Wikipedia provides a simple definition of Quantum Mutual Information:
$$I(\rho^{ab})= S(\rho^{a}) + S(\rho^{b}) - S(\rho^{ab})$$
where in terms of relative information we have:
$$I(\rho^{ab})= S(\rho^{ab}|| \rho^{a} \otimes \rho^{b})$$
what is the best notational approach for these expression when there are more than two systems?
UPDATE
If we take the classical case:
$$I(X;Y) = H(X \cap Y)$$
The form of Pascal's triangle is apparent when the relation is viewed as (1 2 1):
$$0=- H(X\cup Y) + \{ H(X) + H(Y) \}- H(X\cap Y)$$ and (1,3,3,1): $$0=- H(X\cup Y\cup Z) + \{ H(X) + H(Y) + H(Z)\}-\{ H(X\cap Y) + H(X \cap Z) + H(Y \cap Z)\}+ H(X\cap Y\cap Z)$$
Following the inclusion-exclusion principle we find that the number of terms that comprise mutual information are:
$$\sum_{k=2}^{m}(-1)^{k-1}\dbinom{m}{k}$$
Where we could write the complete expression as:
$$H\left(\bigcup_{i=1}^{n}X_i\right)=\sum_{i=1}^{n}H(X_i) - \sum_{i,j:i<j}H(X_i\cap X_j) + \sum_{i,j,k:i<j<k}H(X_i\cap X_j \cap X_k) -\dots + (-1)^{n-1}H\left(\bigcap_{i=1}^{n}X_i\right)$$
or as: $$H\left(\bigcup_{i=1}^{n}X_i\right)=\sum_{i=1}^{n}H(X_i) + \sum_{k=2}^{n}(-1)^{k-1}\sum_{I\subset \{1,\dots,n\}|I|=k} H\left(\bigcap_{i\in I}^{n}X_i\right)$$
which highlights that the mutual information is associated with terms:
$$I\left(\bigcap_{i\in I}^{n}X_i\right)= \sum_{k=2}^{n}(-1)^{k-1}\sum_{I\subset \{2,\dots,n\}|I|=k} H\left(\bigcap_{i\in I}^{n}X_i\right)$$
If we follow the same form then the apparent form for quantum mutual information would would also follow pascal's triangle's binomial structure, so what is the most efficient notation to use?
