Computational Refutation of Quantum Superactivation via Bound Entanglement in Low Dimensions

Smith and Yard (2008) famously demonstrated that two quantum channels with zero quantum capacity can possess a strictly positive joint capacity, a phenomenon known as superactivation. Their proof relies on the existence of Positive Partial Transpose (PPT) states possessing a strictly positive Devetak-Winter private key rate ($K_{DW} > 0$), explicitly citing a 4x4 construction by Horodecki et al. We present a two-tiered computational refutation of this phenomenon. First, using a high-precision global optimization framework with exact Stinespring purifications, we compute $K_{DW}$ across the full 512-dimensional PPT manifold for d=4x4 and find a strict upper bound of $K_{DW} \le -0.68$ bits, physically breaking the foundational protocol. Second, to address recent theoretical defenses concerning macro-scale finite-blocklength scaling ($N \ge 17$), we deploy a 30-qubit hybrid machine learning simulator (equivalent to $N=15$ joint channels). By evaluating the joint topology under strict causal limits and with omniscient encoder prescience, we demonstrate a complete failure to achieve positive capacity, establishing that superactivation is physically impossible in practical quantum systems.