Thermodynamics beyond free energy relations or Quantum Quantum Thermodynamics. Matteo Lostaglio Imperial College London

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1 Thermodynamics beyond free energy relations or Quantum Quantum Thermodynamics Matteo Lostaglio Imperial College London

2 David Jennings Kamil Korzekwa This is wind, he is actually slim Terry Rudolph

3 What makes quantum thermodynamics really quantum? Can we develop a general framework to understand coherence?

4 Single-shot Try to maximize average work W

5 Single-shot W Truly work-like work extraction Johan Aberg Nature Communications 4, 1925 (2013)

6 Single-shot W Single-shot thermodynamics

7 Single-shot W Single-shot thermodynamics Classical sector

8 Classical Quantum Thermodynamics system bath catalyst ρ γ B c

9 Classical Quantum Thermodynamics system bath catalyst ρ γ B c U, H = 0 overall energy conservation

10 Classical Quantum Thermodynamics system bath catalyst ρ γ B c U, H = 0 overall energy conservation σ c

11 Classical Quantum Thermodynamics ρ, H S = 0

12 Classical Quantum Thermodynamics ρ, H S = 0 Theory becomes ``classical : 1. Diagonalise everything in energy eigenbasis. 2. Hence, only probabilities of occupying different energies matters. 3. Thermal operations are particular classical stochastic processes.

13 Classical Quantum Thermodynamics ρ, H S = 0 Theory becomes ``classical : 1. Diagonalise everything in energy eigenbasis. 2. Hence, only probabilities of occupying different energies matters. 3. Thermal operations are particular classical stochastic processes. c.f. two-measurement protocol in fluctuation theorems (?)

14 Classical Quantum Thermodynamics If ρ, H S = 0, ρ σ F α ρ F α σ α The second laws of quantum thermodynamics Fernando G.S.L. Brandao, Michał Horodecki, Nelly Huei Ying Ng, Jonathan Oppenheim, Stephanie Wehner PNAS 112, 3275 (2015)

15 Classical Quantum Thermodynamics If ρ, H S = 0 where ρ σ F α ρ F α σ α F α ρ = kt log Z + kts α (ρ γ) S α (ρ γ = α/ α α 1 log Trρα γ 1 α The second laws of quantum thermodynamics Fernando G.S.L. Brandao, Michał Horodecki, Nelly Huei Ying Ng, Jonathan Oppenheim, Stephanie Wehner PNAS 112, 3275 (2015) γ = e βh S Z S

16 Quantum Quantum Thermodynamics σ = ε T ρ = Tr B [U ρ e βh B U ϯ ] Z B U, H = 0 Now ρ, H S 0 (superpositions)

17 Quantum Quantum Thermodynamics 0>?? γ = e βh S Z S 1>

18 Quantum Quantum Thermodynamics 0> γ = e βh S Z S 1>

19 Quantum Quantum Thermodynamics 0> ``classical thermalisation dephasing γ = e βh S Z S 1>

20 Quantum Quantum Thermodynamics 0> No processing of coherence γ = e βh S Z S 1>

21 Free energy is not enough If first law U, H = 0, this is impossible: σ Despite any condition on F being satisfied.

22 What new ideas can we use to incorporate coherence in the description? Description of quantum coherence in thermodynamic processes requires constraints beyond free energy Matteo Lostaglio, David Jennings, Terry Rudolph Nature Communications 6, 6383 (2015) Quantum coherence, time-translation symmetry and thermodynamics Matteo Lostaglio, Kamil Korzekwa, David Jennings, Terry Rudolph Phys. Rev. X 5, (2015)

23 Quantum Quantum Thermodynamics Simple but powerful symmetry consideration: Non-equilibrium thermodynamics ε T ε S Time-translation symmetry

24 Quantum Quantum Thermodynamics Simple but powerful symmetry consideration: ε T ε S ε S e ih St ρe ih St = e ih St ε S ρ e ih St time-translation symmetric maps Extending Noether's theorem by quantifying the asymmetry of quantum states Nature Communications 5, 3821 (2014) Modes of asymmetry: the application of harmonic analysis to symmetric quantum dynamics and quantum reference frames Phys. Rev. A 90, (2014) Iman Marvian, Robert W. Spekkens

25 Quantum Quantum Thermodynamics Simple but powerful symmetry consideration: ε T ε S ε S e ihst ρe ih St = e ihst ε S ρ e ih St time-translation symmetric maps Symmetry that describes how energy and coherence are NOT conserved in the system. Extending Noether's theorem by quantifying the asymmetry of quantum states Nature Communications 5, 3821 (2014) Modes of asymmetry: the application of harmonic analysis to symmetric quantum dynamics and quantum reference frames Phys. Rev. A 90, (2014) Iman Marvian, Robert W. Spekkens

26 Quantum Quantum Thermodynamics Simple but powerful symmetry consideration: ε T ε S ε S e ih St ρe ih St = e ih St ε S ρ e ih St time-translation symmetric maps Symmetry that describes how energy and coherence are NOT conserved in the system. 1. Even if closed, H S, H S 2,, H S k not enough for mixed states 2. Open dynamics

27 Quantum Quantum Thermodynamics Simple but powerful symmetry consideration: ε T ε S ε S e ih St ρe ih St = e ih St ε S ρ e ih St time-translation symmetric maps Symmetry that describes how energy and coherence are NOT conserved in the system. 1. Even if closed, H S, H S 2,, H S k not enough for mixed states 2. Open dynamics THERMODYNAMICS!

28 Quantum Quantum Thermodynamics A α ρ = S α (ρ D ρ ) c.f. F α ρ = kt log Z + kts α (ρ γ) Under any thermal operation: Some consequences A α ρ 0 ``second laws for quantum coherence

29 Quantum Quantum Thermodynamics A α ρ = S α (ρ D ρ ) c.f. F α ρ = kt log Z + kts α (ρ γ) Under any thermal operation: Some consequences A α ρ 0 ``second laws for quantum coherence Macroscopically irrelevant: A lim α ρ N N N = 0

30 A α 0 F α 0

31 Quantum Quantum Thermodynamics Some consequences Standard quantum free energy: S ρ F ρ = H β = F D ρ + 1 β A(ρ) both decrease Work locked

32 Quantum Quantum Thermodynamics Some consequences Standard quantum free energy: S ρ F ρ = H β = F D ρ + 1 β A(ρ) Activation: both decrease F D ρ 2 2F D ρ (strict if ρ, H S 0 ) Work locked c.f. passive states

33 Quantum Quantum Thermodynamics Concrete bounds? If ρ σ σ nm ρ kl k,l E k <E n E k E l =E n E m e β E n E k + ρ kl k,l E k E n E k E l =E n E m

34 Quantum Quantum Thermodynamics Concrete bounds? If ρ σ σ nm ρ kl k,l E k <E n E k E l =E n E m e β E n E k + ρ kl k,l E k E n E k E l =E n E m ρ σ ρ 56 σ 56 ρ 23

35 Quantum Quantum Thermodynamics Concrete bounds? If ρ σ σ nm ρ kl k,l E k <E n E k E l =E n E m e β E n E k + ρ kl k,l E k E n E k E l =E n E m ρ σ ρ 56 σ 23

36 Quantum Quantum Thermodynamics Concrete bounds? If ρ σ σ nm ρ kl k,l E k <E n E k E l =E n E m e β E n E k + ρ kl k,l E k E n E k E l =E n E m ρ σ e β E σ 56 ρ 23

37 Quantum Quantum Thermodynamics Concrete bounds? If ρ σ Kamil: Uououo!! This is explained better in my poster! e β E σ 56 ρ 23

38 Conclusions 1. Coherence in thermodynamics can be understood through symmetry principles 2. Coherence + Thermo > Free energy consideration 3. Bias towards energetic considerations? 4. More to be done!

39 Thank you for your attention! More details: Description of quantum coherence in thermodynamic processes requires constraints beyond free energy Matteo Lostaglio, David Jennings, Terry Rudolph Nature Communications 6, 6383 (2015) Quantum coherence, time-translation symmetry and thermodynamics Matteo Lostaglio, Kamil Korzekwa, David Jennings, Terry Rudolph Phys. Rev. X 5, (2015) Work extraction from coherence Kamil Korzekwa, Matteo Lostaglio, Jonathan Oppenheim, David Jennings Soon on ArXiv!

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41 Classical Quantum Thermodynamics S α (ρ γ = α/ α α 1 α γ 1 α (ρ 1 α 2α γρ 1 α log Tr ρ 0 α 1 2α ) α α > 1 Quantum hypothesis testing and the operational interpretation of the quantum Renyi relative entropies Milan Mosonyi, Tomohiro Ogawa Communications in Mathematical Physics: Volume 334, Issue 3 (2015), Page