Quantum-information thermodynamicsyitpqip2014.ws/presentation/...Second Law of Thermodynamics A lot...
Transcript of Quantum-information thermodynamicsyitpqip2014.ws/presentation/...Second Law of Thermodynamics A lot...
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Quantum-information thermodynamics
Takahiro Sagawa Department of Basic Science, University of Tokyo
YITP Workshop on Quantum Information Physics (YQIP2014) 4 August 2014, YITP, Kyoto
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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.)
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Outline
• Introduction
• Quantum entropy and information
• Second law with quantum feedback
• Comprehensive framework of quantum-information thermodynamics
• Paradox of Maxwell’s demon
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Outline
• Introduction
• Quantum entropy and information
• Second law with quantum feedback
• Comprehensive framework of quantum-information thermodynamics
• Paradox of Maxwell’s demon
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Thermodynamics in the Fluctuating World
Thermodynamics of small systems
Second law of thermodynamics
Nonequilibrium thermodynamics
Thermodynamic quantities are fluctuating!
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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
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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)
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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
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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
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Outline
• Introduction
• Quantum entropy and information
• Second law with quantum feedback
• Comprehensive framework of quantum-information thermodynamics
• Paradox of Maxwell’s demon
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Classical Entropy
9
10
1
10
Information content with event : 1
lnkp
k
Shannon entropy: 1
lnk
k k
H pp
Average
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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
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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
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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
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Quantum Mutual Information
AB
A B
)()()()( ABBAAB:BA SSSI
0)( AB:BA I
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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
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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
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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
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Outline
• Introduction
• Quantum entropy and information
• Second law with quantum feedback
• Comprehensive framework of quantum-information thermodynamics
• Paradox of Maxwell’s demon
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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, …).
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Quantum Feedback Control
Quantum system Outcome k
Controller
Control protocol can depend on outcome k after measurement
Quantum measurement
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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
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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
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Outline
• Introduction
• Quantum entropy and information
• Second law with quantum feedback
• Comprehensive framework of quantum-information thermodynamics
• Paradox of Maxwell’s demon
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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
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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!
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System and Memory
Memory (Controller)
System (Working engine)
Information
Feedback
Consider the role of memory explicitly
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Entropy Change in System and Memory
System S Memory M
)()()()( SM:MSMSSM ISSS
:MSMSSM ISSS
QISSS :MSMSSMB
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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
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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
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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
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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
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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!
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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
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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
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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
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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
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Generalized Second Law: Summary
Memory M Heat bath B System S Heat bath B
Measurement and feedback
QCSS IQS QCMM IQS
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“Duality” between Measurement and Feedback
Time-reversal transformation
Swap system and memory
Measurement becomes feedback (and vice versa)
S M
Feedback
Measurement
Time Correlation
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Outline
• Introduction
• Quantum entropy and information
• Second law with quantum feedback
• Comprehensive framework of quantum-information thermodynamics
• Paradox of Maxwell’s demon
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What compensates for the entropy decrease here?
Problem
Memory Heat bath System Heat bath
2lnSS QS For Szilard engine,
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Conventional Arguments
Erasure process! (From Landauer principle) Bennett
& Landauer
Measurement process!
Brillouin
Widely accepted since 1980’s…
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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:
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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
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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
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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!