arXiv subjects

Trung Hoa Dinh

Publications and source records attributed to Trung Hoa Dinh.

18 recordsLinked to original sources

arXiv2026

Demailly's Conjecture in Quantum Language and Its Solution

We give an exact translation between fat-point interpolation in complex projective space and a dark-state problem for fixed-particle-number bosons. Under the unitary Fischer--Bargmann realization, homogeneous degree becomes particle number, the Veronese lift becomes a bosonic symmetric product state, and order-$m$ vanishing becomes orthogonality to all coherent jet excitations of orders below $m$. The initial degree of a symbolic power is therefore the first particle number admitting a dark state of a positive constraint Hamiltonian, and the Waldschmidt constant is the asymptotic particle cost per suppression order. After developing both sides of this dictionary from first principles, we prove a finite-scale no-dark-state amplification theorem. Frobenius amplifies the associated full-rank certificate in one good positive-characteristic fiber; a nonzero maximal minor returns it to characteristic zero. Its asymptotic consequence is Demailly's conjectured inequality. The paper is written so that the projective statement, the quantum reformulation, and the arithmetic part of the solution can be read independently and then identified exactly.

quant-ph↗

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

Frobenius dilation of Hilbert function, Demailly's conjecture, and varieties of powers

We prove a Frobenius-dilation inequality for Hilbert functions of ideals generated by powers of fixed homogeneous forms in characteristic zero. Via Emsalem--Iarrobino inverse system duality, it yields quantitative fat-point postulation estimates and a new proof of the recent theorem of Ha--Sivakumar resolving Demailly's conjecture for arbitrary finite point sets. A fixed-characteristic version controls tangent spaces to varieties of powers and, together with plane interpolation and secant-theoretic results, proves complete nondefectivity of $V^k_{2,d}$ for all $k\ge3$ and $d\ge2$; in particular, it settles the ternary Waring problem for generic $k$-rank conjectured by Lundqvist--Oneto--Reznick--Shapiro (suggested by Ottaviani). We also show that, for a prime power exponent $q$, any failure of Nicklasson's conjecture in three variables is confined to $2q-2$ consecutive degrees.

math.AC↗

arXiv2026

Weighted Two-Parameter Quantum Divergences Generated by Matrix Power-Mean Gaps

Let $\mathbb{P}_n$ be the cone of $n\times n$ complex positive definite matrices. For $0<θ<1$ and $r\ge0$, let $M_r^{(θ)}$ denote the weighted matrix power mean, with the logarithmic endpoint at $r=0$, defined by $$ M_r^{(θ)}(A,B) = \begin{cases} \big((1-θ)A^r+θB^r\big)^{1/r}, & r>0,\\[2mm] \exp\big((1-θ)\log A+θ\log B\big), & r=0. \end{cases} $$ For $p>q\ge0$, we introduce $$ Φ_{p,q}^{(θ)}(A,B) = \operatorname{Tr}\big[M_p^{(θ)}(A,B)-M_q^{(θ)}(A,B)\big]. $$ We prove that $Φ_{p,q}^{(θ)}$ is a quantum divergence in the sense of Bhatia-Gaubert-Jain. Moreover, for $$ 0\le q\le1\le p\le2,\qquad p>q, $$ it is jointly convex and satisfies the data processing inequality for completely positive trace-preserving maps, with the output side interpreted by the regularized boundary convention whenever it is singular. The construction recovers Hellinger-type and Jensen-Shannon-type quantities as special cases or limits. We also compute the induced local quadratic form and provide scalar and matrix obstructions showing that the convexity and data-processing range cannot be extended to all parameters.

math.FA↗

arXiv2026

Heron-Wasserstein majorization inequalities for spectral and Kubo-Ando geometric means

We prove sharp Heron-type majorization inequalities for two quadratic matrix expressions associated with the spectral and Kubo-Ando geometric means. For the spectral geometric mean cross term, we show that \[ λ\bigl(a^2A+b^2B+c(A\natural B)\bigr) \prec_w λ\bigl(W_{a,b}(A,B)\bigr), \qquad 0\le c\le 2ab, \] where $W_{a,b}(A,B)$ is the weighted Bures-Wasserstein expression. The coefficient $2ab$ is sharp, and at this endpoint the weak majorization becomes majorization. For the Kubo-Ando geometric mean, we prove the direct comparison \[ λ\bigl(a^2A+b^2B+2ab(A\#B)\bigr) \prec_w λ\bigl(W_{a,b}(A,B)\bigr). \] This settles, in the two-variable setting, Bhatia's question of whether the Heron-type norm inequality of Bhatia-Lim-Yamazaki admits a weak-majorization refinement. More precisely, we prove \[ λ\bigl(a^2A+b^2B+2ab(A\#B)\bigr) \prec_w λ\bigl((aA^{1/2}+bB^{1/2})^2\bigr), \] and consequently obtain the corresponding inequality for all unitarily invariant norms.

math.FA↗

arXiv2026

On the Failure of the Upper Bound in the Refined BMV Conjecture and a Pinching Correction

We analyze why the refined Bessis--Moussa--Villani conjecture fails. The refined conjecture proposed that the normalized trace average over all words with prescribed numbers of letters $A$ and $B$ should be bounded above by the clustered word $\Tr(A^nB^m)$. Recent counterexamples of Cha show that this upper bound is false already for $3\times3$ positive semidefinite matrices when $n=m=5$. We explain the failure from the viewpoint of commutative common parts. The term $\Tr(A^nB^m)$ is not the canonical common part of the pair $(A,B)$; it is only one clustered word. After pinching $B$ relative to $A$, the natural commuting contribution is $\A_{n,m}(A,\EA(B))$. The off-diagonal complement $B-\EA(B)$ creates spectral bridges, and mixed words can distribute the powers of $A$ along closed cycles more efficiently than the clustered word. This gives a mechanism for finding counterexamples. Motivated by this mechanism, we propose a corrected pinching refinement \[ \A_{n,m}(A,B)\ge \A_{n,m}(A,\EA(B)). \] We prove this corrected conjecture in the case of two letters $B$, obtaining a sandwich refinement \[ \A_{n,2}(A,\EA(B)) \le \A_{n,2}(A,B) \le \Tr(A^nB^2). \] Thus, even where the old clustered upper bound remains true, the pinching viewpoint gives a sharper structural decomposition.

math.CO↗

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

Spectral geometric mean and trace characterizations

We use nearly parallel pure states to characterize positive linear functionals $ϕ$ on $\mathbb{M}_n$ as positive multiples of the trace if and only if $ϕ(A \natural B) \leq \sqrt{ϕ(A) ϕ(B)}$ for all positive definite matrices $A$ and $B$. Here $A \natural B = (A^{-1} \# B)^{1/2} A (A^{-1} \# B)^{1/2}$ represents the spectral geometric mean. For further clarification, we establish novel characterizations through the inequality $ϕ(A \natural B) \leq ϕ((A+B)/2)$ for all positive definite matrices $A$ and $B$. We also present a trace inequality related to quantum fidelity that applies to all positive definite matrices, and demonstrate that it does not characterize the trace.

quant-ph↗

arXiv2025

Quantum Algorithms for Computing Maximal Quantum $f$-divergence and Kubo-Ando means

The development of quantum computation has resulted in many quantum algorithms for a wide array of tasks. Recently, there is a growing interest in using quantum computing techniques to estimate or compute quantum information-theoretic quantities such as Renyi entropy, Von Neumann entropy, matrix means, etc. Motivated by these results, we present quantum algorithms for computing the maximal quantum $f$-divergences and the operator-theoretic matrix Kubo--Ando means. Both of them involve Renyi entropies, matrix means as special cases, thus implying the universality of our framework.

quant-ph↗

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↗

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↗

arXiv2020

On the Pólya-Szegö operator inequality

In this paper, we present generalized Pólya-Szegö type inequalities for positive invertible operators on a Hilbert space for arbitrary operator means between the arithmetic and the harmonic means. As applications, we present Operator Grüss, Diaz--Metcalf and Klamkin--McLenaghan inequalities.

math.FA↗

arXiv2019

Functions preserving operator means

Let $σ$ be a non-trivial operator mean in the sense of Kubo and Ando, and let $OM_+^1$ the set of normalized positive operator monotone functions on $(0, \infty)$. In this paper, we study class of $σ$-subpreserving functions $f\in OM_+^1$ satisfying $$f(AσB) \le f(A)σf(B)$$ for all positive operators $A$ and $B$. We provide some criteria for $f$ to be trivial, i.e., $f(t)=1$ or $f(t)=t$. We also establish characterizations of $σ$-preserving functions $f$ satisfying $$f(AσB) = f(A)σf(B)$$ for all positive operators $A$ and $B$. In particular, when $\lim_{t\rightarrow 0} (1σt) =0$, the function $f$ preserves $σ$ if and only if $f$ and $1σt$ are representing functions for weighted harmonic means.

math.FA↗

arXiv2019

Some Geometric Properties of Matrix Means with respect to Different Distance Function

In this paper we study the monotonicity, in-betweenness and in-sphere properties of matrix means with respect to Bures-Wasserstein, Hellinger and Log-Determinant metrics. More precisely, we show that the matrix power means (Kubo-Ando and non-Kubo-Ando extensions) satisfy the in-betweenness property in the Hellinger metric. We also show that for two positive definite matrices $A$ and $B$, the curve of weighted Heron means, the geodesic curve of the arithmetic and the geometric mean lie inside the sphere centered at the geometric mean with the radius equal to half of the Log-Determinant distance between $A$ and $B$.

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↗

arXiv2018

New characterizations of operator monotone functions

If $σ$ is a symmetric mean and $f$ is an operator monotone function on $[0, \infty)$, then $$f(2(A^{-1}+B^{-1})^{-1})\le f(AσB)\le f((A+B)/2).$$ Conversely, Ando and Hiai showed that if $f$ is a function that satisfies either one of these inequalities for all positive operators $A$ and $B$ and a symmetric mean different than the arithmetic and the harmonic mean, then the function is operator monotone. In this paper, we show that the arithmetic and the harmonic means can be replaced by the geometric mean to obtain similar characterizations. Moreover, we give characterizations of operator monotone functions using self-adjoint means and general means subject to a constraint due to Kubo and Ando.

math.FA↗

arXiv2017

Some inequalities for operator (p,h)-convex functions

Let $p$ be a positive number and $h$ a function on $\mathbb{R}^+$ satisfying $h(xy) \ge h(x) h(y)$ for any $x, y \in \mathbb{R}^+$. A non-negative continuous function $f$ on $K (\subset \mathbb{R}^+)$ is said to be {\it operator $(p,h)$-convex} if \begin{equation*}\label{def} f ([αA^p + (1-α)B^p]^{1/p}) \leq h(α)f(A) +h(1-α)f(B) \end{equation*} holds for all positive semidefinite matrices $A, B$ of order $n$ with spectra in $K$, and for any $α\in (0,1)$. In this paper, we study properties of operator $(p,h)$-convex functions and prove the Jensen, Hansen-Pedersen type inequalities for them. We also give some equivalent conditions for a function to become an operator $(p,h)$-convex. In applications, we obtain Choi-Davis-Jensen type inequality for operator $(p,h)$-convex functions and a relation between operator $(p,h)$-convex functions with operator monotone functions.

math.FA↗