A GENERALIZED FANNES’ INEQUALITY

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Generalized Fannes’ Inequality S. Furuichi, K. Yanagi and

K. Kuriyama vol. 8, iss. 1, art. 5, 2007

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A GENERALIZED FANNES’ INEQUALITY

S. FURUICHI

Department of Electronics and Computer Science Tokyo University of Science in Yamaguchi, Sanyo-Onoda City, Yamaguchi, 756-0884, Japan EMail:furuichi@ed.yama.tus.ac.jp

K. YANAGI AND K. KURIYAMA

Dept. of Applied Science, Faculty of Engineering Yamaguchi University, Tokiwadai 2-16-1, Ube City, 755-0811, Japan

EMail:{yanagi,kuriyama}@yamaguchi-u.ac.jp

Received: 02 March, 2006

Accepted: 08 February, 2007

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 94A17, 46N55, 26D15.

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

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K. Kuriyama vol. 8, iss. 1, art. 5, 2007

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Close Abstract: We axiomatically characterize the Tsallis entropy of a finite quantum sys-

tem. In addition, we derive a continuity property of Tsallis entropy. This gives a generalization of the Fannes’ inequality.

Acknowledgements: 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 Ed- ucation, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists (B), 17740068.

Dedicatory: Dedicated to Professor Masanori Ohya on his 60th birthday.

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1. Introduction with Uniqueness Theorem of Tsallis Entropy

Three or four decades ago, a number of researchers investigated some extensions of the Shannon 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))

with one parameter q ∈ R⁺ as an extension of Shannon entropy H₁(X) =

−P

xp(x) logp(x), for any probability distribution p(x) ≡ p(X = x) of a given random variable X, whereq-entropy function is defined by η_q(x) ≡ −x^qln_qx =

x^q−x

1−q and theq-logarithmic functionln_qx ≡ ^x^1−q_1−q⁻¹ is defined forq ≥0,q 6= 1and x≥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 system, introducing the generalized Faddeev’s axiom.

We briefly review the uniqueness theorem for the Tsallis entropy below.

The function I_q(x₁, . . . , x_n) is assumed to be defined on n-tuple (x₁, . . . , x_n) 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.

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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 functionIq : ∆n →R⁺ such that

(1.3) I_q(x₁, . . . , x_n) = µ_qH_q(x₁, . . . , x_n),

whereµ_qis a positive constant that depends on the parameterq >0.

If we further impose the normalized condition on Theorem1.1, it determines the entropy of typeβ(the structurala-entropy), (see [1, p. 189]).

Definition 1.1. For a density operatorρon a finite dimensional Hilbert spaceH, the Tsallis entropy is defined by

S_q(ρ)≡ Tr[ρ^q−ρ]

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

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Note that the Tsallis entropy is a particular case of f-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-normk·k₁.

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

(T3) Generalized mixing condition: Forρ =L_n

k=1λ_kρ_konH =Ln

k=1H_k, where λ_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 Axiom2, thenT_qis uniquely given by the following form T_q(ρ) = µ_qS_q(ρ),

with a positive constant numberµ_qdepending 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

T_q(λ₁, λ₂) =λ^q₁T_q(1) +λ^q₂T_q(1) +T_q(λ₁, λ₂),

which impliesT_q(1) = 0. Moreover, by (T2) and (T3), whenp_n6= 1, we have T_q(p₁, . . . , p_n−1, λp_n,(1−λ)p_n)

=p^q_nTq(λ,1−λ) + (1−pn)^qTq

p₁

1−p_n, . . . , p_n−1 1−p_n

+Tq(pn,1−pn)

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and

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

1−p_n, . . . , p_n−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 tq(x) + (1−x)^qtq

y 1−x

=tq(y) + (1−y)^qtq

x 1−y

.

Taking x = 0 and some y > 0, we have T_q(0,1) = t_q(0) = 0for q 6= 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).

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ThusT_q satisfies all conditions of the generalized Faddeev’s axiom (F1), (F2) and (F3). Therefore we can apply Theorem 1.1 so that we obtain T_q(λ₁, . . . , λ_n) = µ_qH_q(λ₁, . . . , λ_n). Thus we haveT_q(ρ) = µ_qS_q(ρ),for density operatorρ.

Remark 1. For the special case q = 0 in the above theorem, we need the reducing condition as an additional axiom.

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2. A Continuity of Tsallis Entropy

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 haveln_qz ≤z−1forq ≥0andz ≥0, we have ^x−x_1−q^q^y^1−q ≥x−y forx≥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 operators ρ and σ, q ≥ 0 and q 6= 1, is nonnegative.

Then we have 0 ≤ D_q(ρ|¹_dI) = −d^q−1(S_q(ρ)−ln_qd). 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.

Thereforef(t+s)−f(t)≤f(s).

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(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 numberu, v ∈ [0,1]andq ∈ [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 Lemma2.2. Here we set h_q(s)≡η_q(s)−η_q(1−s), s∈[0,1/2], q∈[0,2].

Then we haveh_q(0) = h_q(1/2) = 0andh⁰⁰_q(s) ≤ 0fors ∈ [0,1/2]. Therefore we havehq(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 operatorsρ₁andρ₂on the finite dimensional Hilbert spaceHwithdimH=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 and λ⁽²⁾₁ ≥ λ⁽²⁾₂ ≥ · · · ≥ λ⁽²⁾_d be eigenvalues of two density operatorsρ₁ and ρ₂, respectively. (The degenerate eigenvalues are

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repeated according to their multiplicity.) We setε≡Pd

j=1ε_jandε_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 Lemma2.3, 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 ε ε

)

=ε (

−

d

X

j=1

ε^q_j ε ln_qεj

ε −

d

X

j=1

ε^q_j ε

εj

ε ^1−q

ln_qε )

=ε^q

d

X

j=1

η_qε_j ε

+η_q(ε)

≤ε^qlnqd+η_q(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).

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

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

Corollary 2.5. For two density operatorsρ₁andρ₂on the finite dimensional Hilbert spaceHwithdimH=d <∞, ifkρ₁−ρ₂k₁ ≤ ¹_e, then

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

where S₁ represents the von Neumann entropy S₁(ρ) = Tr[η₁(ρ)] and η₁(x) =

−xlnx.

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[2] R. ALICKIANDM. FANNES, Quantum Dynamical Systems, Oxford Univer- sity Press, 2001.

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[6] S. FURUICHI, On uniqueness theorems for Tsallis entropy and Tsallis relative entropy, IEEE Trans. on Information Theory, 51 (2005), 3638–3645.

[7] M.A. NIELSEN AND I. CHUANG, Quantum Computation and Quantum In- formation, Cambridge Press, 2000.

[8] M. OHYAANDD. PETZ, Quantum Entropy and its Use, Springer-Verlag,1993.

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