Entanglement witnesses for indistinguishable particles

We study the problem of witnessing entanglement among indistinguishable particles. For this purpose, we derive a set of equations which results in necessary and sufficient conditions for probing multipartite entanglement between arbitrary systems of bosons or fermions. The solution of these equations yields the construction of optimal entanglement witnesses for partial and full entanglement in discrete and continuous variable systems. Our approach unifies the verification of entanglement for distinguishable and indistinguishable particles. We provide general solutions for certain observables to study quantum entanglement in systems with different quantum statistics in noisy environments.

Figure 1

Mixing in terms of $p$ for the state (42) depending on $d=dim\mathcal{H}$ is plotted. As long as $p$ is in the gray shaded area, we successfully detected entanglement. The coefficients in Eqs. (31) and (33) for DP and bosons, respectively, are chosen to be equal ${\lambda }_k={\kappa }_k^{\prime }=d^{-1/2} \left(k=1,...,d\right)$. In the case of fermions, we choose ${\kappa }_k={\left(2\lfloor d/2\rfloor \right)}^{-1/2} \left(k=1,...,\lfloor d/2\rfloor \right)$, see Eq. (32), yielding a different behavior for even and odd dimensions $d$.

Figure 2

Expectation value in Eq. (49) is plotted for $N=5$ (solid curve), an amplitude $\left|q\right|=1/\sqrt{3}$, and a uniformly distributed phase in the interval $argq\in [-\delta ,\delta ]$. If $\langle \widehat{L}\rangle$ is above the dashed line $K$, then the state cannot be a $K$-separable one—independent of the quantum statistics.