  A bipartite state $\rho^{AB}$ has a $k$-symmetric extension if there exists a
$k+1$-partite state $\rho^{AB_1B_2\ldots B_k}$ with marginals
$\rho^{AB_i}=\rho^{AB}, \forall i$. The $k$-symmetric extension is called
bosonic if $\rho^{AB_1B_2\ldots B_k}$ is supported on the symmetric subspace of
$B_1B_2\ldots B_k$. Understanding the structure of symmetric/bosonic extension
has various applications in the theory of quantum entanglement, quantum key
distribution and the quantum marginal problem. In particular, bosonic extension
gives a tighter bound for the quantum marginal problem based on seperability.
In general, it is known that a $\rho^{AB}$ admitting symmetric extension may
not have bosonic extension. In this work, we show that when the dimension of
the subsystem $B$ is $2$ (i.e. a qubit), $\rho^{AB}$ admits a $k$-symmetric
extension if and only if it has a $k$-bosonic extension. Our result has an
immediate application to the quantum marginal problem and indicates a special
structure for qubit systems based on group representation theory.
