Experiment 10 · Geometric Entanglement Channel · distributed waveguide

Alice in the cavity.
Bob in the beam.

Alice and Bob share an initial Bell pair. Alice sits at the cavity centre; Bob is placed anywhere outside. His noise is correlated with Alice's only inside the cavity's radiation beam — and that correlation is what protects entanglement in transit. The page asks: which geometry has the most useful beam?

EXPLAIN LIKE I'M 10
· everyday-language one-paragraph version
Alice sits inside a special bowl. Bob walks around outside. The bowl shines a quantum 'spotlight' in one direction. When Bob stands in the spotlight, the two of them can share a quantum signal almost perfectly. When he steps out, the link collapses. Drag Bob around the plot and watch the live readout track him.
WHAT YOU ARE LOOKING AT
Alice sits inside the cavity. Bob is placed anywhere outside — click on the polar plot to move him. The orange lobe is the cavity's radiation pattern: where the cavity actually sends its signal. The line from Alice to Bob is the quantum channel between them.
WHY THIS MATTERS FOR QUANTUM
A quantum computer that fits on one chip is useful. A quantum *network* — qubits across a city, a continent, a satellite link — is the actual prize. The hard part isn't sending information, it's sending *entanglement*. This page asks the geometry question: which cavity shape gives the longest-range, most-protected entanglement waveguide? The Dhara channel's directional beam carries correlated noise to Bob, and correlated noise creates a decoherence-free subspace that protects |Ψ−⟩ all the way across the link.
WHAT TO WATCH
Drag Bob along the polar plot. Inside the Dhara beam (θ near 0°), the distillable-entanglement bar stays high. Move him to the opposite side (θ = 180°) and it collapses. The five-geometry race at the bottom does the same comparison for every shape at Bob's current position — the geometry ordering changes as you move him.
Initial Bell state
Cavity geometry
Bob · angle θ = 0°
W (-180°)S (-90°)E (0°)N (90°)W (180°)
Bob · distance d = 20
channel length Ldha = 40
Noise η · 50%
Radiation pattern · click anywhere to place Bob
dhara · D = 2.82 · 3 dB beamwidth = 56°
R(θ_B)
1.150
beam intensity at Bob
exp(−d/L)
0.607
distance attenuation
γ_corr
0.122
Alice-Bob noise correlation
DFS factor
0.896
0 = uncorrelated · 1 = full DFS
Live state at Bob's location
CHSH |S|2.828 / 2.828
classical bound = 2
Fidelity to target Bell1.000 / 1
Distillable entanglement (F − ½)0.500 / 0.5
Alice (qubit A)
|r| = 0.000
(x, y, z) = (0.00, 0.00, 0.00)
Bob (qubit B)
|r| = 0.000
(x, y, z) = (0.00, 0.00, 0.00)
t = 0.00 · |S| ▲ above classical bound
|S|, F, distillable entanglement over time
Five-geometry race · same Bob position, same noise, same initial state, t = 6.0 simulated
The page tests one specific question: which cavity geometry has the highest distillable entanglement at Bob's position after a fixed transit time? The winner depends on θ. When Bob sits in the Dhara channel mouth (θ ≈ 0°) and the initial state is |Ψ−⟩, the directional beam carries the in-phase noise correlation that gives the antisymmetric state a decoherence-free subspace. Move θ off the channel and the protection vanishes — the page exposes both regimes.
What the model captures
  • 1. The radiation pattern R(θ) is a closed-form heuristic per geometry (cos⁴ for the pyramid apex, Gaussian for the Dhara channel, isotropic for the sphere). It is not derived from the same FDTD as /entangle — that would require a far-field expansion the page doesn't have.
  • 2. The noise correlation between Alice and Bob is γ_corr = R(θ_B) · exp(−d/L) · η. When γ_corr ≈ √(γ_A · γ_B), the dephasing eigenmode is rank-deficient — exactly the decoherence-free subspace that protects |Ψ−⟩.
  • 3. The full Lindblad set is built by eigendecomposing the 2×2 noise-correlation matrix and using each eigenvector as the (a, b) coefficients in L = √λ (a σz_A + b σz_B). This is the standard GKSL form for spatially-correlated noise.
  • 4. The five-geometry race uses identical initial state, identical Bob position, identical noise, identical horizon. The only free variable is the cavity. Whatever the ranking is, it's a geometry verdict.
Predictions to test by moving Bob
  • A. Inside the Dhara beam at |Ψ−⟩(θ ≈ 0°, d < 40): distillable entanglement > 0.3 stays above all other geometries.
  • B. Outside the Dhara beam (θ = 180°): Dhara becomes the worst geometry — no protection, but also no leak — and the isotropic sphere wins on raw distance attenuation.
  • C. Bob far away (d > 50): every geometry collapses to its own attenuation. Directivity stops mattering when the correlation has decayed below the local-noise floor.
  • D. |Φ+⟩ as initial state: the channel-correlated dephasing no longer protects (|Φ+⟩ is in the *bright* subspace), so the geometry race collapses to a near-tie — that's the falsification check.