Entanglement of indistinguishable particles

We present a general criterion for entanglement of $N$ indistinguishable particles decomposed into arbitrary $s$ subsystems based on the unambiguous measurability of correlation. Our argument provides a unified viewpoint on the entanglement of indistinguishable particles, which is still unsettled despite various proposals made mainly for the $s=2$ case. Even though entanglement is defined only with reference to the measurement setup, we find that the so-called independently and identically distributed states form a special class of bosonic states which are universally separable.

Figure 1

Diagrammatical representation of the spaces mentioned in (17) and (22). In the total space ${\mathcal{H}}_{\mathcal{X}}$, we find the subspace ${\mathcal{H}}_{\mathcal{X}}(\Gamma ,V)$ isomorphic to ${\otimes }_i{\mathcal{H}}_{\mathcal{X}}({\Gamma }_i,V_i)$ which has a TPS. The latter is then identified with the space ${\mathcal{H}}^{\mathrm{mes}}$ describing the measurement outcomes.

Figure 2

Concurrence $C$ of $|{\Psi }^{\text{mes}}(\Gamma ,V)\rangle$ as a function of the angle $\theta$. The variation shows that the state undergoes the change from complete separation to maximal entanglement by the adjustment of the setup angle $\theta$.

Figure 3

Distribution of the Schmidt coefficients of the state (36) with the ordering ${\lambda }_1⩽{\lambda }_2⩽{\lambda }_3$ (${\lambda }_3$ is not shown because it can be determined from the other two) obtained under random setups of measurement for the state (33) provided by ${10}^6$ distinct unitaries $U$. The diagonal lines represent ${\lambda }_1={\lambda }_2$ and ${\lambda }_2={\lambda }_3$, whose intersection corresponds to the maximally entangled state. The points fill out basically the entire region of the triangle, although in our random generation the density becomes scarcer for states which are highly entangled.