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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 $\mathcal{H}\bigvee\mathcal{H}$ and antisymmetric space $\mathcal{H}\bigwedge\mathcal{H}$?

The importance for looking at these space is, they represent bosonic and fermionic systems of indistinguishable particles respectively. Hence the above definition, which is for 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. (Also, if possible, and if exists, please give me some reference of (mathematical) papers which work in this direction.)

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Crossposted to – Qmechanic Aug 30 '12 at 8:15
@Qmechanic I was the one who asked it, after not getting reply in this forum :D – RSG Aug 30 '12 at 9:02

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