We introduce a family of positive maps on qudits whose Choi operators combine two competing coherence patterns: independent exchange couplings and a collective set of diagonal couplings. For every mixed member of the family, we derive exact conditions for positivity, complete positivity, and all intermediate levels of k-positivity, and we determine the complete phase structure of decomposability and atomicity. The higher-positivity thresholds depend on the channel multiplicities, whereas the exact decomposability boundary is universal in scale-free coordinates. Its proof combines an explicit positive-plus-copositive decomposition with a matching family of positive-partial-transpose (PPT) states. The latter form PPT edge states whose detection region is an ellipse, or a disk after collective rescaling. An embedded-qutrit certificate shows that indecomposability and atomicity coincide throughout the mixed family. In contrast, the purely collective endpoint is decomposable over its entire positive interval. Finally, at maximal coherence, an explicit positive subtraction produces an atomic optimal entanglement witness with the spanning property and a strictly larger PPT- detection set. The study thus provides an exactly solvable setting in which k-positivity, decomposability, atomicity, PPT-edge-state geometry, and witness optimality can be compared in arbitrary local dimension.
We introduce a family of positive maps on qudits whose Choi operators combine two competing coherence patterns: independent exchange couplings and a collective set of diagonal couplings. For every mixed member of the family, we derive exact conditions for positivity, complete positivity, and all intermediate levels of k-positivity, and we determine the complete phase structure of decomposability and atomicity. The higher-positivity thresholds depend on the channel multiplicities, whereas the exact decomposability boundary is universal in scale-free coordinates. Its proof combines an explicit positive-plus-copositive decomposition with a matching family of positive-partial-transpose (PPT) states. The latter form PPT edge states whose detection region is an ellipse, or a disk after collective rescaling. An embedded-qutrit certificate shows that indecomposability and atomicity coincide throughout the mixed family. In contrast, the purely collective endpoint is decomposable over its entire positive interval. Finally, at maximal coherence, an explicit positive subtraction produces an atomic optimal entanglement witness with the spanning property and a strictly larger PPT- detection set. The study thus provides an exactly solvable setting in which k-positivity, decomposability, atomicity, PPT-edge-state geometry, and witness optimality can be compared in arbitrary local dimension.
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≤w,z≤2/3, complete positivity on the smaller square 0≤w,z≤1/3, and decomposability is lost precisely outside a quarter circle in the corner w,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) 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.
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.
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). 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), obtained as a natural extension of Kye’s indecomposable maps defined for 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⊗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⊗3 and 2⊗4 systems. In addition, we show that generalized Choi maps simply can not detect well known PPT entangled states for 2⊗4 systems.
Let Φ: Md → M3 and Ψ: M3 → Md be completely positive and completely copositive (PPT) maps, and let W be the Choi matrix of Ψ ∘ Φ. The PPT2 conjecture asserts that W is separable. We study the twirl of W by the group of local diagonal unitaries D ⊗ D̄, whose separability is equivalent to the statement that no diagonal-unitary-covariant witness detects W. The twirl is determined by Aij = ⟨ij|W|ij⟩ and the coherence matrix Bik = ⟨ii|W|kk⟩. We prove that for every d the twirl is separable whenever rank B ≤ 2. The proof factors W through Gram data Ta = Ga*Ha, classifies the two-dimensional spaces span{Ta} into five types, and combines a peeling lemma with known separability criteria for low-rank PPT states and, for two of the five types, with the theorem that composites of PPT maps through a qubit are entanglement breaking. A peel at an index c writes Ψ (or Φ) as the sum of an entanglement-breaking part and a PPT remainder whose coherence vanishes at c. Chains of peels, interleaved with the removal of zero rows of B, close further cases at every coherence rank, and for d = 4 a single peel suffices. For every d ≥ 3 the twirl plus the diagonal operator ((d − 3)/2) ∑i Bii|ii⟩⟨ii| is separable. At d = 4, if the compression of the Choi matrix of Φ or of Ψ* to a set of at least two outer indices is a sum of product states with linearly independent outer factors, then Φ, respectively Ψ, is entanglement breaking. At d = 4 and coherence rank 3 with a linear relation ∑a λaTa = 0 in which some λa vanishes, the twirl is separable, and Ψ ∘ Φ is entanglement breaking when exactly one λa vanishes. A cycle criterion for an arbitrary factorization B = FF* yields explicit product-state certificates. Numerically, this criterion certifies the twirl at each of the 42 generic configurations without peels produced by our random semidefinite searches at d = 4. At some pairs of extreme PPT maps it fails for every factorization of B. In particular, for every choice of Kraus operators, it fails when the columns of F are the diagonals of the products of a Kraus operator of Ψ with one of Φ. The twirl is nevertheless separable at these pairs: the Choi matrices of Φ and of Ψ* are supported on S ⊗ ℂ3 and S′ ⊗ ℂ3 for three-dimensional subspaces S, S′ ⊆ ℂd, and the known case M3 → M3 → M3 of the PPT2 conjecture applies. For d ≤ 4 and every coherence rank, the twirl is separable for every composite Φ* ∘ Ξ ∘ Φ in which Ξ is a sum of conjugations by positive semidefinite matrices. This follows from a criterion of Johnston and MacLean for completely positive coherence matrices. For d ≤ 4 we extend their criterion to coherence matrices that become entrywise nonnegative under conjugation by a diagonal unitary, provided that, around each cycle of the support graph of B, the product of the entries of A is the same in both directions. We also show that the number of product terms in a separable decomposition of the twirl can exceed the coherence rank, even for entanglement-breaking pairs through a qutrit. Finally, given the theorem that composites of PPT maps through a qubit are entanglement breaking, we show that W + (1/5) Φ*(I)⊤ ⊗ Ψ(I) is separable for every d, that for d ≥ 4 no pair-independent average of certificates from this theorem lowers the constant 1/5, even with PPT2 for maps on M3 and the separability of PPT states on ℂ2 ⊗ ℂ3, and that certificates adapted to the pair go beyond it: exact rational certificates show that W − (1/20) Φ*(I)⊤ ⊗ Ψ(I) is separable for an explicit pair at d = 4, and that W + (1/10) Φ*(I)⊤ ⊗ Ψ(I) is separable for an explicit Φ: M4 → M3 and every PPT map Ψ. Given the same theorem, for Φ: M3 → M3 a constant below 1/5 works uniformly: there is c < 1/5 such that W + c Φ*(I)⊤ ⊗ Ψ(I) is separable for every such Φ and every PPT map Ψ: M3 → Mn. Version 2 adds Theorem 10.19: no PPT channel M₃ → M₃ is rigid, so Λ(3) < 1/5. Given the theorem that composites of PPT maps through a qubit are entanglement breaking, it follows that a single constant below 1/5 applies to every pair of PPT maps M₃ → M₃ → Mₙ (Corollary 10.20). Exact certificates, checked in Lean, show that the constant of certificates adapted to one factor is at most 1/10 at two explicit points built from the Tiles unextendible product basis (Remark 10.21). The accompanying Lean code now contains 2688 theorems, 1833 of which are checked by Lean's kernel alone. Mathematics Subject Classification (2020): 81P40; 15A69, 47L07.
Entanglement swapping is a fundamental operation in quantum repeaters for establishing entanglement between distant parties. The positive partial transpose (PPT) squared conjecture asks whether two PPT entangled links can generate terminal entanglement through entanglement swapping, or equivalently, whether the composition of two PPT maps is always entanglement breaking. Motivated by this conjecture, we investigate the map-composition problem beyond the PPT setting. For qutrit completely positive (CP) maps, we prove that the composition of any CP map whose Choi matrix is 1-undistillable with any CP map whose Choi matrix has Schmidt number at most two is entanglement breaking in either order. Moreover, we show that the cone of 1-undistillable CP maps is exactly the largest qutrit cone of CP maps whose composition with every CP map whose Choi matrix has Schmidt number at most two is entanglement breaking in both orders. Finally, although map composition captures only the standard maximally entangled outcome in entanglement swapping, we prove that any 1-undistillable two-qutrit state and any state of Schmidt number at most two cannot generate terminal entanglement under an arbitrary selective measurement on the intermediate systems.