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I PRELIMINARIES II. 2 S aa3109 Japan 351-0198 tama ˜ U B opeeydetermines completely steitra energy, internal the is btcaccessibility abatic S N S D T B stenme fparticles. of number the is a enone been has os nthis In does. U , V and V N sthe is Our . S ˜ B 2
(X) ergy. We represent such a set of physical quantities in with πˆ X . Let ∆ denote the set of arrays δX { ~a,δX } ∈X { } terms of vectors as ~a := (a0,a1, ..., aL), where L is a nat- such that limX δX = 0. Then ~a has a one-to-one ural number, e.g., L = 2 in the case of (U,V,N). Here, correspondence→∞ with the following set of arrays of micro- for the simplicity of writing, we express the internal en- canonical state: ergy U as a . Since a macroscopic equilibrium state is 0 (X) uniquely determined by values of macroscopic physical πˆ X δX X ∆ . (7) {{ ~a,δX } ∈X |{ } ∈X ∈ } 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 { } ∈X ∈ uniquely determined as a function of ~a, which we denote fluctuation” of macroscopic physical quantities, any se- by ST D[~a]. quence in the set (7) can be regarded as describing states 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 eral equivalent formulations such as the principle of max- 1 (X) imum work, Clausius inequality and the law of entropy S˜B[~a] := lim log D ↓. (8) X X ~a increase. The law of entropy increase determines the adi- →∞ (X) (X) abatic accessibility, i.e., whether a macroscopic state can ↓ ↓ where D~a is the dimension of the Hilbert space ~a be transformed to another by an adiabatic operation. It defined by H is stated as follows: (X) (X) [l] [l] ~a := span ψ λ Xa , ~a ad ~a′ ST D[~a] ST D[~a′], (4) H n| i ∈ H ∃ ≤ ≺ ⇔ ≤ s.t. A(X)[l] ψ = λ[l] ψ for 0 l L , (9) where ~a ad ~a′ means “an adiabatic transformation | i | i ≤ ≤ o ~a ~a is≺ possible”. ad With concrete calculations, it has been shown that there →Let us introduce the statistical mechanical counterpart exists the limit S˜ as a convex and incresing function for 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- of particles with natural potentials including the van der tem by a Hilbert space (X) depending on a scaling pa- Waars potential [5]. rameter X. The macroscopicH limit is defined as the limit of X . We assume that X takes values in a set N→ ∞ R+ = or = . For each X and l = 0, ,L, III. FORMULATION OF MACROSCOPIC weX denote theX set of the Hermitian∈ operators X on ···(X) as H ADIABATIC ACCESSIBILITY IN TERMS OF 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(X) := Πˆ (X) /D(X) , (5) ~a,δX ~a,δX ~a,δX tion, by “coarse-graining” possibility of microscopic state transformations. We then formulate the macroscopic adi- 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 mension of the following (X) , which is a subspace of of sibility of a microscopic state transformation by unital H~a,δX operations. (X): H Our formulation is based on the following idea: (X) := span ψ (X) λ[l] [X(a[l] δ ), (X) ~a,δX X Basic Idea 1 Suppose a microcanonical state π is H n| i ∈ H ∃ ∈ − ~a,δX [l] (X)[l] [l] transformed by a quantum operation X to another mi- X(a + δX )) s.t. A ψ = λ ψ for 0 l L . (X) E | i | i ≤ ≤ crocanonical state π ′ ′ . From a macroscopic point of o ~a ,δX (6) view, we observe that an equilibrium state ~a is trans- formed to another equilibrium state ~a , for any δ and The parameter δ is a positive function of X, which rep- ′ 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 limX δX = 0. →∞ ~a′ if, for any macroscopically negligible δX and a δX′ , a To describe behavior of a system in the macroscopic (X) (X) state π can be transformed to π ′ ′ . limit (X ), we introduce an array δX X , and ~a,δX ~a ,δX → ∞ { } (X∈X) introduce an array of microcanonical states πˆ X { ~a,δX } ∈X Based on this idea and the concept of an array of mi- for each ~a and δX X . Depending on the choice of crocanonical states introduced in Section II, we propose { } ∈X (X) δX X , there exists many arrays πˆ X corre- a definition of possibility of a macroscopic state transfor- { } ∈X { ~a,δX } ∈X sponding to one ~a. Therefore, a thermodynamic equilib- mation as follows, where denotes a class of quantum rium state ~a does not have a one-to-one correspondence operations (Fig. 1). F 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˜B[~a] exists and is a convex and 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˜B[~a] S˜B[~a′] ~a ad ~a′. (13) and a sequence of class CPTP ≤ ⇔ ≺ which satisfy (10) When S˜B[~a] does not exist and is a convex and strictly increasing function for each element of ~a, Theorem 1 does 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′ →F S˜ [~a] < S˜ [~a′] ~a ~a′, (14) is possible if, for any δX X ∆, there exists B B ad { } ∈X ∈ ⇒ ≺ δ′ X ∆ and an array X X such that X S˜ [~a] S˜ [~a′] ~a ~a′, (15) { } ∈X ∈ {E } ∈X B ≤ B ⇐ ≺ad (X) (X) lim X (ˆπ ) πˆ ′ =0, (10) where we define ~a,δX ~a′,~δ n E − X →∞ 1 X S˜ [~a] := sup lim sup S [~a; δ ], (16) and X is a CPTP map of class on ( ) for all B B X δX X∈X ∆ X X X E . Here, ρ σ is the traceF distanceS H defined by { } ∈ →∞ ∈ X 1 k − k 1 ρ σ := 2 Tr ρ σ . We express as ~a F˜ ~a′ that an S˜B[~a] := sup lim inf SB[~a; δX ], (17) k − k | − | ≺ X adiabatic transformation ~a ˜ ~a′ is possible. δX X∈X ∆ X →F { } ∈ →∞ where, Clearly, if ~a ˜ ~a′ and ~a′ ˜ ~a′′ hold, then ~a ˜ ~a′′ also ≺F ≺F ≺F holds. (X) SB[~a; δX ] := log D . (18) The above definition provides a formulation of various ~a,δX types of possibility of macroscopic state transformations, Theorem 1 states that S˜ provides a necessary and depending on . In this paper, we employ unital CPTP B sufficient condition for the macroscopic adiabatic acces- map for class F to define possibility of adiabatic trans- sibility in the same way as S does, as long as S˜ [~a] formations. Here,F a unital CPTP map is defined as a T D B exists and is a convex and strictly increasing function for completely positive trace preserving map that maps the each element of ~a. Theorem 2 provides a similar relation identity operator to the identity operator, i.e., (1)ˆ = 1.ˆ for the case where S˜ [~a] does not exist as a convex and A unital map does not decrease the von NeumannE en- B strictly increasing function for each element of ~a. The tropy of an arbitrary quantum state[25], that is, we have macroscopic adiabatic accessibility is in this case charac-
S( (ρ)) S(ρ), ρ ( ), (11) terized by two functions S˜B[~a] and S˜B[~a], not by a single 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 feature of the adiabatic transformation in thermodynam- eters, including δX that we have introduced to define the ics that does not decrease the thermodynamic entropy generalized microcanonical stateπ ˆ~a,δX . This is in con- ST D, 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˜B[~a] exists and is a convex and strictly increas- 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 ad ~a′ → tion. It is shown that S˜B[~a] exists and is a convex and is possible if, for any δX X ∆, there exists { } ∈X ∈ increasing function for each element of ~a, in many phys- δX′ X ∆ and an array X X such that { } ∈X ∈ {E } ∈X ical systems, e.g., gases of particles with natural poten- tials including the van der Waars potential [5]. In the lim (ˆπ(X) ) πˆ(X) =0, (12) X ~a,δX ′ ~′ X ~a ,δX region where the phase transition does not occur, the reg- →∞ E − ularized Boltzmann entropy S˜B [~a] is a strictly increasing X and X is a unital CPTP map on ( ) for all X . function for each element of ~a. Therefore, at least in such E S H ∈ X We express as ~a ad ~a′ when an adiabatic transformation a region, the regularized Boltzmann entropy determines ≺ ~a ad ~a′ is possible. the macroscopic adiabatic accessibility. → 4
V. PROOFS Here, f(~a, δX ,X) is a real-valued function which con- 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 only if ρ and σ satisfies the majorizationE relationE ρ σ ≺M Lemma 1 For an arbitrary δ ∆, the following re- defined as { X } ∈ lation holds: m m ρ M σ p↓ q↓ for any m. (26) 1 (X) ≺ ⇐⇒def k ≥ k lim log D ↓ = S˜B[~a] (19) Xk=1 Xk=1 X X ~a(1+δX ) →∞ Here, p↓ and q↓ are eigenvalues of ρ and σ, re- Lemma 2 When the regularized Boltzmann entropy k k k k spectively,{ } sorted{ in decreasing} order [24, 26]. Hence it S˜B[~a] exists for ~a and is a strictly increasing function (0) suffices to show that, for any δX X ∆, there exists for each element of ~a, there exists δ X ∆ such { } ∈X ∈ X,~a another sequence δX′ X ∆ such that we have that { } ∈X ∈ { } ∈X ∈ (X) (X) πˆ πˆ ′ ′ . (27) ~a,δX M ~a ,δ 1 (X) ≺ X lim log D = S˜B[~a] (20) X X ~a,δX →∞ for any sufficiently large X. (0) We firstly show that (27) holds when S˜B[~a] < S˜B[~a′] holds for an arbitrary δX X ∆ satisfying δX,~a <δX (0) { } ∈X ∈ holds. Let us take arrays δ ∆ and δ′′ ∆a for an arbitrary X. { X } ∈ { X } ∈ satisfying δ <δ for arbitrary X. Then, because D(X) X X′′ ~a,δX Lemma 3 Let us consider the case where the regularized is an increasing function for δX , Boltzmann entropy S˜B [~a] exists and is a strictly increas- 1 (X) 1 (X) ing function for each element of ~a. Let us take arbitrary log D~a,δ log D~a,δ′′ (28) X X ≤ X X ~a and ~a′ such that S˜B[~a] = S˜B[~a′]. Let us take an ar- ray δX ∆, and let AX be the following set of real holds. Because of Lemma 2, the righthand-side of the { } ∈ numbers: above converges to S˜B[~a] at the limit of X . Also, → ∞ (0) (X) (X) (X) because of Lemma 2, for an arbitrary array δX′ ∆~a′ , AX := η D ↓ D ′ ↓ D ↓ (21) (X) { } ∈ ~a(1 δX ) ~a (1+η) ~a(1+δX ) 1 ˜ { | − ≤ ≤ } X log D~a′,δ′ converges to SB[~a′] at the limit of X . X → ∞ Let be the set of X such that A is not the empty set, Therefore, because of S˜B[~a] < S˜B[~a′], X0 X and let be 1 the set of X such that AX is the empty set. X (+) ( ) 1 (X) 1 (X) Then, we define two real valued functions ηX and ηX− log D~a,δ log D~a′,δ′ (29) X X X X of X as follows: ≤ holds for sufficient large X. Hence, (27) holds when 1 inf η + 2 sup η (X ) ˜ ˜ (+) 3 η AX 3 η AX 0 SB[~a] < SB[~a′] holds. ηX = ∈1 ∈ ∈ X ,(22) ηX + (X 1) ˜ ˜ X ∈ X Next, let us show that (27) holds when SB[~a]= SB[~a′]. 2 1 For an arbitrary δX ∆, we define ( ) 3 infη AX η + 3 supη AX η (X 0) ηX− = ∈1 ∈ ∈ X ,(23) { } ∈ ηX X (X 1) (+) ( ) − ∈ X δ′ := 4 max η , η − . (30) X {| X | | X |} where ηX is the following real valued function of X 1: ∈ X Clearly, δX′ > supη AX η and infη AX η > δX′ hold for ∈ ∈ − (X) (X) any X 0. Therefore, for any X 0, ηX := sup η D ′ ↓
Because of Lemma 3, δX′ = o(1) also holds. Hence, distance, we have (X) (X) (X) δ ∆ satisfies D < D ′ ′ , because of D = X′ ~a,δX ~a ,δ ~a,δX { } ∈ X (X) (X) (X) (X) (X) (X) X (ˆπ~a,δ ) πˆ~a′,δ′ X′ (ˆπ~a,δ ) πˆ~a′,δ′ D ↓ D ↓ . Therefore, (27) holds. X X X X ~a(1+δX ) ~a(1 δX ) E − ≥ kE − k − − X D( ) Finally, we prove the backward implication of Propo- ~a′,δ′ X sition (13) by showing that an adiabatic transformation 1 1 1 = pi (X) + pi ~a aq ~a′ is not possible if δs := S˜B[~a] S˜B[~a′] > 0. Fix 2 − 2 i=1 D ′ ′ (X) → − X ~a ,δX i>DX (0) ~a′ ,δ′ arbitrary δX X ∆ and δX′ X ∆. We also X ∈X ~a ∈X { } (0)∈ { } ∈ (X) D ′ take δX′′ X ∆~a′ such that δX′ < δX′′ for arbitrary ~a′,δ { } ∈X ∈ 1 X 1 X. From (25), we have p ≥ 2 i − (X) Xi=1 D~a′,δ′ 1 (X) 1 (X) X log D log D ′ ′′ ~a,δX ~a ,δ X − X X Xδs/2 1 (X) 1 e − ˜ ˜ D ′ ′ = (SB [~a] SB[~a′]) + (f(~a, δX ,X) f(~a′,δX′′ ,X)) ~a ,δX (X) (X) − − ≥ 2 D ′ ′ − D ′ ′ ~a ,δX ~a ,δX = δs γX (33) − 1 Xδs/2 = 1 e− . (39) for any X, where we defined γ := f(~a, δ ,X) 2 − X X − f(~a′,δ′′ ,X). Hence we have X (X) Here, we defined pi (i = 1, ,D~a′,δ′ ) as eigenvalues of ··· X 1 (X) 1 (X) (X) (X) log D log D ′ ′ ′ ′ ~a,δX ~a ,δ X′ (ˆπ~a,δ ) on the support ofπ ˆ~a ,δ . Thus we obtain X − X X E X X 1 (X) 1 (X) δs log D log D ′ ′′ > (34) 1 ~a,δX ~a ,δ (X) (X) X (ˆπ ) πˆ ′ ′ (40) ≥ X − X 2 X ~a,δX ~a ,δ E − X ≥ 3 for any sufficiently large X, which leads to for any sufficiently large X and any unital CPTP map Xδs/2 X , which contradicts (12). Therefore, an adiabatic 1 e− 1 E (X) > (X) > (X) . (35) transformation ~a aq ~a′ is not possible. D ′ ′ D ′ ′ D → ~a ,δX ~a ,δX ~a,δX
Note that limX γX = 0 follows from B. Proof of Theorem 2 →∞ limX f(~a, δX ,X) = 0 and limX f(~a′,δX′′ ,X) = 0. →∞(X) →∞ (X) Let P ′ ′ be the projection onto supp[ˆπ ′ ′ ], and ~a ,δX ~a ,δX We first introduce a generalized adiabatic process. (X) define a unital CPTP map ′ ′ by ~a ,δX Definition 3 P Suppose δX X , δX′ X ∆. { } ∈X { } ∈X ∈ An adiabatic transformation (~a, δX X ) ad (X) (X) (X) (X) (X) { } ∈X → ~a′,δ′ (ρ)= P~a′,δ′ ρP~a′,δ′ + (I P~a′,δ′ )ρ(I P~a′,δ′ ). (~a , δ ) is possible if there exists a set P X X X − X − X ′ X′ X X { } ∈X (X) {E } of unital CPTP maps X on ( ) that satisfies (X) E S H Let X′ := ~a′,δ′ X for any unital CPTP map X . E P X ◦E E (X) (X) lim πˆ πˆ ′ ′ =0. (41) Since X′ is a untal CPTP map as well, we have X ~a,δX ~a ,δ E n E − X →∞ (X) (X) (X) πˆ M X (ˆπ ) M ′ (ˆπ ). (36) Definition 2 is then reformulated as follows: ~a,δX ≺ E ~a,δX ≺ EX ~a,δX Definition 2 Consequently, all eigenvalues p (i = 1, 2, ) of An adiabatic transformation ~a ad ~a′ is i ··· → (ˆπ(X) ) satisfies possible if, for any δX X ∆, there exists another X′ ~a,δX { } ∈X ∈ E sequence δX′ X ∆, such that an adiabatic transfor- { } ∈X ∈ mation (~a, δX X ) ad (~a′, δX′ X ) is possible. 1 { } ∈X → { } ∈X pi. (37) D(X) ≥ ~a,δX Theorem 2 is proved by using the following lemma re- garding the possibility and impossibility of generalized From (35) and (37), we obtain adiabatic processes. A proof is given in Appendix B.
Xδs/2 1 e− Lemma 4 Let us define (X) > (X) pi. (38) D ′ ′ D ′ ′ ≥ ~a ,δX ~a ,δX 1 s~a, δX X∈X := lim sup SB[~a, δX ] (42) { } X X (X) (X) →∞ The distance between X (ˆπ~a,δ ) andπ ˆ~a′,δ′ is then E X X 1 s~a, δX X∈X := lim inf SB[~a, δX ] (43) bounded as follows. Due to the monotonicity of the trace { } X X →∞ 6 for arbitrary ~a and δX X ∆. Suppose macroscopic accessibility, in the same way as the thermo- { } ∈X ∈ δX X , δX′ X ∆. Then, an adiabatic trans- dynamic entropy ST D does, by formulating adiabatic op- { } ∈X { } ∈X ∈ formation (~a, δX X ) ad (~a′, δX′ X ) is possible erations in terms of a “coarse-grained” unital operations. ∈X ∈X { } ′ → { } if s ∈X < s ′ . Conversely, an adiabatic The result is applicable for any physical system in which ~a, δX X ~a , δX X∈X { } { } S˜ [~a] exists as a continous and strictly increasing func- transformation (~a, δX X ) ad (~a′, δX′ X ) is pos- B { } ∈X → { } ∈X sible only if s s ′ ′ . tion for each element of ~a. It is shown that S˜B[~a] exists ~a, δX X∈X ~a , δX X∈X { } ≤ { } and is a convex and increasing function for each element 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 region where the phase transition does follows. Due to not occur, the regularized Boltzmann entropy S˜B [~a] is a strictly increasing function for each element of ~a. There- ˜ SB[~a] = sup s~a, δX X∈X , fore, at least in such a region, the regularized Boltzmann δX X∈X ∆ { } { } ∈ entropy determines the macroscopic adiabatic accessibil- ˜ ity. SB[~a] = sup s~a, δX X∈X , (44) δX X∈X ∆ { } { } ∈ we have We also proved that a similar relation holds in general, ˜ ˜ δX X ∆ ; s~a, δX X∈X SB[~a], even for systems in which SB[~a] may not exist as a convex ∀{ } ∈X ∈ { } ≤ and strictly increasing function for each element of ~a. ˜ ′ ′ ǫ> 0, δX′ X ∆ ; SB[~a′] ǫ s~a , δ X∈X (45). ∀ ∃{ } ∈X ∈ − ≤ { X } Possibility of an adiabatic transformation is in this case
characterized by two functions S˜B[~a] and S˜B[~a], not by a By choosing ǫ = (S˜ [~a′] S˜ [~a])/2 > 0, it follows that B − B single function as in the case of thermodynamic theories. for any δX X , there exists δX′ X such that { } ∈X { } ∈X
S˜B[~a′]+ S˜B[~a] ˜ ′ ′ s~a, δX X∈X SB[~a] < s~a , δ X∈X (46), We emphasize that Theorem 1 do not need the stronger { } ≤ 2 ≤ { X } assumptions including the i.i.d. assumption. Also, our due to Lemma 4. Thus an adiabatic transformation a results do not depend on any microscopic parameters, ˜ → a′ is possible. To prove (15), we assume that SB[~a] > including δX that we have introduced to define the gen- eralized microcanonical stateπ ˆ . This is in contrast S˜B[~a′]. Equivalently to (45), we have ~a,δX to previous approaches from quantum information the- ˜ ory [15–24] treating “microscopic accessibility”, in which ǫ> 0, δX X ∆ ; SB[~a] ǫ s~a, δX X∈X ∀ ∃{ } ∈X ∈ − ≤ { } convertibility of states are characterized by functions that ′ ′ ˜ δX′ X ∆ ; s~a , δ X∈X SB[~a′]. (47) depends on microscopic parameters. ∀{ } ∈X ∈ { X } ≤
By choosing ǫ = (S˜ [~a] S˜ [~a′])/2 > 0, we find that B − B there exists δX X , such that for any δX′ X we { } ∈X { } ∈X Our method, presented in Definition 2 to formu- have late possibility of macroscopic transformations in terms coarse-graining, is applicable as well for classes of quan- S˜B[~a]+ S˜B[~a′] ˜ ′ ′ tum operations other than unital operations. To find a s~a, δX X∈X > SB[~a′] s~a , δ X∈X(48), { } ≥ 2 ≥ { X } class of operations that genuinely described adiabatic op- erations, and to see whether the same results as Theorem 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 grateful to Prof. Hal Tasaki, 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 tropy S˜B gives a necessary and sufficient condition for the comments.
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Appendix A: Proof of Lemmas 1, 2 3
Proof of Lemma 1: (X) Because δX = o(1) and D~a ↓ is increasing for each element of ~a, the following inequality holds for an arbitrary ǫ> 0 and sufficiently large X:
1 (X) 1 (X) 1 (X) log D ↓ < log D ↓ < log D ↓ . (A1) ~a(1 ǫ) ~a(1+δX ) ~a(1+ǫ) X − X X Therefore, for an arbitrary ǫ> 0,
1 (X) 1 (X) S˜B[~a(1 ǫ)] lim inf log D ↓ lim sup log D ↓ < S˜B[~a(1 + ǫ)] (A2) − ≤ X ~a(1+δX ) ≤ X ~a(1+δX ) holds. Because S˜B is continuous, Lemma 1 holds.
Proof of Lemma 2: For an arbitrary δ ∆, { X } ∈ 1 (X) 1 (X) (X) log D = log(D ↓ D ↓ ) ~a,δX ~a(1+δX ) ~a(1 δX ) X X − − (X) D ↓ 1 (X) 1 ~a(1 δX ) = log D ↓ + log 1 − . (A3) ~a(1+δX ) (X) X X − ↓ D~a(1+δ ) X
Because of Lemma 1, the first part of the right-hand side converges to S˜B[~a] at the limit of X. Therefore, we only need (0) (0) to show that the second part converges to 0 for an arbitrary δX ∆ such that δX,~a <δX for a δX,~a ∆. Because (X) { } ∈ { } ∈(0) D ↓ is increasing for each element of ~a, it is sufficient to show that the second part vanishes for an δ ∆. ~a { X,~a} ∈ 1 (X) For an arbitrary ~a, the relation log D ↓ S˜ [~a] holds. Therefore, there exists a function γ satisfying X ~a → B ~a,X limX γ~a,X = 0, and →∞ 1 (X) log D ↓ = S˜ [~a]+ γ (A4) X ~a B ~a,X holds. Therefore, we can transform the second part of (A3) as follows:
1 X(S˜B[~a(1 δX )]+γ − S˜B[~a(1+δX )] γ ) log(1 e − ~a(1 δX ),X − − ~a(1+δX ),X ) (A5) X − 8
Because
1 Xg(X) 1 1/√X 0 log(1 e ) log(1 e− ) 0 (A6) ≥ X − ≥ X − → holds for an arbitrary g(X) 1 , we only need to show that ≤− √X
˜ ˜ 1 SB[~a(1 δX )] + γ~a(1 δX ),X SB[~a(1 + δX )] γ~a(1+δX ),X (A7) − − − − ≤−√X holds for an array δ(0) ∆. Because S˜ [~a] is strictly increasing for each element of ~a, for an arbitrary ǫ> 0, there { X,~a} ∈ B exists Xǫ > 0 such that
˜ ˜ 1 SB[~a(1 ǫ)] + γ~a(1 ǫ),X SB[~a(1 + ǫ)] γ~a(1+ǫ),X (A8) − − − − ≤−√X holds for arbitrary X>X . Therefore, the following δ(0) ∆ satisfies (A7): ǫ X,~a ∈
X1 = Xǫ1 , (A9)
Xm = max Xǫm ,Xm 1 +1 , (A10) { − } δ(0) = ǫ for X X X (A11) X,~a m m+1 ≥ ≥ m where ǫm is an array of positive numbers such that ǫm m 0. { } → →∞
Proof of Lemma 3: We firstly consider the case that 1 is bounded. In this case, we only need to consider X 0. By definition, for an arbitrary X , X ∈ X ∈ X0
1 (X) 1 (X) log D ↓ log D ↓ − ~a(1 δX ) ′ ( ) X − ≤ X ~a (1+ηX ) 1 (X) 1 (X) log D ↓ log D ↓ (A12) ′ (+) ~a(1+δX ) ≤ X ~a (1+ηX ) ≤ X
1 (X) (+) ˜ ˜ ↓ ˜ Because of Lemma 1 and SB[~a] = SB[~a′], the relation X log D ′ (+) SB[~a′] holds. If ηX = o(1) would not ~a (1+ηX ) → (+) (+) hold, there exists a subsequence of ηX such that ηX = ǫ +η ˜X for a real number ǫ =0andη ˜X = o(1). Because of { 1 } (X) 6 (+) ↓ ˜ Lemma 1, the subsequence satisfies X log D ′ (+) SB[~a′(1 + ǫ)]. It is a contradiction, and therefore ηX = o(1) ~a (1+ηX ) → ( ) holds. We can show ηX− = o(1) in the same manner. (+) ( ) Next, we consider the case where 1 is unbounded. For the part of X 0, ηX = o(1) and ηX− = o(1) has been X(+) ( ) ∈ X shown. Thus we only need to show η = o(1) and η − = o(1) for X . It is sufficient to show η = o(1) for X X ∈ X1 X X 1. For∈ XX , ∈ X1 1 (X) 1 (X) log D ′ ↓ 1 log D ↓ (A13) ~a (1+ηX ) ~a(1 δX ) X − X ≤ X − 1 (X) 1 (X) log D ↓ log D ′ ↓ 1 (A14) ~a(1+δX ) ~a (1+η + ) X ≤ X X X
(X) (X) (X) holds, where (A14) holds because there is no η satisfying D ↓ D ′ ↓ D ↓ for X 1. If ηX = o(1) ~a(1 δX ) ≤ ~a (1+η) ≤ ~a(1+δX ) ∈ X would not hold, either −
η := lim sup ηX > 0 (A15) X →∞ or
η := lim inf ηX < 0 (A16) X →∞ 9
holds. Note that ηX = η + o(1) holds for a subsequence of ηX X 1 , and that ηX = η + o(1) holds for another subsequence. Because of Lemma 1, for each subsequences, the{following} ∈X relation holds respectively:
1 (X) ˜ log D ′ ↓ 1 SB[~a(1 + η)] (A17) ~a (1+ηX ) X − X → 1 (X) ↓ ˜ log D~a′(1+η + 1 ) SB[~a(1 + η)]. (A18) X X X → Therefore, because of (A13) and (A14),
S˜ [~a′(1 + η)] S˜ [~a′], (A19) B ≤ B 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 the definition). Thus we proveE the first statementE of Lemma 4 by sh≺owing that if s (X) (X) πˆ~a,δ πˆ~a′,δ′ (B1) X ≺ X for any δX X , δX′ X ∆ and sufficiently large X. Define { } ∈X { } ∈X ∈ 1 (X) γ~a, δX X∈X ,X := max S(ˆπ~a,δ ) s~a, δX X∈X , 0 , { } X X − { } 1 (X) γ := min S(ˆπ ) s ′ ′ , 0 ~a δX X∈X ,X ~a,δX ~a , δX X∈X { } − X − { } for any δX X ∆. By definition, we have { } ∈X ∈ s γ ~a, δX X∈X ~a, δX X∈X ,X { } − { } S(ˆπ(X) ) ~a,δX s~a, δX X∈X + γ~a, δX X∈X ,X (B2) ≤ X ≤ { } { } for any ~a and X, as well as lim γ~a, δX X∈X ,X = lim γ = 0 (B3) X { } X ~a, δX X∈X ,X →∞ →∞ { } due to the definitions of s~a, δX X∈X and s~a, δX X∈X . Hence we have { } { } 1 (X) 1 (X) log D~a,δ = S(ˆπ~a,δ ) s~a, δX X∈X + γ~a, δX X∈X ,X , X X X X ≤ { } { } 1 (X) 1 (X) log D ′ ′ = S(ˆπ ′ ′ ) s ′ ′ γ ′ , ~a ,δ ~a ,δ ~a , δX X∈X ~a, δ X∈X ,X X X X X ≥ { } − { X } which leads to 1 (X) 1 (X) log D~a′,δ′ log D~a,δ n X − n X ′ ′ (s~a , δ X∈X s~a, δX X∈X ) ≥ { X } − { } (γ ′ ′ + γ ∈X ). (B4) ~a , δ X∈X ,n ~a, δX X ,X − { X } { } ′ ′ Suppose s~a, δX X∈X 1 (X) 1 (X) 1 (X) 1 (X) log D~a,δ log D~a′,δ′ = S(ˆπ~a,δ ) S(ˆπ~a′,δ′ ) n X − X X n X − X X ′ ′ ′ ′ (s s~a , δ X∈X ) (γ + γ ) ~a, δX X∈X X ~a, δX X∈X ,n ~a , δX X∈X ,X ≥ { } − { } − { } { } = δs γ˜ (B6) − X for any X, where we definedγ ˜X := γ + γ ′ ′ . Hence we obtain ~a, δX X∈X ,X ~a , δX X∈X ,X { } { } Xδs/2 1 e− 1 (X) > (X) (X) (B7) D ′ ′ D ′ ′ ≥ D ~a ,δX ~a ,δX ~a,δX for any sufficiently large n. Note that limn γ˜X = 0 from (B3). Therefore, as proved in Section V A, we have (40) for any sufficiently large n and any unital→∞ CPTP map . This contradicts Condition (41), which implies that an EX adiabatic transformation (a,δ ) (a′,δ′ ) is not possible. X →ad X