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arXiv:2604.27034 · Linear Algebra and its Applications

Some applications of Choi polynomials of linear maps

Minh Toan Ho, Thanh Hieu Le, Cong Trinh Le, Hiroyuki Osaka

quant-phmath.OASubmitted Apr 29, 2026• Announced Apr 30, 2026• 0 citations
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Abstract

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.

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40 pages

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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 Mm(C)⊗Mn(C)M_m(\mathbb{C}) \otimes M_n(\mathbb{C})Mm​(C)⊗Mn​(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.
We construct a family of positive but not completely positive linear maps acting on M4(C)M_4(\mathcal{C})M4​(C), obtained as a natural extension of Kye’s indecomposable maps defined for M3(C)M_3(\mathcal{C})M3​(C). We rigorously prove positivity of these maps on 'X' states and employ them to identify a one-parameter family of bipartite quantum states living in the vicinity of maximally mixed state to be entangled even though states are positive under partial transposition (PPT). This provides an explicit example of a family of quantum states with both bound entangled states and free entangled states in 4⊗44 \otimes 44⊗4 systems, a regime that remains less characterized compared to lower-dimensional cases. The proposed maps detect entanglement and reveal new structural features of the PPT entangled region in higher dimensions. Our results extend the applicability of positive-map–based entanglement detection and contribute to the systematic understanding of bound entanglement beyond the 3⊗33\otimes 33⊗3 and 2⊗42 \otimes 42⊗4 systems. In addition, we show that generalized Choi maps simply can not detect well known PPT entangled states for 2⊗42 \otimes 42⊗4 systems.
We study a family of sparse positive maps on qutrits for which positivity, decomposability, and PPT entanglement can all be analysed explicitly. The block structure of the associated Choi matrices reduces positivity to a Hermitian biquadratic form and leads to exact positivity boundaries for three representative parametric families. For the same families we determine the exact transition between decomposable and non-decomposable maps and construct associated PPT states of two classes. The first consists of witness-adapted deformations naturally tied to the non-decomposability analysis. The second consists of analytically tractable families whose full PPT-entangled branch is detected by fixed positive maps, yielding exact thresholds between separability and bound entanglement. For the trace-preserving subclass, we further compare positivity with a recent eigenvalue bound for 2-positive maps, thereby making the gap between positivity and higher-order positivity fully explicit within this family.
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Power symmetric stochastic matrices introduced by R. Sinkhorn (1981) and their generalization by R.B. Bapat, S.K. Jain and K. Manjunatha Prasad (1999) have been utilized to give positive block matrices with trace one possessing positive partial transpose, the so-called PPT states. Another method to construct such PPT states is given, it uses the form of a matrix unitarily equivalent to to its transpose obtained by S.R. Garcia and J.E. Tener (2012). Evolvement or suppression of separability or entanglement of various levels for a quantum dynamical semigroup of completely positive maps has been studied using Choi-Jamiolkowsky matrix of such maps and the famous Hordeckis criteria (1996). A Trichotomy Theorem has been proved, and examples have been given that depend mainly on generalized Choi maps and clearly distinguish the levels of entanglement breaking.
kkk-Positivity and high-dimensional bound entanglement under symplectic group symmetries
We investigate the structure of kkk-positivity and Schmidt numbers for classes of linear maps and bipartite quantum states exhibiting symplectic group symmetries. Specifically, we consider (1) linear maps on Md(C)M_d(\mathbb{C})Md​(C) which are covariant under conjugation by unitary symplectic matrices SSS, and (2) d⊗dd\otimes dd⊗d bipartite states which are invariant under S⊗SS\otimes SS⊗S or S⊗S‾S\otimes \overline{S}S⊗S actions, each parametrized by two real variables. We provide a complete characterization of all kkk-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/2d/2d/2 and the first explicit constructions of (optimal) kkk-positive indecomposable linear maps for arbitrary k=1,…,d/2−1k=1,\ldots, d/2-1k=1,…,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 symmetries. 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.