## 7 Quantum Correlations in Scattering Dynamics  
After the above remarks and comments on the significance of QoC for the INS context, it is appropriate to consider in some more detail the time-dependence of the correlations accompanying the INS process.  
Obviously, long time before the INS collision we have:  
\[t=-\infty\ \ :\ \ \rho^{n}\otimes\rho^{\rm H,\mathcal{E}} \tag{26}\]  
The H-\(\mathcal{E}\) interaction leads to mixed states for H and for \(\mathcal{E}\)). Let \(\rho_{\lambda}^{H}\) be one of the H-states after the INS collision (with \(\lambda\) indicating possible available states), and \(\rho_{\lambda}^{n}\) be the associated mixed neutron state. Due to the H-\(\mathcal{E}\) interactions and the fragility of entanglement, a fully entangled state for \"neutron + H\" does not exist. But then we may assume that a quantum discordant state exists, which is:  
\[0\approx t\leq\tau_{exp}\sim 10^{-14}\mbox{s}\ \ :\ \ \left(\int\!d\lambda\,p( \lambda)\,\rho^{n}(\lambda)\otimes\rho^{H}(\lambda)\right)\otimes\rho^{ \mathcal{E}} \tag{27}\]  
Here, \(\tau_{exp}>0\) is the very-short neutron-H scattering time window (roughly being in the pico-to femto-second range) due to the delta-like Fermi pseudo-potential [8]. The neutron-H interaction creates quantum correlations (in some exceptional cases, also are extremely short-lived transient neutron-H entanglement). These are represented by \(\rho^{n,H}\). In simpler terms one may say that \(\mathcal{E}\)--or a Maxwell's demon associated with the environment--\"observes\" continuously the neutron-H system, thus causing a continuous change of the quantum correlations between neutron and H.  
These H-\(\mathcal{E}\) interactions will cause strong and complicated time-dependent quantum correlations, during some relaxation time \(\tau_{relax}\), which is a characteristic quantity of the material at the existing temperature and pressure. Then, for times \(\tau_{exp}<t\sim\tau_{relax}\), the complete system \"neutron + H + \(\mathcal{E}\)\" should be in a state described by \(\rho^{n,H,\mathcal{E}}\):  
\[\mbox{excited complete system}:\ \ \ \ \tau_{exp}<t\sim\tau_{relax}\ \ :\ \ \ \rho^{n,H,\mathcal{E}} \tag{28}\]  
At times \(t\gg\tau_{relax}\), the neutron will be far away from the scattering system, and thus the system \"H + \(\mathcal{E}\)\" will evolve towards its earlier equilibrium. Thus we may write:  
\[t\gg\tau_{relax}\ \ :\ \ \sum p_{i}\,\rho_{i}^{H,\mathcal{E}}\otimes\rho_{i}^{n} \tag{29}\]  
(here, the sum can be replaced by an integral, and \(p_{i}\) are classical probabilities). Each term in this discordant state is qualitatively similar to the correlated state of \"H + \(\mathcal{E}\)\" before the collision with the neutron. At the same time, the neutron occupies a mixed state \(\rho_{i}^{n}\).  
Later on, assuming an ideal high-resolution detector which measures momentum, and according to the well-known reduction postulate, an actual neutron detection will produce a pure neutron state and an associated quantum correlated \"H + environment\" state:  
\[\mbox{detector measures}\ \ n\ \ \mbox{at}\ \ \ t=+\infty\ \ :\ \ \rho_{\bf k_{1}}^{H, \mathcal{E}}\otimes|{\bf k_{1}^{\theta}}\rangle\langle{\bf k_{1}^{\theta}}| \tag{30}\]This is the same assumption as conventional theory does--i.e., the plane wave approximation of conventional theory [8].  
For simplicity of notation, \(t\) is not shown explicitly in any time-dependent quantity appearing in several of the above formulas, like, e.g., \(\rho^{H,\mathcal{E}}\).  
From these considerations, it becomes obvious that the amount of quantum discord, and of any other measure of quantum correlations being relevant in this INS process, changes with time--e.g., in the course of time, quantum discord between any two of the three parts of \"neutron + H + \(\mathcal{E}\)\" may be erased and/or created continuously.