Separability of the space of self-adjoint trace class operators over a separable Hilbert space

Question

My interest is to know whether the assertion

the space of self-adjoint trace class operators over a separable Hilbert space is separable with respect to the trace norm

is correct.

The above assertion is claimed (without proof) to be true (and used) in a recent paper: arXiv:quant-ph/0610122 - page 12.

However, so far, I have neither succeeded in finding a formal proof of the above property nor relevant references. I would be grateful for your help in this respect.

(This would help me in clarfying strong measurability - integrability aspects related to some specific problems in the space of self-adjoint trace class operators over a separable Hilbert space).

Thank you

Answer 1

The proof can be divided into three steps:

1. Since $H$ is separable, show that the set of rank 1 operators is separable. Note that every rank 1 operator is of the form $x\otimes y$, where $$ (x\otimes y)(z) = \langle z,y\rangle x $$
2. Show that every finite rank operator is a linear combination of rank one operators. Now conclude that the set $F(H)$ of finite rank operators is separable.
3. Show that $F(H)$ is dense in the set $L^1(H)$ of trace class operators, and so $L^1(H)$ (and any subspace of it) is separable.