Second law of thermodynamics with quantum memory

0.13 MB·

by  Li-Hang Ren

Second law ofthermodynamics with quantum memory
Li-Hang Ren1,2 and Heng Fan1,2,3,∗
1Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
2School of PhysicalSciences, UniversityofChineseAcademyofSciences, Beijing100190, China
3Collaborative Innovation Center of Quantum Matter, Beijing 100190, China
Wedesignaheatenginewithmulti-heat-reservoir, ancillarysystemandquantummemory. Wethenderive
aninequalityrelatedwiththesecond lawofthermodynamics, andgiveanew limitationabout theworkgain
fromtheenginebyanalyzingtheentropychangeandquantummutualinformationchangeduringtheprocess.
In addition and remarkably, by considering two measurements and with the help of the entropic uncertainty
relationwithquantummemory,wefindthatthetotalworkgainsfromtheheatengineshouldbelargerthana
7
quantityrelatedwithquantumentanglementbetweentheancillarystateandthequantummemory. Thisresult
1
0 providesalowerboundfortheworkextracted,incontrastwiththeupperboundintheconventionalsecondlaw
2 ofthermodynamics,andthusshedlightfromanoppositedirectionforthefoundationsofthethermodynamics.

n
PACSnumbers:03.67.-a,05.70.-a,89.70.Cf,03.65.Ta
a
J

6 Introduction.—Maxwell’s demon plays an importantrole immersedinathermalreservoirattemperatureT,andanex-
2 in the history of thermodynamics and information theory ternaldemon.Thedemoninsertsapartitionintothemiddleof

[1,2]. ItisfirstproposedbyMaxwellthatapowerfuldemon thebox,thenmeasuresonwhichsidethemoleculeistrapped
]
h mightconductmicroscopicoperationtobreakthesecondlaw andperformsexpansiontoextractworkWext =kBTln2. In
p ofthermodynamics. However,accordingto Landauer’sprin- SZE, the informationthat the molecule is in the left or right
- ciple, manipulation of information will be accompanied by isexploitedtoextractphysicalwork.Boththeoreticalandex-
t
n energyconsumption[3],whichsavesthesecondlaw,seealso perimental studies on the information heat engine, addition-
a [4]. However,withthisdemon,onecanrelaxtherestrictions ally the extension for quantum case, have been performed
u
imposedbythesecondlawontheenergyexchangedbetween [7–20]. Quantum resources for quantum information pro-
q
asystemandsurroundings,andsomenewthermodynamicin- cessing such as quantum entanglement, quantum discord or
[
equalitiesarestudied,see[5]andareview[6].Inconventional quantumcoherencemayprobablybeconvertedtoextractable

1 thermodynamics,thesecondlawgivesW ≤−△F,where work. It is proved that the work gain may result from the
ext
v
W istheextractableworkfromsystemandF = U −TS entanglementbetweensubsystems[11]becauseofdeepcon-
8 ext
is the free energy during the isothermal process. Due to the nectionsbetweenthermodynamicsandthetheoryofentangle-
2
6 Maxwell’sdemon,thisthermodynamicexpressioncanbeex- ment[12, 13]. Also onecan devise a heatenginewhich can
7 tended to a favorable form with discrete quantum feedback bedrivenbypurelyquantuminformationsuchasthequantum
0 control[7,8]: discord[9].Recently,experimentalinvestigationisperformed
. toshowthatquantumdiscordisnecessaryinenergytransport
1
0 Wext ≤−△F +kBTI, (1) inananoscalealuminum-sapphireinterface[14].
7 We know that for a quantum system, a quantum memory
inwhichk istheBoltzmannconstantandI isthequantum-
1 B canbeavailableandquantumentanglementorsomequantum-
classical mutual information describing the mutual informa-
:
v tion of a fixed quantum system and the outcome classi- ness of correlations may play a critical role in the emergent
i quantumphenomena[21], suchasthatintheentropicuncer-
X cal information obtained by a quantum measurement. This
taintyrelationwitha quantummemory[22, 23]generalizing
quantum-classical mutual information is an extension of the
r
Heisenberguncertaintyprinciple[24]. Aswemaynoticethat
a standard quantum mutual information defined originally be-
entanglementandquantumdiscordappeartoberesourcefor
tween two subsystems. Onemay thenobservethatthe max-
work extraction in thermodynamics. It is desirable to con-
imum work that can be extracted may exceed that in con-
structa heatenginewherequantumcorrelationsorentangle-
ventionalthermodynamics,however, the marginalpart is re-
mentappearinvolvingintheprocess.InthisLetter,wedesign
strictedbytermofthequantum-classicalmutualinformation.
Thisinequalitylaysanextensionofthesecondlawofthermo- a heat engine which includes the system S contacted by in-
dependentheatbathswithpossibledifferenttemperatures,the
dynamics.
The above statement shows information can be exploited ancillarysystemAandaquantummemoryB,seeFig.1.This
setupofheatengineissimilarwiththatinRef.[7],butwithan
toextractphysicalwork,whichmaybecalledaninformation
heat engine[9]. Szilard first proposeda physicalmodel, i.e. ancillarysystemAandaquantummemoryB. Withthehelp
ofthequantummemory,wethencancharacterizetheroleof
the so-called Szilard engine (SZE), to realize Maxwell’s de-
quantumcorrelationinthethermodynamiccircle. Thus,new
mon[10]. ThisSZEinvolvesasingle-moleculegasinabox,
secondlawofthermodynamicinequalitycanbeobtained.
Setupoftheheatengineanditsprocess.—Theheatengine
involvesfourparts:systemS,asetofthermalreservoirsR=
∗Electronicaddress:[email protected] {R1,···Rn} and composite quantum system M consisting
2
R1 R2 R3 Rn canonicaldistribution.Theinitialstatereads
. . .
{ T1 T2 T3 Tn exp(−βH(i)) exp(−β H )
US(1R) S A B ρ(i) = ZS(i) S ⊗ ZR11 R1 ⊗···
exp(−β H )
⊗ n Rn ⊗ρ(i)
Z AB
FIG.1: Setupoftheheatengine. ThesystemScontactswithinde- Rn
pendentheatbathsR1,R2,...,RnattemperaturesT1,T2,...,Tnre- ≡ ρ(i) ⊗ρ(i) (3)
SR AB
spectivelyconstitutingthemulti-heat-reservoirR,thesystemS has
ancillarysystemA,wherewecanmakemeasurement. Particularly, whereβ =1/(k T )isrelatedwithtemperatureT ,andthe
n B n n
thesetup includesaquantum memory B whichispossibly entan- partitionfunctionsareZ(i) = Tr[exp(−βH(i))] andZ =
gledwithancillarystateA. Atstage(ii),aunitaryoperationU(1)is S S Rn
SR Tr[exp(−β H )].
performedgloballyonthesystemandthemulti-heat-reservoirSR. n Rn
Stage(ii).SystemSbeginstointeractwiththesurrounding
toextractwork.Inageneralwayanythermodynamicprocess
betweenthesystemandreservoirscanbeexpressedbyanuni-
R1 R2 R3 Rn
{ T1 T2 T3 . . . Tn taryevolution. WithunitaryoperationU(1) =IAB ⊗US(1R) as
showninFig. 1,theinitialstatebecomes
Uk
SR S A B ρ(1) =U(1)ρ(i)U(1)†, (4)
{
U(2) inwhichIAB istheidentityoperatorforM consistingofan-
cillarystateAandmemoryB. Duringthisprocess,thecom-
positesystemM isleftunchanged.
Instage(iii),thesystemM startstowork,whichplaysthe
k
roleofMaxwell’sdemon.Inordertomakeuseofquantumin-
FIG.2:Processingoftheheatengine.Inthethermodynamicprocess, formationtoextractwork,weletthesystemS interactswith
atstages(iii)and(iv),aunitarytransformationU(2)isperformedon theancillarysystemAsuchthatthereisquantuminformation
system and ancillary state SA, followed by a projective measure- exchange, then we make measurementon A by positive op-
mentonancillarysystemA.Dependingonmeasurementresultsk,a eratorvaluedmeasures(POVMs). Specificallyspeaking,this
feedbackcontrolunitaryoperationUSkR isthenimplementedonthe measurementprocessis performedby Awith rank-1projec-
systemandthemulti-heat-reservoirSR. tor {Πk}. So the measurement process is implemented by
A
performingaunitarytransformationU(2) onSAfollowedby
a projectionmeasurement{Πk : |kihk|} on A. The density
A
matrixisgivenby
ofancillarystateAandquantummemoryB,seeFig. 1. The
totalHamiltonianiswrittenas
ρ(2) = ΠkU(2)ρ(1)U(2)†Πk
A A
Xk
H(t)=HSR(t)+HSinMtAB(t)+HMAB(t), (2) = pk|kiAhk|⊗ρ(B2S)kR (5)
Xk
where HSR(t) = HS(t) + nm=1[HSinRtm(t) + HRm], de- Themeasurementoutcomepk = tr[ΠkAU(2)ρ(1)U(2)†ΠkA] is
scribing the Hamiltonian of tPhe system, reservoirs and their registered by the memory and the post measurementstate is
interaction. During the operating process, system S can ρ(2)k =tr [ΠkU(2)ρ(1)U(2)†Πk/p ].
BSR A A A k
contactR1,R2,...,Rn whichareatrespectivetemperatures Thestage(iv)isthefeedbackcontrol. Itisperformeddis-
T1,T2,...,Tn. By contactingS with R1 atthe startand the cretelydependingonoutcomepk byapplyingcorresponding
endoftheprocess,systemSisinthermodynamicequilibrium operations on the system S and the multi-heat-reservoir R.
in the initial and final state. However, the system may not Feedback control is also a quantum process, the unitary op-
beinthermodynamicequilibriumbetweentheinitialandfinal erator is written as U(3) = I ⊗ |ki hk|⊗Uk . The
B k A SR
state. For convenience, we note the temperature of S in the wholeprocessofstage(iii)andthePfeedbackcontrolofstage
initial and final state as T = T1. At the beginningand last, (iv)areschematicallypresentedinFig. 2. Now,thefinalstate
we assume Hint (t ) = Hint (t ) = 0, H (t ) = H(i) becomes
SRm i SRm f S i S
and H (t ) = H(f). The general process of the engine is
S f S ρ(f) =U(3)ρ(2)U(3)†. (6)
dividedintofourstages,whichissimilarasthatinRef.[7]but
with different operations since we have additional quantum
Inthelastequilibrationprocess,thesystemandheatreser-
systemsAandB.
voirsevolvetoreachthermodynamicequilibrium,whichcan
Stage (i). At time t , the system and reservoirs are in bedescribedbyaunitarytransformation.Asunitarytransfor-
i
thermodynamic equilibrium respectively, that is, they are in mation has no influence on the following calculation. Thus
3
weignorethisprocess,andregardρ(f) asthefinalstate. Al- with△S ≡△S +△S . FromEq.(3)andEq.(7),weknow
A B
thoughthesystemandreservoirsareequilibriumstatesatlast, thatρ(i) andρ(ref)arecanonicaldistribution.Thensubstitut-
SR SR
theymaynotequalthecanonicaldistribution.Inordertoeval- ingtheirspecificexpressionintoinequality(11),weobtain
uatetheenergyofthesystem,weintroducethecanonicaldis-
tributedstateasareferencewhichtakesthestandardform, n T
E(i)−E(f)+ (E(i) −E(f))
S S T Rm Rm
exp(−βH(f)) exp(−β H ) mX=1 m
ρ(ref) = S ⊗ 1 R1 ⊗···
SR ZS(f) ZR1 ≤ FS(i)−FS(f)+kBT[△S−△I], (13)
⊗ exp(−ZβnHRn), (7) where ES(i) = Tr(ρ(i)HS(i)), ER(im) = Tr(ρ(i)HRm), FS(i) =
Rn −k T lnZ(i),E(f) = Tr(ρ(f)H(f)),E(f) = Tr(ρ(f)H )
B S S S Rm Rm
whereZ(f) = Tr[exp(−βH(f))]. Thereferencestatewillbe by comparing the final state to the reference state, also
used to cSorrespond a equilibSrium state which may not be in FS(f) = −kBT lnZS(f). Among them ES(f) −ES(i) ≡ △US
thecanonicaldistributionform. is the change of the internal energy, E(i) − E(f) ≡ Q
Maximumwork extractedfromtheheatengine.—We will is the heat exchange between system anRdmreservoRimr R anmd
m
proceed to calculate the net work gain from the heat engine F(f) −F(i) ≡ △F is the differencein the Helmholtz free
by analyzing the entropy change during the process. The S S S
energyofsystem. Thentheaboveinequalitybecomes
von Neumann entropy of the state ρ is defined as S(ρ) =
−Tr(ρlnρ). Thefollowingdiscussionwillinvolvethetech- n T
niqueofquantuminformation[21]. −△US+ Qm ≤−△FS+kBT[△S−△I]. (14)
T
The difference between states ρ(i) and ρ(1) is a unitary mX=1 m
transformation, so the entropy remains invariant. However,
This inequality is one of the main results in this Letter. Be-
inthethirdstep, measurementisperformed. Thus, weknow
foreinterpretingthis extensionof the second law of thermo-
thatprojectivemeasurementsincreaseentropy,oneobtains
dynamicinequalityinmoredetail,twospecialcaseswillfirst
beillustrated.
S[ρ(i)]≤S[ρ(2)]. (8)
If n = 1, there is only one reservoir with temperature T.
Accordingtoequalities(3)and(5),weknowthatstatesρ(i)is AstheworkextractablefromtheengineisdefinedasWext =
productstate,S[ρ(i)]=S[ρ(SiR)]+S[ρ(Ai)B],alsostateρ(2) can −△US+Q=−(ES(f)−ES(i))+ER(i)−ER(f),thefinalresult
bewrittenasthefollowingform, (14)reducestoasimplecase
S[ρ(2)]=H(p )+ p S[ρ(2)k ]. (9) Wext ≤−△FS +kBT △S−kBT △I. (15)
k k BSR
Xk This inequality is in agreement with the result of Ref. [9]
With the help of the subadditivity of von Neumann entropy, where the quantum-classical mutual information is intro-
S[ρ(2)k ]≤S[ρ(2)k]+S[ρ(2)k],andtheabovefacts,theinitial duced.Hereweusethestandardquantummutualinformation.
BSR SR B Theinequalitymeansthatthe extractedworkcanexceedthe
inequality(8)nowtakestheform,
difference of the free energy which generalizes the conven-
S[ρ(i)]− p S[ρ(2)k]≤H(p )+ p S[ρ(2)k]−S[ρ(i) ] tionalsecondlawofthermodynamics,however,themarginal
SR k SR k k B AB
partisconstrainedbythedifferenceofthechangesforentropy
Xk Xk
(10) andthequantummutualinformationk T(△S−△I).
B
By some calculations, see supplementary material for detail Whenn = 2,theheatenginebecomesananalogueCarnot
[25],wecanfindtheentropychangebetweeninitialstateand cycle. We take two heatbaths for consideration: T > T .
H L
thefinalstateforsystemS andthemulti-heat-reservoirR, After a cycle, we assume that △US = △FS = 0. Because
W =Q +Q ,wefind,
ext H L
S[ρ(i)]−S[ρ(f)] ≤ △S +△S −△I, (11)
SR SR A B
T
L
W ≤ 1− Q +k T (△S−△I) (16)
where△SA = S[ρ(Af)]−S[ρ(Ai)] denotestheentropychange ext (cid:18) TH(cid:19) H B L
forancillarysystemA,and△S denotestheentropychange
B
Theefficiencyoftheheatengineis
ofthequantummemoryB,I denotesquantummutualinfor-
mationdefinedasI(X : Y) = S(ρ )+S(ρ )−S(ρ ),
X Y XY W T k T (△S−△I)
ext L B L
here △I = I(A(2) : B(2))−I(A(i) : B(i)) represents the η = =1− + . (17)
Q T Q
changeofquantummutualinformationintheprocessofheat H H H
engineforcompositesystemM includingbothAandB. IncontrastwiththeefficiencyoftheconventionalCarnotcycle
According to Klein’s inequality S(ρkσ) = tr(ρlnρ) − η =1− TL,theheatenginepresentedherecanexceed
carnot TH
tr(ρlnσ)≥0,wehavetr[ρ(f)lnρ(ref)]≤−S[ρ(f)].There- the conventional Carnot heat engine, but new restrictions is
SR SR SR
fore, stillimposedaspresentedinthelastterm.
Thegeneralncaseheatenginecanbeconsideredassimple
S[ρ(i)]+tr[ρ(f)lnρ(f)can]≤△S−△I (12) extensionforn=1,2,andtheinequalitycanbeconsideredas
SR SR SR
4
thesecondlawofthermodynamicswithquantumcorrelation rewrittenas,
inquantuminformationscience.Wecommentthatthemutual
1
informationcanbedividedintotwopartsasclassicalcorrela- S(ρ(2))+S(ρ(2)′)≥log +S(ρi )+S(ρi ), (21)
tionandquantumdiscord[26],so△I =△J+△δ.Fromthe AB AB 2 c AB B
Eq.(5),wehaveρ(A2B) = kpk|kiAhk|⊗ρ(B2)kwhichisapost- whereρ(2)′indicatesthecorrespondingstatewhenweusethe
AB
measurement density mPatrix, so discord δ(B(2)|A(2)) = 0. measurementbasesof{Πm}.
A
Thusforexamplen=1,Eq.(15)becomes TheextractableworkhasbeenillustratedinEq.(15). Note
that the relation (11) gives△S −△I = S[ρ(2)]−S[ρ(i) ].
Wext ≤−△FS +kBTC, (18) Next we denote Wk as the maximumextraAcBtable worAkBof
max
thefirstenginewithprojector{Πk : |kihk|},andthesimilar
where we use the notation, C = △S −△J +δ(B(i)|A(i)). A
asWm foranothermeasurement.Then
It shows that the initial quantumdiscord can be exploited to max
acquirephysicalwork, in agreementwith the resultsin [14]. Wk = −△F +k T[S(ρ(2))−S(ρ(i) )], (22)
SimilarformcanalsobeobtainedforgeneralnfromEq.(14). max S B AB AB
Lower bound for work gained with different measurement Wmmax = −△FS′ +kBT[S(ρ(A2B)′)−S(ρ(Ai)B)]. (23)
bases.—Theheatengineinthisworkincludesmeasurement
Due to the limitation of the entropic uncertainty relation,
processatthethirdstage.Itiswell-knownthatHeisenberghas
assertedafundamentallimittotheprecisionoftheoutcomes S(ρ(2))+S(ρ(2)′)satisfytheinequality(21). Weputitinto
AB AB
for a pair of incompatible obervables [24]. For a quantum therelationWk +Wm . Thenthefinalresultreads
max max
systemprobablyentangledwithaquantummemory,thereex-
1
iststheentanglement-assistedentropicuncertaintyrelationfor Wk +Wm ≥−△F −△F′+k T[log −S(Ai|Bi)].
max max S S B 2 c
twoincompatiblemeasurements[22],andmoregenerallyfor
(24)
multiplemeasurements[23]. Next,wewillstudytheheaten-
This inequality is also one of our main results. The term
gine,however,byconsideringtwomeasurementsatstage(iii). S(Ai|Bi) appearing on the right-hand side quantifies the
Forconvenience,wejustconsiderthesingle-reservoircase.
amountof entanglementbetween A and B. We remark that
Weemphasizethatwenextconcentrateonthemeasurement thequantityS(Ai|Bi)playsasignificantroleinquantumin-
process. The heat engine will work twice with two sets of
formationscience,seeforexample[27–29]andthereferences
measurementoperators,{Πk : |kihk|}and{Πm : |mihm|}.
A A therein.
We denotethe maximaloverlapof the two sets of projective
Theinequality(24)playsacompletedifferentroleincom-
operatorsas, c = max |hk|mi|2. For state ρi , we have
k,m AB paring with the inequalities of second law of thermodynam-
theentropicuncertaintyrelation,
ics (14-16). Remarkably, the entropic uncertainty which is
a representation of the Heisenberg uncertainty principle im-
1
S(k|B)+S(m|B)≥log +S(A|B), (19) pliesthatthereexistslowerbound,insteadofupperboundas
2 c
shown in (14-16), for the heat engine presented in this Let-
ter. That is to say, the total work gain can be larger than a
in which S(A|B) = S(ρi ) − S(ρi ) is the conditional
AB B bound for two measurement bases. The negative S(Ai|Bi),
entropy for initial state ρi , we would like to point out
AB meaning the existence of entanglement, will result in higher
that S(A|B) can be negativefor entangledinitial state. The
lower bound and let the work extracted larger for two mea-
quantity S(k|B) is the conditionalvon Neumannentropy of
surements. Thisisanewphenomenonwhichisnotobserved
thepost-measurementstateafterperformingthemeasurement
before. Straightforwardly,our results can be generalizedfor
{Πk}onA,
A multiplemeasurements.
Conclusion.—We design a specific heat engine with both
S(k|B)=S[ p |kihk|⊗ρ(i)k]−S(ρi ), (20)
k B B ancillarystateandthequantummemory. Thenewinequality
Xk related with the second law of thermodynamics is obtained.
Forsimplecases,ourresultsextendtheresultsintheisother-
where p = tr(Πkρi Πk) and ρ(i)k =
k A AB A B mal process and the Carnot circle. The changes of entropy
tr (Πkρi Πk)/p . Similarly, S(m|B) can also
A A AB A k and the quantum mutual information lay new limit for the
be defined for measurement {Πm} as follows:
A marginalpartofworkwhichexceedstheconventionalsecond
S(m|B) = S[ mqm|mihm| ⊗ ρ(Bi)m] − S(ρiB), where law of thermodynamics. Surprisingly, if two measurements
q = tr(ΠmρiPΠm) and ρ(i)m = tr (Πmρi Πm)/q . are preformed, we find a new inequality due to the entropic
m A AB A B A A AB A m
We then substitute these equalities into relation (19), the uncertaintyrelation with the assistance of quantummemory,
outcomesshouldsatisfytheuncertaintyrelation. which providesa lower boundfor the work gained from the
By considering the working process of the heat engine, heatengine.Thisresultdescribesanoppositefactofthether-
we know that ρ(2) = Tr ρ(2) = p |ki hk|⊗ ρ(2)k. modynamicsby consideringentanglementin quantuminfor-
AB SR k k A B mationscience. Webelievethatourresultsoffernewcritical
For partial trace, ρ(i)k = tr (ΠkρPi Πk) and ρ(2)k =
B A A AB A B ingredientsforthefoundationsofquantumthermodynamics.
trSR[trA(ΠkAρ(1)′ΠkA)] = trA(ΠkAρiABΠkA). Then we have Acknowledgments: This work was supported by
ρ(i)k = ρ(2)k. Thus the uncertainty relation (19) can be MOST of China (Grants No. 2016YFA0302104 and
B B
5
2016YFA0300600), national natural science foundation of We thank Zheng-An Wang for helping us in drawing the
China NSFC (Grant No. 91536108) and Chinese Academy pictures.
ofSciences(GrantsNo. XDB01010000andXDB21030300).
[1] J.C.Maxwell,TheoryofHeat(Appleton,Lonton,1871). I. SUPPLEMENTARYMATERIAL:SECONDLAWOF
[2] H.S.LeffandA.F.Rex,Maxwell’sdemons2(IOPPublishing, THERMODYNAMICSWITHQUANTUMMEMORY
Bristol,2003).
[3] R.Landauer,IBMJ.Res.Dev.5,183(1961).
Detailedcalculationofthe entropychangebetweeninitial
[4] C.H.Bennett,Int.J.Theor.Phys.21,905(1982).
stateandthefinalstatein theheatengine.—We canextract
[5] S.Lloyd,Phys.Rev.A56,3374(1997).
thefinalstateas
[6] K. Maruyama, F.Nori, and V. Vedral, Rev. Mod. Phys. 81, 1
(2009). ρ(f) =Tr [ρ(f)]= p Uk ρ(2)kUk† (25)
[7] T.SagawaandM.Ueda,Phys.Rev.Lett.100,080403(2008). SR AB k SR SR SR
[8] T.SagawaandM.Ueda,Phys.Rev.Lett.102,250602(2009). Xk
[9] J. J. Park, K.-H. Kim, T. Sagawa and S. W. Kim, Phys. Rev.
Considering the concavity of the von Neumann entropy, we
Lett.111,230402(2013).
have
[10] L.Szilard,Z.Phys.53,840(1929).
[11] K.Funo,Y.WatanabeandM.Ueda,Phys.Rev.A88,052319 S[ρ(f)]≥ p S[ρ(2)k] (26)
(2013). SR k SR
[12] S.Popescu,A.J.Short,andA.Winter,Nat.Phys.2,754(2006). Xk
[13] F.G.S.L.BrandaoandM.B.Plenio,Nat.Phys.4,873(2008). By means of taking partial trace like equation (25), we can
[14] S.Lloyd,V.Chiloyan, Y.J.Hu,S.Huberman, Z.W.Liu,and easilyobtainρ(f),ρ(f)andρ(2),andtheresultsarelistedas,
G.Chen,arXiv:1510.05035. A B AB
[15] S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano,
Nat.Phys.6,988(2010). S[ρ(f)]=S[ρ(2)], S[ρ(f)]=S[ρ(2)], (27)
A A B B
[16] J.M. R.Parrondo, J.M. HorowitzandT. Sagawa, Nat.Phys.
11,131-139(2015).
[17] F.G.S.L.Brandao,M.Horodecki,H.H.Y.Ng,J.Oppenheim, S[ρ(A2B)]=H(pk)+ pkS[ρ(B2)k]. (28)
andS.Wehner,Proc.Natl.Acad.Sci.U.S.A.112,3725(2015). Xk
[18] J.OppenheimandS.Wehner,Science330,1072(2010).
[19] E.HanggiandS.Wehner,Nat.Commun.4,1670(2013). Wethensubstituteequations(26),(27)and(28)intotherela-
[20] L.H.RenandH.Fan,Phys.Rev.A90,052110(2014). tion(10). Itiseasilytopresent
[21] M. A. Nielsen, and I. L. Chuang, Quantum Computation and
QuantumInformation(CambridgeUniversityPress,2000). S[ρ(i)]−S[ρ(f)] ≤ S[ρ(i)]− p S[ρ(2)k]
SR SR SR k SR
[22] M.Berta,M.Christandl,R.Colbeck,J.M.RenesandR.Ren- Xk
ner,Nature6,659-662(2010).
(2)k (i)
≤ H(p )+ p S[ρ ]−S[ρ ]
[23] S.Liu,L.Z.Mu,andH.Fan,Phys.Rev.A91,042133(2015). k k B AB
[24] W.Heisenberg,Z.Phys.43,172(1927). Xk
[25] Somedetailedcalculationsarepresentedinthesupplementary = S[ρ(2)]−S[ρ(i) ]
material. AB AB
≡ △S +△S −△I, (29)
[26] K.Modi,A.Brodutch,H.Cable,T.Paterek,andV.Vedral,Rev. A B
Mod.Phys.84,1655(2012).
[27] M.Horodecki,J.Oppenheim, andA.Winter,Nature436,673 where△SA = S[ρA(f)]−S[ρA(i)]andI denotesquantummu-
(2005). tualinformation,and△I = I(A(2) : B(2))−I(A(i) : B(i))
[28] M.L.HuandH.Fan,Phys.Rev.A87,022314(2013). represents the change of quantum mutual informationin the
[29] A.Streltsov,E.Chitambar,S.Rana,M.N.Bera,A.Winter,and processofheatengine.Thisistheresultinthemaintext.
M.Lewenstein,Phys.Rev.Lett.116,240405(2016). Workgainedfromtheheatenginebyusingentropicuncer-
taintyrelation.—Theentropicuncertaintyrelationiswritten
as,
1
S(K|B)+S(M|B)≥log +S(A|B). (30)
2 c
HereS(·|·)representstheconditionalvonNeumannentropy,
M,K representdifferentmeasurementbases, and c is deter-
minedbytheoverlapbetweenthesetwobases.
For the heat engine studied by us, if there are two sets of
measurement operators, the total work extractable after the
heatengineprocesswillbelimitedduetotheuncertaintyprin-
ciple.
6
Theenginewillworktwice withtwo setsofmeasurement tr (Πmρi Πm)/q . We then substitute these equali-
A A AB A m
operator{Πk : |kihk|} and{Πm : |mihm|}. Therefore,for tiesintorelation(19). Itiseasilytoobtain
A A
stateρi wehavethefollowingrelationduetothelimitation
AB
oftheuncertaintyprinciple, S( p |kihk|⊗ρ(i)k)−S(ρi )
k B B
1 Xk
S(k|B)+S(m|B)≥log2 c +S(A|B), (31) +S( qm|mihm|⊗ρB(i)m)−S(ρiB)
Xm
itniownahlivchonSN(Aeu|Bm)an=neSn(tρroiApBy)f−orSth(eρiBpo)s.tS-m(kea|Bsu)reismtehnetcsotantde,i- ≥ log2 1c +S(ρiAB)−S(ρiB) (34)
S(k|B) = S[ (Πk ⊗I )ρi (Πk ⊗I )]−S(ρi )
A B AB A B B With working process of the heat engine, we find, ρ(2) =
Xk AB
Tr ρ(2) = p |ki hk| ⊗ ρ(2)k. And according to the
= S[ p |kihk|⊗ρ(i)k]−S(ρi ), (32) SR k k A B
k B B conclusionofPthepartialtrace,ρ(i)k = tr (Πkρi Πk)and
Xk B A A AB A
ρ(2)k = tr [tr (Πkρ(1)′Πk)] = tr (Πkρi Πk). Then
B SR A A A A A AB A
where pk = tr(ΠkAρiABΠkA) and ρB(i)k = wehaveρB(i)k = ρ(B2)k. Thustherelation(34)canbesimpli-
trA(ΠkAρiABΠkA)/pk. Similarly, fiedas
S(m|B)=S[ qm|mihm|⊗ρ(Bi)m]−S(ρiB), (33) S(ρ(2))+S(ρ(2)′)≥log 1 +S(ρi )+S(ρi ). (35)
Xm AB AB 2 c AB B
where q = tr(Πmρi Πm) and ρ(i)m = Thisequationistheoneinthemaintext.
m A AB A B