Hello,

Many years before, I had the following problem.

We first give a definition. Given a non-negative definite real-valued definite matrix $n^2\times n^2$ matrix $M$, it is called **separable** if it can be decomposed in the following way:

$$ M=\sum_{i=1}^{k} \:\: \lambda_i E_i\otimes F_i $$

where $k\le n^2$, $\lambda_i>0$, $\otimes$ denotes the tensor product in the usual sense, and $E_i$, $F_i$ are $n\times n$ non-negative matrices of rank one, having unit length and orthogonal with each other:

$$ Tr(E_i E_j^T) =Tr(F_i F_j^T) = \begin{cases} 1& i=j\cr 0& i\ne j \end{cases}\;. $$

Now the problem is: **how to determine whether a non-negative definite matrix is separable or not?**

Recently, I have this problem again, but in a more general context. We define the non-negative definite linear operator $\mu \in \mathcal{S}'(R^{2d})$ ($\mathcal{S}'(R^{2d})$ is the space of Schwartz distributions) over the rapidly decreasing functions $\mathcal{S}(R^d)$ as follows:

$$ \langle\mu,\psi\otimes\psi\rangle\ge 0,\quad\forall \psi\in\mathcal{S}(R^{d}), $$ where $\otimes$ denotes the tensor product.

We call a non-negative definite linear operator $\mu \in \mathcal{S}'(R^{2d})$ ($d$ is even), **separable**, if there exists a sequence of pairs $(\lambda_i,\mu_i,\nu_i)$, with $\lambda_i>0$, and each $\mu_i$ and $\nu_i\in\mathcal{S}'(R^d)$ is non-negative definite, such that

$$ \mu=\sum_i \lambda_i \: \mu_i\otimes\nu_i. $$

Note that the above sum can be integral when the operator has a continuous spectral.

So the similar problem is **how to determine whether a non-negative definite linear operator $\mu \in \mathcal{S}'(R^{2d})$ over $\mathcal{S}(R^d)$ is separable or not?**

I guess that these two problems are still open. Does anyone have any hints or references?

Thank you very much!

Anand