Received02March,2006;accepted08February,2007CommunicatedbyS.S.Dragomir H ( X ) ≡ ( p ( x ) − p ( x ))1 P Threeorfourdecadesago,anumberofresearchersinvestigatedsomeextensionsoftheShan-nonentropy[1].Instatisticalphysics,theTsallisentropy,definedin[10]by 1. I

A GENERALIZED FANNES’ INEQUALITY

S. FURUICHI¹, K. YANAGI², AND K. KURIYAMA²

1DEPARTMENT OFELECTRONICS ANDCOMPUTERSCIENCE

TOKYOUNIVERSITY OFSCIENCE INYAMAGUCHI, SANYO-ONODACITY, YAMAGUCHI, 756-0884, JAPAN

furuichi@ed.yama.tus.ac.jp

2DEPARTMENT OFAPPLIEDSCIENCE, FACULTY OFENGINEERING

YAMAGUCHIUNIVERSITY, TOKIWADAI2-16-1, UBECITY, 755-0811, JAPAN

yanagi@yamaguchi-u.ac.jp kuriyama@yamaguchi-u.ac.jp

Received 02 March, 2006; accepted 08 February, 2007 Communicated by S.S. Dragomir

Dedicated to Professor Masanori Ohya on his 60th birthday.

ABSTRACT. We axiomatically characterize the Tsallis entropy of a finite quantum system. In addition, we derive a continuity property of Tsallis entropy. This gives a generalization of the Fannes’ inequality.

Key words and phrases: Uniqueness theorem, continuity property, Tsallis entropy and Fannes’ inequality.

2000 Mathematics Subject Classification. 94A17, 46N55, 26D15.

1. INTRODUCTION WITH UNIQUENESSTHEOREM OFTSALLIS ENTROPY

Three or four decades ago, a number of researchers investigated some extensions of the Shan- non entropy [1]. In statistical physics, the Tsallis entropy, defined in [10] by

H_q(X)≡ P

x(p(x)^q−p(x))

1−q =X

x

η_q(p(x))

The authors would like to thank referees for careful reading and providing valuable comments to improve the manuscript.

The author (S.F.) was partially supported by the Japanese Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encourage- ment of Young Scientists (B), 17740068.

063-06

with one parameterq∈R⁺as an extension of Shannon entropyH₁(X) = −P

xp(x) logp(x), for any probability distribution p(x) ≡ p(X = x) of a given random variable X, where q- entropy function is defined byη_q(x)≡ −x^qln_qx= ^x_1−q^q−x and theq-logarithmic functionln_qx≡

x^1−q−1

1−q is defined forq≥0,q6= 1andx≥0.

The Tsallis entropyH_q(X)converges to the Shannon entropy−P

xp(x) logp(x)asq →1.

See [5] for fundamental properties of the Tsallis entropy and the Tsallis relative entropy. In the previous paper [6], we gave the uniqueness theorem for the Tsallis entropy for a classical sys- tem, introducing the generalized Faddeev’s axiom. We briefly review the uniqueness theorem for the Tsallis entropy below.

The functionIq(x1, . . . , xn)is assumed to be defined onn-tuple(x1, . . . , xn)belonging to

∆_n≡ (

(p₁, . . . , p_n)

n

X

i=1

p_i = 1, p_i ≥0 (i= 1,2, . . . , n) )

and to take values in R⁺ ≡ [0,∞). Then we adopted the following generalized Faddeev’s axiom.

Axiom 1. (Generalized Faddeev’s axiom)

(F1) Continuity: The functionf_q(x) ≡ I_q(x,1−x)with parameterq ≥ 0is continuous on the closed interval[0,1]andf_q(x₀)>0for somex₀ ∈[0,1].

(F2) Symmetry: For arbitrary permutation{x⁰_k} ∈∆_nof{x_k} ∈∆_n, (1.1) I_q(x₁, . . . , x_n) = I_q(x⁰₁, . . . , x⁰_n).

(F3) Generalized additivity: Forx_n=y+z,y≥0andz >0, (1.2) I_q(x₁, . . . , xn−1, y, z) =I_q(x₁, . . . , x_n) +x^q_nI_q

y x_n, z

x_n

.

Theorem 1.1 ([6]). The conditions (F1), (F2) and (F3) uniquely give the form of the function I_q : ∆_n→R⁺such that

(1.3) Iq(x1, . . . , xn) =µqHq(x1, . . . , xn), whereµ_q is a positive constant that depends on the parameterq >0.

If we further impose the normalized condition on Theorem 1.1, it determines the entropy of typeβ (the structurala-entropy), (see [1, p. 189]).

Definition 1.1. For a density operator ρ on a finite dimensional Hilbert space H, the Tsallis entropy is defined by

S_q(ρ)≡ Tr[ρ^q−ρ]

1−q =Tr[η_q(ρ)], with a nonnegative real numberq.

Note that the Tsallis entropy is a particular case off-entropy [11]. See also [9] for a quasi- entropy which is a quantum version off-divergence [3].

LetT_qbe a mapping on the setS(H)of all density operators toR⁺.

Axiom 2. We give the postulates which the Tsallis entropy should satisfy.

(T1) Continuity: Forρ ∈ S(H), T_q(ρ)is a continuous function with respect to the 1-norm k·k₁.

(T2) Invariance: For unitary transformationU,T_q(U^∗ρU) =T_q(ρ).

(T3) Generalized mixing condition: Forρ =L_n

k=1λ_kρ_k onH= Ln

k=1H_k, whereλ_k ≥ 0, Pn

k=1λ_k = 1, ρ_k ∈S(H_k), we have the additivity:

T_q(ρ) =

n

X

k=1

λ^q_kT_q(ρ_k) +T_q(λ₁, . . . , λ_n),

where(λ₁, . . . , λ_n)represents the diagonal matrix(λ_kδ_kj)k,j=1,...,n.

Theorem 1.2. IfT_qsatisfies Axiom 2, thenT_q is uniquely given by the following form T_q(ρ) = µ_qS_q(ρ),

with a positive constant numberµ_q depending on the parameterq >0.

Proof. Although the proof is quite similar to that of Theorem 2.1 in [8], we present it for readers’

convenience. From (T2) and (T3), we have

Tq(λ1, λ2) =λ^q₁Tq(1) +λ^q₂Tq(1) +Tq(λ1, λ2),

which impliesT_q(1) = 0. Moreover, by (T2) and (T3), whenp_n 6= 1, we have T_q(p₁, . . . , pn−1, λp_n,(1−λ)p_n)

=p^q_nT_q(λ,1−λ) + (1−p_n)^qT_q p₁

1−p_n, . . . , pn−1

1−p_n

+T_q(p_n,1−p_n) and

T_q(p₁, . . . , pn−1, p_n) =p^q_nT_q(1) + (1−p_n)^qT_q p₁

1−p_n, . . . , pn−1

1−p_n

+T_q(p_n,1−p_n). From these equations, we have

(1.4) T_q(p₁, . . . , pn−1, λp_n,(1−λ)p_n) = T_q(p₁, . . . , pn−1, p_n) +p^q_nT_q(λ,1−λ). If we setλp_n =yand(1−λ)p_n=zin (1.4), then forp_n=y+z 6= 0we have

(1.5) T_q(p₁, . . . , pn−1, y, z) = T_q(p₁, . . . , pn−1, p_n) +p^q_nT_q y

p_n, z p_n

.

Then for anyx, y, z ∈Rsuch that0≤x, y <1,0< z≤1andx+y+z = 1, we have T_q(x, y, z) =T_q(x, y+z) + (y+z)^qT_q

y

y+z, z y+z

=T_q(y, x+z) + (x+z)^qT_q x

x+z, z x+z

.

If we sett_q(x)≡T_q(x,1−x), then we have t_q(x) + (1−x)^qt_q

y 1−x

=t_q(y) + (1−y)^qt_q x

1−y

.

Takingx= 0and somey >0, we haveT_q(0,1) =t_q(0) = 0forq6= 0. Again settingλ= 0in (1.4) and using (T2), we have the reducing condition

T_q(p₁, . . . , p_n,0) =T_q(p₁, . . . , p_n).

ThusTqsatisfies all conditions of the generalized Faddeev’s axiom (F1), (F2) and (F3). There- fore we can apply Theorem 1.1 so that we obtainTq(λ1, . . . , λn) = µqHq(λ1, . . . , λn). Thus we

haveT_q(ρ) =µ_qS_q(ρ),for density operatorρ.

Remark 1.3. For the special caseq = 0in the above theorem, we need the reducing condition as an additional axiom.

2. A CONTINUITY OF TSALLISENTROPY

We give a continuity property of the Tsallis entropyS_q(ρ). To do so, we state a few lemmas.

Lemma 2.1. For a density operatorρon the finite dimensional Hilbert spaceH, we have Sq(ρ)≤lnqd,

whered= dimH<∞.

Proof. Since we havelnqz ≤z−1forq≥0andz ≥0, we have ^x−x_1−q^qy1−q ≥x−yforx≥0, y≥0,q≥0andq6= 1, Therefore the Tsallis relative entropy [5]:

D_q(ρ|σ)≡ Tr[ρ−ρ^qσ^1−q] 1−q

for two commuting density operatorsρandσ,q ≥ 0andq 6= 1, is nonnegative. Then we have 0≤Dq(ρ|¹_dI) = −d^q−1(Sq(ρ)−lnqd). Thus we have the present lemma.

Lemma 2.2. Iff is a concave function andf(0) = f(1) = 0, then we have

|f(t+s)−f(t)| ≤max{f(s), f(1−s)}

for anys∈[0,1/2]andt∈[0,1]satisfying0≤s+t ≤1.

Proof.

(1) Consider the function r(t) = f(s) −f(t +s) + f(t). Then r⁰(t) ≥ 0 since f⁰ is a monotone decreasing function. Thus we have r(t) ≥ 0 by r(0) = 0. Therefore f(t+s)−f(t)≤f(s).

(2) Consider the function ofl(t) = f(t+s)−f(t) +f(1−s). Thenl⁰(t) ≤ 0. Thus we havel(t)≥0byl(1−s) = 0. Therefore−f(1−s)≤f(t+s)−f(t).

Thus we have the present lemma.

Lemma 2.3. For any real number u, v ∈ [0,1]and q ∈ [0,2], if |u−v| ≤ ¹₂, then |η_q(u)− η_q(v)| ≤η_q(|u−v|).

Proof. Sinceη_qis a concave function withη_q(0) =η_q(1) = 0, we have

|η_q(t+s)−η_q(t)| ≤max{η_q(s), η_q(1−s)}

fors ∈[0,1/2]andt ∈[0,1]satisfying0≤t+s≤1, by Lemma 2.2. Here we set hq(s)≡ηq(s)−ηq(1−s), s∈[0,1/2], q∈[0,2].

Then we have hq(0) = hq(1/2) = 0 and h⁰⁰_q(s) ≤ 0 for s ∈ [0,1/2]. Therefore we have hq(s)≥0, which implies

max{η_q(s), η_q(1−s)}=η_q(s).

Thus we have the present lemma by lettingu=t+sandv =t.

Theorem 2.4. For two density operators ρ₁ andρ₂ on the finite dimensional Hilbert space H withdimH=dandq∈[0,2], ifkρ₁−ρ₂k₁ ≤q^1/(1−q), then

|S_q(ρ₁)−S_q(ρ₂)| ≤ kρ₁−ρ₂k^q₁ln_qd+η_q(kρ₁−ρ₂k₁), where we denotekAk₁ ≡Tr

(A^∗A)^1/2

for a bounded linear operatorA.

Proof. Letλ⁽¹⁾₁ ≥λ⁽¹⁾₂ ≥ · · · ≥λ⁽¹⁾_d andλ⁽²⁾₁ ≥λ⁽²⁾₂ ≥ · · · ≥λ⁽²⁾_d be eigenvalues of two density operatorsρ₁ andρ₂, respectively. (The degenerate eigenvalues are repeated according to their multiplicity.) We setε≡Pd

j=1ε_j andε_j ≡

λ⁽¹⁾_j −λ⁽²⁾_j

. Then we have ε_j ≤ε≤ kρ₁−ρ₂k₁ ≤q^1/(1−q) ≤ 1

2 by Lemma 1.7 of [8]. Applying Lemma 2.3, we have

|S_q(ρ₁)−S_q(ρ₂)| ≤

d

X

j=1

η_q

λ⁽¹⁾_j

−η_q λ⁽²⁾_j

≤

d

X

j=1

η_q(ε_j).

By the formulaln_q(xy) = ln_qx+x^1−qln_qy, we have

d

X

j=1

η_q(ε_j) = −

d

X

j=1

ε^q_jln_qε_j

=ε (

−

d

X

j=1

ε^q_j

ε ln_qε_j εε

)

=ε (

−

d

X

j=1

ε^q_j ε ln_q ε_j

ε −

d

X

j=1

ε^q_j ε

ε_j ε

1−q

ln_qε )

=ε^q

d

X

j=1

ηq

ε_j ε

+ηq(ε)

≤ε^qlnqd+ηq(ε).

In the above inequality, Lemma 2.1 was used forρ= (ε₁/ε, . . . , ε_d/ε). Therefore we have

|S_q(ρ₁)−S_q(ρ₂)| ≤ε^qln_qd+η_q(ε).

Nowη_q(x)is a monotone increasing function onx∈[0, q^1/(1−q)]. In addition,x^qis a monotone increasing function forq∈[0,2]. Thus we have the present theorem.

By taking the limit asq → 1, we have the following Fannes’ inequality (see pp.512 of [7], also [4, 2, 8]) as a corollary, sincelim_q→1q^1/(1−q) = ¹_e.

Corollary 2.5. For two density operatorsρ₁ andρ₂ on the finite dimensional Hilbert spaceH withdimH=d <∞, ifkρ₁−ρ₂k₁ ≤ ¹_e, then

|S₁(ρ₁)−S₁(ρ₂)| ≤ kρ₁−ρ₂k₁lnd+η₁(kρ₁−ρ₂k₁),

whereS₁ represents the von Neumann entropyS₁(ρ) = Tr[η₁(ρ)]andη₁(x) = −xlnx.

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