Complete freezing of initially maximal entanglement in Schwarzschild black hole

TL;DR

This work studies how gravity via Hawking radiation affects multipartite entanglement, focusing on the tetrapartite fermionic cluster state CL4 in Schwarzschild spacetime. Using Dirac field quantization in Schwarzschild geometry, Bogoliubov transformations, and tracing over interior modes, they compute negativity measures such as $N_{A(BCD)}$ for the CL4, GHZ4, and W4 states. The main result is complete freezing: $N^{CL_{4}}_{A(BCD)}=1$ for all Hawking temperatures, independent of $T$, whereas GHZ4 and W4 decay monotonically in this partition. These findings reveal topology-driven robustness in curved spacetime and suggest the CL4 state as a resilient resource for relativistic quantum information processing in strong gravitational fields.

Abstract

Gravitational effects associated with black holes are widely believed to universally degrade quantum entanglement, with the loss of maximal entanglement being particularly severe and even irreversible for bosonic fields. In this work, we investigate the entanglement properties of the four-qubit cluster state ($CL_4$) for fermionic fields in the curved spacetime of a Schwarzschild black hole. Remarkably, we uncover a counterintuitive phenomenon: as the Hawking temperature increases, quantum entanglement ($1$-$3$ tangle) of the $CL_4$ state remains strictly constant, indicating a ``complete freezing of initially maximal entanglement". This constitutes the first explicit example in which maximal entanglement remains perfectly preserved in a black hole environment, defying the conventional expectation that gravitational effects can only suppress maximal quantum correlations. Moreover, our results indicate that, within a relativistic framework, the $CL_4$ state constitutes a high-quality quantum resource with potential applications in relativistic quantum information processing, and may significantly improve the performance of such protocols.

Figures (3)

• Figure 1: Embedding diagram of the black hole and the detector configuration. Alice, Bob, Charlie and David are stationed at fixed radial positions outside the event horizon, while the anti-David mode resides in the causally inaccessible region inside the event horizon.
• Figure 2: The $N_{A(BCD)}$ of the $CL_{4}$, $GHZ_{4}$ and $W_{4}$ states as a function of the Hawking temperature $T$ for a fixed mode frequency $\omega = 1$.
• Figure 3: The $N_{C(ABD)}$ and $N_{D(ABC)}$ of the $CL_{4}$, $GHZ_{4}$ and $W_{4}$ states as a function of the Hawking temperature $T$ for a fixed mode frequency $\omega = 1$.