What is the analog of HKR theorem in the noncommutative world?

Recall that the well-known theorem by Hochschild-Kostant-Rosenberg says that for a *smooth* *commutative* algebra $A$ of finite type over a field $k$ (let's assume of characteristic zero) there is an isomorphism $$HH_\bullet(A)\simeq \Omega^\bullet(A)$$ where $HH_\bullet(A)$ denotes the Hochschild homology of $A$ with coefficients in itself, and $\Omega^\bullet(A):=\Lambda^\bullet(\Omega^1_k(A))$ is the space of algebraic differential forms on $A$, and $\Omega^1_k(A)$ denotes the module of Kahler differentials.

There is a notion of *smoothness* for any associative algebra $A$, not necessarily commutative (formal smoothness). Moreover, one can define noncommutative differential forms $\Omega^\bullet_{nc}(A)$. So we have all the ingredients for noncommutative HKR, but I can't find the precise statement. I found the page on nLab, but the section is missing.

Thank you very much for your help!

**Edit:** As Mariano pointed out in the comments, $HH_\bullet(A)$ is not the same as $\Omega^\bullet_{nc}(A)$ for formally smooth $A$. I was indeed stupid here. Formal smoothness is equivalent to saying that $\Omega^1_{nc}(A)$ is a projective bimodule. But then the complex $0\to \Omega^1_{nc}(A)\to A\otimes A\to 0$ is a projective resolution of $A$, and so Hochschild homology $HH_n(A)$, defined as $Tor_n^{A^e}(A,A)$, vanish for $n>1$, while noncommutative differential forms do not vanish.

But the question is still valid. What is the NC analog that the nLab page is talking about?