Quantum and Topological Dynamics of the GKSL Equation in the Camel-Like Framework

This paper investigates the properties of the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation within the camel-like framework, with a focus on quantum correlations in the context of open quantum systems. Here, we compute quantum correlations such as quantum discord and quantum steering, analyzing their behavior under decoherence and environmental interaction for three sets of quantum states. Our results indicate that the sign of the entanglement entropy's derivative serves as an indicator of the system's drift toward classical or quantum information exchange—an insight with important implications for quantum error correction and dissipation processes in quantum thermal machines. \\ Moreover, we parametrize quantum states using both single-parameter and Bloch-sphere representations. The Bloch-sphere analysis is particularly developed by examining the topology of the quantum states as elements on the \(\mathbb{S}^2\) sphere, yielding a gradient map and a topological basin map which illustrate how quantum states evolve under the camel-like framework and how this can be partitioned into basins on the Bloch sphere, by stability/instability against decoherence. The decoherence-unstable regions form a Braiding ring around the Bloch sphere at the circle defined by $\theta=\frac{3\pi}{4}$. Notably, these unstable states on the Braiding ring form attraction points for the evolution of states by a constructed Lyapunov function, which gives an illustration of the complex interplay between geometry and dynamics and which shapes the topological geometry on the Bloch sphere. This deeper analysis underscores that quantum dynamics are governed not only by local energetics but also by the global structure of the state space. Moreover, we also derive a testable rudimentary experimental setup from the camel-like entropy, which describes the unitary evolution of pure states on the Bloch surface and add the analytical solutions of selected quantum states.