Volume 8 (2007), Issue 1, Article 5, 6 pp.
A GENERALIZED FANNES’ INEQUALITY
S. FURUICHI1 , K. YANAGI2 , AND K. KURIYAMA2
1
2
D EPARTMENT OF E LECTRONICS AND C OMPUTER S CIENCE
T OKYO U NIVERSITY OF S CIENCE IN YAMAGUCHI ,
S ANYO -O NODA C ITY, YAMAGUCHI , 756-0884, JAPAN
furuichi@ed.yama.tus.ac.jp
D EPARTMENT OF A PPLIED S CIENCE , FACULTY OF E NGINEERING
YAMAGUCHI U NIVERSITY,
T OKIWADAI 2-16-1,
U BE C ITY, 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.
A BSTRACT. 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. I NTRODUCTION WITH U NIQUENESS T HEOREM
OF
T SALLIS E NTROPY
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
P
(p(x)q − p(x)) X
Hq (X) ≡ x
=
ηq (p(x))
1−q
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 Encouragement of Young Scientists (B), 17740068.
063-06
2
S. F URUICHI , K. YANAGI ,
AND
K. K URIYAMA
with one parameter q ∈ R+ as an extension of Shannon entropy H1 (X) = −
P
x
p(x) log p(x),
for any probability distribution p(x) ≡ p(X = x) of a given random variable X, where qq −x
entropy function is defined by ηq (x) ≡ −xq lnq x = x1−q
and the q-logarithmic function lnq x ≡
x1−q −1
1−q
is defined for q ≥ 0, q 6= 1 and x ≥ 0.
The Tsallis entropy Hq (X) converges to the Shannon entropy −
P
x
p(x) log p(x) as q → 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 Iq (x1 , . . . , xn ) is assumed to be defined on n-tuple (x1 , . . . , xn ) belonging to
n
(
)
X
∆n ≡ (p1 , . . . , pn ) pi = 1 , pi ≥ 0 (i = 1, 2, . . . , n)
i=1
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 function fq (x) ≡ Iq (x, 1 − x) with parameter q ≥ 0 is continuous on
the closed interval [0, 1] and fq (x0 ) > 0 for some x0 ∈ [0, 1].
(F2) Symmetry: For arbitrary permutation {x0k } ∈ ∆n of {xk } ∈ ∆n ,
(1.1)
Iq (x1 , . . . , xn ) = Iq (x01 , . . . , x0n ).
(F3) Generalized additivity: For xn = y + z, y ≥ 0 and z > 0,
y z
q
,
.
(1.2)
Iq (x1 , . . . , xn−1 , y, z) = Iq (x1 , . . . , xn ) + xn Iq
xn xn
Theorem 1.1 ([6]). The conditions (F1), (F2) and (F3) uniquely give the form of the function
Iq : ∆n → R+ such that
(1.3)
Iq (x1 , . . . , xn ) = µq Hq (x1 , . . . , xn ),
where µq is a positive constant that depends on the parameter q > 0.
If we further impose the normalized condition on Theorem 1.1, it determines the entropy of
type β (the structural a-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
Sq (ρ) ≡
Tr[ρq − ρ]
= Tr[ηq (ρ)],
1−q
with a nonnegative real number q.
Note that the Tsallis entropy is a particular case of f -entropy [11]. See also [9] for a quasientropy which is a quantum version of f -divergence [3].
Let Tq be a mapping on the set S(H) of all density operators to R+ .
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 5, 6 pp.
http://jipam.vu.edu.au/
A G ENERALIZED FANNES ’ I NEQUALITY
3
Axiom 2. We give the postulates which the Tsallis entropy should satisfy.
(T1) Continuity: For ρ ∈ S(H), Tq (ρ) is a continuous function with respect to the 1-norm
k·k1 .
(T2) Invariance: For unitary transformation U , Tq (U ∗ ρU ) = Tq (ρ).
Ln
Ln
(T3) Generalized mixing condition: For ρ =
k=1 λk ρk on H =
k=1 Hk , where λk ≥ 0,
Pn
k=1 λk = 1, ρk ∈ S(Hk ), we have the additivity:
Tq (ρ) =
n
X
λqk Tq (ρk ) + Tq (λ1 , . . . , λn ),
k=1
where (λ1 , . . . , λn ) represents the diagonal matrix (λk δkj )k,j=1,...,n .
Theorem 1.2. If Tq satisfies Axiom 2, then Tq is uniquely given by the following form
Tq (ρ) = µq Sq (ρ),
with a positive constant number µq depending on the parameter q > 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 ) = λq1 Tq (1) + λq2 Tq (1) + Tq (λ1 , λ2 ),
which implies Tq (1) = 0. Moreover, by (T2) and (T3), when pn 6= 1, we have
Tq (p1 , . . . , pn−1 , λpn , (1 − λ) pn )
=
pqn Tq
q
(λ, 1 − λ) + (1 − pn ) Tq
p1
pn−1
,...,
1 − pn
1 − pn
+ Tq (pn , 1 − pn )
and
Tq (p1 , . . . , pn−1 , pn ) =
pqn Tq
q
(1) + (1 − pn ) Tq
p1
pn−1
,...,
1 − pn
1 − pn
+ Tq (pn , 1 − pn ) .
From these equations, we have
(1.4)
Tq (p1 , . . . , pn−1 , λpn , (1 − λ) pn ) = Tq (p1 , . . . , pn−1 , pn ) + pqn Tq (λ, 1 − λ) .
If we set λpn = y and (1 − λ)pn = z in (1.4), then for pn = y + z 6= 0 we have
y z
q
(1.5)
Tq (p1 , . . . , pn−1 , y, z) = Tq (p1 , . . . , pn−1 , pn ) + pn Tq
,
.
pn pn
Then for any x, y, z ∈ R such that 0 ≤ x, y < 1, 0 < z ≤ 1 and x + y + z = 1, we have
y
z
q
Tq (x, y, z) = Tq (x, y + z) + (y + z) Tq
,
y+z y+z
x
z
q
= Tq (y, x + z) + (x + z) Tq
,
.
x+z x+z
If we set tq (x) ≡ Tq (x, 1 − x), then we have
y
x
q
q
tq (x) + (1 − x) tq
= tq (y) + (1 − y) tq
.
1−x
1−y
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 5, 6 pp.
http://jipam.vu.edu.au/
4
S. F URUICHI , K. YANAGI ,
AND
K. K URIYAMA
Taking x = 0 and some y > 0, we have Tq (0, 1) = tq (0) = 0 for q 6= 0. Again setting λ = 0 in
(1.4) and using (T2), we have the reducing condition
Tq (p1 , . . . , pn , 0) = Tq (p1 , . . . , pn ).
Thus Tq satisfies all conditions of the generalized Faddeev’s axiom (F1), (F2) and (F3). Therefore we can apply Theorem 1.1 so that we obtain Tq (λ1 , . . . , λn ) = µq Hq (λ1 , . . . , λn ). Thus we
have Tq (ρ) = µq Sq (ρ), for density operator ρ.
Remark 1.3. For the special case q = 0 in the above theorem, we need the reducing condition
as an additional axiom.
2. A C ONTINUITY
OF
T SALLIS E NTROPY
We give a continuity property of the Tsallis entropy Sq (ρ). To do so, we state a few lemmas.
Lemma 2.1. For a density operator ρ on the finite dimensional Hilbert space H, we have
Sq (ρ) ≤ lnq d,
where d = dim H < ∞.
Proof. Since we have lnq z ≤ z − 1 for q ≥ 0 and z ≥ 0, we have
x−xq y 1−q
1−q
≥ x − y for x ≥ 0,
y ≥ 0, q ≥ 0 and q 6= 1, Therefore the Tsallis relative entropy [5]:
Dq (ρ|σ) ≡
Tr[ρ − ρq σ 1−q ]
1−q
for two commuting density operators ρ and σ, q ≥ 0 and q 6= 1, is nonnegative. Then we have
0 ≤ Dq (ρ| d1 I) = −dq−1 (Sq (ρ) − lnq d). Thus we have the present lemma.
Lemma 2.2. If f is a concave function and f (0) = f (1) = 0, then we have
|f (t + s) − f (t)| ≤ max {f (s), f (1 − s)}
for any s ∈ [0, 1/2] and t ∈ [0, 1] satisfying 0 ≤ s + t ≤ 1.
Proof.
(1) Consider the function r(t) = f (s) − f (t + s) + f (t). Then r0 (t) ≥ 0 since f 0 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 of l(t) = f (t + s) − f (t) + f (1 − s). Then l0 (t) ≤ 0. Thus we
have l(t) ≥ 0 by l(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| ≤ 21 , then |ηq (u) −
ηq (v)| ≤ ηq (|u − v|).
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 5, 6 pp.
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A G ENERALIZED FANNES ’ I NEQUALITY
5
Proof. Since ηq is a concave function with ηq (0) = ηq (1) = 0, we have
|ηq (t + s) − ηq (t)| ≤ max {ηq (s), ηq (1 − s)}
for s ∈ [0, 1/2] and t ∈ [0, 1] satisfying 0 ≤ 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 h00q (s) ≤ 0 for s ∈ [0, 1/2]. Therefore we have
hq (s) ≥ 0, which implies
max {ηq (s), ηq (1 − s)} = ηq (s).
Thus we have the present lemma by letting u = t + s and v = t.
Theorem 2.4. For two density operators ρ1 and ρ2 on the finite dimensional Hilbert space H
with dim H = d and q ∈ [0, 2], if kρ1 − ρ2 k1 ≤ q 1/(1−q) , then
|Sq (ρ1 ) − Sq (ρ2 )| ≤ kρ1 − ρ2 kq1 lnq d + ηq (kρ1 − ρ2 k1 ),
where we denote kAk1 ≡ Tr (A∗ A)1/2 for a bounded linear operator A.
(1)
(1)
(1)
(2)
(2)
(2)
Proof. Let λ1 ≥ λ2 ≥ · · · ≥ λd and λ1 ≥ λ2 ≥ · · · ≥ λd be eigenvalues of two density
operators ρ1 and ρ2 , respectively. (The degenerate
eigenvalues
are repeated according to their
Pd
(1)
(2) multiplicity.) We set ε ≡ j=1 εj and εj ≡ λj − λj . Then we have
εj ≤ ε ≤ kρ1 − ρ2 k1 ≤ q 1/(1−q) ≤
1
2
by Lemma 1.7 of [8]. Applying Lemma 2.3, we have
d d
X
X
(1)
(2) |Sq (ρ1 ) − Sq (ρ2 )| ≤
ηq (εj ).
η q λ j − ηq λ j ≤
j=1
j=1
By the formula lnq (xy) = lnq x + x1−q lnq y, we have
d
X
ηq (εj ) = −
j=1
d
X
εqj lnq εj
j=1
(
=ε −
d
X
εqj
j=1
(
=ε −
)
d
X
εqj
j=1
= εq
ε j
lnq
ε
ε
ε
d
X
ηq
j=1
d
q
εj X εj εj 1−q
lnq −
lnq ε
ε
ε
ε
ε
j=1
ε j
ε
)
+ ηq (ε)
q
≤ ε lnq d + ηq (ε).
In the above inequality, Lemma 2.1 was used for ρ = (ε1 /ε, . . . , εd /ε). Therefore we have
|Sq (ρ1 ) − Sq (ρ2 )| ≤ εq lnq d + ηq (ε).
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 5, 6 pp.
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6
S. F URUICHI , K. YANAGI ,
AND
K. K URIYAMA
Now ηq (x) is a monotone increasing function on x ∈ [0, q 1/(1−q) ]. In addition, xq is a monotone
increasing function for q ∈ [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, since limq→1 q 1/(1−q) = 1e .
Corollary 2.5. For two density operators ρ1 and ρ2 on the finite dimensional Hilbert space H
with dim H = d < ∞, if kρ1 − ρ2 k1 ≤ 1e , then
|S1 (ρ1 ) − S1 (ρ2 )| ≤ kρ1 − ρ2 k1 ln d + η1 (kρ1 − ρ2 k1 ),
where S1 represents the von Neumann entropy S1 (ρ) = Tr[η1 (ρ)] and η1 (x) = −x ln x.
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J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 5, 6 pp.
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