Experiment 09 · Entanglement Lab · Two-qubit Lindblad

Bell's inequality,
on a clock.

Real density-matrix evolution under the GKSL master equation: dρ/dt = −i[H, ρ] + Σ(LkρLk† − ½{Lk†Lk, ρ}). Two paradigms run side-by-side under matched noise η. The CHSH violation, the linear-entropy entanglement and the Bell-state fidelity update in real time.

EXPLAIN LIKE I'M 10
· everyday-language one-paragraph version
Two coins that always agree, no matter how far apart they are. That spooky link is called entanglement, and it is what makes quantum computers powerful. It also breaks easily. We put two coin-pairs side by side: one pair shares a small room, one pair is locked in separate cages. Which pair stays linked longer? The answer is the central claim of this whole instrument.
WHAT YOU ARE LOOKING AT
Two qubits, side by side. Same noise level. Left panel: they share a cavity and talk to each other (resonance). Right panel: each one is shielded alone (isolation). The CHSH score is the Bell-test scoreboard — if it's above 2, the qubits are entangled in a way no classical pair of objects can be.
WHY THIS MATTERS FOR QUANTUM
Entanglement is what makes a quantum computer worth building. It is also fragile: noise destroys it in microseconds. The cleanest way to test the central hypothesis of this site is to put two entangled qubits in each paradigm and watch which one keeps its CHSH score above 2 the longest. If the resonance side stays above the line while the isolation side falls below, the resonance paradigm is the one to build quantum hardware on.
WHAT TO WATCH
Start with the |Ψ−⟩ Bell state and the Dhara cavity selected. Watch the orange line (resonance) stay flat near 2.8 while the blue line (isolation) decays through the dashed classical bound at 2. That gap is the page's headline claim — and it is computed live, not animated.
Lindblad rates are the heuristic geometry table (Sphere 1:1, Dhara 2.5:0.15, …). Flip the toggle to compute them from the actual wave solver — first run takes ~3 s per geometry, then cached.
Initial Bell state
Cavity geometry
Noise η · 50%
Coupling g · 0.40
Detuning Δω · 0.00
Resonance · coupled cavity
|S| = 2.828 · E = 1.000
|ρ_{ij}| heatmap
CHSH |S|2.828 / 2.828
0classical=22√2
Isolation · independent shielding
|S| = 2.828 · E = 1.000
|ρ_{ij}| heatmap
CHSH |S|2.828 / 2.828
0classical=22√2
Bell-state fidelity · how close ρ is to each maximally entangled state
|Φ+⟩
res0.000
iso0.000
|Φ−⟩
res0.000
iso0.000
|Ψ+⟩
res0.000
iso0.000
|Ψ−⟩ · DFS protected by Dhara
res1.000
iso1.000
Time series · |S| over last 600 ticks
Resonance / Isolation S-ratio at t
1.00×
S_res = 2.828 · S_iso = 2.828
Classical bound crossing
Both still in quantum regime
S > 2 means entanglement is still being detected
DFS protection check
|Ψ−⟩ in DFS ✓
Antisymmetric Bell state is decoherence-free under collective dephasing in the Dhara channel.
What the page is computing
  • 1. The math is the Gorini-Kossakowski-Sudarshan-Lindblad master equation, integrated by RK4 at dt = 0.012, with rotated coupling g and detuning Δω as user inputs.
  • 2. The Lindblad operators encode the cavity-geometry hypothesis. For the Dhara, the symmetric-mode rate (channel) is ~17× the antisymmetric-mode rate (closed cavity). For the sphere both rates are equal. This is the H4 prediction translated into open-quantum-systems language.
  • 3. CHSH S is the Horodecki maximum over all measurement angles. Classical bound is 2; quantum maximum is 2√2 ≈ 2.828. A line above 2 means Bell's inequality is violated.
  • 4. Entanglement is linear entropy of qubit-A's reduced state, scaled to [0, 1]. For pure 2-qubit states this equals the concurrence squared; both reach 1 for Bell states.
  • 5. The numbers depend on the rate ratios I picked for each geometry. They are physically motivated but not derived from a Maxwell solve. The cross-geometry ordering (Dhara > pyramid ≈ dome > sphere > cube for |Ψ−⟩ protection) is the defensible result; the absolute decay times scale with rate magnitudes.
  • 6. Toggle Wave-equation rates · ON at the top to replace these heuristics with rates derived from the actual FDTD field. The bridge confirms the qualitative ordering — Dhara is the only cavity where the antisymmetric mode out-lives the symmetric one — but the wave-derived ratio (~2.6×) is markedly lower than the 17× heuristic suggested. Run on real wave physics, the protection is real but modest. That is the honest finding.
What this page demonstrates · the four hypotheses
  • H1. Run resonance vs isolation at matched noise — the resonance side keeps S above 2 for orders of magnitude longer.
  • H2. Start in |Ψ−⟩ and turn up η — the noise is collective (in-phase) in the resonance paradigm, and the antisymmetric Bell state lives in the decoherence-free subspace. E stays at 1.
  • H3. Slide detuning Δω away from zero — the achievable steady-state entanglement collapses. The frequency gap is the entanglement gap.
  • H4. Switch geometry through the five options — Dhara survives longest because its directional channel breaks the super/subradiant degeneracy, so |Ψ−⟩ is in a decoherence-free sub-mode. No other geometry does this.