Entanglement in the symmetric subspace: mapping multipartite to bipartite states

Abstract

We propose a technique to investigate multipartite entanglement in the symmetric subspace. Our approach is to map an $N$-qubit symmetric state onto a bipartite symmetric state of higher local dimension. We show that this mapping preserves separability and allows to characterize the entanglement of the original multipartite state. In particular, we establish a connection between the border rank and the Schmidt rank, and derive lower bounds on entanglement measures. Finally, we reveal the existence of entangled symmetric subspaces, where all bipartite states are entangled.

Figures (1)

• Figure 1: Pictorial representation of the mapping $\mathcal{M}$ presented in Result \ref{['ob:mapping']}. This paper investigates the entanglement of multipartite states $\ket{\Psi} \in \mathcal{S}_{N}^{(2)}$ by focusing on their mapped bipartite states $\mathcal{M}(\ket{\Psi}) \in \tilde{\mathcal{S}}_{2}^{(N/2 +1)}$. Note that, while all separable states in $\mathcal{S}_{N}^{(2)}$ can be mapped to separable states in $\tilde{\mathcal{S}}_{2}^{(N/2 +1)}$, as shown in Result \ref{['obs:sep']}, not all separable states in $\mathcal{S}_{2}^{(N/2 +1)}$ lie in $\tilde{\mathcal{S}}_{2}^{(N/2 +1)}$.

Theorems & Definitions (8)

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