$k$-Positivity and high-dimensional bound entanglement under symplectic group symmetry

• URL: http://arxiv.org/abs/2602.09860v1
• Date: Tue, 10 Feb 2026 15:03:58 GMT
• Title: $k$-Positivity and high-dimensional bound entanglement under symplectic group symmetry
• Authors: Sang-Jun Park,
• Abstract summary: We study the structure of $k$-positivity and Schmidt numbers for classes of linear maps and bipartite quantum states exhibiting symplectic group symmetry.<n>Our results offer a natural and analytically tractable framework in which both strong forms of positive indecomposability and high degrees of PPT entanglement can be studied systematically.
• Score: 6.52489063705362
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: We investigate the structure of $k$-positivity and Schmidt numbers for classes of linear maps and bipartite quantum states exhibiting symplectic group symmetry. Specifically, we consider (1) linear maps on $M_d(\mathbb{C})$ which are covariant under conjugation by unitary symplectic matrices $S$, and (2) $d\otimes d$ bipartite states which are invariant under $S\otimes S$ or $S\otimes \overline{S}$ actions, each parametrized by two real variables. We provide a complete characterization of all $k$-positivity and decomposability conditions for these maps and explicitly compute the Schmidt numbers for the corresponding bipartite states. In particular, our analysis yields a broad class of PPT states with Schmidt number $d/2$ and the first explicit constructions of (optimal) $k$-positive indecomposable linear maps for arbitrary $k=1,\ldots, d/2-1$, achieving the best-known bounds. Overall, our results offer a natural and analytically tractable framework in which both strong forms of positive indecomposability and high degrees of PPT entanglement can be studied systematically. We present two further applications of symplectic group symmetry. First, we show that the PPT-squared conjecture holds within the class of PPT linear maps that are either symplectic-covariant or conjugate-symplectic-covariant. Second, we resolve a conjecture of Pal and Vertesi concerning the optimal lower bound of the Sindici-Piani semidefinite program for PPT entanglement.

Related papers

• Fractional $k$-positivity: a continuous refinement of the $k$-positive scale [0.0]
  We introduce a real- parameter refinement of the classical integer hierarchies underlying Schmidt number, block-positivity, and $k$-positivity for maps between matrix algebras.<n>We prove a emphfractional Kraus theorem: $$-superpositive maps are precisely the completely positive maps admitting a Kraus decomposition.<n>For non-integer $$ the cones $mathsf P_$ fail stability under CP post-composition, highlighting a sharp structural transition away from the integer theory.
  arXiv  Detail & Related papers  (2026-02-13T08:57:22Z)
• Schwarz maps with symmetry [0.0]
  Theory of symmetry of quantum mechanical systems is applied to study the structure and properties of several classes of relevant maps in quantum information theory.<n>We develop the general structure that equivariant maps $:mathcal A to mathcal B$ between $Cast$-algebras satisfy.<n>We then undertake a systematic study of unital, Hermiticity-preserving maps that are equivariant under natural unitary group actions.
  arXiv  Detail & Related papers  (2026-01-05T17:11:03Z)
• Quantum two-dimensional superintegrable systems in flat space: exact-solvability, hidden algebra, polynomial algebra of integrals [41.99844472131922]
  It includes the Smorodinsky-Winternitz potentials I-II (the Holt potential), the Fokas-Lagerstrom model, the 3-body Calogero and Wolfes (equivalently, $G$ rational, or $I_6$) models.<n>It is shown that all of them are exactly-solvable, thus, confirming the Montreal conjecture.<n>Each model is characterized by infinitely-many finite-dimensional invariant subspaces, which form the infinite flag.
  arXiv  Detail & Related papers  (2025-12-30T07:39:35Z)
• Positive maps and extendibility hierarchies from copositive matrices [4.914288166307964]
  This work introduces and systematically studies a new convex cone of PCOP (pairwise copositive)<n>We establish that this cone is dual to the cone of PCP (pairwise completely positive) and, critically, provides a complete characterization for the positivity of the broad class of covariant maps.
  arXiv  Detail & Related papers  (2025-09-18T17:54:09Z)
• $k$-Positive Maps: New Characterizations and a Generation Method [0.9685837672183747]
  We study $k$-positive linear maps on matrix algebras and address two problems characterizations of $k$-positivity and generation of non-decomposable $k$-positive maps.<n>On the characterization side, we derive optimization-based conditions equivalent to $k$-positivity that (a) reduce to a simple check when $k=d$, (b) reveal a direct link to the spectral norm of certain order-3 tensors.<n>On the generation side, we introduce a Lie-semigroup-based method that, starting from a single $k$
  arXiv  Detail & Related papers  (2025-08-29T06:22:57Z)
• Bridging conformal field theory and parton approaches to SU(n)_k chiral spin liquids [21.876059213677966]
  We employ the $mathrmSU(n)_k$ Wess-Zumino-Witten (WZW) model in conformal field theory to construct lattice wave functions in both one and two dimensions.<n>The spins on all lattice sites are chosen to transform under the $mathrmSU(n)$ irreducible representation with a single row and $k$ boxes in the Young tableau.
  arXiv  Detail & Related papers  (2025-01-16T14:42:00Z)
• 1+1d SPT phases with fusion category symmetry: interface modes and non-abelian Thouless pump [0.0]
  We consider $mathcalC$-SPT phases with finite non-invertible symmetry in 1+1d. We show that there must be a degenerate interface mode between different $mathcalC$-SPT phases. Our invariant is identified with a non-abelian generalization of the Thouless charge pump.
  arXiv  Detail & Related papers  (2024-08-28T17:26:18Z)
• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles [55.41644538483948]
  We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the transformations symmetry and the corresponding generators. Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.
  arXiv  Detail & Related papers  (2023-01-13T16:25:25Z)
• 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)
• Annihilating Entanglement Between Cones [77.34726150561087]
  We show that Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation.
  arXiv  Detail & Related papers  (2021-10-22T15:02:39Z)
• 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)
• SU$(3)_1$ Chiral Spin Liquid on the Square Lattice: a View from Symmetric PEPS [55.41644538483948]
  Quantum spin liquids can be faithfully represented and efficiently characterized within the framework of Projectedangled Pair States (PEPS) Characteristic features are revealed by the entanglement spectrum (ES) on an infinitely long cylinder. Special features in the ES are shown to be in correspondence with bulk anyonic correlations, indicating a fine structure in the holographic bulk-edge correspondence.
  arXiv  Detail & Related papers  (2019-12-31T16:30:25Z)

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.