Matrix trace inequalities on Tsallis relative entropy of negative order

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Matrix trace inequalities on Tsallis relative entropy of negative order Yuki Seo Department of Mathematics Education, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka 582-8582, Japan

a r t i c l e

i n f o

Article history: Received 11 August 2018 Available online xxxx Submitted by M. Mathieu Keywords: Tsallis relative entropy Relative operator entropy Quasi geometric mean Positive definite matrix

a b s t r a c t In this paper, we show matrix trace inequalities on the quantum Tsallis relative entropy of negative order, which includes the quasi geometric mean of positive definite matrices. Moreover, we show an estimate of the difference between two Tsallis relative entropies of negative order. © 2018 Elsevier Inc. All rights reserved.

1. Introduction As a quantum extension of the Shannon entropy [21], von Neumann [18] defined the entropy of the density matrix ρ by the formula S(ρ) = Tr[η(ρ)], where the entropy function η(t) = −t log t. As for the Shannon entropy, it is extremely useful to define a quantum version of the relative entropy. Suppose that ρ and σ are density matrices. The quantum relative entropy of ρ with respect to σ is defined by  SU (ρ|σ) =

Tr[ρ(log ρ − log σ)] +∞

if supp ρ ≤ supp σ, otherwise,

(1.1)

which was firstly introduced in the setting of von Neumann algebra by Umegaki [22] in 1962. This is a quantum generalization of the relative entropy due to Kullback and Leibler [15]. In [1], the Tsallis relative entropy of ρ to σ is defined by E-mail address: [email protected]. https://doi.org/10.1016/j.jmaa.2018.12.005 0022-247X/© 2018 Elsevier Inc. All rights reserved.

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Dα (ρ|σ) =

1 − Tr[ρ1−α σ α ] = Tr[ρ1−α (lnα ρ − lnα σ)] α

(1.2)

for any 0 < α ≤ 1, where lnα ρ = ρ α−1 is the α-logarithmic function. The Tsallis relative entropy (1.2) is one parameter extension of the Umegaki relative entropy (1.1), and Ruskai and Stillinger [20] showed the following relation between the Tsallis relative entropy and the Umegaki relative entropy: α

   1+α −α σ −ρ ρ1−α σ α − ρ ρ ≤ SU (ρ|σ) ≤ Tr = D−α (ρ|σ) Dα (ρ|σ) = −Tr α α 

(1.3)

for all 0 < α ≤ 1. On the other hand, there are another formulation of the quantum relative entropy: Fujii and Kamei [6] introduced the relative operator entropy which is a relative version of the operator entropy defined by Nakamura–Umegaki [17]: For positive definite matrices ρ and σ, the relative operator entropy is defined by S(ρ|σ) = ρ1/2 (log ρ−1/2 σρ−1/2 )ρ1/2 . By virtue of the relative operator entropy, we define the quantum relative entropy as SF K (ρ|σ) = −Tr[S(ρ|σ)].

(1.4)

The quantum quantity Tr[ρ(log ρ1/2 σ −1 ρ1/2 )] is firstly proposed by Belavkin and Staszewski [5] in the framework of C∗ -algebra. Since we treat SF K (ρ|σ) as the minus of the trace of the relative operator entropy S(ρ|σ), we call (1.4) the FK relative entropy, or the BS relative entropy in [19, pp125]. If ρ and σ commute, then we have SU (ρ|σ) = SF K (ρ|σ). Generally, two quantum formulations of the relative entropy are different. In fact, Hiai and Petz [13] showed the following relation: SU (ρ|σ) ≤ SF K (ρ|σ).

(1.5)

Moreover, Yanagi, Kuriyama and Furuichi [23] have been advancing research on the Tsallis relative operator entropy as an operator generalization of the Tsallis relative entropy, which is regarded as a parametric extension of the relative operator entropy by Fujii–Kamei: For positive definite matrices ρ and σ, the Tsallis relative operator entropy is defined by Tα (ρ|σ) =

ρ α σ − ρ α

for 0 < α ≤ 1,

where the matrix α-geometric mean is defined by ρ α σ = ρ1/2 (ρ−1/2 σρ−1/2 )α ρ1/2 , also see [14]. Since S(ρ|σ) ≤

ρ α σ−ρ α

for all 0 < α ≤ 1, we have −Tr[Tα (ρ|σ)] ≤ SF K (ρ|σ).

(1.6)

Compared (1.3) with (1.6), Furuichi, Yanagi and Kuriyama [9] showed the following relation which is an extension of the Hiai–Petz inequality (1.5): Dα (ρ|σ) ≤ −Tr[Tα (ρ|σ)] for all 0 < α ≤ 1. In fact, if we put α → 0, then we have (1.5).

(1.7)

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Since   1+α −α  ρ ρ1+α σ −α − ρ σ −ρ = −Tr Tr α −α 

in (1.3) and −1 ≤ −α < 0, it suggests the formulation of the Tsallis relative operator entropy of negative order α ∈ [−1, 0): We use the notation α for the binary operation ρ α σ = ρ1/2 (ρ−1/2 σρ−1/2 )α ρ1/2

for α ∈ [−1, 0),

(1.8)

whose formula is the same as α . Though ρ α σ for α ∈ [−1, 0) are not matrix mean in the sense of Kubo–Ando theory [14], ρ α σ have matrix mean like properties for positive semidefinite matrices ρ and σ. Thus we call (1.8) the α-quasi geometric mean for α ∈ [−1, 0). Then the Tsallis relative operator entropy of negative order is defined by Tα (ρ|σ) =

ρ α σ − ρ α

for α ∈ [−1, 0).

The α-quasi geometric mean and the Tsallis relative operator entropy of negative order are discussed in detail in [7]. In this paper, we show matrix trace inequalities on the quantum Tsallis relative entropy of negative order, which includes the quasi geometric mean of positive semidefinite matrices. It is a complement of the inequality (1.7) due to Yanagi–Kuriyama–Furuichi. Moreover, we show an estimate of the difference between two Tsallis relative entropies of negative order. 2. Matrix trace inequalities related to Tsallis relative entropies Let Mn = Mn (C) be the algebra of n × n complex matrices and denote the matrix absolute value of any ρ by |ρ| = (ρ∗ ρ)1/2 . First of all, we introduce another quantum Tsallis relative entropy of negative order α ∈ [−1, 0) defined by  N Tα (ρ|σ) = −Tr[Tα (ρ|σ)] = −Tr

ρ α σ − ρ α



for positive definite matrices ρ and σ. Since ρ α (σ + ε) is monotone increasing on ε  0, it follows that Tα (ρ|σ + ε) is monotone decreasing on ε  0 and thus for non-invertible case we can define the quantum Tsallis relative entropy of negative order as N Tα (ρ|σ) = lim N Tα (ρ|σ + ε) ε→0

if the limit exists. We recall the equivalent conditions around existence of ρ α σ for positive semidefinite matrices ρ, σ and α ∈ [−1, 0). The following three conditions are mutually equivalent: (1) range inclusion; ρ ≤ cσ for some c > 0, i.e., ranρ1/2 ⊂ ranσ 1/2 . (2) existence condition; ρ α σ exists as a matrix. (3) kernel inclusion; kerρ ⊃ kerσ. Hence, in the case of kerρ ⊃ kerσ, it follows that N Tα (ρ|σ + ε) has the well-defined limit as ε → 0 for each α ∈ [−1, 0). We have the following properties of the quantum relative entropy N Tα of negative order α ∈ [−1, 0) for non-invertible case, also see [7]: Proposition 2.1. Let ρ and σ be (non-invertible) density matrices in Mn . If Tα (ρ|σ) exists for some α ∈ [−1, 0), then the following properties of the quantum Tsallis relative entropy N Tα hold:

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(1) (Non-negativity) N Tα (ρ|σ) ≥ 0. (2) (Pseudoadditivity) N Tα (ρ1 ⊗ ρ2 |σ1 ⊗ σ2 ) = N Tα (ρ1 |σ1 ) + N Tα (ρ2 |σ2 ) + αN Tα (ρ1 |σ1 )N Tα (ρ2 |σ2 ).    (3) (Joint convexity) N Tα ( j λj ρj | j λj σj ) ≤ j λj N Tα (ρj |σj ). (4) (Monotonicity) For any trace-preserving positive linear map Φ N Tα (Φ(ρ)|Φ(σ)) ≤ N Tα (ρ|σ). Proof. For the sake of convenience, we give a proof. For (1), since ρ α σ exists for some α ∈ [−1, 0), it follows that (1 − α)ρ + ασ ≤ ρ α σ for non-invertible case and so N Tα (ρ|σ) ≥ −Tr[σ − ρ] = 0. For (2), since ρ1 α σ1 and ρ2 α σ2 exist for some α ∈ [−1, 0), it follows that (ρ1 ⊗ ρ2 ) α (σ1 + ε) ⊗ (σ2 + ε) = [ρ1 α (σ1 + ε)] ⊗ [ρ2 α (σ2 + ε)] ≤ [ρ1 α σ1 ] ⊗ [ρ2 α σ2 ] for all ε > 0 and so Tα (ρ1 ⊗ ρ2 |σ1 ⊗ σ2 ) exists. By a similar way of [10, Theorem 4.2], we have Tα (ρ1 ⊗ ρ2 |σ1 ⊗ σ2 ) = αTα (ρ1 |σ1 ) ⊗ Tα (ρ2 |σ2 ) + Tα (ρ1 |σ1 ) ⊗ ρ2 + ρ1 ⊗ Tα (ρ2 |σ2 ) and hence we have the desired equality (2).   For (3), since Tα (ρj |σj ) exist for each j, it follows from [7, Theorem 3.1] that Tα ( j λj ρj | j λj σj ) exists and the Tsallis relative operator entropy of negative order satisfies the sub-additivity and homogeneity for non-invertible case and thus we have (3). For (4), since Tα (ρ|σ) exists for some α ∈ [−1, 0), it follows that Tα (Φ(ρ)|Φ(σ)) exists and Φ(Tα (ρ|σ)) ≤ Tα (Φ(ρ)|Φ(σ)). Hence we have (4). 2 For two quantum relative entropies, we know the following two relations: SU (ρ|σ) ≤ Dα (ρ|σ)

for −1 ≤ α < 0

and SF K (ρ|σ) ≤ N Tα (ρ|σ)

for −1 ≤ α < 0.

(2.1)

In fact, the inequality (2.1) follows from ρ αασ−ρ ≤ S(ρ|σ) for all −1 ≤ α < 0. It is natural to ask the relation between Dα (ρ|σ) and N Tα (ρ|σ) for the case of α ∈ [−1, 0). We have the following result, which is a complement of the inequality (1.7) due to Yanagi–Kuriyama–Furuichi. For this, we recall the Moore–Penrose generalized inverse. For any matrix ρ, there exists a unique matrix ρ† such that the following four properties hold: (1) ρρ† ρ = ρ. (2) ρ† ρρ† = ρ† . (3) (ρρ† )∗ = ρρ† . (4) (ρ† ρ)∗ = ρ† ρ. Then ρ† is called the Moore–Penrose generalized inverse of ρ. If ρ is invertible, then we have ρ† = ρ−1 . Theorem 2.2. Let ρ and σ be positive semidefinite matrices in Mn . If ρ α σ exists for some α ∈ [−1, 0), then

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 −Tr

5

   ρ α σ − ρ ρ1−α (σ † )−α − ρ ≤ −Tr , α α

(2.2)

where σ † is the Moore–Penrose generalized inverse of σ. In particular, if σ is positive definite, then Dα (ρ|σ) ≤ N Tα (ρ|σ)

for all α ∈ [−1, 0).

(2.3)

Proof. Since ρ α σ exists for some α ∈ [−1, 0), the existence condition implies the kernel condition ker ρ ⊃ ker σ. If rank(σ) = r < n, then σ has the matrix form σ = σr ⊕ 0 on ran σ ⊕ ker σ, where σr is positive definite in Mr . Then the Moore–Penrose generalized inverse of σ is σ † = σr−1 ⊕ 0. Since ker ρ ⊃ ker σ holds, we have the matrix form of ρ on ran σ ⊕ ker σ given by ρ = ρr ⊕ 0, where ρr is positive semidefinite in Mr . Then we have  −α ρ α σ = ρ1/2 ρ1/2 σ † ρ1/2 ρ1/2 . (2.4) In fact, it follows that ρ α σ = (ρr ⊕ 0) α (σr ⊕ 0) = (ρr α σr ) ⊕ (0 α 0) = (ρr1/2 (ρ1/2 σr−1 ρr1/2 )−α ρr1/2 ) ⊕ 0  −α = ρ1/2 ρ1/2 σ † ρ1/2 ρ1/2 and so (2.4) holds. Bebiano, Lemos and Providencia in [4, Theorem 2.1] showed that  1+t    t  2 t 1+t    ρ σ ρ 2  ≤ ρ1/2 (ρs/2 σ s ρs/2 ) s ρ1/2 

(2.5)

for all s ≥ t ≥ 0 and any unitarily invariant norm |||·|||. Since |||ρ||| = Tr[|ρ|] for ρ ∈ Mn is unitarily invariant norm, putting s = 1 and t = −α in (2.5) we have Tr[ρ α σ] = Tr[ρ1/2 (ρ1/2 σ † ρ1/2 )−α ρ1/2 ] ≥ Tr[ρ

1−α 2

(σ † )−α ρ

1−α 2

]

= Tr[ρ1−α (σ † )−α ] and hence we have the desired inequality (2.2). If rank(σ) = n, then σ is nonsingular and σ † = σ −1 . Since ρ α σ exists for all α ∈ [−1, 0), we have the desired inequality (2.3): Dα (ρ|σ) = −Tr[

ρ α σ − ρ ρ1−α σ α − ρ ] ≤ −Tr[ ] = −Tr[Tα (ρ|σ)] = N Tα (ρ|σ). α α

2

In [13], Hiai and Petz showed an extension of (1.5): For positive definite matrices ρ and σ, 1 SU (ρ|σ) = Tr[ρ(log ρ − log σ)] ≤ − Tr[ρ(log ρ−q/2 σ q ρ−q/2 )] q

for all q > 0.

(2.6)

Furuichi [8] showed the following extension from viewpoint of the Tsallis relative entropy: For each 0 < α ≤ 1, Dα (ρ|σ) ≤ −Tr[ρ lnα (ρ−q/2 σ q ρ−q/2 )1/q ]

for all q ≥ 1.

(2.7)

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On the other hand, for −1 ≤ α < 0, we have SF K (ρ|σ) ≤ N Tα (ρ|σ) and 1 − Tr[ρ(log ρ−q/2 σ q ρ−q/2 )] ≤ −Tr[ρ lnα (ρ−q/2 σ q ρ−q/2 )1/q ] q

for all q > 0.

We show a complement of the inequality (2.7) for −1 ≤ α < 0: Theorem 2.3. Let ρ and σ be positive definite matrices in Mn and −1 ≤ α < 0. Then Dα (ρ|σ) ≤ −Tr[ρ lnα (ρ−q/2 σ q ρ−q/2 )1/q ]

for all q ≥ −α > 0.

Proof. If we put s = q and t = −α in (2.5), then we have Tr[ρ1−α σ α ] = Tr[ρ

1−α 2

σα ρ

1−α 2

]

≤ Tr[ρ1/2 (ρq/2 σ −q ρq/2 )−α/q ρ1/2 ] = Tr[ρ1/2 (ρ−q/2 σ q ρ−q/2 )α/q ρ1/2 ] for all q ≥ −α > 0. Hence for each α ∈ [−1, 0) Dα (ρ|σ) = −Tr[ ≤ −Tr[

ρ1−α σ α − ρ ] α ρ1/2 (ρ−q/2 σ q ρ−q/2 )α/q ρ1/2 − ρ ] α

= −Tr[ρ lnα (ρ−q/2 σ q ρ−q/2 )1/q ] for all q ≥ −α > 0. 2 Remark 2.4. If we put q = 1 in Theorem 2.3, then we have (2.3). If we put α → 0 in Theorem 2.3, then we have (2.6). Next, we estimate the upper bound of quantum Tsallis relative entropy of negative order: Theorem 2.5. Let ρ and σ be positive semidefinite matrices in Mn . If ρ α σ exists for some −1 ≤ α ≤ −1/2, then N Tα (ρ|σ) ≤ −Tr[

|(σ † )−αq ρ(1−α)q |1/q − ρ ] α

for all q ≥ 1,

(2.8)

where σ † is the Moore–Penrose generalized inverse of σ. In particular, if σ is positive definite, then for each α ∈ [−1, −1/2] N Tα (ρ|σ) ≤ −Tr[

|σ αq ρ(1−α)q |1/q − ρ ] α

for all q ≥ 1.

(2.9)

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Proof. Firstly, suppose that ρ and σ are positive definite. Then we show that for each −1 ≤ α < 0 Tr[ρ α σ] ≤ Tr[(ρq α σ q )1/q ]

for all q ≥ 1.

(2.10)

For this, by the method of the k-fold antisymmetric tensor power, it is suffice to prove that ρ α σ ≤ I implies ρq α σ q ≤ I for all 0 < q ≤ 1, also see [2]. As a matter of fact, ρq α σ q ∞ ≤ (ρ α σ)q ∞ holds for all 0 < q ≤ 1 if and only if (ρq α σ q )1/q ∞ ≤ (ρp α σ p )1/p ∞ holds for all 0 < q ≤ p, where · ∞ denotes the operator norm. Hence we have Tr[(ρp α σ p )1/p ] ≤ Tr[(ρq α σ q )1/q ] for all 0 < p ≤ q, and if we put p = 1, then we have (2.10). Let us assume 1/2 ≤ q ≤ 1 and write q = 1 − ε with 0 ≤ ε ≤ 1/2. Let ω = ρ1/2 σ −1 ρ1/2 . Since ρ α σ ≤ I, then ρ ≤ ω α and we now get ρq α σ q = ρ 2 −ε ρ 2 [ρ− 2 ρ 2 (ρ− 2 ωρ− 2 )1−ε ρ 2 ρ− 2 ]−α ρ 2 ρ 2 −ε 1

ε

ε

1

1

1

1

ε

ε

1

= ρ 2 −ε [ρε −α (ρ 1−ε ω)]ρ 2 −ε 1

1

≤ ρ 2 −ε [ω αε −α (ω α 1−ε ω)]ρ 2 −ε 1

1

= ρ 2 −ε ω α(2ε−1) ρ 2 −ε 1

1

≤ ρ 2 −ε ρ−1+2ε ρ 2 −ε = I 1

1

and so (ρq α σ q )1/q ≤ I. When 0 < q < 1/2, writing q = 2−k (1 − ε) with k ∈ N and 0 ≤ ε ≤ 1/2. We can proceed by induction. Hence it follows that ρ α σ ≤ I implies ρq α σ q ≤ I for all 0 < q ≤ 1, and thus we have (2.10) by the above consideration. Next, suppose that ρ and σ are positive semidefinite and ρ α σ exists for some α ∈ [−1, 0). Then ker ρ ⊃ ker σ and so ker ρq ⊃ ker σ q for all q ≥ 1. Hence ρq α σ q exists for all q ≥ 1. For each ε > 0 we have Tr[ρ α (σ + ε)] ≤ Tr[(ρq α (σ + ε)q )1/q ] and as ε → 0 we have (2.10) for positive semidefinite ρ and σ. Now, suppose that ρ and σ are positive semidefinite and ρ α σ exists for some α ∈ [−1, −1/2]. Since ker ρq ⊃ ker σ q and (σ † )q = (σ q )† for all q ≥ 1, it follows from (2.4) that ρq α σ q = ρq/2 (ρq/2 (σ † )q ρq/2 )−α ρq/2 .

(2.11)

Araki’s inequality [3] says that Tr[(σ 1/2 ρσ 1/2 )st ] ≤ Tr[(σ t/2 ρt σ t/2 )s ]

for all t ≥ 1 and s > 0

(2.12)

for all 0 < t ≤ 1 and s > 0.

(2.13)

or, equivalently Tr[(σ t/2 ρt σ t/2 )s/t ] ≤ Tr[(σ 1/2 ρσ 1/2 )s ] Then for −1 ≤ α ≤ − 12 , we have Tr[ρ α σ] ≤ Tr[(ρq α σ q )1/q ] q/2

= Tr[(ρ

≤ Tr[(ρ−

q/2

(ρ

q(1−α) 2α

by (2.10)

† q q/2 −α q/2 1/q

(σ ) ρ

(σ † )q ρ−

)

q(1−α) 2α

ρ

)

)− q ] α

≤ Tr[(ρq(1−α) (σ † )−2αq ρq(1−α) )1/2q ] = Tr[|(σ † )−αq ρ(1−α)q |1/q ] for all q ≥ 1 and this implies

]

by (2.11) by (2.12) and 1 ≤ − α1 1 by (2.13) and 0 < − 2α ≤1

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ρ α σ − ρ ] α |(σ † )−αq ρ(1−α)q |1/q − ρ ] ≤ −Tr[ α

N Tα (ρ|σ) = −Tr[

for all q ≥ 1 and so we have (2.8). If σ is positive definite, then σ † = σ −1 and ρ α σ exists for all α ∈ [−1, 0). Thus we have the desired inequality (2.9). 2 3. Difference of two Tsallis relative entropies In [11], by using an estimate of the difference between two Tsallis relative entropies Dα(ρ|σ) and N Tα (ρ|σ) for α ∈ (0, 1], Furuta estimated the difference between the Umegaki relative entropy and the FK relative entropy. In this section, we show an estimate of the difference between two Tsallis relative entropies of negative order: As a result, we obtain the result due to Furuta. For this, we recall two important constants. One is the generalized Kantorovich constant, which estimates a converse of Jensen’s inequality for convex functions: If ρ is an Hermitian matrix in Mn such that mI ≤ ρ ≤ M I for some scalars m ≤ M where I stands for the identity matrix in Mn , and h = M m , then it follows from [12, Theorem 3.18] that Φ(ρα ) ≤ K(h, α)Φ(ρ)α

for α ∈ / [0, 1]

(3.1)

for every unital positive linear map Φ on Mn , where the generalized Kantorovich constant K(h, α) is defined by hα − h K(h, α) = (α − 1)(h − 1)



α − 1 hα − 1 α hα − h

α (3.2)

for any real numbers α ∈ R, also see [12, Definition 2.2]. Another is the Specht ratio, which estimates the upper bound of the arithmetic mean by the geometric one for positive numbers: For x1 , . . . , xn ∈ [m, M ] with M ≥ m > 0, √ x1 + x2 + · · · + xn ≤ S(h) n x1 x2 · · · xn n where h =

M m

and the Specht ratio S(h) is defined by 1

(h − 1)h h−1 (h = 1) S(h) = e log h

and S(1) = 1,

(3.3)

also see [12, Page 71]. Furuta [11] showed the following most crucial result on the generalized Kantorovich constant and the Specht ratio: Let h > 1. Then S(h) = e−K



(0)

= eK



(1)

where K(α) = K(h, α) for all α ∈ R, also see [12, Theorem 2.57]. In particular, we have limα→0 1−K(h,α) = α log S(h) since K(0) = 1. We show an estimate of the difference between two Tsallis relative entropies Dα (ρ|σ) and N Tα (ρ|σ) for α ∈ [−1, 0): Theorem 3.1. Let ρ and σ be positive definite matrices in Mn such that m1 I ≤ ρ ≤ M1 I and m2 I ≤ σ ≤ M2 I 1 M2 for some scalars 0 < m1 ≤ M1 and 0 < m2 ≤ M2 . Put h = M m1 m2 . Then for each α ∈ [−1, 0)

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0 ≤ N Tα (ρ|σ) − Dα (ρ|σ) ≤

9

1 − K(h, α) Tr[ρ1−α σ α ], α

where the generalized Kantorovich constant K(h, α) is defined by (3.2). To prove it, we need some preliminaries. Lemma 3.2. Let ρ and σ be positive definite matrices in Mn such that m1 I ≤ ρ ≤ M1 I and m2 I ≤ σ ≤ M2 I 1 M2 for some scalars 0 < m1 ≤ M1 and 0 < m2 ≤ M2 . Put h = M m1 m2 . Let Φ be a unital positive linear map on Mn . Then for each α ∈ [−1, 0) Φ(ρ α σ) ≤ K(h, α) Φ(ρ) α Φ(σ),

(3.4)

where the generalized Kantorovich constant K(h, α) is defined by (3.2). Proof. Put Ψ(ψ) = Φ(ρ)− 2 Φ(ρ 2 ψρ 2 )Φ(ρ)− 2 and so Ψ is a unital positive linear map on Mn . For α ∈ [−1, 0), it follows from (3.1) that 1

1

1

1

Ψ(ψ α ) ≤ K(h, α)Ψ(ψ)α and thus we have Φ(ρ α σ) = Φ(ρ 2 (ρ− 2 σρ− 2 )α ρ 2 ) = Φ(ρ) 2 Ψ((ρ− 2 σρ− 2 )α )Φ(ρ) 2 1

1

1

1

1

1

1

1

≤ K(h, α)Φ(ρ) 2 Ψ(ρ− 2 σρ− 2 )α Φ(ρ) 2 = K(h, α) Φ(ρ) α Φ(σ) 1

1

1

1

as desired. 2 Lemma 3.3. Let ρ and σ be positive definite matrices in Mn . Then for each α ∈ [−1, 0) Tr[ρ]1−α Tr[σ]α ≤ Tr[ρ1−α σ α ]. Proof. Though we prove it by a similar way in [16, Theorem 6], we give a proof for the convenience of 1 1 1 the reader. Put p = 1−α and q = α1 , and so p1 + 1q = 1. Put ψ = σ q /Tr[σ] q and Tr[ψ q ] = 1. Then we 1

see Tr[ρψ] ≥ Tr[ρp ] p . In fact, there exists a unitary matrix U in Mn such that the spectral decomposition  U ∗ ψU = D = diag(λ1 , . . . , λn ). Put Y = (aij ) = U ∗ ρU . Then we have Tr[ρψ] = Tr[Y D] = aii λi and  q  p 1  1 q q p Tr[ψ ] = Tr[D ] = λi = 1. The Hölder inequality implies aii λi ≥ ( aii ) for p ∈ [ 2 , 1] and it is known  p 1 1 1 1 that Tr[ρp ] ≤ aii for p ∈ [ 12 , 1]. Hence it follows that Tr[ρψ] ≥ Tr[Y p ] p = Tr[ρp ] p . Since Tr[ρ p ψ] ≥ Tr[ρ] p by the result above, we have the desired inequality 1

1

1

1

Tr[ρ1−α σ α ] = Tr[ρ p σ q ] ≥ Tr[ρ] p Tr[σ] q = Tr[ρ]1−α Tr[σ]α . Proof of Theorem 3.1. Replacing Φ(ψ) by

1 n Tr[ψ]

2

in (3.4) of Lemma 3.2, we have

Tr[ρ α σ] ≤ K(h, α)Tr[ρ]1−α Tr[σ]α ≤ K(h, α)Tr[ρ1−α σ α ]

by Lemma 3.3

and thus 1 Tr[ρ α σ] − Tr[ρ1−α σ α ] α 1 − K(h, α) 1 K(h, α)Tr[ρ1−α σ α ] − Tr[ρ1−α σ α ] = Tr[ρ1−α σ α ]. ≤− α α

0 ≤ N Tα (ρ|σ) − Dα (ρ|σ) ≤ −

2

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If we put α → 0 in Theorem 3.1, then we have the following theorem due to Furuta [11, Corollary 2]: Theorem 3.4 (Furuta). Let ρ and σ be positive definite matrices in Mn such that m1 I ≤ ρ ≤ M1 I and 1 M2 m2 I ≤ σ ≤ M2 I for some scalars 0 < m1 ≤ M1 and 0 < m2 ≤ M2 . Put h = M m1 m2 . Then 0 ≤ SF K (ρ|σ) − SU (ρ|σ) ≤ log S(h) Tr[ρ]

(3.5)

where the Specht ration S(h) is defined by (3.3). Acknowledgments This work is supported by Grant-in-Aid for Scientific Research (C), JSPS KAKENHI Grant Number JP 16K05253. The author would like to thank the anonymous referee for his or her reading and valuable comments. References [1] S. Abe, Monotone decrease of the quantum nonadditive divergence by projective measurements, Phys. Lett. A 312 (2003) 336–338. [2] T. Ando, F. Hiai, Log-majorization and complementary Golden–Thompson type inequalities, Linear Algebra Appl. 197 (198) (1994) 113–131. [3] H. Araki, On an inequality of Lieb and Thirring, Lett. Math. Phys. 19 (1990) 167–170. [4] N. Bebiano, R. Lemos, J. da Providencia, Inequalities for quantum relative entropy, Linear Algebra Appl. 401 (2005) 159–172. [5] V.P. Belavkin, P. Staszewski, C∗ -algebraic generalization of relative entropy and entropy, Ann. Inst. Henri Poincaré A 37 (1982) 51–58. [6] J.I. Fujii, E. Kamei, Relative operator entropy in noncommutative information theory, Math. Jpn. 34 (1989) 341–348. [7] J.I. Fujii, Y. Seo, Tsallis relative operator entropy with negative parameters, Adv. Oper. Theory 1 (2) (2016) 219–236. [8] S. Furuichi, Matrix trace inequalities on the Tsallis entropies, JIPAM. J. Inequal. Pure Appl. Math. 9 (2008), Article 1, 7 pp. [9] S. Furuichi, K. Yanagi, K. Kuriyama, Fundamental properties of Tsallis relative entropy, J. Math. Phys. 45 (2004) 4868–4877. [10] S. Furuichi, K. Yanagi, K. Kuriyama, A note on operator inequalities of Tsallis relative operator entropy, Linear Algebra Appl. 407 (2005) 19–31. [11] T. Furuta, Reverse inequalities involving two relative operator entropies and two relative entropies, Linear Algebra Appl. 403 (2005) 24–30. [12] T. Furuta, H. Mićić, J. Pečarić, Y. Seo, Mond–Pecaric Method in Operator Inequalities, Element, Zagreb, 2005. [13] F. Hiai, D. Petz, The Golden–Thompson trace inequality is complemented, Linear Algebra Appl. 181 (1993) 153–185. [14] F. Kubo, T. Ando, Means of positive linear operators, Math. Ann. 246 (1980) 205–224. [15] S. Kullback, R. Leibler, On information and sufficiency, Ann. Math. Stat. 22 (1951) 79–86. [16] J.R. Magnus, A representation theorem for (trAp )1/p , Linear Algebra Appl. 95 (1987) 127–134. [17] M. Nakamura, H. Umegaki, A note on the entropy for operator algebras, Proc. Japan Acad. 37 (1961) 149–154. [18] J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, NJ, 1955 (Originally appeared in German in 1932). [19] M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer, 2004, Corrected second printing. [20] M.B. Ruskai, F.H. Stillinger, Convexity inequalities for estimating free energy and relative entropy, J. Phys. A 23 (1990) 2421–2437. [21] C.E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27 (1948) 623–656. [22] H. Umegaki, Conditional expectation in an operator algebra, IV (entropy and information), Kodai Math. Semin. Rep. 14 (1962) 59–85. [23] K. Yanagi, K. Kuriyama, S. Furuichi, Generalized Shannon inequalities based on Tsallis relative operator entropy, Linear Algebra Appl. 394 (2005) 109–118.