Outline of the proof of the additivity of Holevo information

The additivity of Holevo information for the depolarizing channel was proved by Christopher King.^[3] He showed that the maximum output p-norm of the depolarizing channel is multiplicative, which implied the additivity of the minimum output entropy, which is equivalent to the additivity of the Holevo information.

A stronger version of the additivity of the Holevo information is shown for the depolarizing channel ${\displaystyle \Delta _{\lambda }}$. For any channel ${\displaystyle \Psi }$:

${\displaystyle \chi \left(\Delta _{\lambda }\otimes \Psi \right)=\chi \left(\Delta _{\lambda }\right)+\chi \left(\Psi \right)}$

This is implied by the following multiplicativity of maximum output p-norm (denoted as ${\displaystyle v_{p}}$):

${\displaystyle v_{p}\left(\Delta _{\lambda }\otimes \Psi \right)=v_{p}\left(\Delta _{\lambda }\right)v_{p}\left(\Psi \right)}$

The greater than or equal to direction of the above is trivial, it suffices to take the tensor product the states that achieve the maximum p-norm for ${\displaystyle \Delta _{\lambda }}$ and ${\displaystyle \Psi }$ respectively, and input the product state into the product channel to get the output p-norm ${\displaystyle v_{p}(\Delta _{\lambda })v_{p}(\Psi )}$. The proof for the other direction is more involved

The main idea of the proof is to rewrite the depolarizing channel as a convex combination of simpler channels, and use properties of those simpler channels to get the multiplicativity of the maximum output p-norm for the depolarizing channel.

It turns out that we can write the depolarizing channel as follows:

${\displaystyle \Delta _{\lambda }(\rho )=\sum _{n=1}^{2d^{2}(d+1)}c_{n}U_{n}^{*}\Phi _{\lambda }^{(n)}(\rho )Un}$

where ${\displaystyle c_{n}}$'s are positive numbers, ${\displaystyle U_{n}}$'s are unitary matrices, ${\displaystyle \Phi _{\lambda }^{(n)}}$'s are some dephasing channels and ${\displaystyle \rho }$ is an arbitrary input state.

Therefore, the product channel can be written as:

${\displaystyle \left(\Delta _{\lambda }\otimes \Psi \right)(\rho )=\sum _{n=1}^{2d^{2}(d+1)}c_{n}\left(U_{n}^{*}\otimes I\right)\left(\Phi _{\lambda }^{(n)}\otimes \Psi \right)(\rho )\left(U_{n}\otimes I\right)}$

By the convexity and the unitary invariance of the p-norm, it suffices to show the simpler bound:

${\displaystyle \|\left(\Phi _{\lambda }^{(n)}\otimes \Psi \right)(\rho )\|_{p}\leq v_{p}(\Delta _{\lambda })v_{p}(\Psi )}$

One important mathematical tool used in the proof of this bound is the Lieb–Thirring inequality, which provides a bound for p-norm of a product of positive matrices. The details and the calculations of the proof are skipped, interested readers are referred to the paper of C. King mentioned above.

Discussion

The main technique used in this proof, namely rewriting the channel of interest as a convex combination of other simpler channels, is a generalization of the method used earlier to prove similar results for unital qubit channels.^[4]

The fact that the classical capacity of the depolarizing channel is equal to the Holevo information of the channel means that we can't really use quantum effects such as entanglement to improve the transmission rate of classical information. In this sense, the depolarizing channel can be treated as a classical channel.

However the fact that the additivity of Holevo information doesn't hold in general proposes some areas of future work, namely finding channels that violates the additivity, in other words, channels that can exploit quantum effects to improve the classical capacity beyond its Holevo information.