Symmetric versus bosonic extension for bipartite states

Abstract

A bipartite state ${\ensuremath{\rho}}^{AB}$ has a $k$-symmetric extension if there exists a ($k+1$)-partite state ${\ensuremath{\rho}}^{A{B}_{1}{B}_{2}...{B}_{k}}$ with marginals ${\ensuremath{\rho}}^{A{B}_{i}}={\ensuremath{\rho}}^{AB},\ensuremath{\forall}i$. The $k$-symmetric extension is called bosonic if ${\ensuremath{\rho}}^{A{B}_{1}{B}_{2}...{B}_{k}}$ is supported on the symmetric subspace of ${B}_{1}{B}_{2}...{B}_{k}$. Understanding the structure of symmetric and 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 separability. In general, it is known that a ${\ensuremath{\rho}}^{AB}$ 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), ${\ensuremath{\rho}}^{AB}$ 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.