Gravitationally Induced Entanglement

• Gravitationally induced entanglement is a phenomenon where gravity alone creates quantum entanglement between spatially separated masses.
• Experimental setups employ matter-wave interferometry and optomechanical techniques to measure phase shifts and fringe visibility losses as key indicators.
• Protocols quantify entanglement through branch-dependent phases and interference patterns, providing a testbed for distinguishing quantum from classical gravity models.

Gravitationally induced entanglement refers to the phenomenon in which quantum entanglement between two or more spatially separated systems arises solely due to their mutual gravitational interaction. The concept is central to efforts aiming to probe quantum aspects of gravity in laboratory and astrophysical settings, and it occupies a unique intersection between quantum information theory, nonrelativistic quantum mechanics, quantum field theory, and semiclassical gravity. Gravitationally induced entanglement (GIE) is often considered an indirect witness of gravity's nonclassical features, but, as current literature demonstrates, this connection is nuanced and depends critically on the underlying assumptions and dynamical models.

1. Principle and Theoretical Basis

At its core, the possibility of gravitationally induced entanglement emerges from the interplay of superposition and nonlocality in quantum mechanics with the fundamentally universal coupling of mass-energy in gravity. In a canonical GIE protocol (Marletto et al., 2017), two masses—each prepared in spatial superposition states (e.g., using Mach–Zehnder interferometers)—are allowed to interact solely via gravity. If after some interaction time, entanglement is observed between their quantum states, and if no other mediating force is appreciable, this suggests that the gravitational field must itself possess non-commuting observables and thus be quantum in nature.

This assertion relies on a general quantum information theorem: a mediator that can induce entanglement between two quantum systems cannot be a classical system with only commuting observables. The gravitational field must therefore have at least two noncommuting observables if it is to mediate entanglement, providing an operational distinction between quantized and purely classical channels.

For a quantitative description, in the prototypical setup, the gravitationally induced phase accumulated between spatially superposed branches is

$\phi = \frac{G m_1 m_2 \Delta t}{\hbar d},$

where $m_1, m_2$ are the masses, $\Delta t$ the interaction time, and $d$ the separation. The resulting joint quantum state exhibits branch-dependent phases that result in an entangled final state after recombination and measurement operations (Marletto et al., 2017, Krisnanda et al., 2019, Yant et al., 2023).

2. Experimental Protocols and Detection Strategies

Experimental GIE protocols exploit matter-wave interferometry and related techniques in atomic, molecular, optomechanical, or solid-state systems (Krisnanda et al., 2019, Howl et al., 2023, Matsumura et al., 2020, Braccini et al., 2023). Core steps typically include:

1. Preparation: Two systems (atoms, microspheres, mirrors) are initialized in superpositions of spatially separated states—either via beam splitters, Stern-Gerlach splitting, or optical manipulation.
2. Gravitational Interaction: The systems evolve in proximity such that the only significant interaction is gravity (electromagnetic, van der Waals, and Casimir forces are suppressed by shielding or spatial configuration).
3. Phase Accumulation: Gravitational interactions provide branch-dependent phases, converting initial product states into entangled states.
4. Measurement and Witnessing: Entanglement is inferred through measurable quantities such as reduced fringe visibility in interference patterns, atom-number correlations, covariances, or direct entanglement monotones (entropy, negativity, concurrence) (Yant et al., 2023, Ghosal et al., 2023, Rostom, 9 Jan 2024).

Notable experimental designs include:

• Parallel, adjacent atomic interferometers exploiting atom-number correlations as signatures of GIE without the need for macroscopic quantum superpositions (Howl et al., 2023).
• Optomechanical setups where the Newtonian gravitational potential between mirrors in separate cavities modulates quantum correlations between photons (Matsumura et al., 2020).
• Large spin extensions of Stern-Gerlach interferometry enabling spatial superpositions with $2j+1$ branches, enhancing entanglement slightly and enabling detailed parameter scanning via high-performance computing resources (Braccini et al., 2023).

Key performance metrics are the achieved fringe visibility (which diminishes as inter-system entanglement grows) and optimized signal-to-noise ratios, sometimes enhanced via postselection under nonmaximal entanglement conditions (Rostom, 9 Jan 2024).

3. Quantum versus Classical Mediation and Model Dependence

A central issue is whether the observation of GIE uniquely demonstrates the quantization of gravity. Canonical quantum information arguments (Marletto et al., 2017) assert that only a quantum mediator can generate entanglement between quantum systems. However, subsequent analyses highlight important subtleties.

• Hamiltonian Constraint and Non-dynamical Interactions: In the weak-gravity non-relativistic regime, the Newtonian gravitational potential is a solution to a constraint equation (analogous to the Coulomb potential in electromagnetism). The Newtonian interaction can generate entanglement as a consequence of this constraint, independent of whether the mediator is dynamical and quantized (Anastopoulos et al., 2018, Fragkos et al., 2022).
• Classical + Noise Models (Diósi-Penrose): The DP model, featuring classical gravity coupled to quantum matter with a stochastic (collapse-type) term, can also produce GIE under specific configurations—especially for short enough separation distances relative to the DP smearing parameter $\sigma$. Thus, GIE does not generically falsify classical-gravity collapse models unless the predicted spatial dependence and decoherence times disagree with experiment (Trillo et al., 4 Nov 2024).
• Operator Nonlocality and Measurement Ambiguity: The theoretical description of GIE is not unique—mediated interaction Hamiltonians can be recast as nonlocal or direct (constraint-induced) terms, blurring the correspondence between observation of entanglement and inference of local, quantized mediators (Fragkos et al., 2022).
• Amplification and Weak Measurement: Amplification schemes using weak-value measurements and EPR steering can enhance GIE signals well beyond naive measurement reach, providing more sensitive discriminators between quantum and classical mediation (Feng et al., 2022).

4. The Role of Interference, Visibility, and Decoherence

Interference phenomena underlie the operational basis for detecting GIE. Crucially:

• Destructive interference in one system (e.g., as engineered via a phase shifter in one output port) can induce a sign change in the partner state's amplitude, enforcing entanglement via correlated fringe shifts (Rostom, 9 Jan 2024).
• Nonmaximal entanglement, corresponding to weak gravitational phase shifts, allows for postselected recovery of high-contrast signals in complementary channels, reducing the needed experimental mass, interaction time, or spatial separation to meet the "minimum requirements" for GIE detection (Rostom, 9 Jan 2024).
• Loss of fringe visibility in interferometric detection is both a witness of entanglement and a key target for relativistic corrections: for example, in scalar field models in harmonic traps or more general field-theoretic treatments, relativistic (frequency, variance) corrections modulate the rate and timing of visibility decay, enhancing the experimental discriminating power (Yant et al., 2023, Yant et al., 26 Mar 2025).

Relevant formulas in this context include operational metrics, such as: $$\frac{IT\, m_1 m_2 T^2}{d^2} \gg \frac{16 \hbar^2}{G^2}$$ where $I$ is the particle input rate, $T$ is the experiment duration, and $d$ the separation, representing the minimal GIE "threshold" for entanglement visibility (Rostom, 9 Jan 2024).

5. Extensions: Many-Body Systems, Relativistic Effects, and Time-Crystalline Behavior

• Many-Body GIE: Time-dependent I-concurrence and generalized Meyer-Wallach measures extend GIE analysis to systems with $N$ masses/qubits, characterizing multipartite entanglement across all bipartitions. The onset and scale of entanglement are governed by the graph of nonzero gravitational phase connectivities (Ghosal et al., 2023).
• Relativistic and Quantum Field Models: Quantum field-theoretic approaches model each "mass" as a localized excitation of a scalar field in a harmonic trap; GIE arises from gravitational self-interaction terms derived in the static limit of linearized quantum gravity. Relativistic corrections accelerate fringe visibility decay, and observables such as the center-of-mass density probe the operational entanglement signature (Yant et al., 2023, Yant et al., 26 Mar 2025).
• Curved Spacetime and Cosmology: The formation of GIE extends to curved backgrounds (e.g., Schwarzschild or de Sitter spaces), where geodesic deviation and spacetime curvature modulate accumulated phases and, hence, the observable entanglement. Characteristic "spectra" of GIE as a function of, e.g., kinetic energy or cosmological redshift provide targets for astronomical GIE searches (Zhang et al., 2023, Brahma et al., 2023).
• Thermal and Dynamical Effects: Gravitationally induced entanglement in oscillator systems (used as surrogates for, e.g., LIGO mirrors) in contact with environments reveals quantum memory effects and time-crystalline prethermal phases, arising from the interaction between classical gravitational waves, quantum subsystems, and thermal catalysts. Characteristic mixing of Bose-Einstein and Maxwell-Boltzmann statistical behaviors is a haLLMark of these driven, open-system GIE scenarios (Dutta et al., 25 Mar 2025).

6. Experimental Feasibility, Constraints, and Model Discrimination

The realization of GIE in laboratory settings is challenged by:

• The extreme weakness of the gravitational interaction compared to environmental decoherence and technical noise (Marletto et al., 2017, Krisnanda et al., 2019, Howl et al., 2023).
• The requirement for large mass or long integration times, or alternatively, for amplification schemes (e.g., squeezing, weak-value measurement, postselection) to enhance entanglement witness signals (Feng et al., 2022, Cui et al., 2023, Braccini et al., 2023).
• The existence of critical distances and configuration dependence in models such as DP, which provide falsifiable targets—e.g., GIE is only expected for separations $d \lesssim 0.85\sigma$ in horizontal configurations for standard collapse parameters, and this can be extended to larger $d$ in "transversal" arrangements (Trillo et al., 4 Nov 2024).
• The necessity of suppressing non-gravitational forces and maintaining high fringe visibility for operational detection and entanglement verification (Matsumura et al., 2020, Yant et al., 2023).

GIE observation does not constitute a unique test of quantum gravity—models with classical gravity plus specific nonlocal (collapse or noise) dynamics can in some cases mimic the entanglement signature. However, detailed characterization of dependence on parameters such as separation, mass, decoherence time, and spatial configuration can rule out alternative models if experimental results disagree with their predictions (Trillo et al., 4 Nov 2024, Fragkos et al., 2022).

7. Summary Table: Selected Theoretical Models and Experimental Conditions

Model / Scenario	Mediation Type	Entanglement Reach	Model-Specific Predictions
Quantum Field Theoretic GIE (Marletto et al., 2017Yant et al., 26 Mar 2025)	Quantum field	Arbitrary (in absence of decoherence)	GIE is generic; relativistic corrections accelerate decoherence; presence implies quantization of gravity mediator.
Diósi–Penrose (DP) Collapse (Trillo et al., 4 Nov 2024)	Classical + noise	$d < d_c \approx 0.8501\sigma$ (horizontal); $d<2.21\sigma$ (transversal)	GIE possible for limited configurations; long-lived but ultimately decaying entanglement without oscillatory behavior.
Constraint (Coulomb/Newtonian) (Anastopoulos et al., 2018Fragkos et al., 2022)	Non-dynamical (constraint)	No unique mediator needed	GIE agnostic to quantum/classical nature in constraint approximation; can be nonlocal in effect.
Hybrid/Amplified Protocols (Feng et al., 2022Cui et al., 2023)	Quantum or effective	Flexible / enhanced	Weak-value/EPR steering amplification, two-phonon drive, squeezing can boost signal and tolerate weaker coupling.

• "Gravitationally-induced entanglement between two massive particles is sufficient evidence of quantum effects in gravity" (Marletto et al., 2017)
• "Comment on 'A Spin Entanglement Witness for Quantum Gravity' and on 'Gravitationally Induced Entanglement between Two Massive Particles is Sufficient Evidence of Quantum Effects in Gravity'" (Anastopoulos et al., 2018)
• "The Diósi-Penrose model of classical gravity predicts gravitationally induced entanglement" (Trillo et al., 4 Nov 2024)
• "On inference of quantization from gravitationally induced entanglement" (Fragkos et al., 2022)
• "Observable quantum entanglement due to gravity" (Krisnanda et al., 2019)
• "Gravitationally induced entanglement dynamics between two quantum walkers" (Badhani et al., 2019)
• "Gravitational Harmonium: Gravitationally Induced Entanglement in a Harmonic Trap" (Yant et al., 2023)
• "Entanglement induced by quantum gravity in an infinite square well" (Zhang et al., 13 Apr 2025)
• "Amplification of Gravitationally Induced Entanglement" (Feng et al., 2022)
• "Essential role of destructive interference in the gravitationally induced entanglement" (Rostom, 9 Jan 2024)
• "An Operational Quantum Field Theoretic Model for Gravitationally Induced Entanglement" (Yant et al., 26 Mar 2025)
• "Gravitationally induced entanglement at finite temperature: A memory-driven time-crystalline phase?" (Dutta et al., 25 Mar 2025)
• "Gravity-induced entanglement as a probe of spacetime curvature" (Brahma et al., 2023)
• "Gravity-induced entanglement between two massive microscopic particles in curved spacetime: I.The Schwarzschild background" (Zhang et al., 2023)
• "Distribution of quantum gravity induced entanglement in many-body systems" (Ghosal et al., 2023)

Gravitationally induced entanglement thus provides both a theoretical and experimental framework to probe the quantum or effective nonclassical features of gravity, but its ultimate interpretation—whether as definitive evidence of quantum gravity or as a tool to rule out specific classical (collapse, noise) models—depends intricately on the details of the experiment and model under consideration.