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Completely contractive data-processing conjecture for conditional mutual information

Let E:C→C′\mathcal{E}:C\rightarrow C' be a quantum channel with correctable algebra A(E)\mathcal{A}(\mathcal{E}). The channel has trivial correctable algebra when

A(E)=CI.\mathcal{A}(\mathcal{E})=\mathbb{C}I.

For a tripartite state ρABC\rho_{ABC}, let I(A:C∣B)ρI(A:C|B)_\rho denote its conditional mutual information.

Completely contractive data-processing conjecture for conditional mutual information. For any channel E:C→C′\mathcal{E}:C\rightarrow C' with trivial correctable algebra, there exists a constant η<1\eta<1 such that for any tripartite system ABCABC and any state ρABC\rho_{ABC},

I(A:C′∣B)E(ρ)≤ηI(A:C∣B)ρ.I(A:C'|B)_{\mathcal{E}(\rho)}\leq\eta I(A:C|B)_\rho.

A uniform contraction ratio would imply exponential decay of conditional mutual information under iterated Y-shaped channels. The conjecture is open in general quantum systems; even the corresponding mutual-information statement with BB trivial would be a breakthrough. It is known for channels with a forgetful component, and an analogous result is known when AA is classical and BB is trivial.

Sources & referencesView supporting material

Primary source

Chi-Fang Chen, Kohtaro Kato and Fernando G. S. L. Brandão, “Matrix Product Density Operators: when do they have a local parent Hamiltonian?”, arXiv:2010.14682 (2023).

conditional-mutual-informationdata-processingquantum-information

Progress summary

Refreshed
Open

The conjecture remains open in general, with only special cases and no public proof or counterexample.

The conjecture asks whether every quantum channel with trivial correctable algebra uniformly contracts conditional mutual information by a factor η<1\eta<1 for every input state and conditioning system. The general quantum statement remains unresolved; even the corresponding mutual-information case with trivial BB would be a major advance.

Known results

  • Channels with a forgetful component satisfy the conjectured contraction, yielding exponential decay for the associated Y-shaped channels.
  • A single application of any channel with trivial correctable algebra strictly decreases conditional mutual information, but this does not give a uniform contraction factor.
  • The analogous statement is known in the classical setting and when AA is classical with trivial BB.
  • Exponential-decay bounds are available for bistochastic and strictly positive Y-shaped channels.

Current status (as of August 2026): the conjecture is proved only in special cases, while the general quantum statement and its uniform contraction ratio remain open.

Sources

Solutions 0

No solutions have been posted yet.