A bipartite state ${\rho }^{AB}$ has a $k$-symmetric extension if there exists a ($k+1$)-partite state ${\rho }^{A{B}_1{B}_2...B_k}$ with marginals ${\rho }^{A{B}_i}={\rho }^{AB},\forall i$. The $k$-symmetric extension is called bosonic if ${\rho }^{A{B}_1{B}_2...B_k}$ 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 ${\rho }^{AB}$ 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.

• Received 10 November 2018

DOI:https://doi.org/10.1103/PhysRevA.99.012332