## Appendix D Reactant Entanglement  
The A and B subsystems of the reactive system \(\mathrm{R}=\mathrm{A}\)--\(\mathrm{B}\) ultimately represent the bonded (entangled) subsystems in the equilibrium complex \(\mathrm{R}^{*}=(\mathrm{A}^{*}\,\,\mathrm{i}\,\,\mathrm{B}^{*})\) containing the mutually- and internally-open substrates \(\{\mathsf{X}^{*}\}\). Therefore, when brought into temporary interaction and then infinitely separated in \(\mathrm{R}^{*}(\infty)=\mathrm{A}^{*}+\mathrm{B}^{*}\), such dissociated fragments can no longer be described by the individual wavefunctions for each reactant, even after the interaction has utterly ceased and the \"molecular\" Hamiltonian is given by the sum of subsystem Hamiltonians:  
\[\mathrm{H}[\mathrm{R}(\infty)]=\mathrm{H}(\mathrm{A})+\mathrm{H}(\mathrm{B}).\] (A56)  
Indeed, QM deals with a _tensor product_ of subsystem spaces, not with their _sum_[101]. However, for additive parts of the Hamiltonian, the \"separation theorem\" of QM predicts  
\[\Psi[\mathrm{R}(\infty)]=\Psi(\mathrm{A})\;\Psi(\mathrm{B}),\] (A57)  
so that the following natural question arises: What is the molecular trace (\"memory\") of the past interaction left in the entangled separated fragments? As we have already argued elsewhere, it is the phase part of the system wavefunction which preserves a memory of the temporary interaction between subsystems (see also Appendix C). The IT equilibrium phase of the given quantum system as a whole has been previously linked to its probability distribution \(p(\mathbf{r})\)[62; 63], \(\phi_{eq. }(\mathbf{r})=-\left(\frac{1}{2}\right)\,\mathrm{ln}p(\mathbf{r})\), thus predicting the equilibrium current proportional to the negative gradient of probability density [see Equation (37)]:  
\[j_{eq.}(\mathbf{r})=\left(\hbar/m\right)p(\mathbf{r})\;\
abla\phi_{eq.}(\mathbf{r})\equiv p (\mathbf{r})\;\mathbf{V}_{eq.}(\mathbf{r})=-\left[\hbar/(2m)\right]\;\
abla p(\mathbf{r}).\] (A58)  
Therefore, the product function in Equation (A57) of the separated \"dissociated\" subsystems still preserves the memory of the probability distribution \(p(\mathbf{r})\) of the past (interacting) subsystems at a finite separation between reactants, contained in the equilibrium phase for this \"molecular\" separated reactive system. The same phase criterion applies to diagnosing the entanglement distinction between the _bonded_ and _nonbonded_ status of reactants in the equilibrium, \(\mathrm{R^{*}}=(\mathrm{A^{*}}\mid\mathrm{B^{*}})\), and polarized, \(\mathrm{R^{+}}=(\mathrm{A}\mid\mathrm{B})\), complexes, respectively. For the same probability distribution \(p_{\mathrm{R}}(\mathbf{r})\) in both these hypothetical states, the two subsystems are free to exchange electrons in the former, while in the latter this flow of electrons is forbidden. The equilibrium phase of \(\mathrm{\Psi}(\mathrm{R^{*}})\) then reflects the negative of \(\mathrm{ln}p_{\mathrm{R}}\), \(\phi_{eq.}(\mathrm{R^{*}})=\phi_{eq. }[p_{\mathrm{R}}]\), while at a finite separation \(R_{\mathrm{AB}}\) between reactants \(\phi_{eq. }(\mathrm{R^{+}})\) is proportional to the negative of \(\mathrm{ln}[(p_{\mathrm{A}}p_{\mathrm{B}})^{1/2}]\). Therefore, it is also the phase component which distinguishes the bonded (entangled) state of reactants in \(\mathrm{R^{*}}\) from their nonbonded (disentangled) status in \(\mathrm{R^{+}}\), also at finite separations between the two reactants. The mutual opening of the interacting reactants at their finite separation establishes the state global phase \(\phi_{eq.