Table Of ContentRegularized Boltzmann entropy determines macroscopic adiabatic accessibility
Hiroyasu Tajima1 and Eyuri Wakakuwa2
1Center for Emergent Matter Science (CEMS), RIKEN, Wako, Saitama 351-0198 Japan
2Graduate School of Information Systems, The University of Electro-Communications, Chofu, Japan
How the thermodynamic entropy STD is related to the Boltzmann entropy SB has been one
of the central issues since the beginning of statistical mechanics. Today, it is believed that the
thermodynamicentropySTD isequaltoafunctionS˜B thatisdefinedbyregularizingtheBoltzman
entropyinordertoensureextensivity. However,itisnotknownwhetherS˜B completelydetermines
the possibility of a macroscopic adiabatic transformation in the same way as STD does. In this
paper,byformulatingthepossibilityofamacroscopicadiabatictransformationsintermsof“coarse-
graining”ofquantumoperations,weprovethatS˜B providesanecessaryandsufficientconditionfor
6
thepossibility of a macroscopic adiabatic transformation.
1
0
2
I. INTRODUCTION U, V and N [6–13]. However, the the backward im-
t plication of (3) is still left unproven [3]. Recently, in
c
O Thermodynamics[1,3,4] is oneofthe mostsuccessful the field of quantum information theory [15–24], it has
phenomenologiesinphysics,andithasahugeapplication beensucceededinderivingdetailedthermodynamicrela-
2 from chemical reactions [1] to black holes [2]. The ther- tions, which characterize possibility and impossibility of
modynamic entropy S plays a central role in thermo- quantum state transformations by a set of restricted op-
] TD
h dynamics. It is a function of macroscopic variables such erations. However,these conditions for the possibility of
p asU, V andN,denotingtheinternalenergy,thevolume statetransformationsarerepresentednotonlybymacro-
- and the number of particles, respectively. As stated by scopic parameters, but also by microscopic parameters.
t
n thesecondlawofthermodynamics,thethermodynamical In this sence, their results are interpreted as the “micro-
a entropycompletelydeterminesthemacroscopicadiabatic scopic accessibility.” This is in contrast to (3), which is
u
accessibility, [4] i.e., represented only by macroscopic parameters.
q
In this paper, we propose a coarse-graining approach
[
v2 (U,V,N)≺⇔adST(UD′[,UV,V′,,NN′)]≤STD[U′,V′,N′]. (1) pstoirbotivlriyitdytehsoefabanaecmckewascsararordsycimaonppdilcicsauadtffiiioacnbieainnttic(c3ot)nr,adanintsidfoonsrhmfooawrtittohhnea,tpiS.o˜esB.-,
7
where (U,V,N) (U ,V ,N ) means “an adiabatic the macroscopic adiabatic accessibility. Our results hold
8 ≺ad ′ ′ ′
4 transformation from a state (U,V,N) to another state by only assuming that the limit (2) exists and is a con-
0 (U′,V′,N′) is possible”. vexandstrictlyincreasingfunctionforU,V andN. Our
0 HowthethermodynamicentropyS isrelatedtothe resutls do not need stronger assumptions, e.g, the i.i.d.
TD
. BoltzmannentropyS hasbeenoneofthe centralissues feature.
1 B
0 since the beginning of statistical mechanics. The Boltz- This paper is organized as follows. In the section II,
6 mannentropyS isnotequaltothermodynamicentropy we review basic and well-known concepts of thermody-
B
1 S ingeneral. Forexample,S isnotextensiveingen- namics and statistical mechanics. All contents in this
TD B
v: eral, while STD is extensive. Today, it is believed that section are well-known. In the section III, we define the
i STD is equal to the regularized Boltzmann entropy S˜B, macroscopic adiabatic accessibility, based on our coarse-
X
whichisdefinedintermsofS asfollowsandisextensive graining method. In the section IV, we give the main
B
r by definition [5]: results of the present paper. Finally, in the section V,
a
we prove the main results. Several lemmas used in the
S [UX,VX,NX]
S˜ [U,V,N]:= lim B . (2) section V are proven in Appendix.
B
X X
→∞
However, it is not known whether S˜ completely deter-
B
II. PRELIMINARIES
minesthemacroscopicadiabaticaccessibilityinthesame
wayasS . Thatis,itisnotknownwhetherthefollow-
TD
ing statement holds: In this section, we review basic concepts of thermo-
dynamics and statistical mechanics. All contents in this
(U,V,N) ad (U′,V′,N′) section are well-known. See e.g. [3–6] for more details.
≺
S˜ [U,V,N] S˜ [U ,V ,N ]. (3) Inthermodynamics,anequilibriumstateisrepresented
B B ′ ′ ′
⇔ ≤ byvaluesofasetofmacroscopicphysicalquantitiessuch
Inthe fieldof statisticalmechanics,the forwardimpli- as (U,V,N), where U is the internal energy, V is the
cation of (3) has been provenfor certain formulations of volume of the system, and N is the number of particles.
“adiabatic operations,” by only assuming that the limit In this paper, we consider cases where all these physical
(2) exists and is a convex and increasing function for quantities are extensive, the set includes the internalen-
2
ergy. We represent such a set of physical quantities in with πˆ(X) . Let ∆ denote the set of arrays δ
terms ofvectorsas~a:=(a0,a1,...,aL), where L is a nat- such {tha~a,tδXli}mXX∈X δX = 0. Then ~a has a one-to{-oXn}e
ural number, e.g., L = 2 in the case of (U,V,N). Here, correspondence→w∞ith the followingset ofarraysofmicro-
for the simplicity of writing, we express the internal en- canonical state:
ergy U as a . Since a macroscopic equilibrium state is
0
uniquely determined by values of macroscopic physical {{πˆ~a(X,δX) }X∈X|{δX}X∈X ∈∆}. (7)
quantities, we also represent an equilibrium state by ~a.
The thermodynamical entropy of an equilibrium state is Since δX X ∆ represents the range of “negligible
uniquely determined as a function of~a, which we denote fluctua{tion}” ∈oXf m∈acroscopic physical quantities, any se-
by S [~a]. quenceintheset(7)canberegardedasdescribingstates
TD
The second law of thermodynamics is one of the most that are “macroscopically the same.”
imporant assumptions in thermodynamics. It has sev- The regularized Boltzmann entropy is defined as
eralequivalentformulationssuchastheprincipleofmax-
1
imum work, Clausius inequality and the law of entropy S˜B[~a]:=Xlim X logD~a(X)↓. (8)
increase. Thelawofentropyincreasedeterminestheadi- →∞
abaticaccessibility,i.e., whethera macroscopicstate can (X) (X)
whereD ↓ isthe dimensionofthe Hilbertspace ↓
be transformed to another by an adiabatic operation. It ~a H~a
defined by
is stated as follows:
(X) :=span ψ (X) λ[l] Xa[l],
~a ad~a′ STD[~a] STD[~a′], (4) H~a n| i∈H (cid:12)∃ ≤
≺ ⇔ ≤ (cid:12)
s.t. A(X)[l] ψ =λ[l] ψ for(cid:12) 0 l L , (9)
where ~a ad ~a′ means “an adiabatic transformation | i | i ≤ ≤ o
≺
~a ~a is possible”.
→ad With concretecalculations,ithas been shownthatthere
Letusintroducethestatisticalmechanicalcounterpart existsthelimitS˜ asaconvexandincresingfunctionfor
of the thermodynamic equilibrium ~a. Since we are con- B
each element of~a, in many physical systems, e.g., gases
cerning a macroscopic limit, we describe a physical sys-
ofparticleswithnaturalpotentialsincluding the vander
tem by a Hilbert space (X) depending on a scaling pa-
H Waars potential [5].
rameterX. The macroscopiclimitis definedasthe limit
of X . We assume that X takes values in a set
= N→or∞ = R+. For each X and l = 0, ,L,
III. FORMULATION OF MACROSCOPIC
wXe denote tXhe setofthe Hermitian∈oXperatorson ·(·X·) as ADIABATIC ACCESSIBILITY IN TERMS OF
H
A~(X) := (H(X),A(X),[1],...,A(X),[L]). Then, the micro- COARSE-GRAINING
canonical state corresponding to an equilibrium state ~a
is defined by
In this section, we introduce a general method to for-
mulate possibility of a macroscopic state transforma-
πˆ~a(X,δX) :=Πˆ~a(X,δ)X/D~a(X,δX) , (5) tion,by“coarse-graining”possibilityofmicroscopicstate
transformations. Wethenformulatethemacroscopicadi-
where Πˆ(X) and D(X) are the projection and the di- abatic accessibility, based on a coarse-graining of pos-
~a,δX ~a,δX
(X) sibility of a microscopic state transformation by unital
mensionofthe following ,whichis asubspaceofof
H~a,δX operations.
(X):
Our formulation is based on the following idea:
H
H~a(X,δ)X :=spann|ψi∈H(X) (cid:12)∃λ[l] ∈[X(a[l]−δX), Basic Idea 1 Suppose a microcanonical state π~a(X,δX) is
X(a[l]+δ )) s.t. A(X)[l] ψ(cid:12)(cid:12)=λ[l] ψ for 0 l L . transformed by a quantum operation EX to another mi-
X | i | i ≤ ≤ o crocanonical state π~a(X′,δ)′ . From a macroscopic point of
(6) X
view, we observe that an equilibrium state ~a is trans-
formed to another equilibrium state ~a, for any δ and
Theparameterδ isapositivefunctionofX,whichrep- ′ X
X δ within the range of “macroscopically negligible fluc-
resents the negligible fluctuation of macroscopic quanti- X′
tuations”. Therefore, we could say that an equilibrium
ties. Since we are normalizing macroscopic observables
state ~a can be transformed to another equilibrium state
as (6), it is natural to assume that lim δ =0.
To describe behavior of a system inXt→h∞e mXacroscopic ~a′ if, for any macroscopically negligible δX and a δX′ , a
state π(X) can be transformed to π(X) .
liinmtriotd(uXce→an∞ar)r,ayweofinmtircordoucacenoannicaarlrsatyat{eδsX}πˆX(X∈)X, and ~a,δX ~a′,δX′
{ ~a,δX}X∈X Based on this idea and the concept of an array of mi-
for each ~a and δ . Depending on the choice of
X X crocanonical states introduced in Section II, we propose
{δX}X∈X, there{exi}sts∈mXany arrays {πˆ~a(X,δX) }X∈X corre- adefinitionofpossibilityofamacroscopicstatetransfor-
sponding to one~a. Therefore, a thermodynamic equilib- mation as follows, where denotes a class of quantum
F
rium state ~a does not have a one-to-one correspondence operations (Fig. 1).
3
The set (7) The set (7) IV. MAIN RESULTS
characterized by characterized by
The main results of this paper are summarized as fol-
lows.
Theorem 1 When S˜ [~a] exists and is a convex and
B
strictly increasing function for each element of~a, the fol-
lowing holds for arbitrary ~a and~a:
For an arbitrary element ′
there exist an element
S˜ [~a] S˜ [~a] ~a ~a. (13)
B B ′ ad ′
and a sequence of class CPTP ≤ ⇔ ≺
which satisfy (10) WhenS˜ [~a]doesnotexistandisaconvexandstrictly
B
increasingfunctionforeachelementof~a,Theorem1does
not necessarily hold. Even in that case, the following
FIG. 1: Schematic diagram of Definition 1
theorem holds in general:
Theorem 2 We have
Definition 1 A macroscopic transformation ~a ~a
is possible if, for any δ ∆, there→eFx˜ists′ S˜B[~a]<S˜B[~a′] ~a ad~a′, (14)
X X ⇒ ≺
{δX′ }X∈X ∈∆ and an arr{ay {}EX∈}XX∈∈X such that S˜B[~a]≤S˜B[~a′]⇐~a≺ad~a′, (15)
lim (πˆ(X) ) πˆ(X) =0, (10) where we define
n→∞(cid:13)(cid:13)EX ~a,δX − ~a′,~δX′ (cid:13)(cid:13) 1
and is a CPT(cid:13)P map of class (cid:13)on ( X) for all S˜B[~a]:= sup limsup SB[~a;δX], (16)
X X
X E . Here, ρ σ is the traceFdistaSncHe defined by {δX}X∈X∈∆ X→∞
kaρdi−∈abaσXtkic:=tra12nTsfro|kρrm−−atσi|o.knW~ae→eF˜xp~ar′esisspaosss~aib≺leF.˜ ~a′ that an S˜B[~a]:={δX}sXu∈pX∈∆lXim→i∞nf X1 SB[~a;δX], (17)
where,
Clearly,if~a ~a and~a ~a hold, then~a ~a also
˜ ′ ′ ˜ ′′ ˜ ′′
≺ ≺ ≺
holds. F F F
S [~a;δ ]:=logD(X) . (18)
The above definition providesa formulationof various B X ~a,δX
typesofpossibilityofmacroscopicstatetransformations, Theorem 1 states that S˜ provides a necessary and
depending on . In this paper, we employ unital CPTP B
F sufficient condition for the macroscopic adiabatic acces-
mforampaftoironcsla.ssHFeret,oadeufinnietaploCssPibTiPlitymoafpaidsiadbeafitnicedtraasnsa- sibility in the same way as STD does, as long as S˜B[~a]
existsandisaconvexandstrictlyincreasingfunctionfor
completely positive trace preserving map that maps the
eachelementof~a. Theorem2 providesa similar relation
identity operator to the identity operator, i.e., (ˆ1)=ˆ1.
A unital map does not decrease the von NeumEann en- for the case where S˜B[~a] does not exist as a convex and
strictly increasing function for each element of ~a. The
tropyofanarbitraryquantumstate[25], thatis, we have
macroscopicadiabaticaccessibilityis inthis casecharac-
S( (ρ)) S(ρ), ρ ( ), (11) terizedbytwofunctionsS˜B[~a]andS˜B[~a],notbyasingle
E ≥ ∀ ∈S H
function as in the case of thermodynamic theories.
for all ρ ( ). Because this feature is similar to the Our results do not depend on any microscopic param-
∈ S H
featureoftheadiabatictransformationinthermodynam- eters,includingδ thatwehaveintroducedtodefinethe
X
ics that does not decrease the thermodynamic entropy generalized microcanonical state πˆ . This is in con-
~a,δX
STD, many researches have treated the unital operation trast to previous approaches from quantum information
as a quantum counterpart of the adiabatic transforma- theory [15–24], which treat “microscopic accessibility.”
tion in thermodynamics [13, 21–23]. We emphasize that Theorem 1 only needs to assume
We then formulate the macroscopic adiabatic accessi- that S˜ [~a] exists and is a convex and strictly increas-
B
bility as follows: ing function for each element of ~a, i.e., we do not need
the stronger assumptions including the i.i.d. assump-
Definition 2 An adiabatic transformation ~a ~a
is possible if, for any δ ∆, there→aedxists′ tion. It is shown that S˜B[~a] exists and is a convex and
δ ∆ and an arr{ayX}X∈X ∈ such that increasing function for each element of~a, in many phys-
{ X′ }X∈X ∈ {EX}X∈X ical systems, e.g., gases of particles with natural poten-
tials including the van der Waars potential [5]. In the
(X) (X)
lim (πˆ ) πˆ =0, (12)
X→∞(cid:13)(cid:13)EX ~a,δX − ~a′,~δX′ (cid:13)(cid:13) ruelagrioiznedwhBeorletztmheapnhnaesnettrroapnysiSt˜io[n~a]doisesansotrticotclcyuirn,ctrheeasriengg-
and X is a unital(cid:13)CPTP map on ( (cid:13)X) for all X . functionforeachelementof~a.BTherefore,atleastinsuch
E S H ∈X
Weexpressas~a ad~a′ whenanadiabatic transformation a region, the regularized Boltzmann entropy determines
≺
~a ad~a′ is possible. the macroscopic adiabatic accessibility.
→
4
V. PROOFS Here, f(~a,δ ,X) is a real-valued function which con-
X
verges to 0 at the limit of X .
→∞
A. Proof of Theorem 1
Proof of Theorem 1: The forward implication of
In the proof of Theorem 1, we use the following three Proposition (13) is proved as follows. In general, there
lemmas: exists a unital CPTP map satisfying σ = (ρ), if and
E E
onlyifρandσ satisfiesthemajorizationrelationρ σ
M
≺
Lemma 1 For an arbitrary δ ∆, the following re- defined as
X
{ }∈
lation holds:
m m
Xlim X1 logD~a(X(1)+↓δX) =S˜B[~a] (19) ρ≺M σ ⇐de⇒f Xk=1p↓k ≥Xk=1qk↓ for any m. (26)
→∞
Lemma 2 When the regularized Boltzmann entropy Here, {p↓k}k and {qk↓}k are eigenvalues of ρ and σ, re-
spectively, sorted in decreasing order [24, 26]. Hence it
S˜ [~a] exists for ~a and is a strictly increasing function
B suffices to show that, for any δ ∆, there exists
tfohrateach element of ~a, there exists {δX(0,)~a}X∈X ∈ ∆ such another sequence {δX′ }X∈X ∈{∆Xs}uXc∈hXth∈at we have
πˆ(X) πˆ(X) . (27)
lim 1 logD(X) =S˜ [~a] (20) ~a,δX ≺M ~a′,δX′
X X ~a,δX B
→∞ for any sufficiently large X.
We firstly show that (27) holds when S˜ [~a] < S˜ [~a]
holdsforanarbitrary δ ∆satisfyingδ(0) <δ B B ′
for an arbitrary X. { X}X∈X ∈ X,~a X holds. Let us take arrays {δX} ∈ ∆ and {δX′′} ∈ ∆(a0)
satisfyingδ <δ forarbitraryX. Then,becauseD(X)
X X′′ ~a,δX
Lemma 3 Let us consider the case where the regularized is an increasing function for δX,
Boltzmann entropy S˜ [~a] exists and is a strictly increas-
B 1 1
ing function for each element of~a. Let us take arbitrary logD(X) logD(X) (28)
~a and ~a such that S˜ [~a] = S˜ [~a]. Let us take an ar- X ~a,δX ≤ X ~a,δX′′
′ B B ′
ray δX ∆, and let AX be the following set of real holds. Because of Lemma 2, the righthand-side of the
num{bers}: ∈ above converges to S˜ [~a] at the limit of X . Also,
B
→ ∞
becauseofLemma2,foranarbitraryarray δ ∆(0),
AX :={η|D~a(X(1)−↓δX) ≤D~a(X′(1)↓+η) ≤D~a(X(1)+↓δX)} (21) X1 logD~a(X′,δ)X′ convergesto S˜B[~a′] at the limi{t oX′f}X∈→~a∞′ .
Let 0 be the set of X suchthat AX is not the empty set, Therefore, because of S˜B[~a]<S˜B[~a′],
X
and let be the set of X such that A is the empty set.
oTfhXen,awsefodXlleo1fiwnse: two real valued functXions ηX(+) and ηX(−) X1 logD~a(X,δX) ≤ X1 logD~a(X′,δ)X′ (29)
holds for sufficient large X. Hence, (27) holds when
ηX(+) =(cid:26)η31Xin+fη∈X1AXη+ 23supη∈AXη ((XX ∈∈XX01)) ,(22) S˜BN[~ae]x<t,lS˜eBt[u~as′]shhoowldst.hat(27)holdswhenS˜B[~a]=S˜B[~a′].
ηX(−) =(cid:26) η32Xin−fη∈X1AXη+ 13supη∈AXη ((XX ∈∈XX10)) ,(23) For an arbitrarδyX′ {:δ=X}4m∈a∆x,{|wηXe(+d)|e,fi|ηnX(e−)|}. (30)
where η is thefollowing real valued function of X :
X 1
∈X Clearly, δ > sup η and inf η > δ hold for
ηX :=sup{η|D~a(X′(1)↓+η) <D~a(X(1)−↓δX)}. (24) any X ∈XX′0. Therηe∈foArXe, for any Xη∈∈AXX0, − X′
(X) (X)
Then, η(+) =o(1) and η(−) =o(1) hold. D~a(1+↓δX) <D~a′(1↓+δX′ ) (31)
X X D~a(X′(1)↓−δX′ ) <D~a(X(1)−↓δX) (32)
These lemmas are proven in Appendix A.
hold. The inequalities (31) and (32) also hold for any
Hereafter, we refer to the set of δ ∆ such
X X
htfohoraldtasnδ:X(0a,)r~ab<itrδaXry<{δOX(}1X)∈foXr∈ar∆bi~(at0r)a,rtyh{Xe foa}lslo∆∈wX~a(i0n)∈g. rCelleaatriolyn, XXηX∈∈>XX1η1.,⇒(bNecoDateu~a(Xs(t1eh)−↓aδδtX′Xη)>>>ηηXDX~a(aX′⇒(n1)d↓+Dηη)~a(XX′h(1)>o↓+ldη−)f>δoX′rDah~a(noX(1ld)−a↓rδfbXoir)traaannrdyy
X1 logD~a(X,δX) =S˜B[~a]+f(~a,δX,X). (25) XD~a(X(∈1)+↓XδX1),dboeecsaunsoet eηxissattifsofryiXng∈DX~a(1X(1.))−↓δX) ≤ D~a(X′(1)↓+η) ≤
5
Because of Lemma 3, δ = o(1) also holds. Hence, distance, we have
X′
δ ∆ satisfies D(X) < D(X) , because of D(X) =
{Dsit~a(X′FiX(o1i}n)n+↓a∈δ(Xl1ly)3,)−wbDey~a(psX(rh1)o−o↓vwδeXin)t.h~ga,eTδtXhhbaeatrcekafwonr~aaea′r,,ddδX′(i2aim7b)apthliicocaltdtrsiao.nnsfoofrmP~a,raδoXtpioon- (cid:13)(cid:13)(cid:13)1EDX~a((X′,πˆ)δ~X′a(X,δX) )−πˆ~a(X′1,δ)X′ (cid:13)(cid:13)(cid:13)≥k1EX′ (πˆ~a(X,δX) )−πˆ~a(X′,δ)X′ k
= (cid:12)p (cid:12)+ p
~aaXtar→kb.eiFtarr{qaoδ~armX′y′′}i({sX2δ5∈nX)Xo},tXw∈p∈eoX∆shs~a(a∈i0b′v)le∆esu~(ai0fc)hδastn:hd=at{S˜δδBX′X′[}~aX<]−∈δXSX′˜′B∈f[~oa∆r′].a>Wrb0ei.traFalrsixyo 21DX~ia(=X′,)1δX′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i− D~a(1X′,δ)X′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 2i>DX~a(X′,)δX′ i
(cid:12)p (cid:12)
X1 logD~a(X,δX) − X1 logD~a(X′,δ)X′′ ≥ 12 Xi=1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i−1D~a(X′,δ)X′ e(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Xδs/2
=(S˜B[~a]−S˜B[~a′])+(f(~a,δX,X)−f(~a′,δX′′,X)) ≥ 2D~a(X′,δ)X′ D(X) − D−(X)
=δs γX (33) ~a′,δX′ ~a′,δX′
− 1
= 1 e−Xδs/2 . (39)
for any X, where we defined γX := f(~a,δX,X) 2(cid:16) − (cid:17)
−
f(~a,δ ,X). Hence we have
′ X′′ Here, we defined p (i = 1, ,D(X) ) as eigenvalues of
1 1 i ··· ~a′,δX′
logD(X) logD(X) (πˆ(X) ) on the support of πˆ(X) . Thus we obtain
X ~a,δX − X ~a′,δX′ EX′ ~a,δX ~a′,δX′
1 1 δs
logD(X) logD(X) > (34) 1
≥ X ~a,δX − X ~a′,δX′′ 2 (cid:13)EX(πˆ~a(X,δX) )−πˆ~a(X′,δ)X′ (cid:13)≥ 3 (40)
for any sufficiently large X, which leads to (cid:13) (cid:13)
(cid:13) (cid:13)
for any sufficiently large X and any unital CPTP map
1 e Xδs/2 1 X, which contradicts (12). Therefore, an adiabatic
− E
> > . (35) transformation~a ~a is not possible.
D(X) D(X) D(X) →aq ′
~a′,δX′ ~a′,δX′ ~a,δX
Note that lim γ = 0 follows from
X X B. Proof of Theorem 2
lim f(~a,δ ,X)→=∞0 and lim f(~a,δ ,X) = 0.
X X X ′ X′′
Let →P∞(X) be the projection ont→o∞supp[πˆ(X) ], and
~a′,δ′ ~a′,δ′ We first introduce a generalized adiabatic process.
X X
define a unital CPTP map (X) by
P~a′,δX′ DAnefinaidtiiaobnat3icSutprpaonsseform{δaXti}oXn∈X,(~a{,δX′δ}X∈X )∈ ∆.
P~a(X′,δ)X′ (ρ)=P~a(′X,δ)X′ ρP~a(′X,δ)X′ +(I −P~a(′X,δ)X′ )ρ(I −P~a(′X,δ)X′ ). (o~af′u,{nδitX′a}lXC∈PXT)Pismapposssibleonif t(her(Xe))e{xtihXsat}stXsa∈aXtissefites{→EXa}d
X
Let := (X) for any unital CPTP map . E S H
SinceEX′EX′ is Pa~au′,nδtX′al◦CEPXTP map as well, we have EX nlim (cid:13)EX(cid:16)πˆ~a(X,δX) (cid:17)−πˆ~a(X′,δ)X′ (cid:13)=0. (41)
→∞(cid:13) (cid:13)
(X) (X) (X) (cid:13) (cid:13)
πˆ~a,δX ≺M EX(πˆ~a,δX)≺M EX′ (πˆ~a,δX). (36) Definition 2 is then reformulated as follows:
ECX′on(πˆse~a(Xq,δuX)e)ntslayt,isfiaelsl eigenvalues pi (i = 1,2,···) of mpDseoaqesutsfiiieonbnnliecte(ii~aof{,,nδX′fδo2}rXAa∈nnXya∈d{)iδa∆Xb,a}tXsiuc∈(c~aXhtr,at∈hnδas∆tfo,arnmthaaetdrie)ioanibesax~atipisoct→ssstriaabadnlneo~sa.tf′hoerirs-
1 p . (37) { X}X∈X →ad ′ { X′ }X∈X
D(X) ≥ i Theorem 2 is proved by using the following lemma re-
~a,δX
garding the possibility and impossibility of generalized
From (35) and (37), we obtain adiabatic processes. A proof is given in Appendix B.
1 e Xδs/2 Lemma 4 Let us define
−
> p . (38)
D(X) D(X) ≥ i
~a′,δ′ ~a′,δ′ 1
X X s :=limsup S [~a,δ ] (42)
The distance between (πˆ(X) ) and πˆ(X) is then ~a,{δX}X∈X X→∞ X B X
EX ~a,δX ~a′,δX′ s :=liminf 1 S [~a,δ ] (43)
boundedasfollows. Duetothemonotonicityofthetrace ~a,{δX}X∈X X X B X
→∞
6
for arbitrary ~a and δ ∆. Suppose macroscopicaccessibility,inthesamewayasthethermo-
X X
δ , δ {∆. }Th∈eXn, a∈n adiabatic trans- dynamicentropyS does,byformulatingadiabaticop-
f{orXm}aXti∈oXn {(~aX′, }δX∈X ∈) (~a, δ ) is possible erationsintermsofTaD“coarse-grained”unitaloperations.
if s {<Xs}X∈X →a.d Co′n{veX′rs}eXly∈,Xan adiabatic The resultis applicablefor any physicalsystem inwhich
stribalne~as,f{ooδnXrlmy}Xai∈ftiXos~an,{(δ~aX,}~a{X′δ,∈{XδXX′}≤}XX∈∈sXX~a′),{→δX′a}dX(∈~aX′.,{δX′ }X∈X) is pos- Sat˜inBodn[~aif]soeraxecisaotcnshvaeesxleaamncedonntitniocnfroe~aua.ssiInatngidsfussnhtcroitwciotnlnytfhionarctreeSa˜acBshi[~naeg]leefmxuiensnctts-
Theorem 2 is then proved as follows. of ~a, in many physical systems, e.g., gases of particles
with natural potentials including the van der Waars po-
Proof of Theorem 2: The relation (14) is proved as tential [5]. In the regionwhere the phase transition does
follows. Due to not occur, the regularized Boltzmann entropy S˜B[~a] is a
strictly increasingfunction for eachelementof~a. There-
S˜B[~a]={δX}sXu∈pX∈∆s~a,{δX}X∈X, feonrter,oaptyldeaetsetrimnisnuecshtaheremgiaocnr,ostchoepriecgaudlaiarbizaetdicBaoclctezsmsiabniln-
S˜ [~a]= sup s , (44) ity.
B {δX}X∈X∈∆ ~a,{δX}X∈X
we have
Wealsoprovedthatasimilarrelationholdsingeneral,
δ ∆; s S˜ [~a], evenforsystemsinwhichS˜ [~a]maynotexistasaconvex
∀{ǫ>X}0X, ∈Xδ∈ ~a,{∆δX;}XS˜∈X[~a≤] Bǫ s (.45) and strictly increasing funBction for each element of ~a.
∀ ∃{ X′ }X∈X ∈ B ′ − ≤ ~a′,{δX′ }X∈X Possibility of an adiabatic transformation is in this case
characterizedbytwofunctionsS˜ [~a]andS˜ [~a],notbya
By choosing ǫ = (S˜ [~a] S˜ [~a])/2 > 0, it follows that B B
B ′ − B singlefunctionasinthecaseofthermodynamictheories.
for any δ , there exists δ such that
{ X}X∈X { X′ }X∈X
S˜ [~a]+S˜ [~a]
s~a,{δX}X∈X ≤S˜B[~a]< B ′ 2 B ≤s~a′,{δX′ }X∈X(,46) WeemphasizethatTheorem1donotneedthestronger
assumptions including the i.i.d. assumption. Also, our
due to Lemma 4. Thus anadiabatictransformationa results do not depend on any microscopic parameters,
a′ is possible. To prove (15), we assume that S˜B[~a]→> including δX that we have introduced to define the gen-
S˜B[~a′]. Equivalently to (45), we have eralized microcanonical state πˆ~a,δX. This is in contrast
to previous approaches from quantum information the-
ǫ>0, δ ∆; S˜ [~a] ǫ s ory[15–24]treating“microscopicaccessibility”,inwhich
∀ ∃{ X}X∈X ∈ B − ≤ ~a,{δX}X∈X convertibilityofstatesarecharacterizedbyfunctionsthat
∀{δX′ }X∈X ∈∆; s~a′,{δX′ }X∈X ≤S˜B[~a′]. (47) depends on microscopic parameters.
By choosing ǫ = (S˜ [~a] S˜ [~a])/2 > 0, we find that
B B ′
−
there exists δ , such that for any δ we
have { X}X∈X { X′ }X∈X Our method, presented in Definition 2 to formu-
late possibility of macroscopic transformations in terms
coarse-graining,is applicable as well for classes of quan-
S˜ [~a]+S˜ [~a]
s~a,{δX}X∈X ≥ B 2 B ′ >S˜B[~a′]≥s~a′,{δX′ }X∈X(,48) tculamssoopfeorpaetrioantisonosththeratthgaennuuinneitlyaldoepscerriabteidonasd.iaTboatfiicndopa-
erations,andtoseewhetherthesameresultsasTheorem
due to Lemma 4. Thus an adiabatic transformation
1 and 2 hold for that class, are left as future works.
~a ~a is not possible, which completes the proof.
ad ′
→
Acknowledgments: HT is gratefulto Prof. HalTasaki,
VI. CONCLUSION Prof. Gen Kimura, Dr. Yu Watanabe, Dr. Sho Sugiura,
Dr. Kiyoshi Kanazawa, Dr. Jun’ichi Ozaki, Dr. Ryoto
We have proved that the regularized Boltzmann en- Sawada, Dr. Sosuke Ito and Yohei Morikuni for helpful
tropyS˜ givesanecessaryandsufficientconditionforthe comments.
B
[1] E. Fermi, Thermodynamics (Dover Books on Physics, [2] J.M.Bardeen,B.Carter,S.W.Hawking,Comm.Math.
1956). Phys. 31, 161 (1973).
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[3] A. Shimizu, Basis of thermodynamics, (University of [16] O.C.O.Dahlsten,R.Renner,E.Rieper,andV.Vedral,
TokyoPress (in Japanese), ISBN:4130626094, (2007).) New. J. Phys.13, 053015, (2011).
[4] E. H.Lieb and J. Yngvason Phys. Rep.310, 1, (1996). [17] J. Aberg, Nat. Commun.4, 1925 (2013).
[5] H. Tasaki, Statistical mechanics 1,2, ISBN-13: 978- [18] D. Egloff, O. C. O. Dahlsten, R. Renner and V. Vedral,
4563024376 and ISBN-13: 978-4563024383 (Baihukan, arXiv:1207.0434, (2012).
2008 (in Japanese)). [19] F. G. S. L. Brandao, M. Horodeck, N. H. Y. Ng, J. Op-
[6] Landau, L. D. and Lifshitz, E. M. Statistical Physics. penheim, and S. Wehner,PNAS,112,3215(2015).
Vol. 5 of the Course of Theoretical Physics, 3rd edition [20] S. Popescu, arXiv:1009.2536.(2010).
(1976). [21] P.Skrzypczyk,A.J.ShortandS.Popescu,NatureCom-
[7] C. Jarzynski, Phys. Rev.Lett. 78, 2690, (1997). munications 5, 4185, (2014).
[8] H.Tasaki, arXiv:cond-mat/0009244 (2000). [22] H. Tajima and M. Hayashi arXiv:1405.6457 (2014).
[9] J. Kurchan,arXiv:cond-mat/0007360(2000). [23] M. Weilenmann, L. Kr¨amer, P. Faist, and R. Renner,
[10] S. DeLiberato and M. Ueda, Phys. Rev. E 84, 051122 arXiv:1501.06920(2015).
(2011). [24] G. Gour, M. P. Muller, V. Narasimhachar, R. W.
[11] G. Xiao and J. Gong, arXiv:1503.00784, (2015). Spekkens,N. Y.Halpern, arXiv:1309.6586 (2013).
[12] K. Sekimoto, Stochastic Energetics (Lecture Notes in [25] H. P. Breuer and F. Petruccione, The Theory of Open
Physics), Springer, (2010). QuantumSystems(OxfordUniversityPress,USA,2007).
[13] H.Tasaki, arXiv:1511.01999 (2015). [26] M. A. Nielsen and I. L. Chuang, Quantum Computation
[14] H.Touchette, Physics Reports 478, 1 (2009). and Quantum Information(CambridgeUniversityPress,
[15] M.HorodeckiandJ.Oppenheim,Nat.Commun.4,2059 Cambridge, 2000).
(2013).
Appendix A: Proof of Lemmas 1, 2 3
Proof of Lemma 1: Because δX = o(1) and D~a(X)↓ is increasing for each element of ~a, the following inequality
holds for an arbitrary ǫ>0 and sufficiently large X:
1 1 1
logD(X)↓ < logD(X)↓ < logD(X)↓ . (A1)
X ~a(1−ǫ) X ~a(1+δX) X ~a(1+ǫ)
Therefore, for an arbitrary ǫ>0,
1 1
S˜B[~a(1−ǫ)]≤liminf X logD~a(X(1)+↓δX) ≤limsupX logD~a(X(1)+↓δX) <S˜B[~a(1+ǫ)] (A2)
holds. Because S˜ is continuous, Lemma 1 holds.
B
Proof of Lemma 2: For an arbitrary δ ∆,
X
{ }∈
1 1
logD(X) = log(D(X)↓ D(X)↓ )
X ~a,δX X ~a(1+δX)− ~a(1−δX)
1 1 D(X)↓
= logD(X)↓ + log1 ~a(1−δX). (A3)
X ~a(1+δX) X − D(X)↓
~a(1+δX)
BecauseofLemma1,thefirstpartoftheright-handsideconvergestoS˜ [~a]atthelimitofX. Therefore,weonlyneed
B
to showthatthe secondpartconvergesto0 foranarbitrary δ ∆ suchthatδ(0) <δ for a δ(0) ∆. Because
{ X}∈ X,~a X { X,~a}∈
D(X)↓ is increasing for each element of~a, it is sufficient to show that the second part vanishes for an δ(0) ∆.
~a { X,~a}∈
For an arbitrary ~a, the relation X1 logD~a(X)↓ → S˜B[~a] holds. Therefore, there exists a function γ~a,X satisfying
lim γ =0, and
X ~a,X
→∞
1
X logD~a(X)↓ =S˜B[~a]+γ~a,X (A4)
holds. Therefore, we can transform the second part of (A3) as follows:
1
log(1 eX(S˜B[~a(1−δX)]+γ~a(1−δX),X−S˜B[~a(1+δX)]−γ~a(1+δX),X)) (A5)
X −
8
Because
1 1
0 log(1 eXg(X)) log(1 e 1/√X) 0 (A6)
−
≥ X − ≥ X − →
holds for an arbitrary g(X) 1 , we only need to show that
≤−√X
1
S˜ [~a(1 δ )]+γ S˜ [~a(1+δ )] γ (A7)
B − X ~a(1−δX),X − B X − ~a(1+δX),X ≤−√X
holds for an array δ(0) ∆. Because S˜ [~a] is strictly increasingfor eachelement of~a, for an arbitraryǫ>0,there
{ X,~a}∈ B
exists X >0 such that
ǫ
1
S˜ [~a(1 ǫ)]+γ S˜ [~a(1+ǫ)] γ (A8)
B − ~a(1−ǫ),X − B − ~a(1+ǫ),X ≤−√X
holds for arbitrary X >X . Therefore, the following δ(0) ∆ satisfies (A7):
ǫ X,~a ∈
X =X , (A9)
1 ǫ1
X =max X ,X +1 , (A10)
m { ǫm m−1 }
δ(0) =ǫ for X X X (A11)
X,~a m m+1 ≥ ≥ m
where ǫ is an array of positive numbers such that ǫ 0.
m m m
{ } → →∞
Proof of Lemma 3: Wefirstlyconsiderthecasethat isbounded. Inthiscase,weonlyneedtoconsiderX .
1 0
X ∈X
By definition, for an arbitrary X ,
0
∈X
1 1
X logD~a(X(1)−↓δX) ≤ X logD~a(X′(1)↓+ηX(−))
1 1
logD(X)↓ logD(X)↓ (A12)
≤ X ~a′(1+η(+)) ≤ X ~a(1+δX)
X
Because of Lemma 1 and S˜B[~a] = S˜B[~a′], the relation X1 logD~a(X′(1)↓+η(+)) → S˜B[~a′] holds. If ηX(+) = o(1) would not
X
(+) (+)
hold, there exists a subsequence of η suchthatη =ǫ+η˜ for a realnumber ǫ=0 andη˜ =o(1). Becauseof
{ X } X X 6 X
Lemma 1, the subsequence satisfies X1 logD~a(X′(1)↓+η(+)) →S˜B[~a′(1+ǫ)]. It is a contradiction,and therefore ηX(+) =o(1)
X
holds. We can show η(−) =o(1) in the same manner.
X
Next, we consider the case where X1 is unbounded. For the part of X ∈X0, ηX(+) =o(1) and ηX(−) =o(1) has been
shown. Thus we only need to show ηX(+) = o(1) and ηX(−) = o(1) for X ∈ X1. It is sufficient to show ηX = o(1) for
X .
1
∈X
For X ,
1
∈X
1 1
logD(X)↓ logD(X)↓ (A13)
X ~a′(1+ηX−X1) ≤ X ~a(1−δX)
1 1
logD(X)↓ logD(X)↓ (A14)
X ~a(1+δX) ≤ X ~a′(1+ηX+X1)
whooludlsd, wnohterheol(dA,1e4it)hheorlds because there is no η satisfying D~a(X(1)−↓δX) ≤D~a(X′(1)↓+η) ≤D~a(X(1)+↓δX) for X ∈X1. If ηX =o(1)
η :=limsupη >0 (A15)
X
X
→∞
or
η :=liminfη <0 (A16)
X
X
→∞
9
holds. Note that η = η +o(1) holds for a subsequence of η , and that η = η +o(1) holds for another
subsequence. BecauXse of Lemma 1, for each subsequences, the{foXll}oXw∈inXg1relation holdXs respectively:
1
X logD~a(X′(1)↓+ηX−X1) →S˜B[~a(1+η)] (A17)
1
X logD~a(X′(1)↓+ηX+X1) →S˜B[~a(1+η)]. (A18)
Therefore, because of (A13) and (A14),
S˜B[~a′(1+η)] S˜B[~a′], (A19)
≤
S˜ [~a(1+η)] S˜ [~a]. (A20)
B ′ B ′
≥
Hence we have η =η =0. It contradicts the claim that either (A15) or (A16) holds.
Appendix B: Proof of Lemma 4
There exists a unital map satisfying (ρ) = σ, if and only if ρ σ holds, i.e., σ is majorized by ρ (see (26) for
E E ≺
the definition). Thus we prove the first statement of Lemma 4 by showing that if s<s, we have
πˆ(X) πˆ(X) (B1)
~a,δX ≺ ~a′,δX′
for any δ , δ ∆ and sufficiently large X. Define
{ X}X∈X { X′ }X∈X ∈
1
γ :=max S(πˆ(X) ) s , 0 ,
~a,{δX}X∈X,X (cid:26)X ~a,δX − ~a,{δX}X∈X (cid:27)
1
γ := min S(πˆ(X) ) s , 0
~a{δX}X∈X,X − (cid:26)X ~a,δX − ~a′,{δX′ }X∈X (cid:27)
for any δ ∆. By definition, we have
X X
{ } ∈X ∈
s γ
~a,{δX}X∈X − ~a,{δX}X∈X,X
S(πˆ(X) )
~a,δX s +γ (B2)
≤ X ≤ ~a,{δX}X∈X ~a,{δX}X∈X,X
for any~a and X, as well as
lim γ = lim γ =0 (B3)
X→∞ ~a,{δX}X∈X,X X→∞ ~a,{δX}X∈X,X
due to the definitions of s and s . Hence we have
~a,{δX}X∈X ~a,{δX}X∈X
1 1
(X) (X)
logD = S(πˆ ) s +γ ,
X ~a,δX X ~a,δX ≤ ~a,{δX}X∈X ~a,{δX}X∈X,X
1 1
logD(X) = S(πˆ(X) ) s γ ,
X ~a′,δX′ X ~a′,δX′ ≥ ~a′,{δX′ }X∈X − ~a,{δX′ }X∈X,X
which leads to
1 1
logD(X) logD(X)
n ~a′,δX′ − n ~a,δX
(s s )
≥ ~a′,{δX′ }X∈X − ~a,{δX}X∈X
(γ +γ ). (B4)
− ~a′,{δX′ }X∈X,n ~a,{δX}X∈X,X
Suppose s <s . Due to (B3) and (B4), we have
~a,{δX}X∈X ~a′,{δX′ }X∈X
logD(X) logD(X) 0 (B5)
~a′,δX′ − ~a,δX ≥
10
for any sufficiently large n, which implies (B1). This completes the proof of the first part of Theorem 2.
We prove the second statement by showing that an adiabatic transformation (a, δ ) (a, δ ) is
not possible if δs:=s s >0. From (B2), we have { X}X∈X →ad ′ { X′ }X∈X
a′,{δX′ }X∈X − ~a,{δX}X∈X
1 1 1 1
logD(X) logD(X) = S(πˆ(X) ) S(πˆ(X) )
n ~a,δX − X ~a′,δX′ n ~a,δX − X ~a′,δX′
≥(s~a,{δX}X∈X −s~a′,{δX′ }X∈X)−(γ~a,{δX}X∈X,n+γ~a′,{δX′ }X∈X,X)
=δs γ˜ (B6)
X
−
for any X, where we defined γ˜ :=γ +γ . Hence we obtain
X ~a,{δX}X∈X,X ~a′,{δX′ }X∈X,X
1 e−Xδs/2 1
> (B7)
D(X) D(X) ≥ D(X)
~a′,δX′ ~a′,δX′ ~a,δX
for any sufficiently large n. Note that lim γ˜ =0 from (B3). Therefore, as proved in Section VA, we have (40)
n X
for any sufficiently large n and any unital→C∞PTP map . This contradicts Condition (41), which implies that an
X
adiabatic transformation (a,δ ) (a,δ ) is not possEible. (cid:4)
X →ad ′ X′
