How to detect a possible correlation from the information of a subsystem in quantum-mechanical systems

The possibility of detecting correlations between two quantum-mechanical systems from only the information of a subsystem is investigated. For generic cases, we prove that there exist correlations between two quantum systems if the time derivative of the reduced purity is not zero. Therefore, an experimentalist can conclude that correlations between the system and some environment are nonzero if the time derivative of the reduced purity is found not to be zero. A quantitative estimation of the time derivative of the reduced purity with respect to correlations is also given. This clarifies the role of correlations in the mechanism of decoherence in open quantum systems.

Figure 1

How to detect possible correlations between your quantum system $S$ and its environment $E$.

Figure 2

Time evolution of the reduced purity (14) for (a) $p_n=1∕2^n$, $h_n=n∕4$ and (b) $p_n=1∕2^n$, $h_n={25}^n\pi ∕4$, with the unit of time ${\omega }_0\equiv \hslash ∕E_0$. Notice that in both cases the Hamiltonians are unbounded from above. One sees the flat time derivative at $t=0$ in (a), which makes statement B true, while one sees nondifferentiability in (b), which breaks down statement B.