arXiv subjects

Trung Dung Vuong

Publications and source records attributed to Trung Dung Vuong.

7 recordsLinked to original sources

arXiv2026

Three-Point Trace Positivity:Schatten Norm Compression and Interpolating Metrics

Let $(\mathcal N,ν)$ be a finite tracial von Neumann algebra and set $\mathcal{K}_r(P,Q)=ν((P^{1/2}QP^{1/2})^r)$. We prove that $$ [\mathcal{K}_r(P_i,P_j)]_{i,j=1}^3\ge0, \qquad 1/2\le r\le1. $$ The range $1/2<r<1$ is new even for matrices, and order three is sharp for $1/2\le r<1$. This single mechanism resolves two problems. First, if $(\mathcal R,ω)$ is finite tracial and $H=[H_{ij}]_{i,j=1}^3\in M_3(\mathcal R)_+$, then $$ [ω(|H_{ij}|^p)]_{i,j=1}^3\ge0, \qquad 1\le p\le2, $$ settling the Lin--van den Driessche conjecture on Schatten norm compression. Second, for a semifinite tracial von Neumann algebra $(\mathcal M,τ)$, define $$ d_t(A,B)^2=\frac{τ(A)+τ(B)}2- \|B^{t/4}A^{t/4}\|_{2/t}^{2/t}. $$ We prove that $d_t$ is a complete metric on $L^1(\mathcal M,τ)_+$ for $1\le t\le2$, including the previously open range $1<t<2$ between the Hellinger and Bures endpoints, and that $$ d_t\text{ is a metric on }L^1(\mathcal M,τ)_+ \quad\Longleftrightarrow\quad t\in[1,2]\ \text{or}\ \mathcal M\text{ is abelian}. $$ Its topology and Cauchy sequences agree with those of $L^1$. The same classification holds for unital tracial $C^*$-algebras and, with the known $t=0$ obstruction, resolves Problem~2 of Komálovics and Molnár.

math.OA↗

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

Alternative-mean trace divergences: geometry, data processing, and barycenters

Let $f:(0,\infty)\to(0,\infty)$ be a nontrivial normalized operator monotone function and set $s=f'(1)$. We introduce the alternative-mean trace functional $$ \altPhi_f(A,B) :=\Tr(A\nabla_s B) -\Tr\!\left( f(A^{-1}\sharp B)\,A\,f(A^{-1}\sharp B) \right). $$ on the positive definite cone. We prove that $\altPhi_f$ is a quantum divergence in the sense of Bhatia--Gaubert--Jain whose diagonal Hessian induces a positive multiple of the Bures--Wasserstein Riemannian metric. We also establish the sharp comparison $$ s(1-s)d_{\rm BW}(A,B)^2 \le \altPhi_f(A,B) \le (1-s+s^2)d_{\rm BW}(A,B)^2. $$

math-ph↗

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

Sharp Hyperbolic Cutoffs and Dimension-Sharp Counterexamples for Reverse Araki-Type Inequalities

We study reverse Araki-type trace inequalities and log-majorizations beyond the exponent $2$. For arbitrary nonnegative nondecreasing weights, we show that $s=2$ is the sharp dimension-free boundary: for every $s>2$, explicit one-parameter $3\times3$ positive definite examples violate the reverse Liu--Cheng trace inequality and the corresponding dual formulation of Shi--Wei--Wang, whereas the reverse inequality remains valid for every $s\geq1$ in dimension $2$. For power weights, a larger region survives and is bounded by a sharp hyperbola. In normalized variables, for $s>2$, \[ A^{r+s}B^s \succ_{\log}A^r (A^{1/2} BA^{1/2})^s \] holds for all positive semidefinite matrices in every finite dimension if and only if $0\leq r\leq s/(s-2)$; beyond this range, even the associated trace inequality fails for $3\times3$ positive definite matrices. Equivalently, for $0 2p$, the sharp condition is $0\leq r\leq pq/(q-2p)$. Combined with the known all-$r$ regime $p\leq q\leq2p$, this completes the reverse log-majorization phase diagram.

math.FA↗

arXiv2026

Polar Fidelities, Holevo Bases, and Unitary Factors of Generalized Fidelity

Motivated by several open problems in the Afham-Ferrie theory of generalized fidelity, we study polar fidelities, Holevo bases, and unitary factors on $\Pd$, the cone of $d\times d$ complex positive definite matrices. We prove that, for every fixed pair $P,Q\in\Pd$, the one-sided polar fidelities $x\mapsto F_{P^x}(P,Q)$ and $x\mapsto F_{Q^x}(P,Q)$, as well as their symmetrization, are nondecreasing on $(-\infty,1]$ and nonincreasing on $[1,\infty)$. Hence the polar paths realize exactly the interval $[\FM(P,Q),\FU(P,Q)]$ on $[-1,1]$, yielding pointwise, generally pair-dependent realizations of all $z$-fidelity values with $z\ge1/2$ and of the Log-Euclidean fidelity as generalized fidelities. We also show that for $d\ge2$ and $0<z<1/2$, such realization fails in general, even for interior fidelities. We further solve the fixed-pair Holevo-base equation $F_R(P,Q)=\FH(P,Q)$, classify all Holevo bases, and classify exactly which unitary factors of generalized fidelity can arise from a base $R$. These are precisely the unitaries $W$ for which $P^{-1/2}Q^{1/2}W$ is similar to a positive definite matrix. This recovers the special-unitary constraint and disproves the global reverse inclusion $SU(d)$ for $d\ge2$.

quant-ph↗

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↗