Each of the $n$ commitments are of the form $C_i=a_iG+b_iH$, where $a_i$ is the value being committed to, and $b_i$ is the uniformly random blinding factor.
$G$ and $H$ are generator points, arbitrarily and fairly chosen so that $g$ such that $G=gH$ is unknowable (the EC discrete log assumption).
This means that if you treat $C_i$ as a public key on the generator $H$, the corresponding private key will be $a_ig+b_i$. Since $g$ is unknowable, the private key is only known if $a_i\overset{?}{=} 0$, i.e. the commitment is to the value zero.
Therefore, being able to provide a signature (such as a Schnorr signature) proving knowledge of the private key is equivalent to proving that the commitment is to the value zero. The same applies to knowledge of the private key of a sum of commitments.
First, calculate a challenge scalar $r_i$ for each commitment $C_i$. This can be done using an extendable-output function or HKDF, using the concatenation of all commitment EC-point bit strings as the initial keying material.
Calculate $S$ and $S'$ as follows:
$S=\sum\limits_{i=0}^{n-1}{C_i}$
$S'=\sum\limits_{i=0}^{n-1}{r_iC_i}$
Now, create $n$ public keys, as follows:
$P_i=S'-r_iS$
If the commitment to the value $x$ is at index $\pi$, then it will only be possible for $P_{\pi}$ to be a commitment to zero if all other commitments are commitments to zero (because multiplying zero by a random number is still zero). The private key will be $\sum\limits_{i=0}^{n-1}{r_ib_i}-r_{\pi}b_{\pi}$
If we provided a signature on the generator $H$ proving that $P_{\pi}$ is a commitment to zero, we would disclose $\pi$. To avoid disclosing the index $\pi$, we simply generate a ring signature instead.
Essentially, this approach is creating a ring, where each member of the ring contains the sum of $n-1$ commitments. Each ring position excludes a different commitment. The ring can only be signed if in at least one ring position, every commitment involved in the sum is a commitment to zero.
