Schwarzschild-de Sitter black hole as a correlated qubit system via entropic identification

TL;DR

This work develops a framework to treat Schwarzschild–de Sitter black holes with two horizons as correlated two-qubit systems by equating horizon entropies to qubit von Neumann entropies. Using a Dirac-basis density-matrix formalism and non-local unitary transformations, the authors construct a physical ρTotal that matches the reduced density matrices of the black hole and cosmological horizons. A key result is that the positivity constraint on ρTotal imposes a lower bound on the total entropy $S_{Total}$ that is stricter than the Araki–Lieb triangle inequality, implying gravity constrains quantum correlations beyond standard limits. This gravitationally induced bound persists in both 5D and 4D Sch–dS models, with maximal correlation occurring near extremality, and motivates viewing horizon thermodynamics through a running correlation parameter within an effective thermodynamic framework.

Abstract

The thermodynamic behaviours of multi-horizon black holes such as a Schwarzschild-de Sitter black hole have been one of the long-standing mysteries in gravitational physics since they involve quantum natures in gravitational systems and that the search for quantum gravity has not reached its conclusion. In this work, we seeked for a possibility of realising the Schwarzschild-de Sitter black hole as a correlated qubit system, where each of the event horizon is treated as a qubit and both of them are correlated in a way that two qubits could be. By identifying the entropies of subsystems to those of qubits, we successfully constructed the reduced density matrices of the two subsystems as well as the density matrix for the Schwarzschild-de Sitter black hole, modelled as 2-correlating qubits. Moreover, our results suggested that when the gravitational effect has its role in the qubit systems, supposedly like black holes, the correlation between qubits are constrained with a lower bound more stringent than the so-called Araki-Lieb triangle inequality.

Figures (8)

• Figure 1: The horizon structures of the Sch-dS metric function $F(r)$ for several cases of $r_s$ : $r_s=0$, $0<r_s<r_c/2$, $r_s = r_c/2$.
• Figure 2: The total entropy of the correlated Sch-dS system as a function of the dimensionless variable $\epsilon$ at different values of "constant" correlation degree $a$.
• Figure 3: The numerical solutions for $f(\epsilon)$, represented by the blue line, and $g(\epsilon)$, represented by the black line, at $101$ discrete values of $\epsilon$ within the range $[0,1]$.
• Figure 4: The plot shows the behaviour of $a(\epsilon)$ at different values of $\epsilon$ in the case that the qubit system analogous to 5D Sch-dS system is maximally-correlated.
• Figure 5: The above figure shows the entropies in each cases of correlation between subsystems. The red line denotes the entropy of the maximally-correlated 5D Sch-dS black hole system. The below figure demonstrates a zoom-in of the above figure at $\epsilon$ close to unity.