Detecting Entanglement Gravity: Distinguishing Energy-Sourced and Information-Sourced Gravity using High-Q Superconducting Cavities

Detecting Entanglement Gravity: Distinguishing Energy-Sourced and Information-Sourced Gravity using High-Q Superconducting Cavities Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract In the standard framework of General Relativity, the gravitational eld is completely determined by the energy-momentum tensor Tµν ; in the weak-eld limit, as long as the energy density and pressure distribution remain unchanged, gravitational eects are independent of the entanglement structure of quantum states. In contrast, a series of works based on entropy, entanglement, and holographic principles suggest that Einstein's equations can be derived from local entropy balance or vacuum entanglement structure, hinting that spacetime geometry may be fundamentally determined by quantum information. Building on the previous "conservation of optical pathconservation of information volume" framework, this paper introduces a phenomenological "entanglement gravity" term: local von Neumann entropy density or entanglement entropy density ρ ent aects the Shapiro-type phase delay of light by modifying the optical refractive index n(x) . To experimentally distinguish "energy gravity" from "information gravity," we propose a class of table-top experiments: using high-Q superconducting microwave cavities, while keeping the expected electromagnetic eld energy ⟨H⟩ inside the cavity constant, periodically switch between low-entanglement coherent states and highly entangled squeezed states or multi-mode cluster states, and measure the additional optical path length and phase delay via a high-nesse FabryPerot probe beam passing near the cavity. This paper constructs an optical metric model containing an information-gravity coupling constant λ ent , and proves that under weak-eld and paraxial approximations, the phase dierence between modulated states satises |∆Φ ≃λ ent G∆s ent , where ∆s ent is the change in entanglement entropy surface density inside the cavity, and G is a geometric factor determined by cavity geometry and probe beam path. Further combining contemporary superconducting cavity Q-factors and laser interferometry techniques, for typical parameters ( Q∼109  10 , ∆S ent ∼10  102 bit, nesse F ∼ 105 ), we estimate the expected signal magnitude and noise budget. Results show that under reasonable geometric congurations and lock-in schemes, if λ ent is not lower than a critical value, the corresponding phase modulation amplitude can reach ∆Φ ∼10−9 rad scale, approaching the phase sensitivity limit of current laser interferometry and squeezed-light readout chains. Conversely, if no dierential phase signal modulated at the entanglement frequency is observed, direct experimental upper bounds can be imposed on λ ent , thereby providing the rst quantitative constraint on "whether entanglement itself serves as a gravitational source" experimentally. Keywords: Entanglement Gravity; Conservation of Optical Path; Shapiro Delay; Superconducting Microwave Cavities; Quantum Optics; Table-top Gravity Experiments 1 1 Introduction & Historical Context General Relativity describes gravity as four-dimensional spacetime geometry with Lorentzian signature, where the metric gµν is determined by Einstein's equations Gµν =8πG c4Tµν. The right-hand side of this equation is uniquely given by the energy-momentum tensor Tµν , and in this framework, the gravitational source is "energy-momentum" rather than independent information quantities or entanglement structures. In the weak-eld limit, the metric can be written as gµν =ηµν +hµν , where hµν is directly related to the Newtonian potential Φ . Classical Shapiro time delay experiments precisely verify the light speed modication and optical path delay produced by this energy-sourced gravity. On the other hand, research on "whether gravity and spacetime originate from entropy and entanglement" has developed continuously over the past three decades. Jacobson derived Einstein's equations from the thermodynamic equilibrium relation δQ =T δS at local Rindler horizons, interpreting them as an "equation of state." RyuTakayanagi and subsequent works show that in AdS/CFT holographic correspondence, the entanglement entropy of conformal eld theory subregions equals the area of corresponding minimal surfaces in anti-de Sitter space, revealing a precise quantitative connection between quantum entanglement and geometric area. Van Raamsdonk further proposed that macroscopic connected spacetime can be viewed as the "glue" of underlying quantum degrees of freedom's entanglement structure, with reduced entanglement leading to geometric "tearing" or "breaking" of spacetime regions. Verlinde's entropic gravity scheme more directly views gravity as an entropic force related to information, entropy, and holographic screens. These works jointly point to a picture: spacetime geometry and gravitational eld are, in some sense, emergent descriptions of quantum information and entanglement structures, and Einstein's equations can be understood as macroscopic approximations of some "entanglement equilibrium condition." However, most establish indirect connections between "geometry entropy/entanglement" at the theoretical level; experiments directly testing "whether entanglement serves as an independent gravitational source" remain absent. Recently, Bose et al. and MarlettoVedral et al. proposed the famous BMV-type table-top experimental scheme: using gravitational interaction to produce observable spontaneous entanglement between mass superposition states, thereby proving "if two quantum systems become entangled through a eld interaction, then that eld must be a quantum entity." This approach uses "entanglement as witness" to infer the quantum nature of the gravitational eld, but does not claim that entanglement itself contributes additionally to gravitational strength. Subsequent analyses further discuss the feasibility and limitations of this scheme under locality, quantum eld theory modeling, and classical eld reformulations. Dierently, this paper considers a more radical possibility: beyond the energy-momentum tensor, there exists an "information gravity source" related to local entanglement structure or information processing density. In the previous "conservation of information volumeoptical metric" framework, we proposed that in the weak-eld limit, an eective refractive index eld n(x) can encode metric perturbations, where n(x) is determined by local information volume or information processing rate. If this framework is correct, then under conditions of constant energy density, purely changing local entanglement structure should in principle cause observable optical path changes. Experimentally, to decouple "energy" and "entanglement" as two types of gravitational sources under controlled conditions requires satisfying the following: 1. High-precision locking of energy expectation value ⟨H⟩ can be achieved; 2. Entanglement entropy or von Neumann entropy can be modulated signicantly while keeping ⟨H⟩ essentially 2 constant; 3. Extremely weak refractive index or optical path changes can be measured near the target volume. High-Q superconducting microwave cavities and modern quantum optics techniques precisely provide such a platform. In recent years, work based on superconducting RF cavities and threedimensional cavity QED has achieved microwave cavity quality factors Q > 1010 at temperatures around 10 mK, maintaining quantum state coherence over millisecond or longer timescales, providing stable carriers for highly entangled optical elds such as multi-mode squeezed states and cluster states. Meanwhile, in interferometry, relying on high-nesse optical cavities and squeezed light injection, phase sensitivity approaching the quantum limit has been achieved in gravitational wave detectors like LIGO. In this technical context, this paper proposes and analyzes the following question: Near a high-Q superconducting microwave cavity with highly stable total energy, if switching periodically between low-entanglement and high-entanglement states, will a probe beam passing through this region exhibit additional Shapiro-type phase delay, beyond standard GR predictions, modulated at the entanglement frequency? Observing such a dierential signal would support the hypothesis that "information/entanglement serves as an independent gravitational source"; failure to observe it would impose experimental upper limits on relevant coupling constants. The structure below: Section 2 constructs the optical metric model of information gravity and gives basic assumptions; Section 3 presents main results, showing in theorem form that under xed energy conditions, entanglement entropy modulation necessarily leads to optical path modulation; Section 4 combines superconducting microwave cavities with high-nesse optical cavities to provide numerical estimates and parameter constraints; Section 5 discusses experimental engineering implementation and noise budget; Section 6 discusses risks and relationships with existing work; nally concluding with prospects, and detailed derivations of optical metrics, Shapiro phase, and error budget are provided in appendices. 2 Model & Assumptions 2.1 Optical Metric Model of Information Gravity Under weak-eld static approximation, the standard GR line element can be written as ds2=−1 + 2Φ(x) c2c2dt2+1−2Ψ(x) c2dx2, where Φ≃Ψ is the Newtonian potential, satisfying ∇2Φ(x)=4πGρE(x), with ρE being energy density. For paraxial propagating light rays, an eective refractive index can be introduced: n GR (x)≃1−Φ(x) c2, and Shapiro delay comes from increased optical path due to n GR >1 . In the "conservation of optical pathconservation of information volume" framework, we introduce an additional "information gravity potential" Φ ent , and assume the total eective potential Φ e = ΦE+ Φ ent , where ΦE is the conventional potential determined by Tµν , and Φ ent is directly related to local entanglement entropy density. Thus the total refractive index n(x)≃1−Φ e (x) c2= 1 −ΦE(x) c2−Φ ent (x) c2. 3 The phenomenological assumption adopted in this paper is: there exists a coupling compatible with the BekensteinHawking area-entropy relation, linking entanglement entropy surface density s ent (x) to Φ ent : Φ ent (x) = −λ ent c2ℓ2 P L∗ s ent (x), where ℓ P is the Planck length, L∗ is the macroscopic coarse-graining scale (taken as cavity linear size in this experiment), λ ent is a dimensionless coupling constant, and s ent has units of bit/m 2 . Thus, the refractive index correction contributed by information gravity is δn ent (x) = Φ ent (x) c2=−λ ent ℓ2 P L∗ s ent (x). This form is inspired by black hole entropy S=A/(4ℓ2 P ) where "each O(ℓ2 P ) area element corresponds to O(1) bit," using ℓ2 P s ent as dimensionless entanglement "overdensity," while L∗ brings surface density to the appropriate scale for potential. Its physical meaning can be understood as: in a given macroscopic volume, if boundary entanglement entropy surface density exceeds a baseline value, light path must be slowed (equivalent to local time dilation) to "free up computational resources," thereby achieving conservation of information volume. 2.2 Cavity Model and Entanglement Entropy Density Consider a superconducting microwave cavity with length L cav and cross-sectional area A cav , supporting a set of discrete modes {ˆak} , with total Hamiltonian ˆ H=X k ℏωkˆa† kˆak+1 2. In this paper, we only focus on two macroscopically tunable cavity quantities: 1. Energy expectation value E0=⟨ˆ H⟩; 2. Entanglement entropy surface density under a natural partition (e.g., left-right plates, dierent polarization or frequency modes) s ent =S ent A cav , S ent =−tr ˆρAlog ˆρA, where ˆρA is the reduced density operator obtained by tracing over one side's degrees of freedom. Experimentally, we switch between two macroscopically distinguishable states:  State A (baseline state): multi-mode coherent state |A⟩=O k|αk⟩, whose von Neumann entropy and entanglement entropy are approximately zero, S(A) ent ≃0 ;  State B (test state): multi-mode entangled state, such as multiple pairs of two-mode squeezed vacuum states or cluster states |B⟩= M O j=1 | TMSV (rj)⟩, by choosing squeezing parameters rj and mode number M , adjust total photon number ⟨ˆ N⟩=Pk⟨ˆa† kˆak⟩ to match state A, thereby ensuring ⟨A|ˆ H|A⟩=⟨B|ˆ H|B⟩=E0. 4 In these two states, energy density ρE=E0/(A cav L cav ) remains constant, while entanglement entropy surface density jumps from s(A) ent ≈0 to s(B) ent ≫0 . Dene ∆s ent =s(B) ent −s(A) ent ,∆S ent =S(B) ent −S(A) ent . Substituting the above into the information gravity refractive index correction gives the refractive index dierence between states A and B: ∆n ent =nB−nA=−λ ent ℓ2 P L∗ ∆s ent =−λ ent ℓ2 P L∗A cav ∆S ent . In the experimental scenario below, we take L∗ as L cav , and assume cavity refractive index is transversely uniform, so ∆n ent can be viewed as the average correction through a narrow optical path passing through the cavity center. 2.3 Probe Beam Path and Shapiro-Type Phase Delay Consider a probe beam with wavelength λp passing through the cavity eective region along an approximately straight optical path, forming part of a high-nesse FabryPerot optical cavity. Let the geometric length of single-pass through the cavity be L pass (for complete traversal through cavity, take L pass ≃L cav ; for grazing edge passage, take smaller value). Under weak refraction approximation, single-pass additional phase delay δϕ single =kpZ path ∆n ent (l) dl≃kp∆n ent L pass , where kp= 2π/λp . If the probe beam makes N round trips in the F-P cavity, the total additional phase delay is ∆Φ = N δϕ single ≃2πNL pass λp−λ ent ℓ2 P L cav A cav ∆S ent . This gives the basic form of Shapiro-type phase modulation induced by information gravity under xed energy conditions. 3 Main Results (Theorems and Alignments) To facilitate direct correspondence with experimental design, this section presents three core results based on the preceding model, organized in theorem form. 3.1 Theorem 1 (Phase Modulation Induced by Entanglement Entropy Modulation under Fixed Energy) Assumptions: 1. When the electromagnetic eld inside the cavity switches between state A and state B, it satises ⟨A|ˆ H|A⟩=⟨B|ˆ H|B⟩=E0, i.e., the same energy expectation value; 2. Cavity geometry is xed, L cav , A cav unchanged; 3. Information gravity is given by refractive index correction ∆n ent =−λ ent ℓ2 P L cav A cav ∆S ent ; 5 4. Probe beam makes N≃2F/π round trips in the F-P cavity, where F is the nesse, with refractive index correction existing only in the region of length L pass along the optical path. Then: The total phase dierence induced by switching between states A and B is ∆Φ = −λ ent G∆S ent , where the geometric factor is G=2πNL pass λp ℓ2 P L cav A cav . In other words, under xed energy conditions, all phase modulation related to information gravity is linearly proportional to entanglement entropy change ∆S ent , independent of energy itself. 3.2 Theorem 2 (Upper Bound of GR Contribution under Energy Mismatch) Assumptions: 1. In actual experiments, energy locking has residual mismatch δE =EB−EA , satisfying |δE| ≪ E0 ; 2. GR's energy-sourced gravity contribution modies refractive index via Newtonian potential ∆n GR ≃ −∆ΦE c2≃ −G δM c2R, where δM =δE/c2 , R is typical distance from probe beam path to cavity center; 3. Other conditions same as Theorem 1. Then: The upper bound of additional phase dierence introduced by GR is |∆Φ GR |≲2πNL pass λp G|δE| c4R. For typical parameters E0∼10−11 J, |δE|/E0≲10−9 , R∼0.1 m, we obtain |∆Φ GR |≲10−15 rad , far below the information gravity target signal ∆Φ ∼10−9 rad scale. Therefore, as long as energy locking reaches 10−9 relative precision, GR contribution can be neglected in this experimental scheme. 3.3 Theorem 3 (Experimental Constraint on Information Gravity Coupling Constant) Let the noise spectral density for phase modulation of the measurement system during integration time T be S1/2 Φ . Then analytically, under the condition of not observing a signicant spectral line at modulation frequency f mod , the 1σ upper bound on λ ent is |λ ent |≲S1/2 Φ G|∆S ent |T−1/2. For typical parameters ∆S ent ∼10  102,F ∼ 105, L pass ∼L cav ∼0.1 m , A cav ∼10−4 m 2, and laser interferometry chain phase noise S1/2 Φ∼10−10 rad /√ Hz, with eective integration time T∼103 s, |λ ent | can be constrained within a nite range. If a phase signal modulated at ∆S ent frequency is observed, the eective value of λ ent can be tted from data and compared with other experimental and astrophysical constraints. 6 4 Proofs This section derives the above three theorems. Since information gravity itself is a new hypothesis, the starting point of derivation is the standard relation between optical metric and Shapiro delay, and the phenomenological coupling in Section 2. 4.1 Derivation of Theorem 1: Linear Relation between Optical Path and Entanglement Entropy From the denition in Section 2.3, the single-pass phase of probe light is ϕ=kpZ path n(l) dl≃kpL geom +kpZ path δn(l) dl, where L geom is geometric optical path, δn is small deviation from gravitational equivalent refractive index correction. When comparing states A and B, geometric optical path and GR energy source component are the same (completely identical under ideal xed energy assumption), with dierence only from information gravity correction ∆n ent . Thus single-pass phase dierence δϕ single =kpZ path ∆n ent (l) dl. Assuming refractive index in cavity region can be viewed as constant ∆n ent , path length is L pass , then δϕ single ≃kp∆n ent L pass =2π λp−λ ent ℓ2 P L cav A cav ∆S ent L pass . Considering N round trips in F-P cavity, eective phase amplication is ∆Φ = N δϕ single =−λ ent 2πNL pass λp ℓ2 P L cav A cav ∆S ent =−λ ent G∆S ent , consistent with the form stated in Theorem 1. 4.2 Derivation of Theorem 2: Upper Bound of GR Phase from Energy Mismatch For a local energy correction δE , approximately viewing it as a mass perturbation δM =δE/c2 , its Newtonian potential correction at distance R is ∆ΦE≃ −GδM R=−GδE c2R. From refractive index-potential relation n≃1−Φ/c2 , we get ∆n GR =−∆ΦE c2≃GδE c4R. Thus upper bound of single-pass phase dierence is δϕ GR,single ≃kp∆n GR L pass ≲2π λp G|δE| c4RL pass , Total phase dierence |∆Φ GR |≲N|δϕ GR,single |≲2πNL pass λp G|δE| c4R. Substituting typical numerical values gives the order of magnitude stated in Theorem 2. Since |δE| can be locked to extremely low levels through active feedback and SQUID readout, this term's contribution can be safely viewed as secondary systematic error. 7 4.3 Derivation of Theorem 3: Statistical Constraint on Coupling Constant Let the single-sided power spectral density for phase in the detection chain be SΦ(f) , which can be viewed as constant SΦ near modulation frequency f mod . Within integration time T , statistical error of narrow-band spectral line tting follows σΦ≃S1/2 ΦT−1/2. If no signicant spectral line is observed, the 1σ upper bound of actual phase modulation amplitude |∆Φ| can be taken as σΦ . From the linear relation in Theorem 1 |∆Φ|=|λ ent ||G∆S ent |, thus |λ ent |≲σΦ |G∆S ent |≃S1/2 Φ G|∆S ent |T−1/2. This gives the statistical upper bound form of Theorem 3. 5 Model Apply This section, under the above theoretical framework, combines realistic feasible experimental parameters to estimate signal magnitude and discuss potential constraint strength on λ ent . 5.1 Typical Geometry and Cavity Parameters Consider the following representative design:  Superconducting microwave cavity: length L cav = 0.1 m, cross-sectional area A cav = π(1 cm )2≃3.1×10−4 m 2 ;  Cavity quality factor: Q∼109  10 , frequency ωc/2π∼10 GHz, corresponding single-photon energy hν ∼6.6×10−24 J;  Average photon number ⟨ˆ N⟩ ∼ 1012 , then total energy E0∼6.6×10−12 J, energy density ρE∼2×10−9 J / m 3 , conventional GR gravitational eect extremely weak;  Probe light: wavelength λp= 1064 nm, compatible with LIGO-class technology;  Probe beam path: passing through cavity center, single-pass eective length L pass ≃0.1 m;  F-P cavity nesse: F= 105 , then number of round trips N≃2F π≃6.4×104, eective optical path L e =NL pass ≃6.4×103 m. Under such geometry, the geometric factor G=2πNL pass λp ℓ2 P L cav A cav . Substituting numerical values ℓ2 P ≃2.6×10−70 m 2 , λp≃10−6 m, L cav =L pass = 0.1 m, A cav ≃3×10−4 m 2 , 2πNL pass λp∼2π×6.4×104×0.1 10−6∼4×1010, 8 ℓ2 P L cav A cav ∼2.6×10−70 0.1×3×10−4∼10−66, thus G ∼ 4×1010 ×10−66 ∼4×10−56 rad/bit . At this scale, even with ∆S ent ∼102 , if λ ent ∼1 , the resulting phase modulation ∆Φ ∼λ ent G∆S ent ∼4×10−54 rad , far below any feasible experimental sensitivity. Thus if the coupling coecient directly adopts ℓ2 P scale, experiments cannot measure it at laboratory scales. This reects the huge hierarchical gap between Planck scale and laboratory scale, consistent with the general diculty of observing traditional quantum gravity eects in table-top experiments. However, in the emergent picture of information gravity, λ ent need not necessarily be of the same order as Planck scale. If underlying QCA or other discrete ontology exhibits amplied eective information gravity coupling macroscopically (e.g., due to collective eects of manybody entanglement), then λ ent could be much larger than 1, making ∆Φ fall within measurable range. To quantify this, we can write λ ent as λ ent = 10Γ, then ∆Φ ∼4×10−54 ×10Γ×∆S ent 102 rad . If expecting ∆Φ ∼10−9 rad, then need 10−9∼4×10−54 ×10Γ⇒Γ∼45. In other words, this experiment under the above parameters can detect or exclude whether λ ent is around 1045 magnitude. This seems huge, but since λ ent is a completely new eective parameter, currently lacking any experimental evidence constraint, its theoretical natural value is unclear. 5.2 Comparison with Phase Sensitivity Taking LIGO and other interferometric devices as reference, under conditions of introducing frequency-dependent squeezed light, single-frequency phase noise spectral density can reach S1/2 Φ∼10−10 rad /√ Hz or even lower. Under integration time T∼103  104 s, statistical error σΦ∼S1/2 ΦT−1/2∼10−11  12 rad . Therefore, once ∆Φ ≳10−10 rad, it can be detected with >5σ signicance. Conversely, if no spectral line is observed below this threshold, Theorem 3 gives |λ ent |≲1044  45, corresponding to direct experimental upper bound on information gravity coupling. 6 Engineering Proposals This section discusses engineering elements and optional routes needed to implement the above scheme. 9  Typical cavity scale R∼0.1 m. Newtonian potential |ΦE| ∼ GM EM R∼6.7×10−11 ×7.3×10−29 0.1∼5×10−39 J/kg . Relative to c2∼9×1016 m 2/ s 2 , dimensionless potential  ΦE c2∼5×10−56, corresponding refractive index correction ∆n GR ∼10−56. Even accumulating L e ∼104 m optical path in F-P cavity, total phase ∆Φ GR ∼2πL e λp ∆n GR ≲10−42 rad , completely undetectable. Therefore, any phase modulation observable in this experiment, synchronized with state switching frequency, cannot be explained by conventional GR energy gravity, and must originate from non-standard mechanisms (or experimental systematic errors). D Example Construction of Entanglement Entropy To give concrete construction example of ∆S ent ∼10  102 , consider realizing M pairs of two-mode squeezed vacuum states in cavity | TMSV (r)⟩= exp hr(ˆaˆ b−ˆa†ˆ b†)i|0,0⟩, each pair's reduced entropy is S pair (r) = cosh2rlog(cosh2r)−sinh2rlog(sinh2r), average photon number ⟨ˆna+ ˆnb⟩= 2 sinh2r. Taking r∼1 , sinh2r∼1.4 , each pair contributes photon number ∼3 , corresponding entanglement entropy S pair ∼2 bit scale (in appropriate basis). By taking M∼5  50 mode pairs, can realize ∆S ent ∼10  102 range, while total photon number ∼3M matches with same-scale coherent state, thereby signicantly changing entanglement structure while energy locking. More complex cluster states or graph states can introduce higher many-body entanglement degree under same total photon number, but since entanglement entropy denition depends on chosen degrees of freedom partition, its specic contribution to information gravity needs to correspond with "natural partition" of underlying QCA or eld theory ontology. This paper only uses two-mode squeezed state as example, showing that under existing circuit QED technology, ∆S ent ∼10  102 range is fully achievable. 16