I am interested in knowing whether the space of trace class operators is (crudely) finitely representable in an $L^1$-space. I suspect that the answer is negative but I am unable to find any argument confirming my intuition.

As for motivation, I am working on matrix-valued versions of some inequalities coming from harmonic analysis, and I would like to know if the generalisation I seek is non-trivial, if true.

Definition:
A Banach space $X$ is said to be crudely finitely representable in a Banach space $Y$ if there exists a constant $C>0$ such that every finite-dimensional subspace $V$ of $X$ is $C$-isomorphic to a subspace of $Y$, i.e. there exists an isomorphism $T: V \to T(V) \subset Y$ satisfying $\|T\|\cdot \|T^{-1}\| \leqslant C$.