# Quantum mutual information

The quantum mutual information of a bipartite state ρAB is defined as

I(A : B) :  = S(A) + S(B) − S(AB)
where S(A) and S(B) are the Von Neumann entropies of ρA and ρB, obtained by tracing out system "B" and "A" respectively from ρAB. S(AB) is the Von Neumann entropy of the total state.

The quantity is formally equivalent to the classical mutual information with the Shannon entropy changed to its quantum counterpart.

### Properties

The quantum mutual information is always non-negative $I\big(A:B) \geq 0$.

It is monotonic under the action of quantum channels (CPTP maps), that is $I\big(A:\Lambda B\big) \leq I(A:B)$. As the partial trace is a CPTP map, this means that $I\big(A: B\big) \leq I(A:BC)$ for any tripartite state ρABC.

### Uses

When maximized over input states, it gives the entanglement assisted classical capacity of a memoryless quantum channel. That is,

$$C_E\big(\Lambda\big) = \max_{\rho_{RQ}} I(R:\Lambda Q)$$

where $I\big(R:\Lambda Q\big)$ is the quantum mutual information of the state $\big(\mathbb{I}_R \otimes \Lambda_Q\big)\rho_{RQ}$.

When the maximization of the state ρRQ is taken over all separable states, then the maximized quantum mutual information is equivalent to the Holevo capacity, and thus

$$C\big(\Lambda\big) = \lim_{n\rightarrow \infty} \max_{\rho_{RQ} \in \mathcal{D}} \frac{1}{n}I(R:\Lambda^{\otimes n} Q)$$

for D the set of all separable states between systems R and Q. Whether or not the classical capacity of a memoryless channel can be expressed in the unregularized form C(Λ) = maxρRQ ∈ DI(R : ΛQ) is the additivity problem.