What exactly is the relation between the Holevo quantity and the mutual information?

Question

On this page, it is stated that the Holevo quantity is an upper bound to the accessible information of a quantum state. In the scenario where Alice encodes classical information into a quantum state and sends it to Bob, the accessible information is the maximum of the mutual information between Alice and Bob's registers over all possible measurements that Bob makes.

On the other hand, the classical capacity of a quantum channel also looks at the mutual information between Alice and Bob. The maximization of this mutual information is the Holevo quantity.

I do not understand the difference in the two settings. In particular, why is the Holevo quantity only an upper bound in the first linked page but is equal to the maximum mutual information in the second linked page?

Answer 1

Right, they are quite similar. The Holevo bound is a bound on the amount of accessible information between your quantum system and your classical system. The I(X;B) object written in the HSW theorem wikipedia page is actually this bound, while the $\chi$ there is the Holevo rate, or product state capacity. What HSW showed was that if you took many copies of a system, the asymptotic many system rate of accessible information across a quantum channel can be made to approach the single state Holevo bound.

Interestingly, Hastings showed that the many state Holevo bound is actually even higher than the single state bound, meaning that the HSW theorem does not demonstrate the "true" Holevo bound to be saturated.