Class of Bell-diagonal entanglement witnesses in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi mathvariant="double-struck">C</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mo>⊗</mml:mo><mml:msup><mml:mi mathvariant="double-struck">C</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:mrow></mml:math>: Optimization and the spanning property
Anindita Bera, Filip Wudarski, Gniewomir Sarbicki, Dariusz Chruściński
Abstract
Two classes of Bell-diagonal indecomposable entanglement witnesses in ${\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}$ are considered. Within the first class, we find a generalization of the well-known Choi witness from ${\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$, while the second one contains the reduction map. Interestingly, contrary to the ${\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$ case, the generalized Choi witnesses are no longer optimal. We perform an optimization procedure of finding spanning vectors that eventually gives rise to optimal witnesses. Operators from the second class turn out to be optimal, however, without the spanning property. This analysis sheds light onto the intricate structure of optimal entanglement witnesses.