Nadarasa · v0.3.3 · PQP Thm 11.12
Phase-group fingerprint
PQP Thm 11.12 gives a one-line litmus: a ZX-style theory exhibits GHZ-style non-locality iff its phase group contains Z₄ (as opposed to Z₂×Z₂ for Spekkens' toy theory). For each Nadarasa kernel we read the phase angles directly off the emitted gates, reduce each angle θ/(2π) = a/b to lowest terms, and check whether any generator has order 4 (or a multiple of 4) in (R/Z, +).
n = 3 qubits (N = 8)
| Kernel | Max order | Z₄ present? | Fingerprint |
|---|---|---|---|
| G1 · QFT cphase ladder | 4 | yes | quantum-like (Z₄ present) |
| G2-real · two ladders + glue | 8 | yes | quantum-like (Z₄ present) |
| G12 · same skeleton as G1 | 4 | yes | quantum-like (Z₄ present) |
n = 4 qubits (N = 16)
| Kernel | Max order | Z₄ present? | Fingerprint |
|---|---|---|---|
| G1 · QFT cphase ladder | 8 | yes | quantum-like (Z₄ present) |
| G2-real · two ladders + glue | 16 | yes | quantum-like (Z₄ present) |
| G12 · same skeleton as G1 | 8 | yes | quantum-like (Z₄ present) |
n = 5 qubits (N = 32)
| Kernel | Max order | Z₄ present? | Fingerprint |
|---|---|---|---|
| G1 · QFT cphase ladder | 16 | yes | quantum-like (Z₄ present) |
| G2-real · two ladders + glue | 32 | yes | quantum-like (Z₄ present) |
| G12 · same skeleton as G1 | 16 | yes | quantum-like (Z₄ present) |
n = 6 qubits (N = 64)
| Kernel | Max order | Z₄ present? | Fingerprint |
|---|---|---|---|
| G1 · QFT cphase ladder | 32 | yes | quantum-like (Z₄ present) |
| G2-real · two ladders + glue | 64 | yes | quantum-like (Z₄ present) |
| G12 · same skeleton as G1 | 32 | yes | quantum-like (Z₄ present) |
n = 7 qubits (N = 128)
| Kernel | Max order | Z₄ present? | Fingerprint |
|---|---|---|---|
| G1 · QFT cphase ladder | 64 | yes | quantum-like (Z₄ present) |
| G2-real · two ladders + glue | 128 | yes | quantum-like (Z₄ present) |
| G12 · same skeleton as G1 | 64 | yes | quantum-like (Z₄ present) |
Worked example · n = 4
The QFT cphase ladder uses angles 2π · 1/2, 2π · 1/4, 2π · 1/8 at depths k = 1, 2, 3. Their orders in (R/Z, +) are 2, 4, 8 respectively. Z₄ appears starting at depth k ≥ 2, so any Nadarasa kernel with n ≥ 3 qubits inherits the quantum-like fingerprint by construction — the ladder forces it.
Verdict for G1 / G2-real / G12 at any reasonable register size: quantum-like. The kernels live on the right side of Thm 11.12 — they cannot be re-derived inside Spekkens' Z₂×Z₂ toy theory. This is a structural lower bound on what the kernels are computing, not a claim about the verdict in any specific ledger entry.