# From fully quantum thermodynamical identities to a second law equality

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1 From fully quantum thermodynamical identities to a second law equality Alvaro Alhambra, Lluis Masanes, Jonathan Oppenheim, Chris Perry Fluctuating States Phys. Rev. X 6, (2016) Fluctuating Work Phys. Rev. X 6, (2016)

2 Information Theory Physics Lord Spekkens

3 majorization theory open systems optimal control Quantum Information Thermodynamics heat devices finite time thermodynamics stochastic thermodynamics foundations of statistical mechanics

4 Thermodynamics is an information theory Maxwell Szilard Landauer Bennett W=kTlog2 R L

5 What do we mean by W=kTlog2? (consider the limit of perfect erasure) A)W=kTlog2 on average, but there will be fluctuations around this value. B)We can achieve perfect erasure. C)By using slightly more work on average, you can sometimes gain work when you erase. D)None of these statements are true. E)This quiz is undecidable.

6 What do we mean by W=kTlog2? (consider the limit of perfect erasure) A)W=kTlog2 on average, but there will be fluctuations around this value. B)We can achieve perfect erasure. C)By using slightly more work on average, you can sometimes gain work when you erase. D)None of these statements are true. E)This quiz is undecidable.

7 What do we mean by W=kTlog2? (consider the limit of perfect erasure) A)W=kTlog2 on average, but there will be fluctuations around this value. B)We can achieve perfect erasure. C)By using slightly more work on average, you can sometimes gain work when you erase. D)None of these statements are true. E)This quiz is undecidable.

8 Fluctuating work in erasure P(s ', w s)=1 s', w P(s ', w s) e s, w β( E s ' E s+w ) =1

9 Main Results QIT strengthening of Stochastic Thermodynamics Generalisations of doubly-stochastic maps, majorisation Second law of thermodynamics as an equality (fine grained free energy) Fully quantum identity Stochastic Thermodynamics Fluctuations of work and of states Proof and quantification of third law of thermodynamics

10 Outline Review of thermodynamics (Macroscopic, QIT) Equalities for work fluctuations Quantum identities Outlook

11 Outline Review of thermodynamics (Macroscopic, QIT) Equalities for work fluctuations Quantum identities Outlook

12 3 laws of thermodynamics 0) If R1 is in equilibrium with R2 and R3 then R2 is in equilibrium with R3 1) de = dq dw (energy conservation) 2) Heat can never pass from a colder body to a warmer body without some other change occuring. Clausius 3) One can never attain T=0 in a finite number of steps

13 The second law Heat can never pass from a colder body to a warmer body without some other change occurring Clausius

14 The second law W Δ F In any cyclic process

15 Free Energy F = <E> TS <W> rev = F(ρ initial) F(ρ final ) ρ initial ρ final iff <W> ΔF

16 Free Energy F = <E> TS <W> rev = F(ρ initial) F(ρ final ) ρ initial ρ final iff <W> ΔF This is just the first order term of an equality!

17 Resource Theories lass of operations In reversible theories and under minor assumptions, relative entropy distance to free states is the unique measure of the resource (Horodecki et. al. 2011) In thermodynamics, the measure is will turn out to be the Gibbs state But thermodynamics can also be irreversible and

18 What is Thermodynamics?? A resource theoretic approach: Γ (ρs, Hs) adding free states ρb, HB work system ρw, Hw energy conserving unitaries U (1st law) [U, Hs + Hw + HB ] = 0 tracing out can allow changing Hamiltonian by adding switch bit translation invariant on W: [U,ΔW]=0 [ΔW,Hw]=i Streater (1995) Janzig et. al. (2000) Horodecki, JO (2011) Skrzypczyk et. al. (2013) Brandao et. al. (2015)

19 Work or in the micro - regime

20 Broadest definition of thermo Includes other paradigms (Brandao et. al. 2011) Hint H(t) implicit battery: arbitrary U, and take W=trHρ-trHUρU Implemented using very crude control (Perry et. al. 2016) c.f. catalytic transformations (Brandao et. al. 2015) What is the cost of a state transformation? (2 law) nd

21 What do we mean by work? No work: JO (2011) Ruch, Mead (1975); Janzig (2000); Horodecki et. al. (2003); Horodecki, Deterministic or worst case work: Dahlsten et al. (2010); Del Rio et. al. (2011); Horodecki, JO (2011); Aaberg (2011); Faist et. al. (2013), Egloff (2015) Average work: Brandao et. al. (2011); Skrzypczyk et. al. (2013); Korzekwa et. al (2015) Fluctuating work: Jarzynski (1997); Crooks (1999); Tasaki (1999)

22 Outline Review of thermodynamics (Macroscopic, QIT) Equalities for work fluctuations Quantum identities Outlook

23 When can we go from ρ to σ? nd (2 law) Many Second Laws

24 H=0 (Noisy Operations), no work majorisation ρ σ iff ρ σ p (1) p(2) p(3)... p(k ) q (k ) k k k Horodecki et. al. 2003

25 (Thermal Operations), deterministic work thermo-majorisation (β-ordering) Horodecki, JO (2011) Ruch, Mead, (75)

26 Fluctuating work f s := E s +Tlog p (s) β( E P(s ', w s) e s' E s+w ) =1 s, w P(s ', w s)=1 s', w e β(f s ' f s+w) =1 F= f s = E TS 2nd law equality Classical derivation: Seifert (2012)

27 Corrections to second law e β(f s ' f s+w) =1 f s ' f s +w 0 N k =1 k f s := E s +TlogP (s) F= f s = E TS Standard 2nd law β k ( f s ' f s + w) 0 k! W Δ F

28 Fluctuating work in erasure β( E P(s ', w s) e s, w Same considerations apply to non-deterministic case s' E s+w ) =1

29 What do we mean by W=kTlog2? (consider the limit of perfect erasure) A)W=kTlog2 on average, but there will be fluctuations around this value. B)We can achieve perfect erasure. C)By using slightly more work on average, you can sometimes gain work when you erase. D)None of these statements are true. E)This quiz is undecidable.

30 no work to fluctuating work P(s s)=1 s' doubly stochastic maps majorisation P(s ' s)=1 s P(s s)=1 s' Gibbs-stochastic maps P(s ' s)e β( E s' E s) thermo-majorisation =1 s fluctuating work P(s s)=1 linear program s' β( E P(s ', w s) e s, w s' E s+w ) =1

31 Thermo-majorisation curves

32 Outline Review of thermodynamics (Macroscopic, QIT) Equalities for work fluctuations Quantum identities Outlook

33 Quantum identity tr W [F H ' +H ΓSW F S W 1 H S +H W ] 1S ρw =1 S F H (ρ):=e β H 2 ρe β H 2 Γ SW =tr B U ρ SWB U P(s ', w s) e β( E s ' E s+w ) =1 For classical states s, w c.f. Aaberg (2016)

34 Quantum identity Masanes, JO (2014)

35 Fully quantum identities Generalised Gibbsstochastic Second Law equality Generalised Jarzynski equality Crooks equation1 (Petz recovery) 1 c.f. Åberg (2016)

36 Quantum identities w/ diagonal input Generalised Gibbsstochastic Second Law equality Generalised Jarzynski equality Crooks equation (Petz recovery) P(s ', w s) e β( E s ' E s+w ) =1 s, w e e β(f s ' f s+w) β(w f s ) =1 =Z ' S c.f. Sagawa, Ueda (2011) Schumacher (2014) Manzano (2015)

37 Outline Review of thermodynamics (Macroscopic, QIT) Equalities for work fluctuations Quantum identities Outlook

38 Outlook and open questions Fluctuations of states Probabilistic transformation (LOCC: Vidal 1999, Jonathan & Plenio 1999), Alhambra et. al. (2014); Renes (2015); Narasimhachar, Gour (2016) Generalised third laws Reeb, Wolf (2013) Thermal machines: Masanes, JO (2014) Fully quantum fluctuations (non-commuting case) Recovery Maps Alhambra et. al. (2015), Åberg (2016), Alhambra et.al. (2016) Embezzlement of work? (Brandao et. al. 2015) Autonomous machines and clocks Horodecki et. al. (2011), Woods et. al. (2016)

39 Probabilistic state transformations ρ ρ σ W ρ'= p*σ+(1-p*)x l V l (ρ)= p( s) 2 W ρ σ W σ ρ s=1 p 2 * c.f. entanglement theory G. Vidal (1999) Jonathan, Plenio (1999)

40 Probabilistic state transformations 2 W ρ σ W σ ρ p 2 *

41 Probabilistic state transformations β E s β E s ' T ( s ' s) e e s T ( s ' s) p( s) p*p (s ' ) s Renes (2015)

42 Quantitative third law (Masanes, JO; to appear in Nat. Comm.) Heat Theorem (Planck 1911): when the temperature of a pure substance approaches absolute zero, its entropy approaches zero Unattainability Principle (Nernst 1912): any thermodynamical process cannot attain absolute zero in a finite number of steps or within a finite time αt T ' 2d+1 t

43 Thermal Machines Like Turing Machines In a finite time, they interact with a finite volume and inject a finite amount of work 1 1/ d t V v 1 t wmax u Bath of volume V has sub-exponential density of states Ω(E)

44 Outlook and open questions Fluctuations of states Probabilistic transformation (LOCC: Vidal 1999, Jonathan & Plenio 1999), Alhambra et. al. (2014); Renes (2015); Narasimhachar, Gour (2016) Generalised third laws Reeb, Wolf (2013) Thermal machines: Masanes, JO (2014) Fully quantum fluctuations (non-commuting case) Recovery Maps Alhambra et. al. (2015), Åberg (2016), Alhambra et.al. (2016) Embezzlement of work? (Brandao et. al. 2015) Autonomous machines and clocks Horodecki et. al. (2011), Woods et. al. (2016)

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