Circulant quantum channels and its applications

This note introduces a family of circulant quantum channels – a subclass of the mixed-permutation channels – and investigates its key structural and operational properties. We show that the image of the circulant quantum channel is precisely the set of circulant matrices. This characterization facilitates the analysis of arbitrary $n$-th order Bargmann invariants. Furthermore, we prove that the channel is entanglement-breaking, implying a substantially reduced resource cost for erasing quantum correlations compared to a general mixed-permutation channel. Applications of this channel are also discussed, including the derivation of tighter lower bounds for $\ell_p$-norm coherence and a characterization of its action in bipartite systems.

💡 Research Summary

This paper introduces and thoroughly investigates a new class of quantum channels called “circulant quantum channels,” which are a restricted subclass of the mixed‑permutation channels previously studied in the literature. The authors begin by defining the cyclic permutation π₀ = (d,…,2,1) on a d‑dimensional Hilbert space and, for any probability vector λ = (λ₀,…,λ_{d‑1}), construct the linear map

 Φ_λ(X) = Σ_{k=0}^{d‑1} λ_k P_{π₀}^k X P_{π₀}^{‑k}.

When λ is uniform (λ_k = 1/d) the map reduces to Φ, the circulant quantum channel. Several elementary properties are proved: Φ_λ is unital, not self‑adjoint with respect to the Hilbert‑Schmidt inner product, commutes with transposition, preserves Hermiticity, and is idempotent (Φ∘Φ = Φ).

The authors derive an explicit expression for the image of Φ_λ:

 Φ_λ(X) = Σ_{i,j} Tr