Tsallis Relative Operator Entropy Research Articles

Some closer estimations for Tsallis relative operator entropy

In this paper, we find some new and refined bounds for the Tsallis relative operator entropy. We give the lower and upper bounds for the Tsallis relative operator entropy such that the lower bound is an ascending sequence that ascends to the Tsallis relative operator entropy and the upper bound is a descending sequence that descends to the Tsallis relative operator entropy. Moreover, we present some new and refined bounds for the relative operator entropy and the generalized relative operator entropy.

Monotonicity of a trace related to Tsallis relative operator entropy

Monotonicity of a trace related to Tsallis relative operator entropy

On a new generalized Tsallis relative operator entropy

In this paper, we present a generalization of Tsallis relative operator entropy defined for positive operators and we investigate some related properties. Some inequalities involving the generalized Tsallis relative operator entropy are pointed out as well.

Sharp bounds for the Tsallis relative operator entropy

We give the sharp and refined bounds for the Tsallis relative operator entropy by using the improved Hermite–Hadamard inequality. As an application of our result, we give new bounds for the generalized relative operator entropy. We refine the bounds of the relative operator entropy announced by Fujii and Kamei and provide the precise and sharp bounds.

Some operator inequalities via convexity

In this article, we employ a standard convexity argument to obtain new and refined inequalities related to the matrix mean of two accretive matrices, the numerical radius and the Tsallis relative operator entropy.

Further inequalities involving the weighted geometric operator mean and the Heinz operator mean

ABSTRACT In this paper, we first investigate some inequalities involving the p-weighted geometric operator mean where is a real number and A, B are two positive invertible operators acting on a Hilbert space. As applications, we obtain some inequalities about the so-called Tsallis relative operator entropy. We also give some inequalities involving the Heinz operator mean. Our results refine some inequalities existing in the literature. In a second part, we construct iterative algorithms converging to with a high rate of convergence. Some relationships involving are deduced. Numerical examples illustrating the theoretical results are also discussed.

Matrix trace inequalities related to the Tsallis relative entropies of real order

In this paper, we show matrix trace inequalities related to the Tsallis relative entropy of real order: For positive definite matrices ρ and σ, and each 0<α≤1Dα(ρ|σ)≤−Tr[ρ1−qqTαq(ρq|σq)] for all q≥α>0, where the Tsallis relative entropy Dα(ρ|σ) is defined by Dα(ρ|σ)=−Tr(ρ1−ασα−ρα) and the Tsallis relative operator entropy Tα(ρ|σ) is defined by Tα(ρ|σ)=ρ♯ασ−ρα, where ♯α is the matrix α-geometric mean. Moreover, we show estimates of the difference between two Tsallis relative entropies of real order.

Quantifying quantum coherence based on the Tsallis relative operator entropy

Coherence is a fundamental ingredient in quantum physics and a key resource in quantum information processing. The quantification of quantum coherence is of great importance. We present a family of coherence quantifiers based on the Tsallis relative operator entropy. Shannon inequality and its reverse one in Hilbert space operators derived by Furuta [Linear Algebra Appl. 381 (2004) 219] are extended in terms of the parameter of the Tsallis relative operator entropy. These quantifiers are shown to satisfy all the standard criteria for a well-defined measure of coherence and include some existing coherence measures as special cases. Detailed examples are given to show the relations among the measures of quantum coherence.

Some remarks on Tsallis relative operator entropy

This paper intends to give some new estimates for Tsallis relative operator entropy ${{T}_{v}}\left( A|B \right)=\frac{A{{\natural}_{v}}B-A}{v}$. Let $A$ and $B$ be two positive invertible operators with the spectra contained in the interval $J \subset (0,\infty)$. We prove for any $v\in \left[ -1,0 \right)\cup \left( 0,1 \right]$, $$ (\ln_v t)A+\left( A{{\natural}_{v}}B+tA{{\natural}_{v-1}}B \right)\le {{T}_{v}}\left( A|B \right) \le (\ln_v s)A+{{s}^{v-1}}\left( B-sA \right) $$ where $s,t\in J$. Especially, the upper bound for Tsallis relative operator entropy is a non-trivial new result. Meanwhile, some related and new results are also established. In particular, the monotonicity for Tsallis relative operator entropy is improved. Furthermore, we introduce the exponential type relative operator entropies which are special cases of the perspective and we give inequalities among them and usual relative operator entropies.

Inequalities for the generalized Tsallis relative operator entropy

In this paper, we present some inequalities for the generalized relative operator entropy according to the generalized Tsallis relative operator entropy. Our results are generalizations of some existing inequalities.

Some New Karamata Type Inequalities and Their Applications to Some Entropies

Some new inequalities of Karamata type are established with a convex function. The methods of our proof allow us to obtain an extended version of the reverse of Jensen inequality given by PecnaricA and MicAicA. Applying the obtained results, we give reverses for information inequality (Shannon inequality) in different types, namely ratio type and difference type, under some conditions. Also, we provide interesting inequalities for von Neumann entropy and quantum Tsallis entropy which is a parametric extension of von Neumann entropy. The inequality for von Neumann entropy recovers the nonnegativity and gives a refinement for the weaker version of Fannes's inequality for only special cases. Finally, we estimate bounds for the Tsallis relative operator entropy.

CONVEXITY OF PARAMETER EXTENSIONS OF SOME RELATIVE OPERATOR ENTROPIES WITH A PERSPECTIVE APPROACH

AbstractIn this paper, we introduce two notions of a relative operator (α, β)-entropy and a Tsallis relative operator (α, β)-entropy as two parameter extensions of the relative operator entropy and the Tsallis relative operator entropy. We apply a perspective approach to prove the joint convexity or concavity of these new notions, under certain conditions concerning α and β. Indeed, we give the parametric extensions, but in such a manner that they remain jointly convex or jointly concave.Significance Statement. What is novel here is that we convincingly demonstrate how our techniques can be used to give simple proofs for the old and new theorems for the functions that are relevant to quantum statistics. Our proof strategy shows that the joint convexity of the perspective of some functions plays a crucial role to give simple proofs for the joint convexity (resp. concavity) of some relative operator entropies.

A complementary inequality to the information monotonicity for Tsallis relative operator entropy

ABSTRACT We establish a reverse inequality for Tsallis relative operator entropy involving a positive linear map. In addition, we present a converse of Ando's inequality, for each parameter. We give examples to compare our results with the known results by Furuta and Seo. In particular, we establish an extension and a reverse of the Löwner-Heinz inequality under certain conditions. Some interesting consequences in inner product spaces and norm inequalities are also presented.

New sharp inequalities for operator means

ABSTRACTNew sharp multiplicative reverses of the operator means inequalities are presented, with a simple discussion of squaring an operator inequality. As a direct consequence, we extend the operator Pólya-Szegö inequality to arbitrary operator means. Furthermore, we obtain some new lower and upper bounds for the Tsallis relative operator entropy, operator monotone functions and positive linear maps.

Inequalities for Relative Operator Entropies and Operator Means

The main purpose of this article is to study estimates for the Tsallis relative operator entropy, by using the Hermite-Hadamard inequality. We obtain alternative bounds for the Tsallis relative operator entropy and in the process to derive these bounds, we established the significant relation between the Tsallis relative operator entropy and the generalized relative operator entropy. In addition, we study the properties on monotonicity for the weight of operator means and for the parameter of relative operator entropies.

The unique solution of the Karcher equation and the self-adjointness of the Karcher mean

ABSTRACTLawson and Lim showed that the Karcher equation for positive invertible operators on a Hilbert space has a unique solution using the method of the implicit function theorem of a Banach space. In this paper, in the framework of the operator inequality, we show the equivalence of the unique solution of the Karcher equation and the self-adjointness of the Karcher mean. For this, we reform the notion of the operator power means of negative order by virtue of the Tsallis relative operator entropy of negative order.

Estimates for Tsallis relative operator entropy

We give the tight bounds of Tsallis relative operator entropy by using Hermite-Hadamard's inequality. Some reverse inequalities related to Young inequalities are also given. In addition, operator inequalities for normalized positive linear map with Tsallis relative operator entropy are given.

Relative operator entropy related with the spectral geometric mean

We consider the relative operator entropy constructed by the spectral geometric mean and see its properties analogous to those of the Tsallis relative operator entropy by the usual geometric mean. Furthermore, we define the quantum relative entropy constructed by the spectral geometric mean and derive its subadditivity under tensor product.

Operator inequalities associated with Tsallis relative operator entropy

Operator inequalities associated with Tsallis relative operator entropy

Precise estimates of bounds on relative operator entropies

Recently, Zou obtained the generalized results on the bounds for Tsallis relative operator entropy. In this short paper, we give precise bounds for Tsallis relative operator entropy. We also give precise bounds of relative operator entropy.