量子情報と熱力学入門 - 京都大学qisc2014.ws/haihu/sagawa...•2008-2011 東京大学上田研究室（博士） •2011-2012 京都大学白眉センター特定助教

量子情報と熱力学入門

沙川貴大東京大学大学院総合文化研究科

若手のための量子情報基礎セミナー 2014年8月9日，京都大学基礎物理学研究所

WHO

• 2002-2006 京都大学理学部（学部） – 卒業研究：量子光学研究室

• 2006-2008 東京工業大学上田研究室（修士）

• 2008-2011 東京大学上田研究室（博士）

• 2011-2012 京都大学白眉センター特定助教 – 受入先：基礎物理学研究所統計動力学研究室

• 2013- 東京大学大学院総合文化研究科准教授

専門分野

• 非平衡統計力学、量子情報

とくに、情報処理過程の非平衡統計力学

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

• Classical information and entropy

• Quantum information and entropy

• Second law with quantum feedback

• Information thermodynamics

• Paradox of Maxwell’s demon

• Summary

Outline

• Introduction






• Summary

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






• Summary

Claude Shannon （1916-2001）

Father of information science

Established information theory in his 1948 papers (Introduced and analyzed the Shannon information and the mutual information)

Standard textbooks of information theory: - T. M. Cover and J. A. Thomas, “Elements of Information Theory” (John Wiley and Sons, New York, 1991). - M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information” (Cambridge University Press, Cambridge, 2000), chapter 11.

C. Shannon, Bell System Technical Journal 27, 379-423 and 623-656 (1948).

Shannon Information (1)

9

10

1

10

Information content with event : 1

lnkp

k

Shannon information: 1

lnk

k k

H pp

Average

Shannon Information (2)

NH ln0

1ip

0kp

for a single i

for ik

Npk /1for all k

)1ln()1(ln ppppH

2ln0 H

Ex. Binary system “0”: probability p

“1”: probability 1-p

k

kk ppH ln

N: the number of k’s

Mutual Information (1)

System S Memory M

Measurement or communication (with stochastic error, in general)

0

1

0

1

1

1

Ex. Binary symmetric channel

( , ) ( | ) ( )p s m p m s p s

( | )p m s : conditional probability characterizing the error

: joint distribution of S and M

( )p s

( )p m

: distribution of the measured state of S

: distribution of the outcome in M


( : ) ( ) ( ) ( )

( , )( , ) ln

( ) ( )sm

I S M H S H M H SM

p s mp s m

p s p m

( , ) ( | ) ( )p s m p m s p s

( | )p m s : conditional probability characterizing the error

: joint distribution of S and M

( )p s

( )p m

: distribution of the measured state of S

: distribution of the outcome in M

sm

mspmspSMH ),(ln),()(s

spspSH )(ln)()( m

mpmpMH )(ln)()(

System S Memory M I


Ex. Binary channel

)(0 MHI

0

1

0

1 1

0

11

01 p

1-p

S M

psp )0( psp 1)1(

0( 0 | 0) 1p m s

0( 1| 0)p m s

1( 1| 1) 1p m s

1( 0 | 1)p m s

0 1( ) ( ) (1 ) ( )I H M pH p H

( ) ln (1 )ln(1 )H x x x x x

I

2/1p

2ln

0 1


0 ( )I H M

No information Error-free

tt 1)ln(Proof of the left inequality:

011),(

)()(1),(

),(

)()(ln),(

smsm msP

mPsPmsP

msP

mPsPmsP

Equality is achieved if )()(),( mPsPmsP

(S and M are independent)


0 ( )I H M


Proof of the right inequality:

0

)|(ln)|()(

),(

)()(ln),()(ln)()(

s m

smm

smPsmPsP

msP

mPsPmsPmPmPIMH

Equality is achieved if, for every s, there is a unique m such that 1)|( smP

Conditional Shannon information )|( SMH

Mutual Information: Summary

0 ( )I H M


System S Memory M I

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

System S Memory M

(measurement device)

Measurement with stochastic errors

Shannon information: Randomness of the system

Mutual information: Correlation between the system and the memory

Outline

• Introduction






• Summary

Quantum State

Density operator: i iiiq

Statistical mixture of state vector with probability iqi

Note: Such a decomposition to state vectors is not unique!

Non-negativity: 0

Unity: 1][tr i iq

Quantum Dynamics

k

kk MM †)(

dk kk IMM †

Kraus representation of nonunitary dynamics:

’s are Kraus operators with kM

][tr)]([tr Trace-preserving (TP):

Completely positive (CP): dΙ is positive with any ancillary system

Any CPTP map has a Kraus representation

(identity)

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

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

Variety of Quantum Information

• Quantum mutual information

• Quantum discord

• Holevo’s chi

• QC-mutual information (Groenewold information)

Bipartite Quantum System

AB

A B

][tr ABBA ][tr ABAB

Quantum Mutual Information (1)

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

0)( AB:BA I )()()( ABBA SSS

(Subadditivity of von Neumann entropy)

Purely classical correlation ⇒ Classical mutual information

xy yyxxxyq AB with orthonormal bases

xy yx

xy

xyqq

qqI ln)( AB:BA


0000AB

Example: 2 qubits

(no correlation) 0)( AB:BA I

111100002

1AB

(classical correlation)

2ln)( AB:BA I

11000011111100002

1AB

(entanglement)

2ln2)( AB:BA I


Data processing inequality

])[EE()( ABBA:BAAB:BA II

AE BE : CPTP maps on individual systems ,

Straightforward consequence of the monotonicity of the quantum relative entropy

Quantum mutual information is decreased by quantum operations on individual systems

Quantum Discord (1)

xxxP xx : orthonormal basis of A

x xx PP ABAB

)()( AB:BAAB:BA

II (Data processing inequality)

0)()()( AB:BAAB:BAABA|B II

Quantum discord:

)(min)( ABA|BABA|B

Quantum Discord (2)

Example: 2 qubits

111100002

1AB

(classical correlation)

0)( ABA|B

11000011111100002

1AB

(entanglement)

2ln)( ABA|B

Quantifying “quantum correlation”

Quantum Discord (3)

Non-zero discord without entanglement

Ollivier & Zurek, PRL 88, 017901 (2001)

zIz

4

1AB 1100

2

1

Holevo’s chi (1)

xxp : classical probability

: density operator x xxp

Holevo’s chi: x xxSpS )()(

A kind of “quantum-classical correlation”:

x xxxxp AB )( AB:BA I

Zero-discord: 0)( ABA|B

ρx’s are simultaneously diagonalizable ⇒ Holevo’s chi reduces to classical mutual information

Holevo’s chi (2)

For any POVM: I

)(][tr),( xqEyxp xy

xy ypxq

yxpyxpI

)()(

),(ln),(( ) tr[ ] ( , )y

x

p y E p x y

yE : POVM

Holevo’s chi gives an upper bound of the accessible classical information x encoded in ρ

Proof. Straightforwardly obtained from the data processing inequality: Map ρ to a diagonal matrix whose diagonal elements are p(y).

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


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


The QC-mutual information gives an upper bound of the accessible classical information x encoded in ρ

F. Buscemi, M. Hayashi, and M.Horodecki, PRL 100, 210504 (2008).

x

xxq )( : an arbitrary decomposition

x are not necessarily orthogonal

A variant (a “dual”) of the Holevo bound Theorem:

QCII

)(][tr),( xqMMkxp xkk † x

kkk kxpMMp ),(][tr †

xk kpxq

kxpkxpI

)(

),(ln),(

Outline

• Introduction






• Summary

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

Generalized Szilard Engine (1)

Error rates ε0, ε1

0

1

0

1 1

0

11

01 p

1-p

Feedback protocol is determined by (v0, v1)

(v0, v1)=(1,1)

ε0 = ε1 = 0

p=1/2

Szilard engine

T. Sagawa & M. Ueda, PRE 85, 021104 (2012)

Generalized Szilard Engine (2)

Work extraction:

Maximization ( ) and ):

Error rates ε0, ε1

0

1

0

1 10

11

01 p

1-p

0 1( ) ( ) (1 ) ( )I H Y pH p H

ext BW k TI

Langevin System (1)

1 2( )dx

xdt

Overdamped Langevin equation with a harmonic potential

B( ) ( ') 2 ( ')t t k T t t

211 2 2( , , ) ( )

2V x x

Gaussian white noise

A feedback protocol was proposed in D. Abreu and U. Seifert, Europhys Lett. 94, 10001 (2011).

Langevin System (2)

211 2 2( , , ) ( )

2V x x

Step 1: Equilibrium with

1 k

2 0

Step 2: Measurement with Gaussian noise:

B /S k T k

N

: variance of the initial equilibrium

: noise intensity

Step 3: Instantaneous switching to

1 ' (1 / )k k S N

2 ' /( )Sy S N

Step 4: Quasi-static expansion to

1 k 2 ' (unchanged)

Langevin System (3)

N

SI 1ln

2

1

Mutual information

ext BW k TI

Work extraction:

Outline

• Introduction






• Summary

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






• Summary

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

– A misleading argument has been accepted for over 30 years

Outline

• Introduction






• Summary

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!

Documents

量子情報と熱力学入門 - 京都大学qisc2014.ws/haihu/sagawa...•2008-2011 東京大学 上田研究室（博士） •2011-2012 京都大学 白眉センター 特定助教

量子情報と熱力学入門 - 京都大学qisc2014.ws/haihu/sagawa...•2008-2011 東京大学上田研究室（博士） •2011-2012 京都大学白眉センター特定助教