Given an A-algebra B, one can define the cotangent complex $L_{B/A}$ as $\Omega^1_{P/A}\otimes_PB$, where $P$ can be taken as the canonical resolution of $B$ associated to the pair of adjoint functor (forget, free-$A$-algebra).

As I understand it, the cotangent complex is a basic invariant of $A$-algebras --- like the complex of singular chains associated to a space. So I'm wondering if one can recover the original $A$-algebra from a cotangent complex (up to some equivalence).

In other words, if there is a map $B\rightarrow B'$ of $A$-algebras inducing a quasi-isomorphism of cotangent complex of $B'$ with the pull back of that of $B$, does this say something about the two $A$-algebras?

What I have in mind is something like the Whitehead theorem for CW complexes.