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arXiv · 1809.05641

Symmetric vs. bosonic extension for bipartite states

Youning Li·Shilin Huang·Dong Ruan·Bei Zeng
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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.

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Youning Li, Shilin Huang, Dong Ruan, Bei Zeng. 2018-09-15. Symmetric vs. bosonic extension for bipartite states. https://doi.org/10.1103/physreva.99.012332

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