Statistical ensembles without typicality

Transcription

1 Statistical ensembles without typicality Paul Boes, Henrik Wilming, Jens Eisert, Rodrigo Gallego QIP 18

2 A frequent question in thermodynamics Given a (quantum) system of which you only know its average energy e wrt some Hamiltonian H, how will it behave thermodynamically?

3 A frequent question in thermodynamics Given a (quantum) system of which you only know its average energy e wrt some Hamiltonian H, how will it behave thermodynamically? Difficult problem, due to lack of information about underlying state of system.

4 Gibbs trick : Assign canonical ensemble. { :Tr( H) =e} =: (e, H)! e (H) := e (e)h Tr(e (e)h ) 2 (e, H) macrostate microstate

5 Why does this work?

6 Why does this work? Typicality: vast majority of microstates compatible with coarse-grained information behaves like can. ensemble wrt property of interest.

7 Why does this work? Typicality: vast majority of microstates compatible with coarse-grained information behaves like can. ensemble wrt property of interest. E.g.: Canonical Typicality (Popescu et al., Goldstein et al., 06) V µhaar [{ i2h mc D(Tr S( ih ), S) V µhaar [{ i2h mc }] }] apple 0

8 This talk Provide a novel way to motivate the success of Gibbs trick that is independent of any measure or Jaynes-like reasoning. Result: Thermodynamically, any macrostate is operationally equivalent to its corresponding canonical ensemble.

9 Idea

10 Idea 1. Two models of thermodynamic transitions O 1 f (e, H) O 2 f

11 Idea 1. Two models of thermodynamic transitions O 1 f (e, H) O 2 f 2. Compare via reachable state sets (e, H)! f?,! f

12 Idea 1. Two models of thermodynamic transitions O 1 f (e, H) O 2 f 2. Compare via reachable state sets (e, H)! f?,! f 3. Show equivalence between macrostates and canonical ensembles (e, H) e (H)

13 Setting = 1 k B T! %&' )"* ρ (!

14 Setting = 1 k B T! %&' )"* ρ (!

15 Setting = 1 k B T! %&' )"* ρ (!

16 Setting = 1 k B T 1. Bath states: (E i )= e H E i tr(e H Ei )! %&' )"* ρ (!

17 Setting = 1 k B T 1. Bath states: (E i )= e H E i tr(e H Ei )! %&' )"* ρ (!

18 Setting = 1 k B T 1. Bath states: (E i )= e H E i tr(e H Ei ) 2. Evolution:! %&' )"* ρ (! S and E evolve unitarily, such that total average energy preserved

19 Microstate Operations mic.! f if 8, 0 > 0, 9 {H E 1,...,H E N },U s.t. f tr E U NO (H E i) U! i=1 and NO NO! E U (H E i) U! 0 E (H E i) i=1 i=1

20 Setting = 1 k B T! %&' )"* ρ (!

21 Setting = 1 k B T " β " β " β $ " β " β! "! ρ (

22 Setting 1. Bath states: = 1 k B T " β " β " β $ " β (e (H E i),h E i), e (H E i) := E( (H E i)) " β! "! ρ (

23 Setting 1. Bath states: = 1 k B T " β " β " β " β $ (e (H E i),h E i), e (H E i) := E( (H E i)) " β 2. Evolution: S and E evolve unitarily,! "! ρ ( such that total average energy preserved

24 Macrostate Operations (e, H) mac.! f if 8, 0 > 0, 9 {H E 1,...,H E N },U s.t. 8 2 (e, H), (i) 2 (e,h E i) f tr E U NO (i) U! and E U NO (i) U! i=1 0 E NO (i)! i=1 i=1

25 Setting Same: - Fixed bath temperature - Unitary Evolution - Average energy preservation - No initial correlations - Final state is microstate = 1 k B T Different: - Initial states - Constraint on Unitary = 1 k B T " β " β " β $ " β " β! %&' )"* ρ (!! "! ρ (

26 Operational Equivalence (e, H) (e, H) mac.! f, mic.! f

27 Main Result (e, H) e (H), 8 e, H, > 0

28 Motivating the success of Gibbs trick

29 Motivating the success of Gibbs trick The canonical ensemble is the one and only microstate that encodes the possible thermodynamic state transitions of a system whenever one only has partial information about system, bath and evolution.

30 Proof Sketch () ()! "! ' (! (! $%& *'+ ρ, (

31 Proof Sketch Key Lemma 9 f : NO (e, H) l=1 () -mac! f s.t. tr l ( f ) N!1! e (H).! "... (1) (2) (3) (N) U e(h) e(h) e(h) e(h)! γ ( β*'+! $%& (! ' (... ρ,

32 Re-deriving phenomenological TD (e, H) -mac! e (H)

33 Re-deriving phenomenological TD (e, H) -mac! e (H) Work extraction W apple F S, F S := E S T S S cf. Skrzypczyk et al., Nat. Comm. 5, 4185 (2016)

34 Re-deriving phenomenological TD (e, H) -mac! e (H) Work extraction Second Law (e, H) W apple F S, F S := E S T S S -mac! f, F S ( e (H)) F S ( f ). cf. Skrzypczyk et al., Nat. Comm. 5, 4185 (2016)

35 Re-deriving phenomenological TD (e, H) -mac! e (H) Work extraction Second Law (e, H) W apple F S, F S := E S T S S Clausius Inequality -mac! f, F S ( e (H)) F S ( f ). (e, H) -mac! (e, H), Q apple T S cf. Skrzypczyk et al., Nat. Comm. 5, 4185 (2016)

36 Stronger setting: Unitary commutes Exact commutation instead of average preservation. [U, H S + H E ]=0

37 Stronger setting: Unitary commutes Exact commutation instead of average preservation. [U, H S + H E ]=0 Operational equivalence breaks down! H 6= 0, < 1)9e s.t. (e, H) c e(h)

38 Stronger setting: Unitary commutes Exact commutation instead of average preservation. [U, H S + H E ]=0 Operational equivalence breaks down! H 6= 0, < 1)9e s.t. (e, H) c e(h) Can be recovered for special cases, e.g. locally in thermodynamic limit.

39 Generalizable to GGE setting Can generalise all of this to the case of any set of commuting observables (GGEs). (v, Q) v (Q) v(q) := e P j tr(e P j j S (v)qj j S (v)qj ) cf. Guryanova et al./yunger-halpern et al., Nat. Comm. 7, 12049/12051 (2016)

40 Summary Provided novel justification for use of canonical ensembles in (quantum) statistical mechanics by showing operational equivalence wrt possible thermodynamic transitions. Re-derive phenomenological TD without assuming can. ensemble. Operational equivalence breaks down for exactly commuting case. Can be generalised for commuting observables.

41 Thank you arxiv:

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43 S S E vs. E E