[Paper Review] On the additivity conjecture for channels with arbitrary constraints

M. E. Shirokov|ArXiv.org|Aug 29, 2003

Computability, Logic, AI Algorithms7 references3 citations

TL;DR

This paper investigates the additivity conjecture for quantum channels under arbitrary input constraints, generalizing prior results on linear constraints. It establishes a necessary and sufficient condition for additivity of the Holevo capacity using a constrained maximal distance property, and proves that global additivity for unconstrained channels implies strong superadditivity of entanglement of formation via a novel characterization of subadditivity of the χ-function.

ABSTRACT

Recently Shor proved equivalence of several open (sub)additivity problems related to the Holevo capacity and the entanglement of formation [15]. In our previous note [6] equivalence of these to the additivity of the Holevo capacity for channels with arbitrary linear constraints was shown. This note is the development of the previous one in the direction of channels with general constraints. Introducing input constraints provides greater flexibility in the treatment of the additivity conjecture. The Holevo capacity of arbitrarily constrained channel is considered and the characteristic property of an optimal ensemble for such channel is derived, generalizing the maximal distance property of Schumacher and Westmoreland (proposition 1). It is shown that the additivity conjecture for two channels with single linear constraints is equivalent to the similar conjecture for two arbitrarily constrained channels and, hence, to an interesting subadditivity property of the $χ$-function for the tensor product of these channels (theorem 1). We also propose an alternative way of proving that the additivity conjecture for any two unconstrained channels implies strong superadditivity of the entanglement of formation. The arguments from the convex analysis provide another characterization of channels for which subadditivity of the $χ$-function holds (theorem 3). This characterization and some modification of Shor's channel extension provide a simple way of proving that global additivity of the minimum output entropy for unconstrained channels implies global subadditivity of the $χ$-function and strong superadditivity of the entanglement of formation.

Motivation & Objective

• To extend the additivity conjecture for quantum channels beyond linear constraints to arbitrary closed input constraints.
• To derive a generalized maximal distance property for optimal ensembles in constrained channels, generalizing Schumacher and Westmoreland's result.
• To establish a necessary and sufficient condition for the additivity of the Holevo capacity of two constrained channels using the χ-function subadditivity.
• To demonstrate that global additivity of minimum output entropy implies strong superadditivity of entanglement of formation through a new characterization.
• To unify and generalize results from Shor, Holevo, and others by introducing asymptotic additivity and channel extension techniques.

Proposed method

• Introduces a general framework for channels with arbitrary closed input constraints, defined by a set 𝒜 ⊆ 𝒮(ℋ) of allowed average input states.
• Derives the characteristic property of optimal ensembles for constrained channels via a generalized maximal distance inequality involving relative entropy: ∑ⱼ μⱼ S(Φ(ωⱼ) ‖ Φ(ρₐᵥ)) ≤ χ_Φ({πᵢ, ρᵢ}) for all ensembles with average state in 𝒜.
• Applies Shor’s channel extension construction to sequences of channels, introducing the notion of asymptotic additivity to characterize necessary and sufficient conditions for constrained additivity.
• Uses the MSW correspondence and continuity of entanglement of formation to establish continuity and concavity of the χ-function χ_Φ(ρ).
• Employs convex analysis and Legendre transforms to analyze the behavior of the function f(x) = H(Φ(xσ + (1−x)ρ)) − H(Φ(ρ)) and its derivative f’(x), particularly at x=0 and x=1.
• Proves that subadditivity of the χ-function for tensor products of channels is equivalent to the additivity of the Holevo capacity under arbitrary constraints, using perturbation arguments and the sign of Δ = χ_Φ⊗Ψ(σ) − χ_Φ(ρ) − χ_Ψ(ϱ).

Experimental results

Research questions

• RQ1 Under what conditions is the Holevo capacity additive for two quantum channels with arbitrary input constraints?
• RQ2 How does the generalized maximal distance property for optimal ensembles in constrained channels relate to the additivity of the χ-function?
• RQ3 Is the subadditivity of the χ-function for tensor product channels equivalent to the global additivity conjecture for unconstrained channels?
• RQ4 Can the additivity of the minimum output entropy for unconstrained channels imply strong superadditivity of entanglement of formation through the χ-function subadditivity?
• RQ5 What is the role of asymptotic additivity in characterizing the additivity of constrained channels via Shor’s channel extension?

Key findings

• The optimal ensemble for an 𝒜-constrained channel satisfies the generalized maximal distance property: ∑ⱼ μⱼ S(Φ(ωⱼ) ‖ Φ(ρₐᵥ)) ≤ χ_Φ({πᵢ, ρᵢ}) for all ensembles with average state in 𝒜.
• The additivity of the Holevo capacity for two channels with single linear constraints is equivalent to the additivity for two arbitrarily constrained channels, and further equivalent to the subadditivity of the χ-function for their tensor product.
• Subadditivity of the χ-function holds for several classes of channels, as shown in Proposition 4, including those where the optimal ensemble satisfies specific support conditions.
• The necessary and sufficient condition for additivity is given by the inequality f’(1) ≤ Δ, where f(x) is a concave function related to the output entropy and Δ is the difference in Holevo quantities.
• The converse of the main theorem in [6] leads to the notion of asymptotic additivity, which allows reformulating the main result as a necessary and sufficient condition for additivity.
• Global subadditivity of the χ-function is equivalent to strong superadditivity of entanglement of formation, providing an alternative proof to that in [15].