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.
Generalized Fannes’ Inequality
S. Furuichi, K. Yanagi and
K. Kuriyama
vol. 8, iss. 1, art. 5, 2007
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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.
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 Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of
Young Scientists (B), 17740068.
Dedicatory:
Dedicated to Professor Masanori Ohya on his 60th birthday.
Generalized Fannes’ Inequality
S. Furuichi, K. Yanagi and
K. Kuriyama
vol. 8, iss. 1, art. 5, 2007
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Contents
1
Introduction with Uniqueness Theorem of Tsallis Entropy
4
2
A Continuity of Tsallis Entropy
9
Generalized Fannes’ Inequality
S. Furuichi, K. Yanagi and
K. Kuriyama
vol. 8, iss. 1, art. 5, 2007
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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
P
(p(x)q − p(x)) X
Hq (X) ≡ x
=
η q (p(x))
1−q
x
+
with
P one parameter q ∈ R as an extension of Shannon entropy H1 (X) =
− x p(x) log p(x), for any probability distribution p(x) ≡ p(X = x) of a given
random variable X, where q-entropy function is defined by η q (x) ≡ −xq lnq x =
1−q
xq −x
and the q-logarithmic function lnq x ≡ x 1−q−1 is defined for q ≥ 0, q 6= 1 and
1−q
x ≥ 0.
P
The Tsallis entropy Hq (X) converges to the Shannon entropy − 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.
Generalized Fannes’ Inequality
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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) ≡
with a nonnegative real number q.
Tr[ρq − ρ]
= Tr[η q (ρ)],
1−q
Generalized Fannes’ Inequality
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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 of f -divergence [3].
Let Tq be a mapping on the set S(H) of all density operators to R+ .
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
k=1 λk ρk on H =
k=1 Hk , where
Pn mixing condition: For ρ =
λk ≥ 0, k=1 λk = 1, ρk ∈ S(Hk ), we have the additivity:
Tq (ρ) =
n
X
λqk Tq (ρk ) + Tq (λ1 , . . . , λn ),
Generalized Fannes’ Inequality
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k=1
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where (λ1 , . . . , λn ) represents the diagonal matrix (λk δ kj )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 )
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and
Tq (p1 , . . . , pn−1 , pn ) =
pqn Tq
q
(1)+(1 − pn ) Tq
p1
pn−1
,...,
+Tq (pn , 1 − pn ) .
1 − 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
Generalized Fannes’ Inequality
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If we set tq (x) ≡ Tq (x, 1 − x), then we have
x
y
q
q
tq (x) + (1 − x) tq
= tq (y) + (1 − y) tq
.
1−x
1−y
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 ).
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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. For the special case q = 0 in the above theorem, we need the reducing
condition as an additional axiom.
Generalized Fannes’ Inequality
S. Furuichi, K. Yanagi and
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vol. 8, iss. 1, art. 5, 2007
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2.
A Continuity of Tsallis Entropy
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,
Generalized Fannes’ Inequality
S. Furuichi, K. Yanagi and
where d = dim H < ∞.
K. Kuriyama
q 1−q
y
Proof. Since we have lnq z ≤ z − 1 for q ≥ 0 and z ≥ 0, we have x−x1−q
≥x−y
for x ≥ 0, y ≥ 0, q ≥ 0 and q 6= 1, Therefore the Tsallis relative entropy [5]:
vol. 8, iss. 1, art. 5, 2007
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Tr[ρ − ρq σ 1−q ]
Dq (ρ|σ) ≡
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).
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(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| ≤ 12 , then
|η q (u) − η q (v)| ≤ η q (|u − v|).
Proof. Since η q is a concave function with η q (0) = η q (1) = 0, we have
|η q (t + s) − η q (t)| ≤ max η q (s), η q (1 − s)
Generalized Fannes’ Inequality
S. Furuichi, K. Yanagi and
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vol. 8, iss. 1, art. 5, 2007
for s ∈ [0, 1/2] and t ∈ [0, 1] satisfying 0 ≤ t + s ≤ 1, by Lemma 2.2. Here we set
hq (s) ≡ η q (s) − η q (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 η q (s), η q (1 − s) = η q (s).
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Thus we have the present lemma by letting u = t + s and v = t.
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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 + η q (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
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(1)
(2) j=1 εj and εj ≡ λj − λj .
Pd
repeated according to their multiplicity.) We set ε ≡
Then we have
1
εj ≤ ε ≤ kρ1 − ρ2 k1 ≤ q 1/(1−q) ≤
2
by Lemma 1.7 of [8]. Applying Lemma 2.3, we have
d d
X
X
(2) (1)
|Sq (ρ1 ) − Sq (ρ2 )| ≤
η q (εj ).
η q λj − η q λj ≤
j=1
Generalized Fannes’ Inequality
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j=1
vol. 8, iss. 1, art. 5, 2007
By the formula lnq (xy) = lnq x + x1−q lnq y, we have
d
X
η q (εj ) = −
j=1
d
X
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εqj lnq εj
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j=1
(
=ε −
d
X
εqj
j=1
(
=ε −
=ε
d
X
j=1
ε j
ε
ε
d
X
εqj
j=1
q
ε
lnq
ηq
)
d
q
εj X εj εj 1−q
lnq −
lnq ε
ε
ε
ε ε
j=1
ε j
ε
)
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+ η q (ε)
≤ εq lnq d + η q (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 + η q (ε).
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Now η q (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
Generalized Fannes’ Inequality
S. Furuichi, K. Yanagi and
|S1 (ρ1 ) − S1 (ρ2 )| ≤ kρ1 − ρ2 k1 ln d + η 1 (kρ1 − ρ2 k1 ),
K. Kuriyama
vol. 8, iss. 1, art. 5, 2007
where S1 represents the von Neumann entropy S1 (ρ) = Tr[η 1 (ρ)] and η 1 (x) =
−x ln x.
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References
[1] J. ACZÉL AND Z. DARÓCZY, On Measures of Information and their Characterizations, Academic Press, 1975.
[2] R. ALICKI AND M. FANNES, Quantum Dynamical Systems, Oxford University Press, 2001.
[3] I. CSISZÁR, Information type measure of difference of probability distributions and indirect observations, Studia Sci. Math. Hunga., 2 (1967), 299–318.
[4] M. FANNES, A continuity property of entropy density for spin lattice systems,
Commun. Math. Phys., 31 (1973), 291–294.
[5] S. FURUICHI, K. YANAGI AND K. KURIYAMA, Fundamental properties of
Tsallis relative entropy, J. Math. Phys., 45 (2004), 4868–4877.
[6] S. FURUICHI, On uniqueness theorems for Tsallis entropy and Tsallis relative
entropy, IEEE Trans. on Information Theory, 51 (2005), 3638–3645.
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[7] M.A. NIELSEN AND I. CHUANG, Quantum Computation and Quantum Information, Cambridge Press, 2000.
Page 13 of 13
[8] M. OHYA AND D. PETZ, Quantum Entropy and its Use, Springer-Verlag,1993.
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[9] D. PETZ, Quasi-entropies for finite quantum system, Rep. Math. Phys., 23
(1986), 57–65.
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[10] C. TSALLIS, Possible generalization of Bolzmann-Gibbs statistics, J. Stat.
Phys., 52 (1988), 479–487.
[11] A. WEHRL, General properties of entropies, Rev. Mod. Phys., 50 (1978), 221–
260.
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