Symmetric vs. bosonic extension for bipartite states

TL;DR

This paper proves that for qubit systems, symmetric and bosonic extensions are equivalent, providing insights into the structure of bipartite quantum states and implications for quantum marginal problems.

Contribution

It establishes that for qubit subsystems, symmetric and bosonic extensions coincide, revealing a special structure and simplifying analysis in quantum information theory.

Findings

Symmetric and bosonic extensions are equivalent for qubit systems.

The result has implications for the quantum marginal problem.

Provides a group-theoretic perspective on bipartite states.

Abstract

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.