Approximate broadcasting of quantum correlations

Broadcasting quantum and classical information is a basic task in quantum information processing, and is also a useful model in the study of quantum correlations including quantum discord. We establish a full operational characterization of two-sided quantum discord in terms of bilocal broadcasting of quantum correlations. Moreover, we show that both the optimal fidelity of unilocal broadcasting of the correlations in an arbitrary bipartite quantum state and that of broadcasting an arbitrary set of quantum states can be formulized as semidefinite programs (SDPs), which are efficiently computable. We also analyze some properties of these SDPs and evaluate the broadcasting fidelities for some cases of interest.

Figure 1

Unilocal (left) and bilocal (right) broadcasting of quantum correlations in initial state ${\rho }_{AB}^{}$. The objective is for the quantum channels $\mathrm{\Lambda },\mathrm{\Gamma }$ to make the states on $A_{i}B$ or $A_i{B}_i$ as close to ${\rho }_{AB}^{}$ as possible measured in some way.

Figure 2

The solid line depicts the optimal unilocal two-broadcasting fidelity $f_2\left({\psi }_{\theta }\right)$, the dashed line depicts the fidelity of unilocal two-broadcasting via channel $\mathrm{\Xi }, f_{2,\mathrm{\Xi }}\left({\psi }_{\theta }\right)$, which almost coincides with $f_2\left({\psi }_{\theta }\right)$ except when $\theta$ is close to $\pi /4$, and the dotted line depicts the fidelity of unilocal two-broadcasting via channel $\mathrm{{\rm Y}}, f_{2,\mathrm{{\rm Y}}}\left({\psi }_{\theta }\right)$.

Figure 3

Broadcasting states ${\rho }_1,{\rho }_2$ via the same channel $\mathrm{\Lambda }$.