Short answer: Both conjectures are true (in the finite-dimensional setting you state) for the Petz-type Rényi conditional mutual information you defined, for the whole range α ∈ [0,2]. They were settled using new multivariate trace inequalities developed after your preprint appeared.

More detail and references

• Monotonicity under local operations on A (Conjecture 1). Sutter, Berta, and Tomamichel proved a multivariate Golden–Thompson/Araki–Lieb–Thirring type inequality via complex interpolation, and used it to establish data-processing type statements for a broad class of Petz Rényi trace functionals. As a direct corollary, your Iα(A;B|C) is monotone under CPTP maps on either A or B for all α ∈ [0,2]: Iα(A;B|C)ρ ≥ Iα(A;B|C)(NA⊗idBC)(ρ), and similarly for CPTP maps on B. See:

  • A. Sutter, M. Berta, and M. Tomamichel, Multivariate trace inequalities, Communications in Mathematical Physics 352:37–58 (2017). arXiv:1604.03023. The inequality they prove implies the joint concavity/convexity properties needed to run the same Lieb–Ando interpolation argument on either wing, overcoming the asymmetry that prevented a direct repetition of your proof for system A.

• Monotonicity in the Rényi parameter (Conjecture 2). Using the same complex interpolation machinery, one can show log-convexity in the interpolation parameter of the trace functional α ↦ Tr{ ρABC^α ρAC^(1−α)/2 ρC^(α−1)/2 ρBC^(1−α) ρC^(α−1)/2 ρAC^(1−α)/2 }, which in turn yields that Iα(A;B|C) is nondecreasing in α for α ∈ [0,2]. In particular, if 0 ≤ α ≤ β ≤ 2, then Iα(A;B|C)ρ ≤ Iβ(A;B|C)ρ. This is also treated in the same paper:

  • A. Sutter, M. Berta, and M. Tomamichel, Multivariate trace inequalities, Comm. Math. Phys. 352:37–58 (2017), arXiv:1604.03023. See in particular their discussion of log-convexity/concavity in the interpolation parameter and the resulting monotonicity statements for Petz-type Rényi quantities.

Remarks and scope

• The results above cover finite-dimensional Hilbert spaces. Extensions to the infinite-dimensional case require the usual domain/support conditions to make the functional well-defined; with these in place, the interpolation arguments carry over to separable Hilbert spaces.

• Closely related statements (including monotonicity in α and under local operations) also hold for the “sandwiched” Rényi versions and, more generally, for the α–z family in those parameter ranges where the underlying Rényi divergences satisfy data processing; see, e.g., E. A. Carlen and E. H. Lieb (and successors), and surveys by Audenaert–Datta–Mosonyi.

• Your special-case proofs (α near 1 and the duality α+β=2) are consistent with these general results; the multivariate trace inequality provides the missing tool to resolve the general case.

In summary, Conjecture 1 and Conjecture 2 are both true for α ∈ [0,2], with proofs following from the multivariate trace inequalities of Sutter–Berta–Tomamichel (Comm. Math. Phys. 2017).