arXiv subjects

Cong Trinh Le

Publications and source records attributed to Cong Trinh Le.

11 recordsLinked to original sources

arXiv2026

Real radical branch recovery for second order semialgebraic optimization

We study parabolic second order tangent sets in semialgebraic optimization when the supplied polynomial representation is singular. Our focus is the representation problem: formal differentiation of singular defining equations may lose the intrinsic second order geometry. The main result is a real radical branch recovery theorem. If the local real minimal prime branches of the equality ideal are smooth, then the branches selected by a direction and their exact parabolic second order tangent sets can be recovered from a local real radical decomposition and first- and second order polynomial data, without a preassigned stratification. The formula is invariant under changes of regular branch generators, while counterexamples delimit the roles of real radicalization, branch regularity, and constraint qualification. In optimization, the recovered models reduce each directional second order necessary test to finitely many linear programs in the parabolic correction and hence to branchwise multiplier certificates. A parameterized example shows that the formal second order system of the original singular representation may falsely reject a strict local minimizer, whereas branch recovery gives the correct test. Intrinsically singular branches are separated from representation singularities and can be incorporated when an exact higher order model is available.

math.AG↗

arXiv2026

Determinantal Kernel Schemes of Matrix Nets and Applications to Positive Maps

For a linear map on a matrix space, we study the projective schemes obtained by intersecting its projectivized kernel with determinantal rank loci. For matrix nets, that is, three dimensional matrix spaces, we classify every positive dimensional intersection with the rank one Segre variety in arbitrary rectangular size. The possibilities are a ruling plane, a smooth conic, two Segre lines from opposite rulings, a reduced Segre line, or a Segre line with one reduced or embedded residual point. The smooth conic case is contained in a $2\times2$ compression. For $a,b\ge3$, the corresponding locus in $\mbox{Gr}(3,M_{a,b})$ has exactly three irreducible components, whose geometry and intersections are determined explicitly. For nets in $M_3(\mathbb C)$, every finite rank one scheme has length at most three; the adjugate identity gives an intrinsic determinantal obstruction to the length four case allowed for general systems of plane quadrics. As an application, a four parameter family arising from the merging construction admits exact positivity and decomposability criteria. After normalization, positivity is the unit square and decomposability is the quarter disk. The remaining region is atomic and carries explicit PPT entangled states of birank $(5,5)$ and Schmidt number two. The point of this application is that the exact phase boundary is realized on a smooth conic determinantal kernel stratum selected independently by the projective classification.

math.AG↗

arXiv2026

A Weighted Spectral Quantum Fidelity

We introduce and study a one-parameter family of fidelity-type quantities based on the weighted spectral geometric mean, which we call the \emph{weighted spectral fidelity} $ \mathsf{F}_t^{\mathrm{spec}}(ρ,σ):=\Tr\!\big[ρ(ρ^{-1}\sharpσ)^{2t}\big],\ t\in[0,1]. $ This family interpolates smoothly between the trivial overlap ($t=0,1$) and the Uhlmann (root) fidelity at $t=\tfrac12$, and it is distinct from the sandwiched Rényi family except at this midpoint. We establish core structural features-unitary invariance, tensor stabilization and multiplicativity, flip symmetry, endpoint behavior, and a orthogonality criterion. We further show explicit \emph{violations of DPI} for generic $t\neq\tfrac12$. For concavity in the state variables we obtain concavity in each variable separately. Closed forms are obtained for pure states and for qubits in Bloch coordinates. We also extend the first Fuchs--van de Graaf inequality to $\mathsf{F}_t^{\mathrm{spec}}$ for all $t\in[0,1]$, while the second inequality fails away from the midpoint.

math.FA↗

arXiv2026

Some applications of Choi polynomials of linear maps

This paper investigates the properties of Choi polynomials and their fundamental role in the theory of positive linear maps between matrix algebras. By focusing on Hermitian symmetric biquadratic forms, we establish a connection between the positivity of these forms and the structure of positive maps. We specifically explore the construction of indecomposable positive maps in matrix algebras, and their application as entanglement witnesses. Our analysis extends to the detection of Positive Partial Transpose (PPT) entangled states and the classification of edge PPT states in $M_m(\mathbb{C}) \otimes M_n(\mathbb{C})$. Our results provide a refined framework for identifying non-separable states that escape the standard PPT criterion, contributing to the broader understanding of entanglement distillation and quantum information theory.

quant-ph↗

arXiv2026

Perturbation of monic matrix polynomials

In this paper, we study the stability of matrix polynomials under structured perturbations of their coefficients. More precisely, we consider a family of matrix polynomials \[ P_u(λ)=A_d(u)λ^d+A_{d-1}(u)λ^{d-1}+\cdots+A_0(u), \] whose matrix coefficients depend continuously and semialgebraically on a parameter vector $u\in\mathbb{C}^p$. Assuming that the matrix polynomial is monic, we show that the spectrum, the $\varepsilon$-pseudospectrum, the numerical range, and the joint numerical range associated with $P_u(λ)$ define set-valued maps that are Hölder continuous with respect to the parameter $u$. Moreover, the parameter space $\mathbb{C}^p$ can be decomposed into a finite union of analytic semialgebraic submanifolds such that, on each submanifold, the eigenvalues and the Jordan pairs of $P_u(λ)$ depend analytically on $u$. We also note that most of the results remain valid if the monicity assumption is replaced by the local nonsingularity of the leading coefficient matrix $A_d(u)$. However, the monic setting is adopted throughout the paper in order to simplify the exposition and to avoid additional technical assumptions, which are required in particular for results concerning numerical ranges.

math.RA↗

arXiv2026

Optimization of maximal quantum f-divergences between unitary orbits

Maximal quantum $f$-divergences, defined via the commutant Radon--Nikodym derivative, form a fundamental class of distinguishability measures for quantum states associated with operator convex functions. In this paper, we study the optimization of maximal quantum $f$-divergences along unitary orbits of two quantum states. For any operator convex function $f:(0,+\infty)\to\mathbb{R}$, we determine the exact minimum and maximum of $$ U \longmapsto \widehat S_f(ρ\|U^*σU) $$ over the unitary group, and derive explicit spectral formulas for these extremal values together with complete characterizations of the unitaries that attain them. Our approach combines the integral representation of operator convex functions with majorization theory and a unitary-orbit variational method. A key step is to show that any extremizer must commute with the reference state, which reduces the noncommutative optimization problem to a spectral permutation problem. As a consequence, the minimum is achieved by pairing the decreasing eigenvalues of $ρ$ and $σ$, while the maximum corresponds to pairing the decreasing eigenvalues of $ρ$ with the increasing eigenvalues of $σ$. Hence, the range of the maximal quantum $f$-divergence along the unitary orbit is exactly the closed interval determined by these two extremal configurations. Finally, we compare our results with recent unitary-orbit optimization results for quantum $f$-divergences defined via the quantum hockey-stick divergence, highlighting fundamental structural differences between the two frameworks. Our findings extend earlier extremal results for Umegaki, Rényi, and related quantum divergences, and clarify the distinct operator-theoretic nature of maximal quantum $f$-divergences.

math.QA↗

arXiv2021

Matrix power means and new characterizations of operator monotone functions

For positive definite matrices $A$ and $B$, the Kubo-Ando matrix power mean is defined as $$ P_μ(p, A, B) = A^{1/2}\left(\frac{1+(A^{-1/2}BA^{-1/2})^p}{2}\right )^{1/p} A^{1/2}\quad (p \ge 0). $$ In this paper, for $0\le p \le 1 \le q$, we show that if one of the following inequalities \begin{align*} f(P_μ(p, A, B)) \le f(P_μ(1, A, B)) \le f(P_μ(q, A, B))\nonumber \end{align*} holds for any positive definite matrices $A$ and $B$, then the function $f$ is operator monotone on $(0, \infty).$ We also study the inverse problem for non-Kubo-Ando matrix power means with the powers $1/2$ and $2$. As a consequence, we establish new charaterizations of operator monotone functions with the non-Kubo-Ando matrix power means.

math.FA↗

arXiv2019

Tracial moment problems on hypercubes

In this paper we introduce the "tracial $K$-moment problem" and the "sequential matrix-valued $K$-moment problem" and show the equivalence of the solvability of these problems. Using a Haviland's theorem for matrix polynomials, we solve these $K$-moment problems for the case where $K$ is the hypercube $[-1,1]^n$.

math.FA↗

arXiv2019

Two trace inequalities for operator functions

In this paper we show that for a non-negative operator monotone function $f$ on $[0, \infty)$ such that $f(0)= 0$ and for any positive semidefinite matrices $A$ and $B$, $$ Tr((A-B)(f(A)-f(B))) \le Tr(|A-B|f(|A-B|)). $$ When the function $f$ is operator convex on $[0, \infty)$, the inequality is reversed.

math.FA↗

arXiv2019

Some applications of Scherer-Hol's theorem for polynomial matrices

In this paper we establish some applications of the Scherer-Hol's theorem for polynomial matrices. Firstly, we give a representation for polynomial matrices positive definite on subsets of compact polyhedra. Then we establish a Putinar-Vasilescu Positivstellensatz for homogeneous and non-homogeneous polynomial matrices. Next we propose a matrix version of the Pólya-Putinar-Vasilescu Positivstellensatz. Finally, we approximate positive semi-definite polynomial matrices using sums of squares.

math.AG↗

arXiv2008

On deformations of maps and curve singularities

We study several deformation functors associated to the normalization of a reduced curve singularity $(X,0) \subset (\c^n,0)$. The main new results are explicit formulas, in terms of classical invariants of (X,0), for the cotangent cohomology groups $T^i, i = 0,1,2,$ of these functors. Thus we obtain precise statements about smoothness and dimension of the corresponding local moduli spaces. We apply the results to obtain explicit formulas resp. estimates for the $\hoa{A}_e$-codimension of a parametrized curve singularity, where $\hoa{A}_e$ denotes the Mather-Wall group of left-right equivalence.

math.AG↗