On the Rate of Convergence in the Quantum Central Limit Theorem

Transcription

1 On the Rate of onvergence in the Quantum entra Limit Theorem M. ramer Um University on work with F.G.S.L. Brandão Microsoft Research and University oege London M. Guta University of Nottingham

2 NX The Rate of onvergence in the entra Limit Theorem X = i=1 X i X i : weaky correated entra Limit Theorem: [X appe x] = F (x) N!1! G(x) = 1 p 2 2 Z x dy e 1 (y µ)2 2 2 µ = hxi, 2 = (X µ) 2

3 NX The Rate of onvergence in the entra Limit Theorem X = i=1 X i X i : weaky correated entra Limit Theorem: [X appe x] = F (x) N!1! G(x) = 1 p 2 2 Z x dy e 1 (y µ)2 2 2 Berry Esseen: sup x F (x) G(x) appe p N µ = hxi, 2 = (X µ) 2

4 The Rate of onvergence in the Quantum entra Limit Theorem ˆX = X i2 ˆX i = X k x k kihk oca  L ˆB ˆ% appe N z e L/ ˆX i : hâ ˆBi hâih ˆBi kâkk ˆBk Quantum entra Limit Theorem: = {1,...,n} d,n= n d X x k appex hk ˆ% ki = F (x) N!1! G(x) = 1 p 2 2 Z x dy e 1 (y µ)2 2 2 Goderis, Vets (1989); Hartmann, Maher, Hess (2004) Berry Esseen: sup x F (x) G(x) appe og2d (N) p N ramer, Brandão, Guta, in prep. (2015) µ = h ˆXi, 2 = ( ˆX µ) 2

5 The Rate of onvergence in the Quantum entra Limit Theorem ˆX = X i2 ˆX i = X k x k kihk oca  L ˆB ˆ% appe N z e L/ ˆX i : hâ ˆBi hâih ˆBi kâkk ˆBk Quantum entra Limit Theorem: = {1,...,n} d,n= n d X x k appex hk ˆ% ki Z x 1 = F (x) N!1 (y µ)2 reation! G(x) to = density p of states: dy e 2 2 ˆ% / 2 2 k : E 1 E<E k appe E Goderis, Vets (1989); Hartmann, Maher, Hess (2004) Berry Esseen: sup x F (x) G(x) appe / F (E) F (E E) og2d (N) p N ramer, Brandão, Guta, in prep. (2015) µ = h ˆXi, 2 = ( ˆX µ) 2

6 The Rate of onvergence in the Quantum entra Limit Theorem: Proof Idea main ingredient (aso for (quantum) centra imit): sup x F (x) G(x) appe 1 Z T T + dt (t) e t2 /2 0 t Esseen (1945)

7 The Rate of onvergence in the Quantum entra Limit Theorem: Proof Idea main ingredient (aso for (quantum) centra imit): sup x F (x) G(x) appe 1 Z T T + dt (t) e t2 /2 0 t Esseen (1945) need to bound (t) e t2 /2 (t) = e i ˆXt : characteristic function :

8 The Rate of onvergence in the Quantum entra Limit Theorem: Proof Idea main ingredient (aso for (quantum) centra imit): sup x F (x) G(x) appe 1 Z T T + dt (t) e t2 /2 0 t Esseen (1945) need to bound (t) e t2 /2 (t) = e i ˆXt : characteristic function ˆX = Ĥ : pure state: Loschmidt echo, return probabiity ˆ% = 2 N : Fourier transform of d.o.s

9 The Rate of onvergence in the Quantum entra Limit Theorem: Proof Idea main ingredient (aso for (quantum) centra imit): sup x F (x) G(x) appe 1 Z T T + dt (t) e t2 /2 0 t Esseen (1945) need to bound (t) e t2 /2 (t) = e i ˆXt : characteristic function ˆX = Ĥ : pure state: Loschmidt echo, dynamica phase transitions return probabiity ˆ% = 2 N : Fourier transform of d.o.s

10 The Rate of onvergence in the Quantum entra Limit Theorem: Proof Idea need to bound (t) e t2 /2 set up differentia equation for char. function and bound its derivative cf. Tikhomirov (1980), Sunkodas (1984)

11 The Rate of onvergence in the Quantum entra Limit Theorem: Appication Ĥ = X i2 Ĥ i = X k E k kihk oca A L B ˆ% T : hâ ˆBi hâih ˆBi kâkk ˆBk appe N z e L/ d =1: Araki (1969) d>1, T>T c : Kiesch, Gogoin, Kastoryano, Riera, Eisert (2014) Ĥ i = {1,...,n} d,n= n d canonica state ˆ% T =e Ĥ/T /Z

12 The Rate of onvergence in the Quantum entra Limit Theorem: Appication Ĥ = X i2 Ĥ i = X k E k kihk oca A L B ˆ% T : hâ ˆBi hâih ˆBi kâkk ˆBk appe N z e L/ d =1: Araki (1969) d>1, T>T c : Kiesch, Gogoin, Kastoryano, Riera, Eisert (2014) Ĥ i = {1,...,n} d,n= n d canonica state ˆ% =e Ĥ/T T /Z u(t )= tr[ĥˆ% T ] with energy density ( ) specific heat capacity c(t ( u(t ) = 2 NT 2 N = µ N

13 The Rate of onvergence in the Quantum entra Limit Theorem: Appication Ĥ = X i2 Ĥ i = X k E k kihk oca A L B ˆ% T : hâ ˆBi hâih ˆBi kâkk ˆBk appe N z e L/ d =1: Araki (1969) d>1, T>T c : Kiesch, Gogoin, Kastoryano, Riera, Eisert (2014) Ĥ i = {1,...,n} d,n= n d canonica state ˆ% T =e Ĥ/T /Z ˆ% : state on microcanonica subspace M = ki : E k Nu(T ) appe p N, og2d (N) p N.. 1 quantum Berry Esseen S(ˆ%kˆ% T ). og( M ) S(ˆ%) + og 2d (N)

14 ˆ% T =e Ĥ/T /Z Why Do Systems Thermaize?

15 Why Do Systems Thermaize? ack of knowedge, ignorance Jaynes principe ˆ% T =e Ĥ/T /Z

16 Why Do Systems Thermaize? Kinematics and Dynamics part of a arge (cosed) system ˆ% =tr \ [ˆ%]

17 Why Do Systems Thermaize? Kinematics and Dynamics part of a arge (cosed) system ˆ% =tr \ [ˆ%] e Ĥ/T /Z

18 Why Do Systems Thermaize? Kinematics and Dynamics part of a arge (cosed) system ˆ% =tr \ [ˆ%] tr \ e Ĥ/T /Z

19 Why Do Systems Thermaize? Kinematics and Dynamics part of a arge (cosed) system ˆ% tr \ e Ĥ/T /Z in contact with heat bath ˆ% (0) ˆ% B

20 Why Do Systems Thermaize? Kinematics and Dynamics part of a arge (cosed) system ˆ% tr \ e Ĥ/T /Z in contact with heat bath, unitary evoution e itĥ ˆ% (0) ˆ% B e itĥ

21 Why Do Systems Thermaize? Kinematics and Dynamics part of a arge (cosed) system ˆ% tr \ e Ĥ/T /Z in contact with heat bath, unitary evoution tr \ e itĥ ˆ% (0) ˆ% B e itĥ = ˆ% (t)

22 Why Do Systems Thermaize? Kinematics and Dynamics part of a arge (cosed) system ˆ% tr \ e Ĥ/T /Z in contact with heat bath, unitary evoution tr \ e itĥ ˆ% (0) ˆ% B e itĥ = ˆ% (t) t!1! tr \ e Ĥ/T /Z

23 Why Do Systems Thermaize? Kinematics and Dynamics part of a arge (cosed) system ˆ% tr \ e Ĥ/T /Z quantum quench tr \ e itĥ ˆ% (0) ˆ% B e itĥ = ˆ% (t) t!1! tr \ e Ĥ/T /Z

24 Why Do Systems Thermaize? Kinematics and Dynamics part of a arge (cosed) system ˆ% tr \ e Ĥ/T /Z anonica Typicaity Godstein, Lebowitz, Tumuka, Zanghi, Phys. Rev. Lett. (2006); arxiv:cond-mat/ Entangement and the foundations of statistica mechanics Popescu, Short, Winter, Nature Physics (2006); arxiv:quant-ph/ Thermaization in Nature and on a Quantum omputer Riera, Gogoin, Eisert, Phys. Rev. Lett. (2012); arxiv: Thermaization and anonica Typicaity in Transation-Invariant Quantum Lattice Systems Mueer, Adam, Masanes,Wiebe, arxiv: Equivaence of Statistica Mechanica Ensembes for Non-ritica Quantum Systems Brandão, ramer, arxiv: quantum quench ˆ% (t) t!1! tr \ e Ĥ/T /Z Time-dependence of correation functions foowing a quantum quench aabrese, ardy, Phys. Rev. Lett. (2006); arxiv:cond-mat/ Reaxation in a ompetey Integrabe Many-Body Quantum System Rigo, Dunjko, Yurovsky, Oshanii, Phys. Rev. Lett. (2007); arxiv:cond-mat/ Effect of suddeny turning on interactions in the Luttinger mode azaia,phys. Rev. Lett. (2006); arxiv:cond-mat/ Quenching, Reaxation, and a entra Limit Theorem for Quantum Lattice Systems ramer, Dawson, Eisert, Osborne, Phys. Rev. Lett. (2008); arxiv:cond-mat/ Thermaization and its mechanism for generic isoated quantum systems Rigo, Dunjko, Oshanii, Nature (2008); arxiv: Foundation of Statistica Mechanics under Experimentay Reaistic onditions Reimann, Phys. Rev. Lett. (2008); arxiv: Quantum mechanica evoution towards therma equiibrium Linden, Popescu, Short, Winter, Phys. Rev. E (2009); arxiv:

25 Why Do Systems Thermaize? Kinematics and Dynamics part of a arge (cosed) system ˆ% tr \ e Ĥ/T /Z anonica Typicaity Godstein, Lebowitz, Tumuka, Zanghi, Phys. Rev. Lett. (2006); arxiv:cond-mat/ Entangement and the foundations of statistica mechanics Popescu, Short, Winter, Nature Physics (2006); arxiv:quant-ph/ Thermaization in Nature and on a Quantum omputer Riera, Gogoin, Eisert, Phys. Rev. Lett. (2012); arxiv: Thermaization and anonica Typicaity in Transation-Invariant Quantum Lattice Systems Mueer, Adam, Masanes,Wiebe, arxiv: Equivaence of Statistica Mechanica Ensembes for Non-ritica Quantum Systems Brandão, ramer, arxiv: quantum quench ˆ% (t) t!1! tr \ e Ĥ/T /Z Time-dependence of correation functions foowing a quantum quench aabrese, ardy, Phys. Rev. Lett. (2006); arxiv:cond-mat/ Reaxation in a ompetey Integrabe Many-Body Quantum System Rigo, Dunjko, Yurovsky, Oshanii, Phys. Rev. Lett. (2007); arxiv:cond-mat/ Effect of suddeny turning on interactions in the Luttinger mode azaia,phys. Rev. Lett. (2006); arxiv:cond-mat/ Quenching, Reaxation, and a entra Limit Theorem for Quantum Lattice Systems ramer, Dawson, Eisert, Osborne, Phys. Rev. Lett. (2008); arxiv:cond-mat/ Thermaization and its mechanism for generic isoated quantum systems Rigo, Dunjko, Oshanii, Nature (2008); arxiv: Foundation of Statistica Mechanics under Experimentay Reaistic onditions Reimann, Phys. Rev. Lett. (2008); arxiv: Quantum mechanica evoution towards therma equiibrium Linden, Popescu, Short, Winter, Phys. Rev. E (2009); arxiv:

26 Why Do Systems Thermaize? Kinematics and Dynamics part of a arge (cosed) system ˆ% tr \ e Ĥ/T /Z anonica Typicaity Godstein, Lebowitz, Tumuka, Zanghi, Phys. Rev. Lett. (2006); arxiv:cond-mat/ Entangement and the foundations of for statistica random mechanics Popescu, Short, Winter, Nature Physics (2006); arxiv:quant-ph/ Thermaization in Nature and on a Quantum with omputer high probabiity Riera, Gogoin, Eisert, Phys. Rev. Lett. (2012); arxiv: Thermaization and anonica Typicaity in Transation-Invariant Quantum Lattice Systems Mueer, Adam, Masanes,Wiebe, arxiv: therma? Equivaence of Statistica Mechanica Ensembes for Non-ritica Quantum Systems Brandão, ramer, arxiv: gen. can. pricnipe quantum quench ˆ% (t) i 2H R H H B t!1! tr \ e Ĥ/T /Z Time-dependence of correation functions foowing a quantum quench aabrese, ardy, Phys. Rev. Lett. (2006); arxiv:cond-mat/ Reaxation in a ompetey Integrabe Many-Body Quantum System Rigo, Dunjko, Yurovsky, Oshanii, Phys. Rev. Lett. (2007); arxiv:cond-mat/ Effect of suddeny turning on interactions in the Luttinger mode azaia,phys. Rev. Lett. (2006); arxiv:cond-mat/ Quenching, Reaxation, and a entra Limit Theorem for Quantum Lattice Systems ramer, Dawson, Eisert, Osborne, Phys. Rev. Lett. (2008); arxiv:cond-mat/ Thermaization and its mechanism for generic isoated quantum systems Rigo, Dunjko, Oshanii, Nature (2008); arxiv: Foundation of Statistica Mechanics under Experimentay Reaistic onditions Reimann, Phys. Rev. Lett. (2008); arxiv: Quantum mechanica evoution towards therma equiibrium Linden, Popescu, Short, Winter, Phys. Rev. E (2009); arxiv: ˆ% tr \ [ R/d R ]

27 Why Do Systems Thermaize? Kinematics and Dynamics part of a arge (cosed) system ˆ% tr \ e Ĥ/T /Z anonica Typicaity Godstein, Lebowitz, Tumuka, Zanghi, Phys. Rev. Lett. (2006); arxiv:cond-mat/ Entangement and the foundations of for statistica random mechanics Popescu, Short, Winter, Nature Physics (2006); arxiv:quant-ph/ Thermaization in Nature and on a Quantum with omputer high probabiity Riera, Gogoin, Eisert, Phys. Rev. Lett. (2012); arxiv: Thermaization and anonica Typicaity in Transation-Invariant Quantum Lattice Systems Mueer, Adam, Masanes,Wiebe, arxiv: therma? Equivaence of Statistica Mechanica Ensembes for Non-ritica Quantum Systems Brandão, ramer, arxiv: gen. can. pricnipe quantum quench ˆ% (t) i 2H R H H B t!1! tr \ e Ĥ/T /Z Time-dependence of correation functions foowing a quantum quench aabrese, ardy, Phys. Rev. Lett. (2006); arxiv:cond-mat/ Reaxation in a ompetey Integrabe Many-Body Quantum System Rigo, Dunjko, Yurovsky, Oshanii, Phys. Rev. Lett. (2007); arxiv:cond-mat/ Effect of suddeny turning on interactions in the Luttinger mode azaia,phys. Rev. Lett. (2006); arxiv:cond-mat/ ensembe Quenching, Reaxation, and a entra Limit Theorem for Quantum Lattice Systems ramer, Dawson, Eisert, Osborne, Phys. Rev. Lett. (2008); arxiv:cond-mat/ Thermaization and its mechanism for generic isoated quantum systems Rigo, Dunjko, Oshanii, Nature (2008); arxiv: Foundation of Statistica Mechanics under Experimentay Reaistic onditions Reimann, Phys. Rev. Lett. (2008); arxiv: Quantum mechanica evoution towards therma equiibrium Linden, Popescu, Short, Winter, Phys. Rev. E (2009); arxiv: Integrabe non-integrabe ˆ% tr \ [ R/d R ] no thermaization instead: generaized Gibbs equiibrium state, cose to it for most times therma? time scae?

28 Quench: Quasi-Free Bosons Ĥ = P ij ˆb i A ijˆb j + ˆb i B ijˆb j +h.c. oca, t.i. ˆ%(0) 2 H H B sufficienty custering (not necessariy Gaussian) n n A Quantum entra Limit Theorem for Non-Equiibrium Systems: Exact Loca Reaxation of orreated States ramer, Eisert, New J. Phys. (2010) N = n d

29 Quench: Quasi-Free Bosons Ĥ = P ij ˆb i A ijˆb j + ˆb i B ijˆb j +h.c. oca, t.i. ˆ%(0) 2 H H B sufficienty custering (not necessariy Gaussian) kˆ% (t) Ĝ(t)k tr appe for a t 2 [t 1 (,N),t 2 (,N)] Ĝ(t) : Gaussian with same n second moments as ˆ% (t) n A Quantum entra Limit Theorem for Non-Equiibrium Systems: Exact Loca Reaxation of orreated States ramer, Eisert, New J. Phys. (2010) N = n d

30 Quench: Quasi-Free Bosons Ĥ = P ij ˆb i A ijˆb j + ˆb i B ijˆb j +h.c. oca, t.i. ˆ%(0) 2 H H B sufficienty custering (not necessariy Gaussian) kˆ% (t) Ĝ(t)k tr appe for a t 2 [t 1 (,N),t 2 (,N)] Ĝ(t) : Gaussian with same n second moments as ˆ% (t) maximum entropy state equiibration, non-therma: Tegmark, Yeh (1994) n A Quantum entra Limit Theorem for Non-Equiibrium Systems: Exact Loca Reaxation of orreated States ramer, Eisert, New J. Phys. (2010) N = n d

31 Quench: Quasi-Free Bosons, Proof Idea characteristic function (FT of Wigner function, Bochner s theorem) ˆ% (t)( )=tr [ˆ% (t) ˆD( )] ˆD( )= Y i2 e i ˆb i i ˆb i n n A Quantum entra Limit Theorem for Non-Equiibrium Systems: Exact Loca Reaxation of orreated States ramer, Eisert, New J. Phys. (2010) N = n d

32 Quench: Quasi-Free Bosons, Proof Idea characteristic function (FT of Wigner function, Bochner s theorem) ˆ% (t)( )=tr [ˆ% (t) ˆD( )] =tr ˆ%(0) ˆD( (t, )) ˆD( )= Y i2 e i ˆb i i ˆb i n i = X j2 j e ita ij n A Quantum entra Limit Theorem for Non-Equiibrium Systems: Exact Loca Reaxation of orreated States ramer, Eisert, New J. Phys. (2010) N = n d

33 Loca oseness A Lemma for which states ˆ%, ˆ (and which ) is kˆ% ˆ k tr appe? N = n d Equivaence of Statistica Mechanica Ensembes for Non-ritica Quantum Systems Brandão, ramer, arxiv:

34 Loca oseness A Lemma for which states ˆ%, ˆ (and which ) is kˆ% ˆ k tr appe? non-t.i.: [ ] N = n d Equivaence of Statistica Mechanica Ensembes for Non-ritica Quantum Systems Brandão, ramer, arxiv:

35 Loca oseness A Lemma ˆ : hâ ˆBi hâih ˆBi kâkk ˆBk appe N z e L/ A L B for which states ˆ%, (and which ) is kˆ% ˆ k tr appe? N = n d Equivaence of Statistica Mechanica Ensembes for Non-ritica Quantum Systems Brandão, ramer, arxiv:

36 Loca oseness A Lemma ˆ : hâ ˆBi hâih ˆBi kâkk ˆBk appe N z e L/ A L B for which states ˆ%, (and which ) is kˆ% ˆ k tr appe? N = n d for those with S(ˆ%kˆ ) d d. ( 2 N) 1 n(n) Equivaence of Statistica Mechanica Ensembes for Non-ritica Quantum Systems Brandão, ramer, arxiv:

37 Loca oseness A Lemma ˆ : hâ ˆBi hâih ˆBi kâkk ˆBk appe N z e L/ A L B for which states ˆ%, (and which ) is kˆ% ˆ k tr appe? N = n d for those with S(ˆ%kˆ ) d d. ( 2 N) 1 n(n) Quantum Substate Theorem Lemma S 2p (ˆ%kˆ ) appe S 2p max(ˆ%kˆ ) appe S(ˆ%kˆ )+1 + og( 1 Pinsker s inequaity Super-additivity Jain, Radhakrishnan, Sen (2009); Jain, Nayak (2011) 1 ) Equivaence of Statistica Mechanica Ensembes for Non-ritica Quantum Systems Brandão, ramer, arxiv:

38 Loca oseness A Lemma ˆ : hâ ˆBi hâih ˆBi kâkk ˆBk appe N z e L/ A L B for which states ˆ%, (and which ) is kˆ% ˆ k tr appe? N = n d for those with S(ˆ%kˆ ) d d. ( 2 N) 1 n(n) Quantum Substate Theorem Lemma Pinsker s S max (ˆ%kˆ ) appeinequaity S p 8appe max (ˆ%kˆ ) appe + og( 1 appe =2 kˆ ˆ k tr Super-additivity Jain, Radhakrishnan, Sen (2009); Jain, Nayak (2011) Datta, Renner (2009); Brandão, Penio (2010); Brandão, Horodecki (2012) 1 appe ) Equivaence of Statistica Mechanica Ensembes for Non-ritica Quantum Systems Brandão, ramer, arxiv:

39 Loca oseness A Lemma ˆ : hâ ˆBi hâih ˆBi kâkk ˆBk appe N z e L/ A L B for which states ˆ%, (and which ) is kˆ% ˆ k tr appe? N = n d for those with S(ˆ%kˆ ) d d. ( 2 N) 1 n(n) Quantum Substate Theorem Lemma kˆ 1 M Pinsker s inequaity Super-additivity Jain, Radhakrishnan, Sen (2009); Jain, Nayak (2011) Datta, Renner (2009); Brandão, Penio (2010); Brandão, Horodecki (2012) ˆ 1 ˆ 1 kappe P M j=2 cov(â 1 Â j 1, Â j ) Equivaence of Statistica Mechanica Ensembes for Non-ritica Quantum Systems Brandão, ramer, arxiv:

40 Loca oseness A Lemma ˆ : hâ ˆBi hâih ˆBi kâkk ˆBk appe N z e L/ A L B for which states ˆ%, (and which ) is kˆ% ˆ k tr appe? N = n d for those with S(ˆ%kˆ ) d d. ( 2 N) 1 n(n) Quantum Substate Theorem Lemma Pinsker s inequaity kˆ% Jain, Radhakrishnan, Sen (2009); Jain, Nayak (2011) Datta, Renner (2009); Brandão, Penio (2010); Brandão, Horodecki (2012) ˆ k tr appe n(4)s(ˆ%kˆ ) Super-additivity P M j=1 S(ˆ% j kˆ j ) appe S(ˆ%kˆ 1 ˆ M ) Equivaence of Statistica Mechanica Ensembes for Non-ritica Quantum Systems Brandão, ramer, arxiv:

41 Loca Equivaence of Micro- and Macrocanonica Ensembes canonica state ˆ =e Ĥ/T /Z for which states ˆ%, (and which ) is kˆ% ˆ k tr appe?

42 Loca Equivaence of Micro- and Macrocanonica Ensembes canonica state ˆ =e Ĥ/T /Z for which states ˆ%, (and which ) is kˆ% ˆ k tr appe? which states ˆ% are ocay therma?

43 Loca Equivaence of Micro- and Macrocanonica Ensembes canonica state ˆ =e Ĥ/T /Z for which states ˆ%, (and which ) is kˆ% ˆ k tr appe? which states ˆ% are ocay therma? for microcanonica states ˆ% = M M this question goes back to Botzmann and Gibbs previous work: Thermodynamica functions [Lebowitz, Lieb (1969); Lima (1971/72); Touchette (2009)] States [Mueer, Adam, Masanes, Wiebe (2013)] Popescu, Short, Winter (2005); Riera, Gogoin, Eisert (2011) thermodynamica imit, t.i.

44 Loca Equivaence of Micro- and Macrocanonica Ensembes canonica state ˆ =e Ĥ/T /Z for which states ˆ%, (and which ) is kˆ% ˆ k tr appe? which states ˆ% are ocay therma? for microcanonica states ˆ% = M M this question goes back to Botzmann and Gibbs here: Finite size, expicit bounds Not necessariy transationa invariant More genera than microcanonica

45 Loca Equivaence of Micro- and Macrocanonica Ensembes canonica state ˆ =e Ĥ/T /Z for which states ˆ%, (and which ) is kˆ% ˆ k tr appe? which states ˆ% are ocay therma? microcanonica states ˆ% = M M with M = ki : E k Nu(T ) appe p N, og2d (N) p N.. 1 and such that d. ( 2 N) d+1 1 n(n)

46 Loca Equivaence of Micro- and Macrocanonica Ensembes canonica state ˆ =e Ĥ/T /Z for which states ˆ%, (and which ) is kˆ% ˆ k tr appe? which states ˆ% are ocay therma? microcanonica states ˆ% = M M with M = ki : E k Nu(T ) appe p N, og2d (N) p N.. 1 and such that d. ( 2 N) d+1 1 n(n) =0: Eigenstate Thermaization

47 anonica Typicaity canonica state ˆ =e Ĥ/T /Z for which states ˆ%, (and which ) is kˆ% ˆ k tr appe? which states ˆ% are ocay therma? pure states ˆ% drawn from the subspace spanned by : M kˆ%c (m.c.) k tr appe p +2 d / p M 1 2e M Popescu, Short, Winter (2005)

48 anonica Typicaity canonica state ˆ =e Ĥ/T /Z for which states ˆ%, (and which ) is kˆ% ˆ k tr appe? which states ˆ% are ocay therma? pure states ˆ% drawn from the subspace spanned by : M kˆ%c (m.c.) k tr appe p +2 d / p M 1 2e M Popescu, Short, Winter (2005) 1 2 exp exp S(ˆ ) og 2d (N) p N =: p QBE

49 anonica Typicaity canonica state ˆ =e Ĥ/T /Z for which states ˆ%, (and which ) is kˆ% ˆ k tr appe? which states ˆ% are ocay therma? pure states ˆ% drawn from the subspace spanned by : M ˆ,M,, as before with probabiity at east p kˆ% ˆ k tr appe +2 d exp (S(ˆ ) og 2d (N) p N cf. Riera, Gogoin, Eisert (2011); Mueer, Adam, Masanes, Wiebe (2013)

50 Sufficient onditions for Loca Thermaization: Summary canonica state ˆ =e Ĥ/T /Z for which states ˆ% is kˆ% ˆ k tr appe? which states ˆ% are ocay therma? ˆ, as before then those with sma free energy F T (ˆ%). F T (ˆ )+ T 2 ( 2 N) 1 d+1 n(n) in microcanonica subspace with arge entropy F T (ˆ%) =tr[ĥˆ%] TS(ˆ%)

51 Sufficient onditions for Loca Thermaization: Summary canonica state ˆ =e Ĥ/T /Z for which states ˆ% is kˆ% ˆ k tr appe? which states ˆ% are ocay therma? ˆ,M,, as before then those with sma free energy F T (ˆ%). F T (ˆ )+ T 2 ( 2 N) 1 d+1 n(n) or in microcanonica subspace with arge entropy S(ˆ%) og( M ) 2 ( 2 N) 1 d+1 n(n) (in fact, amost a states in this subspace)

52 Loca Thermaization after Quantum Quench ˆ%(t) =e itĥ ˆ% 0 e itĥ Ĥ = P k E k kihk ˆ! = im T!1 1 T Z T 0 dt ˆ%(t)

53 Loca Thermaization after Quantum Quench ˆ%(t) =e itĥ ˆ% 0 e itĥ Ĥ = P k E k kihk ˆ! = im T!1 1 T Z T 0 dt ˆ%(t) = P k hk ˆ% 0 ki kihk non-degen. energy gaps im T!1 1 T Z T 0 dt kˆ% (t) ˆ! k tr appe 2 p tr[ˆ! 2 ] Linden, Popescu, Short, Winter (2008)

54 Loca Thermaization after Quantum Quench ˆ%(t) =e itĥ ˆ% 0 e itĥ Ĥ = P k E k kihk ˆ! = im T!1 1 T Z T 0 dt ˆ%(t) = P k hk ˆ% 0 ki kihk non-degen. energy gaps im T!1 1 T Z T 0 dt kˆ% (t) ˆ! k tr appe 2 p tr[ˆ! 2 ] fraction of times for which kˆ% (t) ˆ! k tr appe is at east 1 2 p tr[ˆ! 2 ]/ Linden, Popescu, Short, Winter (2008)

55 Loca Thermaization after Quantum Quench ˆ%(t) =e itĥ ˆ% 0 e itĥ Ĥ = P k E k kihk ˆ! = im T!1 1 T Z T 0 dt ˆ%(t) = P k hk ˆ% 0 ki kihk non-degen. energy gaps im T!1 1 T Z T 0 dt kˆ% (t) ˆ! k tr appe 2 p tr[ˆ! 2 ] Geometry irreevant fraction of times for which Even goba observabes kˆ% (t) ˆ! k tr appe is at Aso oca quenches east 1 2 p tr[ˆ! 2 ]/ Linden, Popescu, Short, Winter (2008)

56 Loca Thermaization after Quantum Quench ˆ%(t) =e itĥ ˆ% 0 e itĥ Ĥ = P k E k kihk ˆ! = im T!1 1 T Z T 0 dt ˆ%(t) = P k hk ˆ% 0 ki kihk non-degen. energy gaps im T!1 1 T Z T 0 dt kˆ% (t) ˆ! k tr appe 2 p tr[ˆ! 2 ] Purity? fraction of times for which Therma? kˆ% (t) ˆ! k tr appe is at Time scae? east 1 2 p tr[ˆ! 2 ]/ Linden, Popescu, Short, Winter (2008)

57 Loca Thermaization after Quantum Quench: Summary QBE Purity oca Hamitonian, sufficienty weaky correated initia state: tr[ˆ! 2 ]. n2d p (N) N

58 Loca Thermaization after Quantum Quench: Summary QBE Purity oca Hamitonian, sufficienty weaky correated initia state: tr[ˆ! 2 ]. n2d p (N) N integrabe: no thermaization (instead generaized Gibbs ensembe) Thermaization

59 Loca Thermaization after Quantum Quench: Summary QBE Purity oca Hamitonian, sufficienty weaky correated initia state: tr[ˆ! 2 ]. n2d p (N) N integrabe: no thermaization (instead generaized Gibbs ensembe) QBE Thermaization most Hamitonians that are unitariy equivaent to a oca Hamitonian ead to fast thermaization* ramer, Thermaization under randomized oca Hamitonians (2012) *the subsystem spends most of the times in [0,N 1 5d 1 2 ] cose to the maximay mixed state

60 Loca Thermaization after Quantum Quench: Summary QBE Purity oca Hamitonian, sufficienty weaky correated initia state: tr[ˆ! 2 ]. n2d p (N) N integrabe: no thermaization (instead generaized Gibbs ensembe) QBE Thermaization most Hamitonians that are unitariy equivaent to a oca Hamitonian ead to fast thermaization* ramer, Thermaization under randomized oca Hamitonians (2012) trans. inv., thermodynamic imit: entropic condition on initia state impies thermaization Mueer, Adam, Masanes, Wiebe, Thermaization and canonica typicaity in transation-invariant quantum attice systems (2013) QBE non-t.i., finite size *the subsystem spends most of the times in [0,N 1 5d 1 2 ] cose to the maximay mixed state