Theorem: Commuting-coupling obstruction

Suppose

for every pair of couplings. For any insertion isometry $W$, the accessible reservoir channel is entanglement breaking. Every deterministic decoder therefore satisfies

For a qubit diary, the lower bound is $1/2$.

Proof. Choose a common source eigenbasis, with spectral projectors $P_{\lambda }$. The complete propagator is block diagonal:

Tracing the source removes terms between different projectors. Tracing the carrier does not restore them. The accessible channel therefore has measure-and-prepare form

where

are positive effects summing to $I_C$.

Such a channel breaks entanglement with every reference. A decoder preserves that property. A separable state on $Q\otimes C$ has overlap at most $1/d$ with $|{\Phi }_d⟩$: this is immediate for pure product states by Cauchy–Schwarz and extends by convexity. Testing the Bell projector gives trace distance at least $1-1/d$, hence the stated diamond lower bound.