# Is the set of separable quantum states closed

Let $$\mathcal H,\mathcal H’$$ be Hilbert spaces (not necessarily separable).

A “separable state” is a trace-class operator of the form $$\sum_i \rho_i\otimes\rho_i’$$ where $$\rho_i,\rho_i’$$ are positive trace-class operators over $$\mathcal H,\mathcal H’$$, respectively. (Convergence of the sum is with respect to the trace norm. $$\otimes$$ represents the tensor product.)

Is the set of separable states closed with respect to the trace norm?