Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.09675

Determinantal Kernel Schemes of Matrix Nets and Applications to Positive Maps

Trung Hoa Dinh, Ho Minh Toan, Công Trình Lê, Trung-Dung Vuong

Published Sep 9, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2609.09675

Related papers

Sufficient separability criteria and linear maps

Maciej Lewenstein, Remigiusz Augusiak, Dariusz Chruściński +2 more

We study families of positive and completely positive maps acting on a bipartite system ${\mathbb{C}}^{M}\ensuremath{\bigotimes}{\mathbb{C}}^{N}$ (with $M\ensuremath{\le}N$). The maps have a property that, when applied to any state (of a given entanglement class), result in a separable state or, more generally, a state of another certain entanglement class (e.g., Schmidt number $\ensuremath{\le}\phantom{\rule{0.16em}{0ex}}k$). This allows us to derive useful families of sufficient separability criteria. Explicit examples of such criteria have been constructed for arbitrary $M,N$, with a special emphasis on $M=2$. Our results can be viewed as generalizations of the known facts that in the sufficiently close vicinity of the completely depolarized state (the normalized identity matrix), all states are separable (belong to ``weakly'' entangled classes). Alternatively, some of our results can be viewed as an entanglement classification for a certain family of states, corresponding to mixtures of the completely polarized state with pure states, partial transposes, and/or local transformations thereof.

arXiv:1512.08278•Physical Review A•Apr 25, 2016•17 citations

Sparse anisotropic positive maps for qutrit entanglement: exact indecomposability and PPT geometry

Massimiliano F. Sacchi

Positive but not completely positive maps provide one of the most direct ways to detect entanglement beyond the positive-partial-transpose (PPT) criterion. We introduce and analyze an exactly solvable two-parameter family of sparse bistochastic positive maps on qutrits, in which two coherence channels are independently tuned by parameters $w$ and $z$. The sparse structure makes the full phase diagram analytic: positivity holds exactly on the square $0\le w,z\le2/3$, complete positivity on the smaller square $0\le w,z\le1/3$, and decomposability is lost precisely outside a quarter circle in the corner $w,z\ge1/3$. The indecomposable region is certified by explicit PPT entangled state adapted to the same witness geometry. At the endpoint $W_*=W(2/3,2/3)$ we construct a four-parameter family of PPT edge states of rank type $(5,5)$, derive their analytic detection region, and show that the corresponding rays are exposed faces of the PPT cone. Finally, although $W_*$ is not optimal, we give an explicit optimal refinement whose detection region on this family is strictly larger. The result is an analytically tractable qutrit setting in which positivity, indecomposability, PPT entanglement, optimality, and exposed convex geometry can be studied in a single framework.

quant-pharXiv:2607.12470•Physical Review A•Jul 15, 2026•0 citations

A Class of Extremal Positive Maps in $3 \times 3$ Matrix Algebras

Hiroyuki Osaka

We shall provide a large class of extremal positive maps in M_3(\mathbf C) which are neither 2-positive nor 2-copositive and study the algebraic structure of the set of all positive linear maps in M_3(\mathbf C) .

Publications of the Research Institute for Mathematical Sciences•Oct 31, 1992•37 citations

On the minimal dimension of range-criterion counterexamples to the PPT² conjecture

Tejasvi Singh Tomar

The PPT² conjecture asserts that the composition of two completely positive and completely copositive (PPT) maps is entanglement breaking. A recent counterexample uses two maps through a six-dimensional middle algebra between ten-dimensional outer algebras, and certifies entanglement by the range criterion: the range of the composite Choi matrix contains no product vector. We ask how small such a certificate can be. For PPT maps Φ: Mₘ → Mₖ and Ψ: Mₖ → Mₙ with Ψ ∘ Φ ≠ 0 we prove, with computer-assisted steps, that the range of the Choi matrix of Ψ ∘ Φ contains a product vector whenever max(m, k, n) ≤ 6 and either k ≤ 5 or an outer dimension is at most five. If both outer dimensions are four, this holds for every middle dimension k. In dimension six only the reduced type (6, 6, 6) remains, and we bound its Choi ranks. The proofs use that, after a reduction, a product-free composite range forbids Kraus operators of rank at most two, that the rank-loss loci of a PPT map are finite unions of linear subspaces with generically constant range on each subspace, a rank inequality for PPT maps at product vectors in the kernel of their Choi matrices, top Chern classes at middle dimensions four and five, an excess intersection on Gr(3, 6) at middle dimension six, and, for the type (5, 6, 5), the Euler characteristic of a sheaf on a multiple line in ℙ². For the known example we show that the quadratic Veronese images of the rank-loss directions of its base map and of their kernel vectors are Gale dual by the Cayley–Bacharach theorem, and that the kernel conditions determine the Kraus spaces of both factors. If the rank-loss scheme of the base map is reduced and of the expected dimension zero, then among the lifts through arbitrary Schur functors with nonzero composite only those through exterior powers give an unextendible product set of kernel product vectors, and these only when one of the three dimensions is at least ten. For a general base map, the member of the family with outer dimension six is one product vector short of the range criterion, and its factors are rigid. For such base maps on M₄ that are not entanglement breaking, a compression of the lift to outer dimensions 4 ≤ D ≤ 9 keeps at most 2D − 4 of the twenty kernel product vectors. Finally, symmetry under a compact connected group acting irreducibly on both outer spaces, or under SU(2) acting without multiplicities on one of them, puts a product vector into the range. Mathematics Subject Classification (2020): 81P40; 15A69, 14M15, 14C17, 47L07.

•Oct 24, 2026•0 citations

A Class of Linear Positive Maps in Matrix Algebras II

Gen Kimura, Andrzej Kossakowski

We provide a class of linear trace preserving positive maps on matrix algebras which is a generalization of that in [7]. A systematic construction by means of spectra of generators of SU (n) is discussed.

Open Systems & Information Dynamics•Dec 1, 2004•29 citations

On the minimal dimension of range-criterion counterexamples to the PPT² conjecture

Tejasvi Singh Tomar

The PPT² conjecture asserts that the composition of two completely positive and completely copositive (PPT) maps is entanglement breaking. A recent counterexample uses two maps through a six-dimensional middle algebra between ten-dimensional outer algebras, and certifies entanglement by the range criterion: the range of the composite Choi matrix contains no product vector. We ask how small such a certificate can be. For PPT maps Φ: Mₘ → Mₖ and Ψ: Mₖ → Mₙ with Ψ ∘ Φ ≠ 0 we prove, with computer-assisted steps, that the range of the Choi matrix of Ψ ∘ Φ contains a product vector whenever max(m, k, n) ≤ 6 and either k ≤ 5 or an outer dimension is at most five. If both outer dimensions are four, this holds for every middle dimension k. In dimension six only the reduced type (6, 6, 6) remains, and we bound its Choi ranks. The proofs use that, after a reduction, a product-free composite range forbids Kraus operators of rank at most two, that the rank-loss loci of a PPT map are finite unions of linear subspaces with generically constant range on each subspace, a rank inequality for PPT maps at product vectors in the kernel of their Choi matrices, top Chern classes at middle dimensions four and five, an excess intersection on Gr(3, 6) at middle dimension six, and, for the type (5, 6, 5), the Euler characteristic of a sheaf on a multiple line in ℙ². For the known example we show that the quadratic Veronese images of the rank-loss directions of its base map and of their kernel vectors are Gale dual by the Cayley–Bacharach theorem, and that the kernel conditions determine the Kraus spaces of both factors. If the rank-loss scheme of the base map is reduced and of the expected dimension zero, then among the lifts through arbitrary Schur functors with nonzero composite only those through exterior powers give an unextendible product set of kernel product vectors, and these only when one of the three dimensions is at least ten. For a general base map, the member of the family with outer dimension six is one product vector short of the range criterion, and its factors are rigid. For such base maps on M₄ that are not entanglement breaking, a compression of the lift to outer dimensions 4 ≤ D ≤ 9 keeps at most 2D − 4 of the twenty kernel product vectors. Finally, symmetry under a compact connected group acting irreducibly on both outer spaces, or under SU(2) acting without multiplicities on one of them, puts a product vector into the range. Version 3 adds a description of a Lean formalization that derives the main theorem from nine stated algebro-geometric hypotheses (code: doi:10.5281/zenodo.23643757); the mathematics is otherwise the same as in version 2. Mathematics Subject Classification (2020): 81P40; 15A69, 14M15, 14C17, 47L07.

•Oct 25, 2026•0 citations
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 2×22\times22×2 compression. For a,b≥3a,b\ge3a,b≥3, the corresponding locus in \mboxGr(3,Ma,b)\mbox{Gr}(3,M_{a,b})\mboxGr(3,Ma,b​) has exactly three irreducible components, whose geometry and intersections are determined explicitly. For nets in M3(C)M_3(\mathbb C)M3​(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)(5,5)(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.
We study families of positive and completely positive maps acting on a bipartite system ${\mathbb{C}}^{M}\ensuremath{\bigotimes}{\mathbb{C}}^{N}$ (with $M\ensuremath{\le}N$). The maps have a property that, when applied to any state (of a given entanglement class), result in a separable state or, more generally, a state of another certain entanglement class (e.g., Schmidt number $\ensuremath{\le}\phantom{\rule{0.16em}{0ex}}k$). This allows us to derive useful families of sufficient separability criteria. Explicit examples of such criteria have been constructed for arbitrary M,NM,NM,N, with a special emphasis on M=2M=2M=2. Our results can be viewed as generalizations of the known facts that in the sufficiently close vicinity of the completely depolarized state (the normalized identity matrix), all states are separable (belong to ``weakly'' entangled classes). Alternatively, some of our results can be viewed as an entanglement classification for a certain family of states, corresponding to mixtures of the completely polarized state with pure states, partial transposes, and/or local transformations thereof.
Positive but not completely positive maps provide one of the most direct ways to detect entanglement beyond the positive-partial-transpose (PPT) criterion. We introduce and analyze an exactly solvable two-parameter family of sparse bistochastic positive maps on qutrits, in which two coherence channels are independently tuned by parameters www and zzz. The sparse structure makes the full phase diagram analytic: positivity holds exactly on the square 0≤w,z≤2/30\le w,z\le2/30≤w,z≤2/3, complete positivity on the smaller square 0≤w,z≤1/30\le w,z\le1/30≤w,z≤1/3, and decomposability is lost precisely outside a quarter circle in the corner w,z≥1/3w,z\ge1/3w,z≥1/3. The indecomposable region is certified by explicit PPT entangled state adapted to the same witness geometry. At the endpoint W∗=W(2/3,2/3)W_*=W(2/3,2/3)W∗​=W(2/3,2/3) we construct a four-parameter family of PPT edge states of rank type (5,5)(5,5)(5,5), derive their analytic detection region, and show that the corresponding rays are exposed faces of the PPT cone. Finally, although W∗W_*W∗​ is not optimal, we give an explicit optimal refinement whose detection region on this family is strictly larger. The result is an analytically tractable qutrit setting in which positivity, indecomposability, PPT entanglement, optimality, and exposed convex geometry can be studied in a single framework.
A Class of Extremal Positive Maps in 3×33 \times 33×3 Matrix Algebras
The PPT² conjecture asserts that the composition of two completely positive and completely copositive (PPT) maps is entanglement breaking. A recent counterexample uses two maps through a six-dimensional middle algebra between ten-dimensional outer algebras, and certifies entanglement by the range criterion: the range of the composite Choi matrix contains no product vector. We ask how small such a certificate can be. For PPT maps Φ: Mₘ → Mₖ and Ψ: Mₖ → Mₙ with Ψ ∘ Φ ≠ 0 we prove, with computer-assisted steps, that the range of the Choi matrix of Ψ ∘ Φ contains a product vector whenever max(m, k, n) ≤ 6 and either k ≤ 5 or an outer dimension is at most five. If both outer dimensions are four, this holds for every middle dimension k. In dimension six only the reduced type (6, 6, 6) remains, and we bound its Choi ranks. The proofs use that, after a reduction, a product-free composite range forbids Kraus operators of rank at most two, that the rank-loss loci of a PPT map are finite unions of linear subspaces with generically constant range on each subspace, a rank inequality for PPT maps at product vectors in the kernel of their Choi matrices, top Chern classes at middle dimensions four and five, an excess intersection on Gr(3, 6) at middle dimension six, and, for the type (5, 6, 5), the Euler characteristic of a sheaf on a multiple line in ℙ². For the known example we show that the quadratic Veronese images of the rank-loss directions of its base map and of their kernel vectors are Gale dual by the Cayley–Bacharach theorem, and that the kernel conditions determine the Kraus spaces of both factors. If the rank-loss scheme of the base map is reduced and of the expected dimension zero, then among the lifts through arbitrary Schur functors with nonzero composite only those through exterior powers give an unextendible product set of kernel product vectors, and these only when one of the three dimensions is at least ten. For a general base map, the member of the family with outer dimension six is one product vector short of the range criterion, and its factors are rigid. For such base maps on M₄ that are not entanglement breaking, a compression of the lift to outer dimensions 4 ≤ D ≤ 9 keeps at most 2D − 4 of the twenty kernel product vectors. Finally, symmetry under a compact connected group acting irreducibly on both outer spaces, or under SU(2) acting without multiplicities on one of them, puts a product vector into the range. Mathematics Subject Classification (2020): 81P40; 15A69, 14M15, 14C17, 47L07.
The PPT² conjecture asserts that the composition of two completely positive and completely copositive (PPT) maps is entanglement breaking. A recent counterexample uses two maps through a six-dimensional middle algebra between ten-dimensional outer algebras, and certifies entanglement by the range criterion: the range of the composite Choi matrix contains no product vector. We ask how small such a certificate can be. For PPT maps Φ: Mₘ → Mₖ and Ψ: Mₖ → Mₙ with Ψ ∘ Φ ≠ 0 we prove, with computer-assisted steps, that the range of the Choi matrix of Ψ ∘ Φ contains a product vector whenever max(m, k, n) ≤ 6 and either k ≤ 5 or an outer dimension is at most five. If both outer dimensions are four, this holds for every middle dimension k. In dimension six only the reduced type (6, 6, 6) remains, and we bound its Choi ranks. The proofs use that, after a reduction, a product-free composite range forbids Kraus operators of rank at most two, that the rank-loss loci of a PPT map are finite unions of linear subspaces with generically constant range on each subspace, a rank inequality for PPT maps at product vectors in the kernel of their Choi matrices, top Chern classes at middle dimensions four and five, an excess intersection on Gr(3, 6) at middle dimension six, and, for the type (5, 6, 5), the Euler characteristic of a sheaf on a multiple line in ℙ². For the known example we show that the quadratic Veronese images of the rank-loss directions of its base map and of their kernel vectors are Gale dual by the Cayley–Bacharach theorem, and that the kernel conditions determine the Kraus spaces of both factors. If the rank-loss scheme of the base map is reduced and of the expected dimension zero, then among the lifts through arbitrary Schur functors with nonzero composite only those through exterior powers give an unextendible product set of kernel product vectors, and these only when one of the three dimensions is at least ten. For a general base map, the member of the family with outer dimension six is one product vector short of the range criterion, and its factors are rigid. For such base maps on M₄ that are not entanglement breaking, a compression of the lift to outer dimensions 4 ≤ D ≤ 9 keeps at most 2D − 4 of the twenty kernel product vectors. Finally, symmetry under a compact connected group acting irreducibly on both outer spaces, or under SU(2) acting without multiplicities on one of them, puts a product vector into the range. Version 3 adds a description of a Lean formalization that derives the main theorem from nine stated algebro-geometric hypotheses (code: doi:10.5281/zenodo.23643757); the mathematics is otherwise the same as in version 2. Mathematics Subject Classification (2020): 81P40; 15A69, 14M15, 14C17, 47L07.