We are using entanglement entropy approach to study how the entropy depends upon the location of degrees of freedom near and far from the horizon. We consider quantum fields in the vicinity of black holes space-time, the entropy receives log- arithmic and power law corrections in area law. The power law corrections arises when the fields are in excited or mixed state (superposition of ground and excited state)a but logarithmic corrections are present in ground state. In this paper, we try to understand physically the deviation from the area law for the excited and mixed state. We show that the degrees of freedom near to the horizon contribute the total entropy, but when degrees of freedom far away from the horizon are more significant for excited/mixed state instead of ground state.
We elucidate the Bures metric in quantum state space near a rank-changing point of the density matrix and show contrasting behavior for two-level (
N=2) systems versus higher-level systems. Due to the smooth pure-state boundary for
N=2, we prove the apparent metric divergences to be merely coordinate artifacts and present three Lindblad processes exhibiting qualitatively different evolution near rank-changing points, showing geodesic approach, power-law scaling, and pure-state escape law. For higher-dimensional (
N≥3) systems, the geometry near a rank-changing point differs fundamentally. Under suitable restrictions of the density matrix and its approach towards a pure state, the Bures metric reduces to a conical metric with the pure state at the cone tip. Such a conic geometry leads to genuine curvature singularities: A two-dimensional cone exhibits a Dirac delta-function curvature near the tip while a higher-dimensional cone shows a power-law divergence of the curvature towards the cone tip. A construction of Lindblad evolution for
N=3 systems with conic singularities is presented, along with possible implications for future experimental and theoretical research.