS1

+ ∆FS2

+ T IS1S2

For any thermal state: ∆F = T S(ρ||τ(β)

)Hence: βW = S(ρ

S1||τS1

) + S(ρS2||τS2

) + S(ρB||τB

) + IS1S2

Maximal correlation for fixed temperature and work

IS1S2≤ β W

For arbitrarily large bath ∆FB

= 0, ISB

= 0 achievable

(see Aberg, Nat. Commun. 4, 1925 (2013) [arXiv:1110.6121])

) = S(ρS1

) + S(ρS2

) − S(ρS

W = ∆FS

+ ∆FB

+ T ISB

Similarly: ∆FS

= ∆FS1

+ ∆FS2

+ T IS1S2

)Hence: βW = S(ρ

S1||τS1

) + S(ρS2||τS2

) + S(ρB||τB

) + IS1S2

IS1S2≤ β W

= 0, ISB

= 0 achievable

) = S(ρS1

) + S(ρS2

) − S(ρS

W = ∆FS

+ ∆FB

+ T ISB

Similarly: ∆FS

= ∆FS1

+ ∆FS2

+ T IS1S2

)Hence: βW = S(ρ

S1||τS1

) + S(ρS2||τS2

) + S(ρB||τB

) + IS1S2

IS1S2≤ β W

= 0, ISB

= 0 achievable

) = S(ρS1

) + S(ρS2

) − S(ρS

W = ∆FS

+ ∆FB

+ T ISB

Similarly: ∆FS

= ∆FS1

+ ∆FS2

+ T IS1S2

)Hence: βW = S(ρ

S1||τS1

) + S(ρS2||τS2

) + S(ρB||τB

) + IS1S2

IS1S2≤ β W

= 0, ISB

= 0 achievable

) = S(ρS1

) + S(ρS2

) − S(ρS

W = ∆FS

+ ∆FB

+ T ISB

Similarly: ∆FS

= ∆FS1

+ ∆FS2

+ T IS1S2

)Hence: βW = S(ρ

S1||τS1

) + S(ρS2||τS2

) + S(ρB||τB

) + IS1S2

IS1S2≤ β W

= 0, ISB

= 0 achievable

) = S(ρS1

) + S(ρS2

) − S(ρS

W = ∆FS

+ ∆FB

+ T ISB

Similarly: ∆FS

= ∆FS1

+ ∆FS2

+ T IS1S2

)Hence: βW = S(ρ

S1||τS1

) + S(ρS2||τS2

) + S(ρB||τB

) + IS1S2

IS1S2≤ β W

= 0, ISB

= 0 achievable

) = S(ρS1

) + S(ρS2

) − S(ρS

W = ∆FS

+ ∆FB

+ T ISB

Similarly: ∆FS

= ∆FS1

+ ∆FS2

+ T IS1S2

)Hence: βW = S(ρ

S1||τS1

) + S(ρS2||τS2

) + S(ρB||τB

) + IS1S2

IS1S2≤ β W

= 0, ISB

= 0 achievable

) = S(ρS1

) + S(ρS2

) − S(ρS

W = ∆FS

+ ∆FB

+ T ISB

Similarly: ∆FS

= ∆FS1

+ ∆FS2

+ T IS1S2

)Hence: βW = S(ρ

S1||τS1

) + S(ρS2||τS2

) + S(ρB||τB

) + IS1S2

IS1S2≤ β W

= 0, ISB

= 0 achievable

Optimal protocol for the generation of mutual information

Without loss of generality we may split the protocol into two steps

(I) Cooling: Lower temperature of S from T to TI ≤ T

Minimal energy cost WI = ∆FS

= F(τS

(βI))− F

(II) Correlating: Isolate system from bath

Correlate via unitary US

such that S1 and S2 are locally thermal attemperature TII ≥ TI,

⇒ ensures correlation at minimal energy cost (see Huber et al., arXiv:1404.2169)

Energy cost WII = E(τS

(βII))− E

(βI))

= F(τS

(βI))− F

(βII))− E

(βI))

= F(τS

(βI))− F

(βII))− E

(βI))

= F(τS

(βI))− F

(βII))− E

(βI))

= F(τS

(βI))− F

(βII))− E

(βI))

= F(τS

(βI))− F

(βII))− E

(βI))

Overall work cost:

W = WI + WII =T S(τS(βII)||τS(β)

Optimal conversion of energy into correlations when

βII = β, TII = T ⇒ S(τS(βII)||τS(β)

)= 0 I

Overall work cost:

W = WI + WII =T S(τS(βII)||τS(β)

Optimal conversion of energy into correlations when

βII = β, TII = T ⇒ S(τS(βII)||τS(β)

)= 0 I

Optimal protocol: distinguish two regimes

Low energy: IS1S2

= βW if W ≤ T S(τS

(β)),

High energy: IS1S2= S

(βII))

if W > T S(τS

(β)),

βII implicitly given by E(τS

(βII))

= W + F(τS

Low energy: IS1S2

(β)),

(τS(βII)

)if W > T S

(τS(β)

(βII))

= W + F(τS

Low energy: IS1S2

(β)),

(τS(βII)

)if W > T S

(τS(β)

(βII))

= W + F(τS

Low energy: IS1S2

(β)),

(τS(βII)

)if W > T S

(τS(β)

(βII))

= W + F(τS

High energy regime: W > T S(τS(β)

)2 fermionic modes at frequency ω

Fermi-Dirac statistics: partition function ZFD = 1 + e−β

Thermal state: average particle number NS1= NS2

=(eβ + 1

Pauli exclusion principle: 0 ≤ NSi≤ 1 for any state

⇒ maximally useful energy for protocol: Wmax = 2T ln(eβ + 1

)− ω

Notation: β = ω/T , temperatures specified in units of ~ω/kB, T =0, 0.1, . . .,0.9, 1

=(eβ + 1

)− ω

=(eβ + 1

)− ω

=(eβ + 1

)− ω

=(eβ + 1

)− ω

=(eβ + 1

)− ω

)2 bosonic modes at frequency ω

Bose-Einstein statistics: partition function ZBE =(1− e−β

=(eβ − 1

Energy of thermal state: E(τS

= ω[coth

)− 1]

Entropy of thermal state: S(τS

coth(β/2

))where f(x) = x+1

2ln(x+12

)− x−1

2ln(x−12

)For W � T S

(β)): all energy of step II for correlations

WII = E(τS

(βII))

and IS1S2

= S(τS

(βII))

= 2 + 2 ln(

12WIIω

(ωWII

)Notation: β = ω/T , temperatures specified in units of ~ω/kB

=(eβ − 1

= ω[coth

)− 1]

coth(β/2

))where f(x) = x+1

2ln(x+12

)− x−1

2ln(x−12

)For W � T S

WII = E(τS

(βII))

and IS1S2

= S(τS

(βII))

= 2 + 2 ln(

12WIIω

(ωWII

=(eβ − 1

= ω[coth

)− 1]

coth(β/2

))where f(x) = x+1

2ln(x+12

)− x−1

2ln(x−12

)For W � T S

WII = E(τS

(βII))

and IS1S2

= S(τS

(βII))

= 2 + 2 ln(

12WIIω

(ωWII

=(eβ − 1

= ω[coth

)− 1]

coth(β/2

))where f(x) = x+1

2ln(x+12

)− x−1

2ln(x−12

)For W � T S

WII = E(τS

(βII))

and IS1S2

= S(τS

(βII))

= 2 + 2 ln(

12WIIω

(ωWII

=(eβ − 1

= ω[coth

)− 1]

coth(β/2

))where f(x) = x+1

2ln(x+12

)− x−1

2ln(x−12

)For W � T S

WII = E(τS

(βII))

and IS1S2

= S(τS

(βII))

= 2 + 2 ln(

12WIIω

(ωWII

=(eβ − 1

= ω[coth

)− 1]

coth(β/2

))where f(x) = x+1

2ln(x+12

)− x−1

2ln(x−12

)For W � T S

WII = E(τS

(βII))

and IS1S2

= S(τS

(βII))

= 2 + 2 ln(

12WIIω

(ωWII

=(eβ − 1

= ω[coth

)− 1]

coth(β/2

))where f(x) = x+1

2ln(x+12

)− x−1

2ln(x−12

)For W � T S

WII = E(τS

(βII))

and IS1S2

= S(τS

(βII))

= 2 + 2 ln(

12WIIω

(ωWII

=(eβ − 1

= ω[coth

)− 1]

coth(β/2

))where f(x) = x+1

2ln(x+12

)− x−1

2ln(x−12

)For W � T S

WII = E(τS

(βII))

and IS1S2

= S(τS

(βII))

= 2 + 2 ln(

12WIIω

(ωWII

Finite-dimensions: fermionic modesInfinite-dimensions: bosonic modes

Optimal creation of entanglement

Entanglement of Formation

Entanglement of formation

EoF = infD(ρS)

∑ipi E

(|ψi 〉

)where E

(|ψi 〉

(|ψi 〉〈ψi |

))and D(ρS) =

{pi, |ψi 〉 |

∑i pi |ψi 〉〈ψi | = ρS

}Fermions

Superselection rules: take infimum only over allowed states |ψi 〉

No superpositions of even and odd fermions numbers

Entanglement of Formation computable

BosonsHilbert space infinite-dimensional ⇒ restrict problem: Gaussian states

Only second moments relevant: use covariance matrix

Symmetric states: Entanglement of Formation computable

EoF = infD(ρS)

∑ipi E

(|ψi 〉

)where E

(|ψi 〉

(|ψi 〉〈ψi |

))and D(ρS) =

{pi, |ψi 〉 |

}Fermions

EoF = infD(ρS)

∑ipi E

(|ψi 〉

)where E

(|ψi 〉

(|ψi 〉〈ψi |

))and D(ρS) =

{pi, |ψi 〉 |

}Fermions

EoF = infD(ρS)

∑ipi E

(|ψi 〉

)where E

(|ψi 〉

(|ψi 〉〈ψi |

))and D(ρS) =

{pi, |ψi 〉 |

}Fermions

EoF = infD(ρS)

∑ipi E

(|ψi 〉

)where E

(|ψi 〉

(|ψi 〉〈ψi |

))and D(ρS) =

{pi, |ψi 〉 |

}Fermions

EoF = infD(ρS)

∑ipi E

(|ψi 〉

)where E

(|ψi 〉

(|ψi 〉〈ψi |

))and D(ρS) =

{pi, |ψi 〉 |

}Fermions

EoF = infD(ρS)

∑ipi E

(|ψi 〉

)where E

(|ψi 〉

(|ψi 〉〈ψi |

))and D(ρS) =

{pi, |ψi 〉 |

}Fermions

EoF = infD(ρS)

∑ipi E

(|ψi 〉

)where E

(|ψi 〉

(|ψi 〉〈ψi |

))and D(ρS) =

{pi, |ψi 〉 |

}Fermions

EoF = infD(ρS)

∑ipi E

(|ψi 〉

)where E

(|ψi 〉

(|ψi 〉〈ψi |

))and D(ρS) =

{pi, |ψi 〉 |

}Fermions

Numerical Optimization

Entanglement of Formation for two fermionic modes

Previous Protocol Optimal Protocol

Notation: β = ω/T , temperatures specified in units of ~ω/kB, T =0, 0.1, . . .,0.9, 1 and∞

Entanglement of Formation for two fermionic modes

Previous Protocol Optimal Protocol

Notation: β = ω/T , temperatures specified in units of ~ω/kB, T =0, 0.1, . . .,0.9, 1 and∞

Entanglement of Formation for two bosonic modes

Gaussian Protocol Optimal Protocols

High energies:

Gaussian protocol optimal

Low energies:

Outperformed by non-Gaussianprotocol:

Rotation to ”Bell state” in 2-dimsubspace

High energies:

Low energies:

High energies:

Low energies:

High energies:

Low energies:

Thank you for your attention.

D. E. Bruschi, M. Perarnau-Llobet, N. Friis, K. V. Hovhannisyan, and M. Huber,

arXiv:1409.4647 [quant-ph].

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