Pedagogical Tool

Ekert91 (E91) Protocol Simulator

Interactively verifying quantum non-locality (CHSH test) & cryptographic key sifting

Bell State: |Φ⁺⟩
Target CHSH Bound: 2√2 ≈ 2.828

Control Console

ID: E91-SYS-CTRL
50
N = 10 N = 100 N = 200

Launches projective intercept-resend attack

Medium
Step-by-Step Slow Normal Instant (Bulk)

Orientation Bases Map

Alice Angles (a)

  • a₁ = 0° (0 rad)
  • a₂ = 45° (π/4 rad) (Key Setting)
  • a₃ = 90° (π/2 rad) (Key Setting)

Bob Angles (b)

  • b₁ = 45° (π/4 rad) (Key Setting)
  • b₂ = 90° (π/2 rad) (Key Setting)
  • b₃ = 135° (3π/4 rad)

* Perfect alignment matches occur when Alice uses basis a₂ & Bob uses b₁ (both at 45°), or Alice uses basis a₃ & Bob uses b₂ (both at 90°). CHSH tests use non-matching combinations (e.g., a₁, a₃ with b₁, b₃).

Quantum Channel Visualizer

IDLE
Alice's Lab (Rx-A) Quantum Channel Bob's Lab (Rx-B)
Emitted Pairs
0 / 0
Sifted Key Length
0 bits
QBER: 0.00%
CHSH Index (S)
0.0000
No sample
Channel Status
STANDBY
Awaiting sequence

Alice's Sifted Secret Key

Awaiting simulation run...

Bob's Sifted Secret Key

Awaiting simulation run...

Protocol Ledger

0 records
ID Alice Basis (Angle) Bob Basis (Angle) Alice Bit Bob Bit Allocation Type Status
No telemetry data. Configure parameters and click "Run Simulation".

E91 Physics & Pedagogical Reference

1. Entangled Source State

The simulator emits maximally entangled photon pairs in the singlet-like Bell State: |Φ⁺⟩ = (|00⟩ + |11⟩) / √2 This configuration represents perfect quantum correlation. If measured in matching polarization angles, Alice and Bob observe 100% correlation.

2. State Space Alignment

To align polarizer orientations with state space vectors: θ_A = φ_A / 2, θ_B = φ_B / 2 The projection measurement correlation reduces to: E(φ_A, φ_B) = cos(2(θ_A - θ_B)) = cos(φ_A - φ_B) The probability of Alice and Bob measuring matching bits is: P_same = cos²(θ_A - θ_B)

3. CHSH Inequality Verification

The CHSH correlation coefficient index S is calculated as: S = E(a₁, b₁) - E(a₁, b₃) + E(a₃, b₁) + E(a₃, b₃) With Alice's settings {0°, 90°} and Bob's settings {45°, 135°}: S = cos(45°) - cos(135°) + cos(45°) + cos(-45°) Evaluating each term: S = 1/√2 - (-1/√2) + 1/√2 + 1/√2 = 4/√2 = 2√2 ≈ 2.828 This violates the local hidden variables classical limit of |S| ≤ 2.0.

4. Eavesdropper (Eve) Impact

When Eve is active, she executes a projective measurement on Alice's photon at angle θ_E chosen randomly from {0°, 22.5°, 45°, 67.5°} (display angles: {0°, 45°, 90°, 135°}). This measurement collapses the entanglement state, forcing the system into local realism. The correlation collapses to: E(a, b) = 1/4 Σ_E cos(a - e)cos(b - e) As a result, the maximum CHSH value S collapses to ≤ 2.0 (typically ≈ 1.414) and a 25% QBER (Quantum Bit Error Rate) is introduced on the matching bases, exposing the security breach.