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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 calledbosonic 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 extensionhas various applications in the theory of quantum entanglement, quantum keydistribution and the quantum marginal problem. In particular, bosonic extensiongives a tighter bound for the quantum marginal problem based on seperability.In general, it is known that a $\rho^{AB}$ admitting symmetric extension maynot have bosonic extension. In this work, we show that when the dimension ofthe subsystem $B$ is $2$ (i.e. a qubit), $\rho^{AB}$ admits a $k$-symmetricextension if and only if it has a $k$-bosonic extension. Our result has animmediate application to the quantum marginal problem and indicates a specialstructure for qubit systems based on group representation theory.

Symmetric vs. bosonic extension for bipartite states