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!