Detecting entanglement of quantum channels

Chaojian Li,Bang-Hai Wang,Bujiao Wu,Xiao Yuan

Table 3. Quantum game with ${W}_{\mathrm{SWAP},1}$ for the noisy SWAP gate.

α ${\rho }_{A}^{{\rm{T}}}$ ${\rho }_{B}^{{\rm{T}}}$ ${O}_{A^{\prime} }$ ${O}_{B^{\prime} }$ α ${\rho }_{A}^{{\rm{T}}}$ ${\rho }_{B}^{{\rm{T}}}$ ${O}_{A^{\prime} }$ ${O}_{B^{\prime} }$ 1 ${\mathbb{I}}$ ${\mathbb{I}}$ ${\mathbb{I}}$ ${\mathbb{I}}$ 1 ${\mathbb{I}}$ ${\sigma }_{z}$ ${\mathbb{I}}$ ${\mathbb{I}}$ 1 ${\mathbb{I}}$ ${\mathbb{I}}$ ${\sigma }_{z}$ ${\mathbb{I}}$ 1 ${\mathbb{I}}$ ${\sigma }_{z}$ ${\sigma }_{z}$ ${\mathbb{I}}$ −1 ${\sigma }_{x}$ ${\mathbb{I}}$ ${\mathbb{I}}$ ${\sigma }_{x}$ −1 ${\sigma }_{x}$ ${\sigma }_{z}$ ${\mathbb{I}}$ ${\sigma }_{x}$ −1 ${\sigma }_{x}$ ${\mathbb{I}}$ ${\sigma }_{z}$ ${\sigma }_{x}$ −1 ${\sigma }_{x}$ ${\sigma }_{z}$ ${\sigma }_{z}$ ${\sigma }_{x}$ 1 ${\sigma }_{y}$ ${\mathbb{I}}$ ${\mathbb{I}}$ ${\sigma }_{y}$ 1 ${\sigma }_{y}$ ${\sigma }_{z}$ ${\mathbb{I}}$ ${\sigma }_{y}$ 1 ${\sigma }_{y}$ ${\mathbb{I}}$ ${\sigma }_{z}$ ${\sigma }_{y}$ 1 ${\sigma }_{y}$ ${\sigma }_{z}$ ${\sigma }_{z}$ ${\sigma }_{y}$ −1 ${\sigma }_{z}$ ${\mathbb{I}}$ ${\mathbb{I}}$ ${\sigma }_{z}$ −1 ${\sigma }_{z}$ ${\sigma }_{z}$ ${\mathbb{I}}$ ${\sigma }_{z}$ −1 ${\sigma }_{z}$ ${\mathbb{I}}$ ${\sigma }_{z}$ ${\sigma }_{z}$ −1 ${\sigma }_{z}$ ${\sigma }_{z}$ ${\sigma }_{z}$ ${\sigma }_{z}$