I have asked the question in math.stackexchange, but perhaps it should be more relevant here. Hence I am re-posting it with necessary reediting.
Let $\mathcal{H}=\mathbb{C}^n$ be a Hilbert space. A state $\rho\in\mathcal{B(H)}$ is a positive semi-definite operator with unit trace. $\rho\in \mathcal{B(H)}$, where $\mathcal{H}=\mathcal{H}_1\otimes\mathcal{H}_2=\mathbb{C}^{n_1}\otimes\mathbb{C}^{n_2}$, is called entangled, if it can not be written as convex sum of one dimensional projections - like $P_x\otimes P_y$, where $|x\rangle\in\mathcal{H}_1$ and $|y\rangle\in\mathcal{H}_2$.
In a similar spirit can we define entanglement in the symmetric space $\mathrm{Sym}^2\mathcal{H}$ and antisymmetric space $\bigwedge^2\mathcal{H}$?
As you have already understood, I am looking for entanglement in the indistinguishable particles. Hence the above definition for the entanglement of distinguishable particles does not work. A quick googling gave me a few papers, which refer to different definitions (for multipartite systems, and some are based on certain entropic conditions). Hence I want to know whether there is mathematical definitions for entanglement in such systems which is closest to the spirit of the definition mentioned above (for distinguishable particle). Advanced thanks for any help in this direction.