Q07 — CHSH Game and Bell Inequality Violation

Field Value
Domain Quantum Computing
System Two-party game with shared entanglement vs shared randomness
Group SU(2) × SU(2) (Alice’s and Bob’s local quantum strategies)
H^k tier H²
ISA Meld (β→0)
Status Validated
Opcodes ORBIT · TWIST · BIND · LABEL
Papers Paper 421, Paper 469, Paper 325

Physical system

The CHSH game (Clauser-Horne-Shimony-Holt 1969 inequality; game formulation by Tsirelson 1980) is a cooperative two-player game testing whether nature is classical or quantum:

  • A referee sends Alice input bit x ∈ {0,1} and Bob input bit y ∈ {0,1}, each uniformly random
  • Alice outputs bit a ∈ {0,1}; Bob outputs bit b ∈ {0,1}
  • Win condition: a ⊕ b = x ∧ y (XOR of outputs = AND of inputs)
    • Alice and Bob win if (x=1 and y=1 and a≠b) or (otherwise and a=b)
  • No communication allowed after inputs are received

Classical bound: any classical strategy (shared randomness allowed) wins with probability at most 3/4 = 0.75.

Quantum bound (Tsirelson): with a shared Bell pair and optimal local measurements, Alice and Bob win with probability cos²(π/8) = (2+√2)/4 ≈ 0.854.

Tsirelson’s bound is tight: no quantum strategy (no matter how much entanglement) can exceed cos²(π/8). This is the ISA β* snap: the quantum advantage is precisely bounded by the geometry of the BIND (entangled state) and the TWIST (measurement angles).

The CHSH inequality violation (observed first by Aspect et al. 1982) is the experimental proof that nature cannot be explained by local hidden variable theories — the BIND (quantum entanglement) is real and irreducible to any classical H⁰/H¹ description.

Target category

GameCat(CHSH) — the category of non-signalling correlation matrices p(a,b|x,y) for the CHSH game. Objects:

  • Classical strategies: convex combinations of deterministic strategies; form the local polytope ℒ; best win probability 3/4
  • Quantum strategies: correlations from measurements on entangled states; form the quantum set 𝒬; best win probability cos²(π/8) ≈ 0.854
  • Non-signalling strategies: all correlations consistent with no communication; allow “super-quantum” (PR box) correlations; best win probability 1

The CHSH game maps to the I₃₃₂₂ Bell inequality; its violation certifies BIND. Morphisms are local operations (Alice acts on her subsystem; Bob on his).

Interpretation functor

F: C → GameCat(CHSH) defined by:

Opcode F(opcode)        
BIND Bell pair resource: Φ⁺⟩ = ( 00⟩+ 11⟩)/√2; the shared entangled state; the H² BIND is what separates quantum from classical; without it, the classical bound 3/4 cannot be exceeded  
ORBIT Alice’s qubit rotation: Alice measures in basis {cos(α) 0⟩+sin(α) 1⟩, −sin(α) 0⟩+cos(α) 1⟩}; she rotates by α₀ = 0 if x=0 or α₁ = π/4 if x=1; the ORBIT angle determines her output correlations
TWIST Bob’s qubit rotation: Bob measures in basis {cos(β) 0⟩+sin(β) 1⟩, …}; optimal angles β₀ = π/8 if y=0, β₁ = −π/8 if y=1; the TWIST angle is π/8 relative to Alice; the TWIST-ORBIT angle difference = π/8 is the ISA β* snap    
LABEL Win probability: p_win = cos²(angle/2) where angle = α_x − β_y for each (x,y) pair; averaged over inputs: p_win = (3 + cos(π/4))/4 = (2+√2)/4 ≈ 0.854; the LABEL eigenvalue of the optimal quantum strategy    

ISA programme 

PREPARE: BIND[|Phi+> = (|00>+|11>)/sqrt(2) | shared entanglement]
INPUT_A: LABEL[x in {0,1} | Alice's bit from referee]
INPUT_B: LABEL[y in {0,1} | Bob's bit from referee]
ROTATE_A:ORBIT[Alice measures at angle alpha_x: 0 if x=0, pi/4 if x=1]
ROTATE_B:TWIST[Bob measures at angle beta_y: pi/8 if y=0, -pi/8 if y=1]
CORREL:  LABEL[E(x,y) = <A_x B_y> = cos(alpha_x - beta_y) | correlator]
CHSH:    LABEL[S = E(0,0)+E(0,1)+E(1,0)-E(1,1) <= 2sqrt(2) | Tsirelson]
WIN:     LABEL[p_win = (2+sqrt(2))/4 ~ 0.854 | quantum win rate]
CLASSIC: LABEL[p_win <= 3/4 | classical upper bound, no BIND]

Computable output

  • Quantum win probability: cos²(π/8) = (2+√2)/4 ≈ 0.8536. This is the ISA LABEL eigenvalue of the optimal quantum strategy. It exceeds the classical bound 3/4 = 0.75 by 0.1036 — the “quantum advantage margin.”

  • CHSH correlator S = E(0,0)+E(0,1)+E(1,0)−E(1,1) where E(x,y) = ⟨a_x⊗b_y⟩:
    • Classical bound: S ≤ 2 (CHSH inequality)
    • Quantum optimum: S = 2√2 ≈ 2.828 (Tsirelson bound)
    • No-signalling PR box: S = 4 (super-quantum, unrealised in nature)

    The Tsirelson bound S = 2√2 is the ISA β* snap: it is the maximum correlation achievable with BIND but without non-local signalling. The factor √2 = cos(π/4) arises from the optimal angle difference π/8 = π/(4·2) between Alice’s and Bob’s measurement bases — a TWIST at exactly 22.5°.

  • Device-independent certification: if Alice and Bob observe S > 2 (classical violation), they can certify that their shared state contains genuine BIND (entanglement), without trusting their devices. This “device-independent” LABEL output is the foundation of quantum cryptography: a violation of S > 2 proves that eavesdroppers cannot have copied the BIND (no-cloning theorem enforces it).

  • Loophole-free Bell test (Hensen et al. 2015): S = 2.42 ± 0.20 observed in loophole-free test at 1.3 km separation (NV centres in diamond). The detection-loophole and locality-loophole were simultaneously closed; the classical explanation is ruled out with p < 0.039. The BIND (entanglement) is confirmed as irreducibly non-local.

Why H² (not H¹)

The CHSH violation is the defining H² signature:

  • Classical strategies (H⁰): shared randomness (a pre-agreed function a=f(x), b=g(y)) achieves at most 3/4. No ORBIT or TWIST is needed — just H⁰ LABEL (a classical correlation table). The CHSH inequality is the H⁰ bound.

  • H¹ strategies (shared Clifford / stabiliser): a stabiliser state (e.g., |+⟩|+⟩ = product state, or a stabiliser entangled state with classical Bell correlations) achieves at most 3/4. Clifford operations + Pauli measurements on stabiliser states cannot exceed the classical bound — H¹ TWIST is not sufficient. This is the Gottesman-Knill theorem for correlations: stabiliser circuits are classically simulable, so their correlations lie in the local polytope.

  • H² (Bell pair BIND): the |Φ⁺⟩ Bell pair is a non-stabiliser entangled state with a non-trivial BIND holonomy. The optimal CHSH measurement at angle π/8 requires a rotation that is not a Clifford gate (π/8 = half the T gate angle π/4; the T gate is the TWIST gate that generates magic). The Tsirelson bound cos²(π/8) is achievable only with genuine H² BIND — the Bell pair entanglement is a magic-class resource (in the language of Paper 469/470).

The magic connection: the π/8 optimal CHSH angle is the T gate angle divided by 2. The T gate (π/8 rotation on the Bloch sphere) is the generator of the H² magic resource (Paper 469). The CHSH violation is thus a direct measurement of the H² BIND content of the Bell pair — winning cos²(π/8) ≈ 0.854 requires exactly the same non-Clifford resource as the T gate that elevates stabiliser computation to universal quantum computation.

The β-plane interpretation

The CHSH game sweeps through the β-plane as the measurement angle θ varies:

Angle θ (Alice–Bob difference) S = CHSH correlator β-plane ISA
θ = 0 S = 4 (PR box, non-physical) β→0 (full Meld)  
θ = π/8 (optimal) S = 2√2 (Tsirelson) β* snap BIND eigenvalue
θ = π/4 (Clifford) S = 2√2·sin(π/4)=2 β=1 classical boundary
θ = π/2 S = 0 β→∞ (Origami) uncorrelated

The Tsirelson bound S = 2√2 at θ = π/8 is the β* snap: it is the maximum of S(θ) = 2√2 cos(2θ), achieved at θ* = 0 for the reduced correlator but at θ* = π/8 for the full CHSH game structure. The β* snap occurs at the measurement angle where the BIND resource is most efficiently converted into a classical observable (the winning probability).

Paper 597 (Soft Thresholds) connection: the Tsirelson bound is another instance of the MGE soft threshold. The CHSH correlator S(θ) = 2√2 cos(2θ) is a soft function with maximum at θ=0 and β* snap where dS/dθ = 0 → the classical threshold S=2 is the β=∞ “frozen” limit, and the quantum optimum S=2√2 is the Meld β→0 limit of the same MGE saddle point.

Connections to other entries

  • Q05 Teleportation / Q06 Superdense coding: all three use the same Bell pair BIND resource; CHSH tests its non-classicality; Q05/Q06 use it for communication tasks
  • Paper 469 (ISA Completeness): the T gate is the generator of magic (H²); CHSH at θ=π/8 requires exactly this T gate resource; CHSH violation ↔ magic state (TV < 1 in the language of Paper 470)
  • Paper 470 (Hot Logic): the Tsirelson bound cos²(π/8) is a dark-magic threshold: stabiliser (H¹) achieves ≤ 3/4; dark-magic states can exceed 3/4 but may not reach the Tsirelson bound; genuine magic (Bell pair BIND) achieves exactly cos²(π/8)
  • K01/K04 (Cryptography): device-independent quantum key distribution (DI-QKD) uses CHSH violation as a security certificate; the LABEL output S > 2 guarantees that eavesdroppers hold no information about the key

Validation

  • CHSH (1969): original Bell inequality; classical bound 2; measurement setting.
  • Tsirelson (1980): quantum bound 2√2; optimal strategy proved; angle π/8.
  • Aspect, Grangier & Roger (1982): first loophole-free (in practice) photonic Bell test; S = 2.697±0.015; classical explanation ruled out.
  • Hensen et al. (2015), Nature 526, 682: loophole-free Bell test; NV centres 1.3 km apart; S = 2.42±0.20; detection + locality loopholes closed simultaneously.
  • Giustina et al. (2015), PRL 115, 250401: photonic loophole-free; S > 2 with p < 3.74×10⁻³¹.
  • Shalm et al. (2015), PRL 115, 250402: NIST loophole-free; multiple entangled photon pairs; S > 2 with p < 2.3×10⁻⁷.

Part of the ISA Zoo. Categorical foundations: Paper 591.