We present the positive-partial-transpose squared conjecture introduced by M. Christandl at Banff International Research Station Workshop: Operator Structures in Quantum Information Theory (Banff International Research Station, Alberta, 2012). We investigate the conjecture in higher dimensions and offer two novel approaches (decomposition and composition of quantum channels) and correspondingly, several schemes for finding counterexamples to this conjecture. One of the schemes involving the composition of PPT quantum channels in unsolved dimensions yields a potential counterexample.
We study mapping cones of positive maps of
B(H) into itself, i.e., cones which are closed under composition with completely positive maps. As applications we obtain characterizations of linear functionals with strong positivity properties with respect to so-called symmetric mapping cones, with special emphasis on separable and PPT states.
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.
We present the positive-partial-transpose (PPT) square conjecture introduced by M. Christandl Banff International Research Station Workshop: Operator Structures in Quantum Information Theory (Banff International Research Station, Alberta, 2012). We prove the conjecture in the case
n=3 as a consequence of the fact that two-qutrit PPT states have Schmidt number of at most 2. The PPT square conjecture in the case of $n\ensuremath{\ge}4$ is still open. We present an example to support the conjecture for
n=4.
The PPT2 conjecture asserts that the composition of any two PPT maps is entanglement breaking. It is proven for maps on matrices of size up to
3times3 and for several structured families, but the general case remains open
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.