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?