We construct a class of quantum bipartite $d\ensuremath{\bigotimes}d$ states which are positive under partial transposition (PPT states). This class is invariant under the maximal commutative subgroup of
U(d) and contains as special cases many well-known examples of PPT states. States from our class provide criteria for testing the indecomposability of positive maps. Such maps are crucial for constructing entanglement witnesses.
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 investigate the structure of
k-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) which are covariant under conjugation by unitary symplectic matrices
S, and (2)
d⊗d bipartite states which are invariant under
S⊗S or
S⊗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,…,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.
Abstract The problem of bound entanglement detection is a challenging aspect of quantum information theory for higher dimensional systems. Here, we propose an indecomposable positive map for two-qutrit systems, which is shown to detect a class of positive partial transposed (PPT) states. A corresponding witness operator is constructed and shown to be weakly optimal and locally implementable. Further, we perform a structural physical approximation of the indecomposable map to make it a completely positive one, and find a new PPT-entangled state which is not detectable by certain other well-known entanglement detection criteria.