Quantum-information thermodynamicsyitpqip2014.ws/presentation/...Second Law of Thermodynamics A lot of “derivations” have been known (Positivity of the relative entropy, Its monotonicity,

Quantum-information thermodynamics

Takahiro Sagawa Department of Basic Science, University of Tokyo

YITP Workshop on Quantum Information Physics (YQIP2014) 4 August 2014, YITP, Kyoto

Collaborators on Information Thermodynamics

• Masahito Ueda (Univ. Tokyo)

• Shoichi Toyabe (LMU Munich)

• Eiro Muneyuki (Chuo Univ.)

• Masaki Sano (Univ. Tokyo)

• Sosuke Ito (Univ. Tokyo)

• Naoto Shiraishi (Univ. Tokyo)

• Sang Wook Kim (Pusan National Univ.)

• Jung Jun Park (National Univ. Singapore)

• Kang-Hwan Kim (KAIST)

• Simone De Liberato (Univ. Paris VII)

• Juan M. R. Parrondo (Univ. Madrid)

• Jordan M. Horowitz (Univ. Massachusetts)

• Jukka Pekola (Aalto Univ.)

• Jonne Koski (Aalto Univ.)

• Ville Maisi (Aalto Univ.)

Outline

• Introduction

• Quantum entropy and information

• Second law with quantum feedback

• Comprehensive framework of quantum-information thermodynamics

• Paradox of Maxwell’s demon

Outline

• Introduction

• Quantum entropy and information

• Second law with quantum feedback

• Comprehensive framework of quantum-information thermodynamics

• Paradox of Maxwell’s demon

Thermodynamics in the Fluctuating World

Thermodynamics of small systems

Second law of thermodynamics

Nonequilibrium thermodynamics

Thermodynamic quantities are fluctuating!

Information Thermodynamics

Information processing at the level of thermal fluctuations

Foundation of the second law of thermodynamics

Application to nanomachines and nanodevices

System Demon

Information

Feedback

Szilard Engine (1929)

Heat bath

T

Initial State Which? Partition

Measurement

Left

Right

Feedback

ln 2

F E TS Free energy: Decrease by feedback Increase

Isothermal, quasi-static expansion

B ln 2k T

Work

Can control physical entropy by using information

L. Szilard, Z. Phys. 53, 840 (1929)

Information Heat Engine

B ln 2k TB ln 2k T

Controller

Small system

Can increase the system’s free energy even if there is no energy flow between the system and the controller

Information

Feedback

Experimental Realizations (yet classical)

• With a colloidal particle Toyabe, TS, Ueda, Muneyuki, & Sano, Nature Physics (2010)

Efficiency: 30% Validation of

• With a single electron Koski, Maisi, TS, & Pekola, PRL (2014)

Efficiency: 75% Validation of

( )W Fe

( ) 1W F Ie

Outline

• Introduction

• Quantum entropy and information

• Second law with quantum feedback

• Comprehensive framework of quantum-information thermodynamics

• Paradox of Maxwell’s demon

Classical Entropy

9

10

1

10

Information content with event : 1

lnkp

k

Shannon entropy: 1

lnk

k k

H pp

Average

Quantum Entropy

lntr)( S

: density operator

Von Neumann entropy:

Characterizes the randomness of the classical mixture in the density operator

ii i qqS ln)( i iiiq

with an orthonormal basis

Von Neumann and Thermodynamic Entropies

The von Neumann entropy is consistent with thermodynamic entropy in the canonical distribution

Canonical distribution:

Free energy: Average energy:

)(

can

EFe

EeTkF trlnB EE cancan

tr

thermcanTSFE Cf.

Thermodynamic entropy

cancanBcanlntr TkFE

E: Hamiltonian

Mutual Information

0 ( )I H M

No information No error

System S Memory M I

( : ) ( ) ( ) ( )I S M H S H M H SM

System S Memory M (measurement device)

Measurement with stochastic errors

0

1

0

1

1

1

Ex. Binary symmetric channel

Correlation between S and M

I

)1ln()1(ln2ln I

Quantum Mutual Information

AB

A B

)()()()( ABBAAB:BA SSSI

0)( AB:BA I

Quantum Measurement

Projection operators { }kP

tr( )k kp P

Projection measurement (error-free)

1k k

k

P Pp

Probability

Post-measurement state

k k

k

A PObservable ,{ }k iMKraus operators

tr( )k kp E

General measurement

k

k

E I

†1ki ki

ik

M Mp

†

k ki ki

i

E M M

Probability

Post-measurement state

POVM：

k : measurement outcome

{ }kE

QC-mutual Information (1)

H. J. Groenewold, Int. J. Theor. Phys. 4, 327 (1971). M. Ozawa, J. Math. Phys. 27, 759 (1986). TS and M. Ueda, PRL 100, 080403 (2008).

k

kk SpSI )()(QC

][tr kkk MMp † : probability of obtaining outcome k

†kk

k

k MMp

1

: post-measurement state with outcome k

: measured density operator

Assumed a single Kraus operator for each outcome

Information flow from Quantum system to Classical outcome by quantum measurement

QC-mutual Information (2)

QC0 I H

No information Error-free & classical

Any POVM element is identity operator

Classical measurement, QCI reduces to the classical mutual information

If the measured state is a pure state: 0QC I

Any POVM element is projection and commutable with measured state

k

kk ppH ln

Outline

• Introduction

• Quantum entropy and information

• Second law with quantum feedback

• Comprehensive framework of quantum-information thermodynamics

• Paradox of Maxwell’s demon

The Second Law of Thermodynamics (without Feedback)

FW ext

With a cycle, 0F holds, and therefore

0ext W (Kelvin’s principle)

Heat bath (temperature T)

Volume V Work

(the equality is achieved in the quasi-static process)

We extract the work by changing external parameters (volume of the gas, frequency of optical tweezers, …).

Quantum Feedback Control

Quantum system Outcome k

Controller

Control protocol can depend on outcome k after measurement

Quantum measurement

Generalized Second Law with Feedback

Engine Heat bath

F

extWWork

IInformation

Feedback

The upper bound of the work extracted by the demon is bounded by the QC-mutual information.

K. Jacobs, PRA 80, 012322 (2009) J. M. Horowitz & J. M. R. Parrondo, EPL 95, 10005 (2011) D. Abreu & U. Seifert, EPL 94, 10001 (2011) J. M. Horowitz & J. M. R. Parrondo, New J. Phys. 13, 123019 (2011) T. Sagawa & M. Ueda, PRE 85, 021104 (2012) M. Bauer, D. Abreu & U. Seifert, J. Phys. A: Math. Theor. 45, 162001 (2012)

The equality can be achieved:

TS and M. Ueda, PRL 100, 080403 (2008)

QCBext TIkFW

Information Heat Engine Conventional heat engine: Heat → Work

ext L

H H

1W T

eQ T

TH TL Heat engine

QH QL

Wext

Heat efficiency

Carnot cycle

Szilard engine

Information heat engine: Mutual information → Work and Free energy

QCBext TIkFW

Outline

• Introduction

• Quantum entropy and information

• Second law with quantum feedback

• Comprehensive framework of quantum-information thermodynamics

• Paradox of Maxwell’s demon

Entropy Production in Nonequilibrium Dynamics

System S

Heat bath B

(inverse temperature β)

Heat Q

Entropy production in the total system: QSS SSB

Change in the von Neumann entropy of S

If the initial and the final states are canonical distributions: FWS SB

Free-energy difference

W Work

Second Law of Thermodynamics

A lot of “derivations” have been known (Positivity of the relative entropy, Its monotonicity, Fluctuation theorem & Jarzynski equality, …)

0SB S

If the initial state is the canonical distribution: FW

Holds true for nonequilibrium initial and final states

Entropy production in the whole universe is nonnegative!

System and Memory

Memory (Controller)

System (Working engine)

Information

Feedback

Consider the role of memory explicitly

Entropy Change in System and Memory

System S Memory M

)()()()( SM:MSMSSM ISSS

:MSMSSM ISSS

QISSS :MSMSSMB

Memory Structure

“0” “1”

Symmetric memory

“0” “1”

Asymmetric memory

)(

MM

k

kHH

Total Hilbert space of memory is the direct sum of subspaces corresponding to outcome k

Measurement Process

Initial state: )0(

MSSM )0(

M is on )0(

MH

CPTP map of measurement process: measE

)(

M'k is on

)(

M

kH (projection postulate)

kp : probability of obtaining outcome k

k

kk

kp )(

M

)(

SSM

meas

SM '')(E'

Post-measurement state: Assume

†kk

k

k MMp

S

)(

S

1' :post-measurement state of S

Entropy Change during Measurement

)'()'()'()'( SM:MSMSSM ISSS

After measurement:

Before measurement: )()()( MSSM SSS

k

kSSI )'()'()'( )(

SSSM:MS Mutual information:

)'( SM:MS

meas

M

meas

S

meas

SM ISSS

Back-action of measurement

Second Law for Measurement

MSM:MS

meas

M

meas

S

meas

SMB )'( QISSS

0meas

SMB S

k

k

k SpSSQS )'()'( )(

SS

meas

SM

meas

M

)'( SM:MS

meas

SM

meas

M ISQS

QC-mutual information

QCM

meas

M IQS

Assume: heat is absorbed only by memory during measurement

Energy Cost for Measurement

TS and M. Ueda, PRL 102, 250602 (2009); 106, 189901(E) (2011).

ΔF=0 (symmetric memory) & H=IQC (classical and error-free measurement)

0M W

QCM

meas

M IQS

QCB

meas

MBMM TIkSTkEW

Same bound as that without measurement

Additional energy cost to obtain information

Information is not free!

Feedback Process

CPTP map of feedback process: k

)(

M

)( fb

S

fb PEE kk

Projection super-operator onto )(

M

kH

Use only classical outcome for feedback

Post-feedback state:

k

kk

kp )(

M

)(

SSM

fb

SM ''')'(E''

Assume

Unchanged

]'[E'' )(

S

)( fb

S

)(

S

kkk

Entropy Change during Feedback

)'()'()'()'( SM:MSMSSM ISSS Before feedback:

)''()'( SM:MSSM:MS

fb

S

fb

SM IISS

k

kSSI )'()'()'( )(

SSSM:MS

After feedback: )''()'()''()''( SM:MSMSSM ISSS

Unchanged

Second law for Feedback

0fb

SMB S )''()'( SM:MSSM:MSS

fb

S IIQS

)'( SM:MSS

fb

S IQS

SSM:MSSM:MS

fb

S

fb

SMB )''()'( QIISS

Entropy decrease by feedback

Entire Entropy Change in Engine

)'( SM:MSS

fb

S IQS By adding to the both-hand sides of meas

SS

QCSS IQS

)'( SM:MS

meas

SSS ISQS

During measurement and feedback,

QCBSext TIkFW

QC-mutual information

Generalized Second Law: Summary

Memory M Heat bath B System S Heat bath B

Measurement and feedback

QCSS IQS QCMM IQS

“Duality” between Measurement and Feedback

Time-reversal transformation

Swap system and memory

Measurement becomes feedback (and vice versa)

S M

Feedback

Measurement

Time Correlation

Outline

• Introduction

• Quantum entropy and information

• Second law with quantum feedback

• Comprehensive framework of quantum-information thermodynamics

• Paradox of Maxwell’s demon

What compensates for the entropy decrease here?

Problem

Memory Heat bath System Heat bath

2lnSS QS For Szilard engine,

Conventional Arguments

Erasure process! (From Landauer principle) Bennett

& Landauer

Measurement process!

Brillouin

Widely accepted since 1980’s…

Total Entropy Production

If the quantum mutual information is taken into account, the total entropy production is always nonnegative for each process of measurement or feedback.

0:MSMS QISS

0SMB S

Generalized second laws have been derived from the second law for the total system:

What compensates for the entropy decrease here?

Revisit the Problem

Memory Heat bath System Heat bath

2lnSS QS For Szilard engine,

The quantum mutual information compensates for it.

02ln2lnSSSMB IQSS

Resolution of the “Paradox”

• Maxwell’s demon is consistent with the second law for measurement and feedback processes individually

– The quantum mutual information is the key

• We don’t need the Landauer principle to understanding the consistency

Summary

Unified theory of quantum-information thermodynamics

Minimal energy cost for quantum information processing

Paradox of Maxwell’s demon

Thank you for your attentions!

Theory: T. Sagawa & M. Ueda, Phys. Rev. Lett. 100, 080403 (2008) . T. Sagawa & M. Ueda, Phys. Rev. Lett. 102, 250602 (2009); 106, 189901(E) (2011). T. Sagawa & M. Ueda, Phys. Rev. Lett. 109, 180602 (2012). T. Sagawa & M. Ueda, New Journal of Physics 15, 125012 (2013). Experiment: S. Toyabe, T.Sagawa, M. Ueda, E. Muneyuki, & M. Sano, Nature Physics 6, 988 (2010). J. V. Koski, V. F. Maisi, T. Sagawa, & J. P. Pekola, Phys. Rev. Lett. 113, 030601 (2014). Review: J. M. R. Parrondo, J. M. Horowitz, & T. Sagawa, Nature Physics, Coming soon!