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

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][1] - 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

Yes, this is easy. Let $H$ be a separable Hilbert space and let $(x_i)$ be a dense sequence in $H$. Then there are countably many operators of the form $x \mapsto \langle x,x_i\rangle x_j$ and these operators are dense, for the trace norm, in the rank 1 operators on $H$. Taking finite linear combinations with complex rational coefficients gives a countable set which is trace norm dense in the set of finite rank operators on $H$. The latter is dense in the set of all trace class operators so we are done.