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Bound Entanglement Detection in 4 ⊗ 4 Systems via Generalized Choi Maps

Mazhar Ali

Published Feb 6, 2026• 0 citations
Publisher

Abstract

We construct a family of positive but not completely positive linear maps acting on $M_4(\mathcal{C})$, obtained as a natural extension of Kye’s indecomposable maps defined for $M_3(\mathcal{C})$. 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 $4 \otimes 4$ 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 $3\otimes 3$ and $2 \otimes 4$ systems. In addition, we show that generalized Choi maps simply can not detect well known PPT entangled states for $2 \otimes 4$ systems.

DOI
10.20944/preprints202602.0393.v1

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Bound entanglement detection in 4⊗44 \otimes 44⊗4 systems via generalized Choi maps
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.
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.
Abstract In this work, we concentrate on identifying bound entangled states with positive partial transposition in tripartite system. Since directly extending PPT-BE states from bipartite to tripartite systems is nontrivial, we adopt unextendible product bases (UPBs) as a tool to generate tripartite PPT-BE states. First, a set of UPB with cardinalities d 3 − 4 d + 3 is presented to characterize the extremal states of PPT set in the C d ⨂ C d ⨂ C d system. Using this basis, the missing states are introduced and employed to identify a 2 d − 1 -parameter family of PPT entangled edge states. Additionally, a generation circuit is provided which transforms the states in Z-basis to the missing states in three-qudit composite system. This circuit intuitively reflects the structure of the missing states and prepares for the generation of PPT entangled states. Our work demonstrates a different construction method of PPT entangled edge states in higher-dimensional multipartite system, providing a general insight into the relationship between nonlocality and entanglement in such system.