Emergent Fluctuation Theorem
for Pure Quantum States
Takahiro Sagawa
Department of Applied Physics, The University of Tokyo
16 June 2016, YITP, Kyoto
YKIS2016: Quantum Matter, Spacetime and Information

Kazuya Kaneko
arXiv:1603.07857

Eiki Iyoda

Have been working on…
• Nonequilibrium statistical physics• Quantum information theory

In particular, thermodynamics of information

Maxwell’s demon

Thermodynamics of Information
Information processing at the level of thermal fluctuations
E. Coli chemotaxis

Experimental realizationof Maxwell’s demon:
Toyabe, Sagawa, Ueda,
Muneyuki, Sano,
Nature Physics (2010)Ito & Sagawa,Nature Communications (2015)

Review: J. M. R. Parrondo, J. M. Horowitz, & T. Sagawa, Nature Physics 11, 131-139 (2015).

A related fundamental issue:

How does thermodynamics (and its connection to information)
emerge in purely quantum systems?
Today’s topic!

Outline• Introduction• Review of fluctuation theorem
Our results:• Second law• Fluctuation theorem• Numerical check• SummaryOutline• Introduction• Review of fluctuation theorem
Our results:• Second law• Fluctuation theorem• Numerical check• SummaryOrigin of macroscopic irreversibility
micro (Quantum mechanics)
reversible (unitary)

MACRO (Thermodynamics)
irreversible DS > 0“How does the macroscopic irreversibility
emerge from microscopic dynamics?”→ Fundamental question since Boltzmann

Relaxation in isolated quantum systems
Microscopically reversible unitary dynamics→ Relaxes towards a macroscopic steady state（Recurrence time is very long: almost irreversible!）

Experiment：Ultracold atoms

Y(0) Non-steady pure stateex. 1d Bose-Hubbard, 87RbÛ = exp(-iĤt) ：UnitaryY(t) Macroscopically steadypure stateRigorous proof forarbitrary initial states
Von Neumann, 1929 (arXiv:1003.2133)

S. Trotzky et al., Nature physics 8, 325 (2012)

Info. entropy vs thermo. entropy
Macroscopically irreversible relaxation emerges frommicroscopically reversible unitary dynamics
Information entropy

DS = 0？

Thermodynamic entropy

DSthermo > 0

S(t) = tr [-r̂ (t)ln r̂(t)]Sthermo = kB lnW

von Neumann entropy: invariant underunitary time evolution

Increases underirreversible processes

W : determined by
Hamiltonian

Fundamental GAP between
information/thermodynamics entropy

Iyoda, Kaneko, Sagawa,
arXiv:1603.07857

Our results

For pure states under reversible unitary dynamics, within small errors
 2nd Law

DSS ³ b Q

relates von Neumann entropy
to thermodynamic heat
→ Information-thermodynamics link
 The fluctuation theorem

PF (s )= es
PR (-s )bath B(pure state)system Ss : entropy production

characterizes fundamental symmetry of entropy production→ Thermal fluctuation emerges from quantum fluctuation

Mathematically rigorous proof + Numerical check
Key idea: Lieb-Robinson bound, based on locality of interactions

Outline• Introduction• Review of fluctuation theorem
Our results:• Second law• Fluctuation theorem• Numerical check• SummarySecond law and fluctuation theorem
2nd laws ³0

Entropy production is non-negative on average

Fluctuation theorem

PF (s )
s=e
PR (-s )Universal relation far from equilibrium

Probabilities of Positive/negative entropy productions
Second law as an equality!

Theory（1990’s-）
Dissipative dynamical systems,
Classical Hamiltonian systems,
Classical Markov (ex. Langevin),
Quantum Unitary, Quantum Markov, …
Experiment（2000’s-）
Colloidal particle, Biopolymer,
Single electron, Ion trap, NMR, …

ClassicalQuantum (Ion-trap)

RNA

J. Liphardt et al.,Science 296, 1832 (2002)

A. An et al.,Nat. phys. 11, 193 (2015)

Setup for previous studies
By J. Kurchan, H. Tasaki, C. Jarzynski, …

Total system: system S and bath B
S+B obeys unitary dynamics

bath B(Inv. Temp. b )r̂(t) = Û r̂ (0)Û †, Û = exp(-iĤt)system S

 Initial state of S: arbitrary
 Initial state of B: Canonical→ This assumption effectively breaks time reversal symmetry.

 No initial correlation between S and B.

r̂(0) = r̂S (0) Ä r̂B (0), r̂B (0) = e- b Ĥ B/ ZB

Second law (Clausius inequality)

DSS ³ b Q
heat absorbed by system S
Change in the von Neumann entropyof system S (Information entropy)

bath B(Inv. Temp. b )system S

SS (t) = trS [-r̂S (t)ln r̂S (t)], r̂S (t) = trB [ r̂ (t)]Q = -trB éë(r̂ (t) - r̂ (0))Ĥ B ùû ：heat absorbed by system S

Information entropy and Heat are linked!
（if the initial state of bath B is canonical）

Fluctuation theorem  SS   Q  0 : entropy production on average(non-negative) ：stochastic entropy production (fluctuates)
Let ˆ (t )   ln S (t )   Hˆ B
Projection measurements of ˆ (t ) at initial and final time
Difference of outcomes:

Fluctuation theorem universally characterizes the ratio between
the probabilities of positive/negative entropy productions

PF (s )
s=e
PR (-s )Forward process
Reverse processTimeFluctuation theorem
Another representation with characteristic function(moment generating function)

GF (u) = GR (-u + i)

Fourier transf.

PF (s )= es
PR (-s )GF/R (u )   de iu PF/R ( ) : Fourier transf. ofprobability distribution
Cf. Fluctuation theorem leads to several important relations
Fluctuation theorem

Jarzynski identity

Second law

PF (s ) = PR (-s )es

exp(-s ) =1

s ³0

Integrate

Jensen inequality (convexity)

Also reproduces the Green-Kubo formula in the linear response regime,and its higher order generalization

Second law with pure state bath?
In the conventional argument, the initial canonical distribution of the bath isassumed, which effectively breaks the time-reversal symmetry.
The origin of irreversibility was not fully understood,and thus we should consider pure state baths.
A few previous works (on the second law):→ Assumption of “random waiting time”：similar effect to dephasing(unitary V̂

exp -iĤ tt)t

H. Tasaki, arXiv:0011321 (2000): random time
S. Goldstein, T. Hara, and H. Tasaki, arXiv:1303.6393 (2013)
T. N. Ikeda, N. Sakumichi, A. Polkovnikov, and M. Ueda, Ann. Phys. 354, 338 (2015)

Information-thermodynamics link and the fluctuation theorem
for pure state baths were open problems.

tOutline• Introduction• Review of fluctuation theorem
Our results:• Second law• Fluctuation theorem• Numerical check• SummarySetup: system and bath・ Bath B: quantum many body system on a lattice・ Interaction: local and translational invariant
・ Correlation in B is exponential decaying
・ System S contacts with a part of bath Bbath B(pure state)system S

・ Initial state of B: a typical pure state̂ B   ・ No initial correlation between system and bathˆ (0)  ˆ S (0)  ˆ B

・Temperature of bath B
is define by the temperature of the canonical distribution
whose energy density is equal to the pure stateSetup: time evolution
・ Unitary time evolution:(r̂ (t) = Û r̂ (0)Û †, Û = exp -iĤt)bath B(pure state)system S・ Relaxation after quench:

Hamiltonian of S changes quickly at t = 0
and is time-independent for t  0

Outline• Introduction• Review of fluctuation theorem
Our results:• Second law• Fluctuation theorem• Numerical check• SummarySecond law (Clausius inequality)SS   Q   2ndSS (t )  trS  ˆ S (t ) ln ˆ S (t ) ：von Neumann entropy of system SQ   trB ( ˆ (t )  ˆ (0)) Hˆ B

 ：heat absorbed by system S

e 2nd ：Error term, vanishing in the large bath limit
For any  2nd  0 , for any t, there exists a sufficiently large bath, such that 2nd law holds.

→ Mathematically rigorous

Even though the state of B is pure,
information and thermodynamics are linked!

Key of the proof: typicality
Reduced density operator ofa typical pure state (with respect to the uniform measure in theHilbert space of the microcanonical energy shell) B  trB'    00

is nearly equal to canonical distribution（When B’0 is large, the error is small）

S

B0

B’0

S. Popescu et al., Nature physics (2006)
A. Sugita, RIMS Kokyuroku (2006)
S. Lloyd, Ph.D. Thesis

Almost all pure states arelocally thermal!
H. Tasaki, arXiv:1507.06479,(2015)→ Origin of thermodynamic entropy of B0 is the entanglement entropy

Key of the proof: Lieb-Robinson boundThe velocity of “information propagation”
in B is finite, due to locality of interaction
Effective “light-cone” like structureS

S is not affected by B’0 in the short time regime

B0 B’0∂B0boundary

Lieb-Robinson boundOˆ (t ), Oˆ   C Oˆ  Oˆ
S

B 0S

B 0v / m ：Lieb-Robinson velocityt Lieb-Robinson time

 S  B0  exp dist(S, B0)exp(v t )  1t << t º mdist(S,¶B0 ) / v → small
E. Lieb and D. Robinson, Commun. Math. Phys. 28, 251 (1972)
M. Hastings and T. Koma, Commun. Math. Phys. 265, 781 (2006)

Outline• Introduction• Review of fluctuation theorem
Our results:• Second law• Fluctuation theorem• Numerical check• SummaryFluctuation theorem: setup  SS   Q  0 : entropy production on average(non-negative) ：stochastic entropy production (fluctuates）
Let ˆ (t )   ln S (t )   Hˆ BProjection measurements of ˆ (t ) at initial and final time
Difference of outcomes: Moment generating function
for entropy production（F: Forward, R: Reverse）

ForwardReverse
Time

GF/R (u) = ò ds e PF/R (s )
+¥ius

-¥

Initial state of the reverse process: r̂S (t) Ä r̂B
Same as the initial stateof forward process

Result: Fluctuation theorem
GF (u) -GR (-u +i) £ eFT
Fourier
Transf.

PF ( ) ePR ( )For any  FT  0 , for any time t, there exists a sufficiently large bath, such that…

→ Mathematically rigorous
In addition, [ H S  H B , H I ]  0 is assumed.
If this commutator is not zero but small,a small correction term is needed.

Universal property of thermal fluctuation far from equilibrium
emerges from quantum fluctuation of pure states!

Outline• Introduction• Review of fluctuation theorem
Our results:• Second law• Fluctuation theorem• Numerical check• SummaryNumerical check
Second law   2ndFluctuation theorem

GF (u )  GR (u  i )   FT

Error estimation is mathematically rigorous（For any error and time, 2nd law and FT hold for sufficiently large bath.）→ However, it is not trivial whether the errors are small in realistic situations.
→ We confirm that the second law and FT holdeven with a very small bath (16 sites)

System and Hamiltonian
Hard core bosons with n.n. repulsion on 2-dim square latticeĤ = åei ĉi†ĉi + å(-g ij ) ( ĉi† ĉ j + ĉ†j ĉi ) + å gij ĉi† ĉi ĉ†j ĉ ji, ji, ji
Potential

Repulsion

Hopping

gij = g, g ij = g(i, j Î B)

gij = 0, g ij = g ¢(otherwise)

ei = e ← unit of energy
Bath: X  Y sites, #particles=NY

Bath B(Pure state)

System: 1 site
Initial state:r̂S (0) = 1 1

Method: Exact diagonalization (full)

Xsystem S

Second law (1)LiebRobinsontimes

 ~1t ~1etparameters: e =1, g =1, g ¢ =1, g = 0.1, (X,Y, N) = (4, 4, 5)

Average entropy production is always non-negative→ Second law holds (even beyond the Lieb-Robinson time!)

Second law (2)g ¢ = 0.1g ¢ =10setet
LiebRobinsontime

 ~1BathRabi oscillation betweensystem S and BathSystemparameters: e =1, g =1, g = 0.1, (X,Y, N) = (4, 4, 5)

Fluctuation theorem (1)1.00000000

1.000000

Forward
Reverse1.00000000
1.00000000

1.000000

0.99999999

0.999999

0.99999999

0.999999

0.99999999

0.999999

0.99999999

0.999999

t = 0.01

0.99999999
0.99999998

012

3

456

LiebRobinson
Time1.000000

t = 0.1

0.999999
0.999998

70

1

2

3

45u
GF (u), GR (-u +i)

1.00000
0.99999
0.99998
0.99997

1t~
267

are almost the same in the short timeregime (imaginary part is also the same)

0.999960.99995
0.99994
0.99993
0.99992

t =10.99991→ Deviation comes fromu“bare” quantum fluctuation（Dynamical crossover between
thermal fluctuation and bareparameters: e =1, g =2, g ¢ = 0.01, g = 0.1, (X,Y, N) = (3, 5, 4)
quantum fluctuation）
0.99990

012

3

4

567

Fluctuation theorem (2)1.000000

Forward
Reverse1.000000
1.000000
0.999999
0.999999
0.999999
0.999999t  0 .1

0.999999
0.999998

01Lower ordermoments

23u456nth moment
7Higher ordermomentsnG u , t n cn
 iu  u 0→ Higher order moments deviate faster.
parameters: e =1, g =2, g ¢ = 0.01, g = 0.1, (X,Y, N) = (3, 5, 4)

Fluctuation theorem: error estimation
Integrated error:

2pdG º ò 0 du GF (u) - GR (-u + i)

100

1dG

10-2

10-4

LiebRobinson
Time0.1

g ¢ = 0.01

1t~
2

10-6

10-8

10t2-10

10-12
0.001

0.01

t~
0.1

0.01

1120.1time t

101

100

1000

10000

parameters: e =1, g =2, g = 0.1, (X,Y, N) = (3, 5, 4)

Agrees with t2 dependence predicted by our theory

Outline• Introduction• Review of fluctuation theorem
Our results:• Second law• Fluctuation theorem• Numerical check• SummarySummary

Iyoda, Kaneko, Sagawa,
arXiv:1603.07857

For pure states under reversible unitary dynamics,
 Second law

SS   Q   2nd

B’0

B0S

relates thermodynamic heat and the von Neumann entropy→ Information-thermodynamics link
1.000000
1.000000
1.000000
0.999999
0.999999

 Fluctuation theorem

GF (u) -GR (-u +i) £ eFT

0.999999
0.999999
0.999999
0.999998

012

3

4

Fundamental symmetry of entropy production far from equilibrium→ Emergence of thermal fluctuation from quantum fluctuation

Mathematically rigorous proof＋ Numerical check (Exact disgonalization)
Key idea: Lieb-Robinson bound567

Perspectives
Possible experiments→ Ultracold atoms?

Possible connection to quantum gravity→ Unruh & Hawking radiation?
→ “Fast scrambling” conjecture?

M. Cheneau et al., Nature (2012)

Thank you for your attention!