In addition, some matrix trace inequalities related to the Tsallis relative entropy are studied

MATRIX TRACE INEQUALITIES ON THE TSALLIS ENTROPIES

SHIGERU FURUICHI

DEPARTMENT OFELECTRONICS ANDCOMPUTERSCIENCE

TOKYOUNIVERSITY OFSCIENCE

YAMAGUCHI, SANYO-ONODACITY

YAMAGUCHI, 756-0884, JAPAN

furuichi@ed.yama.tus.ac.jp

Received 28 June, 2007; accepted 29 January, 2008 Communicated by F. Zhang

Dedicated to Professor Kunio Oshima on his 60th birthday.

ABSTRACT. Maximum entropy principles in nonextensive statistical physics are revisited as an application of the Tsallis relative entropy defined for non-negative matrices in the framework of matrix analysis. In addition, some matrix trace inequalities related to the Tsallis relative entropy are studied.

Key words and phrases: Matrix trace inequality, Tsallis entropy, Tsallis relative entropy and maximum entropy principle.

2000 Mathematics Subject Classification. 47A63, 94A17, 15A39.

1. INTRODUCTION

In 1988, Tsallis introduced the one-parameter extended entropy for the analysis of a physi- cal model in statistical physics [10]. In our previous papers, we studied the properties of the Tsallis relative entropy [5, 4] and the Tsallis relative operator entropy [17, 6]. The problems on the maximum entropy principle in Tsallis statistics have been studied for classical systems and quantum systems [9, 11, 2, 1]. Such problems were solved by the use of the Lagrange multipli- ers formalism. We give a new approach to such problems, that is, we solve them by applying the non-negativity of the Tsallis relative entropy without using the Lagrange multipliers formalism.

In addition, we show further results on the Tsallis relative entropy.

The author would like to thank the reviewer for providing valuable comments to improve the manuscript. The author would like to thank Professor K.Yanagi and Professor K. Kuriyama for providing valuable comments and constant encouragement. This work was supported by the Japanese Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists (B), 17740068. This work was also partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (B), 18300003.

216-07

In the present paper, the set ofn ×n complex matrices is denoted by M_n(C). That is, we deal withn×nmatrices because of Lemma 2.2 in Section 2. However some results derived in the present paper also hold for the infinite dimensional case. In the sequel, the set of all density matrices (quantum states) is represented by

D_n(C)≡ {X ∈M_n(C) :X ≥0,Tr[X] = 1}.

X ∈M_n(C)is called by a non-negative matrix and denoted byX ≥0, if we havehXx, xi ≥0 for all x ∈ Cⁿ. That is, for a Hermitian matrixX, X ≥ 0means that all eigenvalues of X are non-negative. In addition,X ≥ Y is defined by X −Y ≥ 0. For−I ≤ X ≤ I and λ ∈ (−1,0)∪(0,1), we denote the generalized exponential function byexp_λ(X) ≡(I+λX)^1/λ. As the inverse function ofexp_λ(·), forX ≥0andλ∈(−1,0)∪(0,1), we denote the generalized logarithmic function byln_λX ≡ ^Xλ_λ^−I. Then the Tsallis relative entropy and the Tsallis entropy for non-negative matricesXandY are defined by

D_λ(X|Y)≡Tr

X^1−λ(ln_λX−ln_λY)

, S_λ(X)≡ −D_λ(X|I).

These entropies are generalizations of the von Neumann entropy [16] and of the Umegaki rela- tive entropy [14] in the sense that

λ→0limS_λ(X) =S₀(X)≡ −Tr[XlogX]

and

λ→0limDλ(X|Y) =D0(X|Y)≡Tr[X(logX−logY)].

2. MAXIMUM ENTROPYPRINCIPLE INNONEXTENSIVE STATISTICALPHYSICS

In this section, we study the maximization problem of the Tsallis entropy with a constraint on theλ-expectation value. In quantum systems, the expectation value of an observable (a Hermit- ian matrix)Hin a quantum state (a density matrix)X ∈D_n(C)is written asTr[XH]. Here, we consider theλ-expectation valueTr[X^1−λH]as a generalization of the usual expectation value.

Firstly, we impose the following constraint on the maximization problem of the Tsallis entropy:

Cf_λ ≡

X ∈D_n(C) : Tr[X^1−λH] = 0 ,

for a given n ×n Hermitian matrix H. We denote a usual matrix norm by k·k, namely for A∈M_n(C)andx∈Cⁿ,

kAk ≡ max

kxk=1kAxk. Then we have the following theorem.

Theorem 2.1. LetY =Z_λ⁻¹exp_λ(−H/kHk), whereZλ ≡Tr[exp_λ(−H/kHk)], for ann×n Hermitian matrixH andλ ∈ (−1,0)∪(0,1). IfX ∈ Cf_λ, thenS_λ(X) ≤ −c_λln_λZ_λ⁻¹,where c_λ ≡Tr[X^1−λ].

Proof. SinceZ_λ ≥0and we haveln_λ(x⁻¹Y) = ln_λY + (ln_λx⁻¹)Y^λfor a non-negative matrix Y and scalarx, we calculate

Tr[X^1−λlnλY] = Tr[X^1−λlnλ

Z_λ⁻¹exp_λ(−H/kHk) ]

= Tr[X^1−λ

−H/kHk+ lnλZ_λ⁻¹(I−λH/kHk) ]

= Tr[X^1−λ

ln_λZ_λ⁻¹I−Z_λ^−λH/kHk ] =c_λln_λZ_λ⁻¹, sinceln_λZ_λ⁻¹ = ^Z

−λ λ −1

λ by the definition of the generalized logarithmic functionln_λ(·). By the non-negativity of the Tsallis relative entropy:

(2.1) Tr[X^1−λln_λY]≤Tr[X^1−λln_λX], we have

S_λ(X) = −Tr[X^1−λln_λX]≤ −Tr[X^1−λln_λY] =−c_λln_λZ_λ⁻¹.

Next, we consider the slightly changed constraint:

C_λ ≡

X ∈D_n(C) : Tr[X^1−λH]≤Tr[Y^1−λH] and Tr[X^1−λ]≤Tr[Y^1−λ]

for a givenn×nHermitian matrixH, as the maximization problem for the Tsallis entropy. To this end, we prepare the following lemma.

Lemma 2.2. For a givenn×nHermitian matrixH, ifnis a sufficiently large integer, then we haveZλ ≥1.

Proof.

(i) For a fixed0< λ <1and a sufficiently largen, we have

(2.2) (1/n)^λ ≤1−λ.

From the inequalities− kHkI ≤H ≤ kHkI, we have

(2.3) (1−λ)^1λI ≤exp_λ(−H/kHk)≤(1 +λ)^λ1I.

By inequality (2.2), we have 1

nI ≤(1−λ)^1λI ≤exp_λ(−H/kHk), which impliesZ_λ ≥1.

(ii) For a fixed−1< λ <0and a sufficiently largen, we have

(2.4) (1/n)^λ ≥1−λ.

Analogously to (i), we have inequalities (2.3) for−1< λ < 0. By inequality (2.4), we

have 1

nI ≤(1−λ)^1λI ≤exp_λ(−H/kHk), which impliesZ_λ ≥1.

Then we have the following theorem by the use of Lemma 2.2.

Theorem 2.3. Let Y = Z_λ⁻¹exp_λ(−H/kHk), where Z_λ ≡ Tr[exp_λ(−H/kHk)], for λ ∈ (−1,0)∪(0,1)and ann×nHermitian matrixH. IfX ∈ C_λ andnis sufficiently large, then S_λ(X)≤S_λ(Y).

Proof. Due to Lemma 2.2, we have lnλZ_λ⁻¹ ≤ 0 for a sufficiently large n. Thus we have lnλZ_λ⁻¹Tr[X^1−λ] ≥lnλZ_λ⁻¹Tr[Y^1−λ]forX ∈ Cλ. Similarly to the proof of Theorem 2.1, we have

Tr[X^1−λln_λY] = Tr[X^1−λln_λ

Z_λ⁻¹exp_λ(−H/kHk) ]

= Tr[X^1−λ

−H/kHk+ ln_λZ_λ⁻¹(I−λH/kHk) ]

= Tr[X^1−λ

ln_λZ_λ⁻¹I−Z_λ^−λH/kHk ]

≥Tr[Y^1−λ

ln_λZ_λ⁻¹I−Z_λ^−λH/kHk ]

= Tr[Y^1−λ

−H/kHk+ ln_λZ_λ⁻¹(I−λH/kHk) ]

= Tr[Y^1−λln_λ

Z_λ⁻¹exp_λ(−H/kHk) ]

= Tr[Y^1−λln_λY].

By Eq.(2.1) we have

S_λ(X) =−Tr[X^1−λln_λX]≤ −Tr[X^1−λln_λY]≤ −Tr[Y^1−λln_λY] =S_λ(Y).

Remark 2.4. Since−x^1−λln_λxis a strictly concave function,S_λ is a strictly concave function on the setC_λ. This means that the maximizingY is uniquely determined so that we may regard Y as a generalized Gibbs state, since an original Gibbs statee^−βH/Tr[e^−βH], whereβ ≡ 1/T andT represents a physical temperature, gives the maximum value of the von Neumann entropy.

Thus, we may define a generalized Helmholtz free energy by F_λ(X, H)≡Tr[X^1−λH]− kHkS_λ(X).

This can be also represented by the Tsallis relative entropy such as

F_λ(X, H) = kHkD_λ(X|Y) + ln_λZ_λ⁻¹Tr[X^1−λ(kHk −λH)].

The following corollary easily follows by taking the limit asλ →0.

Corollary 2.5 ([12, 15]). LetY = Z₀⁻¹exp (−H/kHk), whereZ₀ ≡ Tr[exp (−H/kHk)], for ann×nHermitian matrixH.

(i) IfX ∈fC₀, thenS₀(X)≤logZ₀. (ii) IfX ∈C₀, thenS₀(X)≤S₀(Y).

3. ONSOME TRACEINEQUALITIESRELATED TO THE TSALLIS RELATIVEENTROPY

In this section, we consider an extension of the following inequality [8]:

(3.1) Tr[X(logX+ logY)]≤ 1

pTr[XlogX^p/2Y^pX^p/2] for non-negative matricesXandY, andp >0.

For the proof of the following Theorem 3.3, we use the following famous inequalities.

Lemma 3.1 ([8]). For any Hermitian matricesAand B, 0 ≤ λ ≤ 1and p > 0, we have the inequality:

Trh

e^pA]_λe^pB1/pi

≤Tr

e^{(1−λ)A+λB} , where theλ-geometric mean for positive matricesAandBis defined by

A]_λB ≡A^1/2 A^−1/2BA^−1/2λ A^1/2.

Lemma 3.2 ([7, 13]). For any Hermitian matricesG and H, we have the Golden-Thompson inequality:

Tr e^G+H

≤Tr e^Ge^H

.

Theorem 3.3. For positive matricesX andY,p≥1and0< λ≤1, we have (3.2) D_λ(X|Y)≤ −Tr[Xln_λ(X^−p/2Y^pX^−p/2)^1/p].

Proof. First of all, we note that we have the following inequality [3]

(3.3) Tr[(Y^1/2XY^1/2)^rp]≥Tr[(Y^r/2X^rY^r/2)^p]

for non-negative matricesXandY, and0≤r ≤1, p >0. Similar to the proof of Theorem 2.2 in [5], inequality (3.2) easily follows by settingA = logXandB = logY in Lemma 3.1 such that

Tr[(X^p]_λY^p)^1/p]≤Tr[e^{logX1−λ+logYλ}]

≤Tr[e^logX1−λe^logYλ]

= Tr[X^1−λY^λ], (3.4)

by Lemma 3.2. In addtion, we have

(3.5) Tr[X^rY^r]≤Tr[(Y^1/2XY^1/2)^r], (0≤r ≤1), on takingp= 1of inequality (3.3). By (3.4) and (3.5) we obtain:

Tr[(X^p]_λY^p)^1/p] = Trh

X^p/2(X^−p/2Y^pX^−p/2)^λX^{p/2 1/p}i

≥Tr[X(X^−p/2Y^pX^−p/2)^λ/p].

Thus we have,

D_λ(X|Y) = Tr[X−X^1−λY^λ] λ

≤ Tr[X−X(X^−p/2Y^pX^−p/2)^λ/p] λ

=−Tr[X

((X^−p/2Y^pX^−p/2)^1/p)^λ −I ] λ

=−Tr[Xln_λ(X^−p/2Y^pX^−p/2)^1/p].

Remark 3.4. For positive matrices X and Y, 0 < p < 1 and 0 < λ ≤ 1, the following inequality does not hold in general:

(3.6) D_λ(X|Y)≤ −Tr[Xln_λ(X^−p/2Y^pX^−p/2)^1/p].

Indeed, the inequality (3.6) is equivalent to

(3.7) Tr[X(X^−p/2Y^pX^−p/2)^λ/p]≤Tr[X^1−λY^λ].

Then we have many counter-examples. If we setp= 0.3,λ = 0.9andX =

10 3

3 9

, Y = 5 4

4 5

,then inequality (3.7) fails. (R.H.S. minus L.H.S. of (3.7) approximately becomes -0.00309808.) Thus, inequality (3.6) is not true in general.

Corollary 3.5.

(i) For positive matricesX andY, the trace inequality

D_λ(X|Y)≤ −Tr[Xln_λ(X^−1/2Y X^−1/2)]

holds.

(ii) For positive matricesX andY, andp≥1, we have inequality (3.1).

Proof.

(i) Putp= 1in (1) of Theorem 3.3.

(ii) Take the limit asλ→0.

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