# Quantum mutual information

The **quantum mutual information** of a bipartite state *ρ**A**B* is defined as

*I*(*A* : *B*) : = *S*(*A*) + *S*(*B*) − *S*(*A**B*) where *S*(*A*) and *S*(*B*) are the Von Neumann entropies of *ρ**A* and *ρ**B*, obtained by tracing out system "B" and "A" respectively from *ρ**A**B*. *S*(*A**B*) 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 *ρ**A**B**C*.

### 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 *ρ**R**Q* 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) = max*ρ**R**Q* ∈ D*I*(*R* : Λ*Q*) is the additivity problem.

Category:Quantum Information Theory Category:Quantum Communication Category:Handbook of Quantum Information