Metrics on completely positive maps via noncommutative geometry

We study methods of inducing metrics on unital completely positive maps by employing seminorms arising in noncommutative geometry. Our main approach relies on the development of an infinite-dimensional $C^*$-algebraic analogue of the Choi-Jamiołkowski isomorphism. Under suitable conditions, we show that the induced metrics satisfy the quantum information theoretic properties of stability and chaining. Moreover, we show how to generate such metrics using constructions native to noncommutative geometry, by for example using external Kasparov products of spectral triples.

💡 Research Summary

The paper develops a systematic framework for defining metrics on the space of unital completely positive (UCP) maps between possibly infinite‑dimensional C*‑algebras by exploiting seminorms that arise naturally in noncommutative geometry. The authors begin by recalling two desiderata from quantum information theory: stability (the distance should be invariant under tensoring with the identity channel on an ancillary system) and chaining (the distance should satisfy a triangle‑type inequality under composition of channels). While these properties have been studied mainly for finite‑dimensional matrix algebras, the present work extends them to a broad C*‑algebraic setting.

The first construction is a “pull‑back” method. Given a unital C*‑algebra A equipped with a densely defined seminorm L, one obtains a Monge–Kantorovich extended metric mk_L on the state space of A. For a UCP map F: A→B, the seminorm L can be transferred to B via the pull‑back FL, and the distance between two channels F and G is defined as mk_{FL}(F,G). The authors prove that if L satisfies certain tensor‑product compatibility and continuity conditions (essentially that L is stable under maximal tensor products), then the induced metric enjoys both stability and chaining. Theorem 4.5 and Corollary 4.7 give precise sufficient conditions, and the authors show that seminorms coming from external Kasparov products of spectral triples meet these requirements.

The second construction tackles the lack of a canonical Choi–Jamiołkowski isomorphism for general C*‑algebras. For a C*‑algebra B carrying a faithful trace τ, they define a linear functional μ_τ on the algebraic tensor product B⊙B^op by μ_τ(b₁⊗b₂^op)=τ(b₁b₂). Kirchberg’s theorem guarantees that μ_τ is continuous for the minimal tensor norm precisely when τ is amenable. Using this, the authors build a map ω_τ : Hom(A,B) → S(A⊗_max B^op) that embeds each completely positive map into the state space of the maximal tensor product of A with the opposite algebra of B. Theorem 5.15 characterizes the image of this embedding as “trace channels,” i.e., those UCP maps that preserve the trace in a suitable sense.

With the embedding in hand, any seminorm L on A⊗_max B^op yields an “embedding‑induced” metric Δ_L(F,G)=mk_L(ω_τ(F),ω_τ(G)). The paper then investigates when Δ_L satisfies stability and chaining. The key insight is that if L originates from an external Kasparov product of spectral triples (𝔄₁,ℋ₁,D₁) and (𝔄₂,ℋ₂,D₂), then the resulting metric automatically respects both properties. Theorem 7.2 establishes stability for all such L, while chaining is proved for a large class of examples involving group C*‑algebras.

In particular, the authors consider twisted group C*‑algebras C*_r(G,σ) associated with a countable amenable group G equipped with a proper length function ℓ. The corresponding spectral triple (C_c(G),ℓ²(G),D_ℓ) yields a seminorm L_ℓ that, when applied to Fourier‑multiplier channels, satisfies the chaining inequality. Example 7.8 demonstrates that for amenable groups with natural length functions, the induced metric simultaneously enjoys stability (under tensoring with identities) and chaining (under composition).

Overall, the paper bridges noncommutative geometry and quantum information theory by providing a robust, infinite‑dimensional metric theory for completely positive maps. It introduces the novel notion of trace channels, extends the Choi–Jamiołkowski correspondence beyond nuclear algebras, and shows that geometric constructions such as external Kasparov products naturally produce metrics with the desired quantum‑information‑theoretic properties. The results open avenues for applying noncommutative geometric tools to problems like channel discrimination, optimal transport on quantum channels, and the analysis of quantum dynamical semigroups in infinite‑dimensional settings.