02 / OUR INQUIRY · 03
Quantum entanglement.
Spacetime questions.
Can quantum information reveal how geometry is organized—and what would it take for a connection to become a traversable path?
Enter the first modelEXPLORATORY RESEARCH PROGRAM · GRAVITON
A FIXED-TIME SLICE OF AdS₃
THE RESEARCH AIM
From a shared state
to a physical geometry.
Graviton seeks a quantitative account of when quantum information can describe geometric structure. The long-term motivation includes new ways to understand distance, transport, and the possibility of traversable spacetime. A useful first contribution must identify what can be measured and what competing explanation that measurement would exclude.
“Folding space and time” expresses an ambition. In a physical model it must become a specified metric, a source of stress-energy, boundary conditions, and a causal analysis. Faster-than-light communication would require a demonstrable departure from established constraints; ordinary entanglement does not supply it.
Three reproducible models establish the starting point. A fourth study tests a specific Graviton diagnostic hypothesis after matching an energy scale. It is an exploratory calculation using a published model, with no claim to new gravitational physics.
CONCEPT GUIDE
Eight ideas behind the calculations.
Open the short definitions
- Entanglement
- A state shared by subsystems that cannot be written as a probabilistic mixture of independent product states. Strong correlation alone is not sufficient to establish it.
- Reduced state
- The quantum state describing observations on one subsystem when the other is ignored. It is obtained by a partial trace.
- CHSH inequality
- A bound, |S| ≤ 2, on a particular combination of correlations in a local hidden-variable model with the usual independence assumptions. Each observer chooses between two settings.
- Negativity
- The sum of the absolute values of the negative eigenvalues of a partially transposed density matrix. For this two-qubit family it detects entanglement exactly.
- Holographic duality
- A proposed equivalence between certain gravitational theories in a bulk spacetime and quantum theories on its boundary. The AdS/CFT setting below is a specific example, with restrictive assumptions.
- Geodesic
- A curve determined by a metric that locally extremizes length. On the spatial slice used here, the relevant boundary-anchored curves are semicircles.
- Mutual information
- I(A:B) = S(A) + S(B) − S(A∪B), a measure of total correlations between two regions. Here S is von Neumann entropy, evaluated through a leading gravitational approximation.
- Autocorrelation
- A comparison of an observable with its time-evolved version. A small average can arise from cancellation among large positive and negative contributions.
01 / EXACT TWO-QUBIT MODEL
Change the correlations.
Inspect each observer’s record.
Mix an electron-like singlet with a completely mixed state. Two thresholds separate separability, entanglement, and violation of the stated CHSH inequality. [1–3]
What can Bob read locally?
The selected groups require Alice’s classical record. Alice cannot choose her random outcome to encode a message. Without that record, Bob’s local distribution remains 50:50.
THE STATE AND ITS LOCAL MARGINAL
Swipe the equation horizontally if needed.
The singlet is |ψ⁻⟩ = (|01⟩ − |10⟩)/√2; I₄ is the identity on the four-dimensional joint space. Outcomes s and t are ±1. The angle θ is between the two spin-measurement axes.
Thresholds, measurement settings, and scope
Use a₀ = z, a₁ = x, b₀ = (z + x)/√2, and b₁ = (z − x)/√2. Then S = E₀₀ + E₀₁ + E₁₀ − E₁₁ = −2√2 p. The partial-transpose eigenvalues are (1 − 3p)/4 and three copies of (1 + p)/4.
The state is entangled strictly above p = ⅓. The chosen CHSH expression exceeds 2 strictly above p = 1/√2. Equality at the first threshold is separable; equality at the second saturates the bound. Preset buttons retain the full numerical values. Failure to violate CHSH does not rule out every other Bell test or protocol.
02 / STATIC AdS₃ GEOMETRY
Let entropy choose
the geodesic pairing.
Take two equal boundary intervals of length ℓ, separated by a gap d. In this gravitational approximation, the shorter allowed geodesic pairing determines their joint entropy. [4,5]
A METRIC, THEN AN ENTROPY
Swipe the equation horizontally if needed.
L is the AdS curvature radius; ε is a short-distance boundary cutoff. The dimensionless c is the boundary theory’s central charge, not the speed of light. Natural logarithms give entropy in nats. Units use ℏ = 1.
Derivation and limits of the picture
The Ryu–Takayanagi prescription gives S(A∪B) = min(Sdis, Scon), subject to the homology condition. Substituting S(A) = S(B) = (c/3) ln(ℓ/ε) gives the expression plotted above. The two candidates exchange order when r(2 + r) = 1, so r = √2 − 1. The cross-ratio x = 1/(1 + r)² then equals ½.
This calculation assumes the vacuum on an infinite boundary line, a static semiclassical AdS₃ dual, and ε ≪ min(ℓ,d). Here ℓ sets the unit, ε/ℓ = 10⁻⁴, and r ≥ 0.05. The result is independent of that cutoff at this order. The plotted coordinate plane is conformal: the metric, rather than Euclidean arc length on the screen, determines the physical lengths.
A connected entanglement wedge does not by itself supply a traversable wormhole. The Werner-state parameter above has no assigned mapping to this interval geometry.
03 / EXACT FOUR-QUBIT DYNAMICS
What does an average hide?
Follow eight operator autocorrelations in a small, closed quantum system. Compare individual curves, their signed average, and the average of their magnitudes before assigning a physical interpretation.
Loading the independently checked calculation…
The perturbation changes both commutation structure and energy scales; neither case is rescaled. This comparison does not isolate the effect of noncommutation. The next study matches centered RMS energy and states a quantitative test; further controls remain necessary.
THE OBSERVABLE
Swipe the equation horizontally if needed.
Eight Majorana operators act on four qubits. We use ψᵢ = γᵢ/√2, {ψᵢ,ψⱼ} = δᵢⱼ I, and Gᵢ(0) = 1. These operators are represented by Pauli strings; the demonstration does not create physical Majorana particles.
The Hamiltonian, conventions, and exact calculation
H₀ is the sum of the following five quartic terms. The coefficients and perturbation come from the published discussion of the quantum-processor model. All five baseline terms commute. [10,11]
| Coefficient | Ordered operator product |
|---|---|
| −0.36 | ψ₁ ψ₂ ψ₄ ψ₅ |
| +0.19 | ψ₁ ψ₃ ψ₄ ψ₇ |
| −0.71 | ψ₁ ψ₃ ψ₅ ψ₆ |
| +0.22 | ψ₂ ψ₃ ψ₄ ψ₆ |
| +0.49 | ψ₂ ψ₃ ψ₅ ψ₇ |
The Jordan–Wigner representation is γ₂ⱼ₋₁ = Z⊗(j−1) ⊗ X ⊗ I⊗(4−j), with Y replacing X for γ₂ⱼ, j = 1,…,4. The 16 × 16 Hamiltonian is diagonalized exactly. Evolution follows from its eigenvalues and eigenvectors; representative values are checked independently using direct matrix evolution and a Taylor expansion of the exponential.
This β = 0, one-sided autocorrelation calculation is a diagnostic example. It is not a reproduction of the processor’s finite-temperature teleportation experiment. Decay of an average here establishes neither thermalization nor a gravitational dual.
Why include this diagnostic?
The 2022 quantum-processor experiment investigated a model intended to exhibit aspects of traversable-wormhole dynamics. Later criticism and the authors’ reply disagree about which features are distinctive and how broadly they survive changes of model. A 2025 correction clarified that particular plotted correlators were averages over fermions. [9–12]
Our calculation makes that averaging explicit at β = 0. Its narrower lesson is methodological: preserve the individual observables, inspect cancellations, and test alternative models before treating a collective curve as evidence for a gravitational mechanism.
The verification summary will appear when the calculation loads.
04 / GRAVITON COMPUTATIONAL STUDY
Fix the energy scale.
Make a quantitative prediction.
Does the perturbation reduce the active operators’ mean absolute autocorrelation when both Hamiltonians have the same centered RMS energy? This is a narrower question than whether the system has a gravitational interpretation.
Loading the matched-energy calculation…
The fixed-window result
| λ | D(λ) | D(λ) − D(0) |
|---|---|---|
| 0.0 | 0.188286 | +0.000000 |
| 0.1 | 0.171843 | -0.016443 |
| 0.2 | 0.152175 | -0.036111 |
| 0.3 | 0.153447 | -0.034838 |
All three tested changes are negative. The effect is not monotonic: λ = 0.3 has a higher D than λ = 0.2. This is a deterministic calculation, not experimental evidence.
THE MATCHING RULE
H̄ = Tr(H)/16 and σ(H)² = Tr[(H − H̄I)²]/16. Matching this energy scale does not match the complete spectrum or every conserved quantity. Time remains in inverse baseline coefficient-energy units, with ℏ = 1.
THE PREDICTION
Swipe the equation horizontally if needed.
tₙ = 12 + 0.1n, N = 121, β = 0, and λ ∈ {0.1, 0.2, 0.3}. The prediction is the final inequality; the preceding equality defines the measured quantity. A nonnegative contrast for any tested λ would defeat the stated “all three” prediction.
What passed, what was fixed, and what remains open
The directional prediction holds on the specified grid and retains its sign at time spacing 0.05. The largest change in a contrast after refinement is 0.000232, rounded upward. This measures sensitivity to the time grid; it is not a statistical confidence interval or a rigorous continuum error bound.
The choices were fixed before this matched calculation. Earlier unscaled λ = 0 and 0.3 curves had already been inspected. This is an exploratory computational study, not a preregistered or blind validation. Neither the Hamiltonian nor the autocorrelation formula is claimed as a new discovery.
RMS matching removes one overall energy-scale difference. The perturbation still changes detailed couplings, spectral shape, and conserved quantities. The result does not isolate noncommutation as a cause, demonstrate thermalization, or establish gravity.
A next validation should fix additional windows and alternative quartic perturbations before computing them, then compare model classes matched on more than a single energy statistic. If the sign reverses, retain and explain the failure rather than changing the endpoint.
THE LONGER HORIZON
What would make
a shortcut physical?
ER=EPR proposes relationships between entanglement and Einstein–Rosen bridges in specific settings; extending that idea to arbitrary entangled systems is conjectural. Gao, Jafferis, and Wall constructed a traversable example using coupled AdS boundaries and negative averaged null energy. Their construction respects causality. [6,7]
A proposed transport mechanism must identify the spacetime, its matter and energy requirements, the signal trajectory, stability, and causal structure. A quantum simulation can test a model’s dynamics without producing a navigable region of spacetime in the laboratory.
Our ambition remains open to new physics. A claim that established communication limits can be exceeded must state the changed physical assumption and predict a measurable departure. The no-signalling result above is a constraint for that work to confront.
GRAVITON / A PROPOSED RESEARCH ROUTE
Which signature survives
the competing explanation?
The first research target is to identify a combination of observables that distinguishes a controlled gravitational interpretation from ordinary finite-system quantum transport. Successful state transfer alone is insufficient: established quantum protocols also transfer information. [8]
- Choose a model and a claim.
Specify the Hamiltonian, state, coupling protocol, and claimed gravitational correspondence. Derive the observable from that correspondence before fitting a convenient small model.
- Build matched alternatives.
Compare commuting and noncommuting systems, scrambled and unscrambled controls, and ordinary transport models. Match Hilbert-space size, relevant symmetries, energy scale, and measurement resources.
- Keep predictions out of the fit.
Reserve operators, states, coupling signs, and time windows for validation. Report individual correlators alongside averages and quantified uncertainties.
- State the result that would change our view.
If matched nongravitational controls reproduce the proposed signature, it does not uniquely support the gravitational interpretation. If a prediction fails outside its fitted examples, revise the correspondence.
- Treat traversal as a separate test.
Only after a specified spacetime model is justified should transport claims be evaluated through causal curves, energy conditions, backreaction, and stability.
This is a proposed program. The page reproduces established calculations and an explicit diagnostic, not a completed test of the research hypothesis.
METHODS / PRIMARY SOURCES
Follow the argument
back to its assumptions.
The first two models evaluate the displayed formulas in the browser. Independent density-matrix calculations check the Werner-state probabilities, spectra, and CHSH expression. The geometry calculation is checked against both entropy pairings and a common rescaling of lengths. The third model uses a downloadable grid with 241 times for each of two Hamiltonians.
The numerical generator requires Python and NumPy. A fourth calculation compares four Hamiltonians at matched centered RMS energy under a fixed protocol. Sources and interpretations were reviewed on 18 September 2026. Earlier public preprints are linked where they provide accessible detail for the later discussion.
- Werner (1989) · Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model
- Horodecki, Horodecki & Horodecki (1996) · Separability of mixed states
- Horodecki, Horodecki & Horodecki (1995) · Violating Bell inequality by mixed spin-½ states
- Ryu & Takayanagi (2006) · Holographic derivation of entanglement entropy
- Headrick (2010) · Entanglement Rényi entropies in holographic theories
- Maldacena & Susskind (2013) · Cool horizons for entangled black holes
- Gao, Jafferis & Wall (2017) · Traversable wormholes via a double trace deformation
- Schuster et al. (2022) · Many-body quantum teleportation via operator spreading in the traversable wormhole protocol
- Jafferis et al. (2022) · Traversable wormhole dynamics on a quantum processor
- Kobrin, Schuster & Yao (2025) · comment on the quantum-processor interpretation · earlier open preprint
- Jafferis et al. (2025) · reply · earlier open response
- Jafferis et al. (2025) · author correction concerning averaged correlators
Updated
THIRD INQUIRY / OPEN RESEARCH
Author: Graviton. The fixed-protocol calculation is reproducible. The broader information–geometry connection remains a research question.
Return to the first model