arXiv subjects

Mohsen Kian

Publications and source records attributed to Mohsen Kian.

At least 19 recordsLinked to original sources

arXiv2026

Sharpness Loss for Radon--Nikodym Derivatives of Completely Positive Maps

The Radon--Nikodym theorem for completely positive maps identifies the order interval below a map with positive contractions in the commutant of its minimal Stinespring representation. In this paper, we study the behavior of projection-valued Radon--Nikodym derivatives (sharp submaps) under composition. We prove that sharpness loss is determined by the multiplicative defect of the induced pullback map. In the finite-dimensional case, we show that this defect is determined by the orthogonal complement of the composite Kraus relation space, denoted $\mathcal{E}_{Λ,Ω}$. A sharp submap remains sharp if and only if $\mathcal{E}_{Λ,Ω}$ reduces the lifted Radon--Nikodym projection. Furthermore, we establish a tensor-factor criterion for the global preservation of sharpness and explicitly quantify the sharpness-loss defect.

math.FA↗

arXiv2026

Finite-Corner Reconstruction and Separation for Cones of Normal Maps

We develop a finite-corner framework for convex cones of normal maps between $\mathcal B(\mathcal H)$ and $\mathcal B(\mathcal K)$. The basic structural assumption is stability under finite cut--pad operations. We prove that every point-ultraweakly closed cut--pad stable cone is completely determined by its finite-dimensional corner cones, and conversely that every coherent family of closed finite-dimensional corner cones admits a unique global realization of this type. For a cut--pad stable cone that is not necessarily point-ultraweakly closed, reconstruction from the norm closures of its corner cones yields exactly its point-ultraweak closure. Combining this reconstruction principle with finite-dimensional separation and the Choi representation, we show that non-membership in a global cone is always detected on a single finite-dimensional corner by a finite-dimensional witness. We illustrate the framework for completely positive and decomposable maps; in the decomposable case, the Hermitian parts of the finite-corner witnesses are PPT.

math-ph↗

arXiv2026

Matrix-Test Duality: A Support-Function Characterization for $C^*$-Convex Families of CP Maps

We develop a matrix-test dual framework for $C^*$-convex families of completely positive maps $\CP(\mathscr S,\mathscr T)$, where $\mathscr S$ is an operator system and $\mathscr T$ is a unital $C^*$-algebra. Matrix tests $(k,f,s)$ induce evaluation functionals $Φ\mapsto f(Φ_k(s))$ and generate a natural weak topology $τ=σ(\mathcal E,\mathcal F)$ on $\mathcal E=\mathrm{span}_{\mathbb C}(\CP(\mathscr S,\mathscr T))$. Our main result provides a support-function/separation characterization of the $τ$-closed $C^*$-convex hull $\overline{\cconv(\mathcal K)}^{\,τ}$ of a family $\mathcal K\subseteq \CP(\mathscr S,\mathscr T)$ in terms of matrix-test inequalities. A key technical tool is a finite-dimensional folding procedure that compresses finite linear combinations of test functionals into a single higher-level matrix test. As consequences, we obtain a single-test witness for non-membership, support-function criteria for inclusion and equality of $τ$-closed $C^*$-convex hulls, and, under $0\in\overline{\cconv(\mathcal K)}^{\,τ}$, an exact normalized bipolar-type reconstruction statement. We also show that $τ$ is already generated by level-$1$ tests, although higher matrix levels remain essential in the geometric test inequalities.

math.OA↗

arXiv2026

Relative Kubo-Ando Means of Completely Positive Maps

We develop a Kubo--Ando theory on order intervals of completely positive maps. Using Arveson's Radon--Nikodym theorem as a structural tool, we define relative Kubo--Ando means $Φσ_ΩΨ$ for completely positive maps dominated by a common ambient map $Ω$. The special choice $Ω=Φ+Ψ$ yields an intrinsic mean of two completely positive maps. We prove that these means are independent of the chosen Stinespring representation and satisfy the expected order-theoretic properties, including monotonicity, transformer inequalities, Jensen-type inequalities, data processing, and monotonicity with respect to the ambient map. For the geometric mean, we obtain a block-positivity characterization and show that the intrinsic geometric mean vanishes exactly when the two maps have no nonzero common completely positive submap. Finally, we compare the construction with existing finite-dimensional and form-theoretic approaches: for maps between matrix algebras it agrees with the Choi-matrix mean, and in the geometric case it agrees with Okayasu's Pusz--Woronowicz mean on their common domain.

math.OA↗

arXiv2026

Kubo-Ando Means and Rigidity of Quantum Positivity Cones

We investigate the stability of quantum positivity cones under nonlinear operator means. Specifically, we examine how Kubo--Ando means interact with the separable, positive partial transpose (PPT), and Schmidt-number cones. By analyzing the curvature of operator monotone functions at the identity, we give a strict rigidity phenomenon: weighted arithmetic means are the only Kubo--Ando means that preserve the separable cone in all dimensions. We show that the strictly positive curvature of any non-arithmetic mean explicitly forces a violation of the PPT condition, even in the foundational two-qubit setting, and can strictly increase the Schmidt number of the resulting operator. Finally, using the Choi--Jamiołkowski correspondence, we translate these geometric obstructions to the map-theoretic setting, concluding that convex mixing is the uniquely permissible Kubo--Ando operation for preserving entanglement-breaking quantum channels.

quant-ph↗

arXiv2026

Fractional $k$-positivity: a continuous refinement of the $k$-positive scale

We introduce a real-parameter refinement of the classical integer hierarchies underlying Schmidt number, block-positivity, and $k$-positivity for maps between matrix algebras. Starting from a compact family of $α$-admissible unit vectors ($α\in[1,d]$), we define closed cones $\mathsf K_α$ of bipartite positive operators that interpolate strictly between successive Schmidt-number cones, together with their dual witness cones. Via the Choi--Jamiołkowski correspondence this yields a matching filtration of map cones $\mathsf P_α$, recovering the usual $k$-positive/$k$-superpositive classes at integer parameters and complete positivity at the top endpoint. Two results show that the fractional levels capture genuinely new structure. First, we prove a \emph{fractional Kraus theorem}: $α$-superpositive maps are precisely the completely positive maps admitting a Kraus decomposition whose Kraus operators satisfy an explicit singular-value (Ky--Fan) constraint, extending the classical rank-$k$ characterization. Second, for non-integer $α$ the cones $\mathsf P_α$ fail stability under CP post-composition, highlighting a sharp structural transition away from the integer theory. Finally, we derive sharp thresholds on canonical symmetric families (including the depolarizing ray and the isotropic slice), turning familiar stepwise criteria into continuous, computable profiles.

math.FA↗

arXiv2026

A Hierarchy of Deviation from Complete Positivity and Optimal Entanglement Witnesses

We introduce the \emph{CP-distance} to quantify the deviation of Hermitian linear maps from complete positivity, defined as the minimal depolarizing noise required to render a map completely positive. We derive a closed spectral formula for this distance and extend the framework to \emph{directional robustness} against arbitrary completely positive maps, establishing stability and tensor-product properties. Expanding this to the intermediate cones of $k$-positive maps, we introduce a \emph{hierarchy of deviation}, $d_k(Φ)$. We derive a spectral formula for $d_k$ based on entanglement depth and demonstrate that it serves as an optimal threshold for certifying Schmidt numbers, allowing for the universal construction of dimension-sensitive entanglement witnesses.

quant-ph↗

arXiv2025

A Hierarchy of Entanglement Cones via Rank-Constrained $C^*$-Convex Hulls

This paper systematically investigates the geometry of fundamental quantum cones, the separable cone ($\mathscr{P}_+$) and the Positive Partial Transpose (PPT) cone ($\mathcal{P}_{\mathrm{PPT}}$), under generalized non-commutative convexity. We demonstrate a sharp stability dichotomy analyzing $C^*$-convex hulls of these cones: while $\mathscr{P}_+$ remains stable under local $C^*$-convex combinations, its global $C^*$-convex hull collapses entirely to the cone of all positive semidefinite matrices, $\operatorname{MCL}(\mathscr{P}_+) = \mathscr{P}_0$. To gain finer control and classify intermediate structures, we introduce the concept of ``$k$-$C^*$-convexity'', by using the operator Schmidt rank of $C^*$-coefficients. This constraint defines a new hierarchy of nested intermediate cones, $\operatorname{MCL}_k(\cdot)$. We prove that this hierarchy precisely recovers the known Schmidt number cones for the separable case, establishing a generalized convexity characterization: $\operatorname{MCL}_k(\mathscr{P}_+) = \mathcal{T}_k$. Applied to the PPT cone, this framework generates a family of conjectured non-trivial intermediate cones, $\mathcal{C}_{\mathrm{PPT}, k}$.

math-ph↗

arXiv2025

Hermitian Maps: Approximations and Completely Positive Extensions

This study investigates Hermitian linear maps, focusing on their decomposition into completely positive (CP) maps and their extensions to CP maps using auxiliary spaces. We derive a precise lower bound on the Hilbert-Schmidt norm of the negative component in any CP decomposition, proving its attainability through the Jordan decomposition. Additionally, we demonstrate that the positive part of this decomposition provides the optimal CP approximation in the Hilbert-Schmidt norm. We also determine the minimal dimension of an auxiliary space required to extend a Hermitian map to a CP map, with explicit constructions provided. Practical examples illustrate the application of our results.

math.FA↗

arXiv2025

A new estimation of the quantum Chernoff bound

Relating to finding possible upper bounds for the probability of error for discriminating between two quantum states, it is well-known that \begin{align*} \mathrm{tr}(A+B) - \mathrm{tr}|A-B|\leq 2\, \mathrm{tr}\big(f(A)g(B)\big) \end{align*} holds for every positive-valued matrix monotone function $f$, where $g(x)=x/f(x)$, and all positive definite matrices $A$ and $B$. In this paper, we introduce a new class of functions that satisfy the above inequality. As a consequence, we derive a novel estimation of the quantum Chernoff bound. Additionally, we characterize matrix decreasing functions and establish matrix Powers-Störmer type inequalities for perspective functions.

quant-ph↗

arXiv2024

Decomposition of tracial positive maps and applications in quantum information

Every positive multilinear map between $C^*$-algebras is separately weak$^*$-continuous. We show that the joint weak$^*$-continuity is equivalent to the joint weak$^*$-continuity of the multiplications of $C^*$-algebras under consideration. We study the behavior of general tracial positive maps on properly infinite von Neumann algebras and by applying the Aron--Berner extension of multilinear maps, we establish that under some mild conditions every tracial positive multilinear map between general $C^*$-algebras enjoys a decomposition $Φ=φ_2 \circ φ_1$, in which $φ_1$ is a tracial positive linear map with the commutative range and $φ_2$ is a tracial completely positive map with the commutative domain. As an immediate consequence, tracial positive multilinear maps are completely positive. Furthermore, we prove that if the domain of a general tracial completely positive map $Φ$ between $C^*$-algebra is a von Neumann algebra, then $Φ$ has a similar decomposition. As an application, we investigate the generalized variance and covariance in quantum mechanics via arbitrary positive maps. Among others, an uncertainty relation inequality for commuting observables in a composite physical system is presented.

math.OA↗

arXiv2024

Matrix Inequalities between $f(A)σf(B)$ and $AσB$

Let $A$ and $ B$ be $n\times n$ positive definite complex matrices, let $σ$ be a matrix mean, and let $f : [0,\infty)\to [0,\infty)$ be a differentiable convex function with $f(0)=0$. We prove that $$f^{\prime}(0)(A σB)\leq \frac{f(m)}{m}(AσB)\leq f(A)σf(B)\leq \frac{f(M)}{M}(AσB)\leq f^{\prime}(M)(AσB),$$ where $m$ represents the smallest eigenvalues of $A$ and $B$ and $M$ represents the largest eigenvalues of $A$ and $B$. If $f$ is differentiable and concave, then the reverse inequalities hold. We use our result to improve some known subadditivity inequalities involving unitarily invariant norms under certain mild conditions. In particular, if $f(x)/x$ is increasing, then $$|||f(A)+f(B)|||\leq\frac{f(M)}{M} |||A+B|||\leq |||f(A+B)|||$$ holds for all $A$ and $B$ with $M\leq A+B$. Furthermore, we apply our results to explore some related inequalities. As an application, we present a generalization of Minkowski's determinant inequality.equality.

math.FA↗

arXiv2024

Some Generalizations of Mercer inequality and its operator extensions

We study the Mercer inequality and its operator extension for superquadratic functions. In particular, we give a more general form of the Mercer inequality by replacing some constants by positive operators. As some consequences, our results produce a Jensen operator inequality for superquadratic functions. Moreover, we present some Mercer inequalities of Hermite-Hadamard's type.

math.FA↗

arXiv2024

Improved eigenvalue inequalities via two major subclasses of superquadratic functions

There exist two major subclasses in the class of superquadratic functions, one comprises concave and decreasing functions, while the other consists of convex and monotone increasing functions. Leveraging this distinction, we introduce eigenvalue inequalities for each case. The characteristics of these functions allow us to advance our findings in two ways: firstly, by refining existing results related to eigenvalues for convex functions, and secondly, by deriving complementary inequalities for other function types. To bolster our claims, we will provide illustrative examples.

math.FA↗

arXiv2021

How type of Convexity of the Core function affects the Csiszár $f$-divergence functional

We investigate how the type of Convexity of the Core function affects the Csiszár $f$-divergence functional. A general treatment for the type of convexity has been considered and the associated perspective functions have been studied. In particular, it has been shown that when the core function is \rm{MN}-convex, then the associated perspective function is jointly \rm{MN}-convex if the two scalar mean \rm{M} and \rm{N} are the same. In the case where $\mathrm{M}\neq\mathrm{N}$, we study the type of convexity of the perspective function. As an application, we prove that the \textit{Hellinger distance} is jointly \rm{GG}-convex. As further applications, the matrix Jensen inequality has been developed for the perspective functions under different kinds of convexity.

cs.IT↗

arXiv2020

Jointly convex mappings related to the Lieb's functional and Minkowski type operator inequalities

Employing the notion of operator log-convexity, we study joint concavity$/$ convexity of multivariable operator functions: $(A,B)\mapsto F(A,B)=h\left[ Φ(f(A))\ σ Ψ(g(B))\right]$, where $Φ$ and $Ψ$ are positive linear maps and $σ$ is an operator mean. As applications, we prove jointly concavity$/$convexity of matrix trace functions $\Tr\left\{ F(A,B)\right\}$. Moreover, considering positive multi-linear mappings in $F(A,B)$, our study of the joint concavity$/$ convexity of $(A_1,\cdots,A_k)\mapsto h\left[ Φ(f(A_1),\cdots,f(A_k))\right]$ provides some generalizations and complement to results of Ando and Lieb concerning the concavity$/$ convexity of maps involving tensor product. In addition, we present Minkowski type operator inequalities for a unial positive linear map, which is an operator version of Minkowski type matrix trace inequalities under a more general setting than Carlen and Lieb, Bekjan, and Ando and Hiai.

math.FA↗

arXiv2020

Klein's trace inequality and superquadratic trace functions

We show that if $f$ is a non-negative superquadratic function, then $A\mapsto\mathrm{Tr}f(A)$ is a superquadratic function on the matrix algebra. In particular, \begin{align*} \tr f\left( {\frac{A + B}{2}} \right) +\tr f\left(\left| {\frac{A - B}{2}}\right|\right) \leq \frac{{\tr {f\left( A \right)} + \tr {f\left( B \right)} }}{2} \end{align*} holds for all positive matrices $A,B$. In addition, we present a Klein's inequality for superquadratic functions as $$ \mathrm{Tr}[f(A)-f(B)-(A-B)f'(B)]\geq \mathrm{Tr}[f(|A-B|)] $$ for all positive matrices $A,B$. It gives in particular an improvement of the Klein's inequality for non-negative convex function. As a consequence, some variants of the Jensen trace inequality for superquadratic functions have been presented.

math.FA↗

arXiv2020

Asymmetric Choi--Davis inequalities

Let $Φ$ be a unital positive linear map and let $A$ be a positive invertible operator. We prove that there exist partial isometries $U$ and $V$ such that \[ |Φ(f(A))Φ(A)Φ(g(A))|\leq U^*Φ(f(A)Ag(A))U \] and \[\left|Φ\left(f(A)\right)^{-r}Φ(A)^rΦ\left(g(A)\right)^{-r}\right|\leq V^*Φ\left(f(A)^{-r}A^rg(A)^{-r}\right)V\] hold under some mild operator convex conditions and some positive numbers $r$. Further, we show that if $f^2$ is operator concave, then $$ |Φ(f(A))Φ(A)|\leq Φ(Af(A)).$$ In addition, we give some counterparts to the asymmetric Choi--Davis inequality and asymmetric Kadison inequality. Our results extend some inequalities due to Bourin--Ricard and Furuta.

math.FA↗