The geometry of absolute separability and other convex matrix properties from spectrum

• URL: http://arxiv.org/abs/2608.03390v1
• Date: Tue, 04 Aug 2026 09:40:14 GMT
• Title: The geometry of absolute separability and other convex matrix properties from spectrum
• Abstract summary: We investigate the geometric structure of the spectra of bipartite absolute separable states ($mathrmASEP_m,n$) and absolute positive partial states ($mathrmAPPT_m,n$)<n>We provide a complete characterization of the faces and extreme points of $mathrmAPPT_m,n$ and demonstrate that the dimension of a face is determined by the rank of a certain matrix.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We investigate the geometric structure of the set of spectra of bipartite absolute separable states ($\mathrm{ASEP}_{m,n}$) and absolute positive partial transpose states ($\mathrm{APPT}_{m,n}$), i.e., bipartite quantum states that remain separable or PPT respectively, under all global unitary transformations. First, we establish general geometric properties of absolute convex sets of matrices, their spectra and extreme points. Regarding absolute separability, we present a permutation-symmetric reformulation of the absolute PPT criterion and use it to demonstrate that $\mathrm{APPT}_{m,n}$ is a spectrahedron for all $m\leq n$: in particular, all its faces are exposed. In contrast, while $\mathrm{ASEP}_{2,n}$ is also a spectrahedron, we prove that in general $\mathrm{ASEP}_{m,n}$ is a semialgebraic set for all $m\leq n$. Furthermore, we provide a complete characterization of the faces and extreme points of $\mathrm{APPT}_{m,n}$ and demonstrate that the dimension of a face is determined by the rank of a certain matrix, with maximal proper faces having dimension $(mn-m-1)$. In the quantitative setting, we provide a rigorous lower bound on the maximal attainable purity of $\mathrm{APPT}_{m,n}$ via an inscribed polytope $\mathcal{P}_{m,n}$ and conjecture that the maximal purity of $\mathrm{APPT}_{m,n}$ (along with its spectra) coincides with the polytope for arbitrary dimensions except when $m=n=2$. Additionally, we also provide a rigorous upper bound on the minimal von Neumann entropy of $\mathrm{APPT}_{m,n}$ and demonstrate numerically that the minimum entropy eventually coincides with the polytope $\mathcal{P}_{m,n}$ as the local system dimension $n$ increases. Finally, we show that the relative spectral volume of $\mathrm{APPT}_{m,n}$ decays exponentially in $n$ by a constant multiplicative factor of the relative volume of the inscribed polytope $\mathcal{P}_{m,n}$.

Related papers

• Isospectral majorization and isoperimetric inequalities for coherent states on the Bloch sphere [0.0]
  We prove an isospectral version of Lieb-Solovej inequality.<n>Spherical caps maximize every partial sum of the eigenvalues of Toeplitz operators with symbol $1_$.
  arXiv  Detail & Related papers  (2026-08-12T16:46:26Z)
• On the convergence of graph Laplacians with a symmetric divergence [0.0]
  A key estimate for the geodesic distance $d_g$ is that there exists $K > 0$ such that $0 leq d_g(p, q)2 - |p-q|2 leq K d_g(p, q)4$ for all $p, q in mathcalM$.<n>We demonstrate that this is sufficient for the pointwise convergence of graph Laplacians constructed with $D$ and discuss examples where $D$ is given by the Sinkhorn divergence on a family of probability measures para
  arXiv  Detail & Related papers  (2026-07-07T06:41:21Z)
• Spectral-angular parametrization of open qudit dynamics [0.0]
  A generic density matrix $rho_mathbfr,pmb$, characterized by $n2-1$ real parameters, naturally decomposes into two sets.<n>A key observation is that the spectral parameters $mathbfr = (r_1, ldots, r_n-1)$ admit a natural Lie-algebraic interpretation.
  arXiv  Detail & Related papers  (2026-04-13T16:02:24Z)
• Optimal Scalar Quantization for Matrix Multiplication: Closed-Form Density and Phase Transition [50.36362492608702]
  We study entrywise scalar quantization of two matrices prior to multiplication.<n>We obtain a closed-form optimal point density [ star(u) propto exp!left(-fracu26right)bigl( (1-2)+2u22bigr), qquad u=fracx_X,, and prove a correlation-driven phase transition.
  arXiv  Detail & Related papers  (2026-03-20T01:53:44Z)
• Approximating the operator norm of local Hamiltonians via few quantum states [53.16156504455106]
  Consider a Hermitian operator $A$ acting on a complex Hilbert space of $2n$.<n>We show that when $A$ has small degree in the Pauli expansion, or in other words, $A$ is a local $n$-qubit Hamiltonian.<n>We show that whenever $A$ is $d$-local, textiti.e., $deg(A)le d$, we have the following discretization-type inequality.
  arXiv  Detail & Related papers  (2025-09-15T14:26:11Z)
• Wigner quasi-probability distribution for symmetric multi-quDit systems and their generalized heat kernel [0.0]
  We analyze the phase-space structure of Schr"odinger $U(D)$-spin cat states.<n>We compute the generalized heat kernel relating two quasi-probability distributions $mathcalF(s)_rho$ and $mathcalF(s')_rho$.
  arXiv  Detail & Related papers  (2025-07-20T08:26:28Z)
• Leading and beyond leading-order spectral form factor in chaotic quantum many-body systems across all Dyson symmetry classes [8.105213101498085]
  We show the emergence of random matrix theory (RMT) spectral correlations in the chaotic phase of generic periodically kicked interacting many-body systems.<n>We analytically calculating spectral form factor (SFF), $K(t)$, up to two leading orders in time, $t$.<n>Our derivation only assumes random phase approximation to enable ensemble average.
  arXiv  Detail & Related papers  (2025-02-06T15:37:18Z)
• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Near-optimal fitting of ellipsoids to random points [68.12685213894112]
  A basic problem of fitting an ellipsoid to random points has connections to low-rank matrix decompositions, independent component analysis, and principal component analysis. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some $n = Omega(, d2/mathrmpolylog(d),)$. Our proof demonstrates feasibility of the least squares construction of Saunderson et al. using a convenient decomposition of a certain non-standard random matrix.
  arXiv  Detail & Related papers  (2022-08-19T18:00:34Z)
• Monogamy of entanglement between cones [43.57338639836868]
  We show that monogamy is not only a feature of quantum theory, but that it characterizes the minimal tensor product of general pairs of convex cones.<n>Our proof makes use of a new characterization of products of simplices up to affine equivalence.
  arXiv  Detail & Related papers  (2022-06-23T16:23:59Z)
• Beyond the Berry Phase: Extrinsic Geometry of Quantum States [77.34726150561087]
  We show how all properties of a quantum manifold of states are fully described by a gauge-invariant Bargmann. We show how our results have immediate applications to the modern theory of polarization.
  arXiv  Detail & Related papers  (2022-05-30T18:01:34Z)
• Low-degree learning and the metric entropy of polynomials [44.99833362998488]
  We prove that any (deterministic or randomized) algorithm which learns $mathscrF_nd$ with $L$-accuracy $varepsilon$ requires at least $Omega(sqrtvarepsilon)2dlog n leq log mathsfM(mathscrF_n,d,|cdot|_L,varepsilon) satisfies the two-sided estimate $$c (1-varepsilon)2dlog
  arXiv  Detail & Related papers  (2022-03-17T23:52:08Z)
• Random matrices in service of ML footprint: ternary random features with no performance loss [55.30329197651178]
  We show that the eigenspectrum of $bf K$ is independent of the distribution of the i.i.d. entries of $bf w$. We propose a novel random technique, called Ternary Random Feature (TRF) The computation of the proposed random features requires no multiplication and a factor of $b$ less bits for storage compared to classical random features.
  arXiv  Detail & Related papers  (2021-10-05T09:33:49Z)
• Spectral properties of sample covariance matrices arising from random matrices with independent non identically distributed columns [50.053491972003656]
  It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$. Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
  arXiv  Detail & Related papers  (2021-09-06T14:21:43Z)
• Stochastic behavior of outcome of Schur-Weyl duality measurement [45.41082277680607]
  We focus on the measurement defined by the decomposition based on Schur-Weyl duality on $n$ qubits. We derive various types of distribution including a kind of central limit when $n$ goes to infinity.
  arXiv  Detail & Related papers  (2021-04-26T15:03:08Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.