Black Holes, Entanglement and Decoherence

Abstract: It was recently shown that a black hole (or any Killing horizon) will decohere any quantum superposition in their vicinity. I review three distinct but equivalent arguments that illustrate how this phenomenon arises: (1) entanglement with "degrees of freedom" in the interior (2) the absorption of soft, entangling radiation emitted by the superposition and (3) interactions with the quantum, fluctuating multipole moments of a black hole arising from ultra low frequency Hawking quanta. The relationship between "soft hair" and interactions with "internal degrees of freedom" is emphasized and some implications for the nature of horizons in a quantum theory of gravity are discussed.

• The paper establishes that black holes induce quantum decoherence via entanglement with internal degrees of freedom, soft radiation absorption, and local vacuum interactions.
• It derives quantitative decoherence timescales for gravitational and electromagnetic superpositions, setting a fundamental bound on quantum coherence.
• The research unifies three equivalent perspectives, linking soft hair, horizon memory, and vacuum fluctuations to the loss of coherence near horizons.

Black Holes, Entanglement, and Decoherence: Mechanisms and Implications

Introduction

This paper provides a rigorous analysis of the mechanisms by which black holes, and more generally any Killing horizon, induce decoherence in quantum superpositions situated in their exterior. The author synthesizes three distinct but physically equivalent perspectives: (1) entanglement with internal degrees of freedom, (2) absorption of soft, entangling radiation, and (3) local interactions with quantum fluctuating multipole moments. The work establishes that the presence of a horizon fundamentally limits the coherence of quantum superpositions, with implications for both experimental quantum physics and the theoretical understanding of quantum gravity.

Entanglement with Internal Degrees of Freedom

The first argument is constructed via a gedankenexperiment involving two agents, Alice and Bob. Alice attempts to maintain a quantum superposition in her laboratory, shielded from environmental decoherence. However, the gravitational field sourced by her superposition cannot be shielded, and Bob, even if located inside a black hole, can in principle measure the gravitational field and become entangled with Alice's superposition. Due to the causal structure of the black hole, Alice cannot detect Bob's actions, yet the mere possibility of entanglement across the horizon necessitates that the black hole itself acts as an environment, decohering Alice's superposition at a rate commensurate with the entanglement that could be established with internal degrees of freedom.

This argument is robustly grounded in quantum mechanics and causality: the horizon acts as a one-way membrane, and the decoherence rate is determined by the amount of "which-path" information that could be extracted by hypothetical observers inside the black hole. The analysis generalizes to any spacetime with a Killing horizon, indicating a universal property of horizons in quantum gravity.

Absorption of Soft Radiation and Horizon Memory

The second perspective focuses on the emission and absorption of soft radiation. When a massive particle is placed in a spatial superposition near a black hole, the superposed gravitational field leads to the emission of soft gravitons, which are absorbed by the horizon. The quantum state of the horizon becomes entangled with the superposition, and the overlap of the radiation states associated with different branches of the superposition determines the decoherence rate.

A key result is the derivation of the decoherence timescale for gravitational and electromagnetic superpositions:

• For a mass $m$ superposed over a distance $d$ at a distance $D$ from a black hole of mass $M$, the expected number of entangling gravitons emitted into the horizon grows linearly with the experiment duration $T$:

$$\langle N \rangle \sim \frac{M^5 m^2 d^4}{D^{10}} T$$

• The corresponding decoherence timescale for gravity is:

$$T_D^{GR} \sim \frac{\hbar c^{10} D^{10}}{G^6 M^5 m^2 d^4}$$

• For electromagnetism:

$$T_D^{EM} \sim \frac{\epsilon_0 \hbar c^6 D^6}{G^3 M^3 q^2 d^2}$$

These expressions demonstrate a strong dependence on the distance from the horizon and the mass/charge and separation of the superposed object. For realistic laboratory conditions, decoherence due to astrophysical black holes is negligible, but the effect sets a fundamental bound on quantum coherence in the presence of horizons.

The analysis also connects the horizon memory effect ("soft hair") to decoherence: permanent changes in the horizon metric due to the motion of external bodies correspond to the absorption of soft quanta, which are responsible for entanglement and loss of coherence. The equivalence between "which-path" information accessible inside the horizon and the decoherence observed outside is established quantitatively.

Local Interactions and Fluctuating Multipole Moments

The third approach reframes the decoherence process in terms of local interactions with quantum vacuum fluctuations. The black hole, due to its reservoir of soft Hawking quanta, exhibits randomly fluctuating multipole moments (mass quadrupole for gravity, electric dipole for electromagnetism) with constant power spectra at low frequencies. These fluctuations stimulate the emission of soft quanta from the superposed object, leading to decoherence.

Explicitly, the expected number of entangling gravitons is given by:

$$\langle N \rangle = \langle \Omega_U | [h^{in}_{ab}(T_1^{ab} - T_2^{ab})]^2 | \Omega_U \rangle$$

where $h^{in}_{ab}$ is the free field operator and $\Omega_U$ is the Unruh vacuum.

The magnitude of the black hole's fluctuating quadrupole and dipole moments is:

• Gravitational quadrupole:

$$\Delta |Q_U|(\omega) \sim \frac{\sqrt{\hbar} G^2 M^{5/2}}{c^5}$$

• Electromagnetic dipole:

$$\Delta |\vec{P}_U|(\omega) \sim \frac{\sqrt{\epsilon_0 \hbar} G^{3/2} M^{3/2}}{c^3}$$

These results demonstrate that, quantum mechanically, black holes possess an infinite set of fluctuating multipole moments, in contrast to the classical "no-hair" theorem. The decoherence induced by these fluctuations can be mimicked by suitably engineered material bodies, but only under extreme physical conditions (e.g., high resistivity or viscosity), especially for the gravitational case.

Implications and Future Directions

The findings have several important implications:

• Fundamental Bound on Quantum Coherence: The presence of a horizon imposes a universal, irreducible source of decoherence for quantum superpositions, independent of environmental noise.
• Equivalence of Interpretations: The three perspectives—entanglement with internal degrees of freedom, absorption of soft radiation, and local interactions with vacuum fluctuations—are quantitatively equivalent and provide complementary insights into the mechanism of horizon-induced decoherence.
• Connection to Soft Hair and Memory: The horizon memory effect and soft hair are directly linked to the entanglement and decoherence process, suggesting a deeper relationship between infrared structure, information loss, and quantum gravity.
• Experimental Relevance: While the effect is negligible for current laboratory experiments, it sets a theoretical limit for future macroscopic quantum superpositions and motivates precision tests near strong gravitational fields.
• Theoretical Extensions: The analysis generalizes to arbitrary Killing horizons, including cosmological spacetimes, and is consistent with microscopic models of black holes in string theory. The connection between horizon memory and internal degrees of freedom warrants further investigation, particularly in the context of holography and quantum information transfer across horizons.

Conclusion

This work establishes that black holes and horizons universally decohere quantum superpositions in their exterior, acting as environments with effective internal degrees of freedom. The decoherence mechanism is robustly supported by three equivalent physical arguments, each elucidating different aspects of the process. The results provide a fundamental bound on quantum coherence in the presence of horizons, clarify the role of soft hair and horizon memory, and open avenues for further research into the quantum nature of gravity and spacetime.