On Channels with Finite Holevo Capacity

On Properties of Quantum Channels Related to Their Classical Capacity

“…Thus it is natural to introduce the notion of generalized optimal ensemble as a probability measure on the set of all states, at which the supremum in definition (5) of the Holevo capacity is achieved. In [10] the sufficient condition of existence of optimal measure for arbitrary constrained channel is obtained, but examples in [10], [32] shows that optimal measure does not exist in some cases.…”

“…Note that there exist examples of the A-constrained channels Φ with finite Holevo capacity with no input optimal averages while the output optimal average Ω(Φ, A) is explicitly determined and plays important role in analysis of these channels (see. examples in [3], [32]).…”

“…The definition of the Holevo capacity of the A-constrained channel Φ implies C(Φ, A) = sup ρ∈A χ Φ (ρ). In [10] it is shown that the question of attainability of the above supremum is closely related with the question of existence of optimal measure for the A-constrained channel Φ and due to examples in [32] it can have negative answer.…”

“…If Φ and Ψ are finite dimensional channels then in [21] it is shown that validity of inequality (27) for all states ω is equivalent to validity of the following inequality (32) H…”

“…As a result for infinite dimensional constrained channel with finite Holevo capacity generally there exist no optimal ensembles, which plays important role in study of finite dimensional channels [29]. In [10] the sufficient condition of existence of an optimal measure (generalized ensemble) for arbitrary constrained channel is obtained, but in [10], [32] the examples of constrained channels with no optimal measure are constructed. In section 3 of this paper it is shown that despite possible nonexistence of optimal measure for arbitrary channel constrained by convex set of states there exists the unique state, called output optimal average, which inherits properties of the image of the average state of an optimal ensemble for a finite dimensional constrained channel (proposition 1).…”