Equality of conditional channel min-entropy and Choi-state bound for noisy two-qubit unitaries

Determine whether, for every two-qubit unitary channel U_{A'B'→AB} and every mixing parameter p ∈ [0,1], the white-noise mixture N_p := p R^π + (1 − p) U satisfies the exact identity S_∞[A|B]_{N_p} = S_∞(R_AA|R_BB)_{Φ^{N_p}} − log|A'|, where Φ^{N_p} is the Choi state of N_p and R^π denotes the bipartite uniformly mixing (replacer) channel outputting the maximally mixed state. This asks whether the conditional channel min-entropy of such noisy two-qubit unitaries is always exactly determined by the conditional min-entropy of their Choi states.

Background

The paper proves that for any two-qubit unitary channel U_{A'B'→AB}, the conditional channel min-entropy equals the conditional min-entropy of its Choi state minus log|A'|. This provides an exact characterization for the noiseless two-qubit case.

Numerical evidence in the paper indicates that this equality appears to extend to noisy channels formed by mixing a two-qubit unitary with white noise, N_p := p R^π + (1 − p) U. Establishing this equality in full generality would clarify whether the conditional channel min-entropy of such noisy two-qubit unitaries is completely determined by the corresponding Choi-state quantity.