Symmetric vs. bosonic extension for bipartite states

Abstract

A bipartite state ${\rho }^{AB}$ has a $k$-symmetric extension if there exists a $k+1$-partite state ${\rho }^{A{B}_1{B}_2\dots B_k}$ with marginals ${\rho }^{A{B}_i}={\rho }^{AB},\forall i$. The $k$-symmetric extension is called bosonic if ${\rho }^{A{B}_1{B}_2\dots B_k}$ is supported on the symmetric subspace of $B_1{B}_2\dots B_k$. 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.