Output Side Information Changes the Entanglement-Breaking Threshold

Discarding an output subsystem is a local post-processing operation, so an entanglement-breaking channel remains entanglement breaking after output reduction. The converse need not hold: a reduced channel can be entanglement breaking even when a richer output still carries reference entanglement. This preprint gives an exactly solvable two-parameter benchmark that quantifies this asymmetry. On two qubits, let (Π_±=(I± Z⊗ I)/2), let ( L(X)=∑_sΠ_sXΠ_s) be sector dephasing, let ( M(X)=∑_sTr(Π_sX)Π_s/2) be measure-and-prepare within the two sectors, and let ( D_4(X)=Tr(X)I_4/4). The channel family is [Φt,η=(1-η)[(1-t) L+t M]+η D_4, 0≤ t,η≤1.] For the reduced task, a variable second-qubit input is embedded with an arbitrary fixed first-qubit state and the first output qubit is discarded. The induced qubit channel is depolarizing with parameter (q=(1-η)(1-t)), and is entanglement breaking exactly when (q≤1/3). For the full four-dimensional output, the partially transposed normalized Choi state has one potentially negative eigenvalue, [λ_- = -4(1-η)+6(1-η)t+η/16,] with multiplicity two. The block structure makes PPT sufficient for separability in this family. Hence, for (η<1), [tred(η)=max!{0,1-1/3(1-η)},full(η)=max!{0,4-5η/6(1-η)}.] The exact region in which the reduced channel is already entanglement breaking while the full channel is not is (tred(η)≤ t<tfull(η)). Its width is [w(η)={cases}η/[6(1-η)],&0≤η≤2/3,\(4-5η)/[6(1-η)],&2/3≤η<4/5,\0,&4/5≤η≤1,{cases}] and reaches the exact maximum (wₘₐₓ=1/3) at (η=2/3). The Choi negativities are [ Nred=[3(1-η)(1-t)-1]+/4, N{ full}=[4-5η-6(1-η)t]_+/8.] Therefore the maximal full-output Choi negativity compatible with an already entanglement-breaking reduced output is (1/12), attained at ((η,t)=(2/3,0)). The result is a controlled benchmark for output-access dependence: the same physical channel family can cross the entanglement-breaking boundary at different noise values solely because an output subsystem is retained or discarded.