6 problems

Antisymmetric-state bound conjecture. The bound $\frac{1}{2r+2}$ cannot be improved. This concerns quantitative lower bounds on the Schmidt number of optimal PPT states and would s…

Strengthened Sanpera–Bruß–Lewenstein conjecture. In dimension $d$, the Choi matrix of every map that is both completely positive and completely co-positive has Schmidt number at mo…

Nonmaximal Schmidt number conjecture. The Schmidt number of a PPT-entangled state cannot be maximal.

Let $\mathcal{M}_d$ denote the algebra of complex $d\times d$ matrices, and let $T:\mathcal{M}_d\rightarrow\mathcal{M}_d$ be a linear map. Its Choi matrix is denoted by $C_T$, and…

Let $\rho_{AB}$ be a quantum state on $\mathds{C}^{d_A}\otimes\mathds{C}^{d_B}$, and suppose that it has positive partial transpose, meaning…

Sanpera–Bruß–Lewenstein conjecture. All bound entangled states with positive partial transpose in $\mathbb{C}^3\otimes\mathbb{C}^3$ have Schmidt number $2$.