IIT 4.0 analysis of quantum circuit measurement statistics
We computed IIT Phi for 5 quantum states: Bell state (2 qubits), GHZ state (3 qubits), product state (3 qubits), W state (3 qubits), and a random circuit (3 qubits, depth 5). The question: does IIT's measure of "integrated information" agree with standard entanglement measures?
Every quantum state — including maximally entangled Bell and GHZ states — has Φ = 0.
| State | Entanglement | Phi | EI (bits) | Emergence | Eff. Rank |
|---|---|---|---|---|---|
| Product | ψ⟩ = | 000⟩ | None | 0 | 2.08 |
| W state | Bipartite | 0 | 1.75 | 0.07 | 7/8 |
| GHZ | ψ⟩ = | 000⟩+ | 111⟩ | Maximal (3-way) | 0 |
| Bell | ψ⟩ = | 00⟩+ | 11⟩ | Maximal (2-way) | 0 |
| Random circuit | Variable | 0 | 0.41 | 0.12 | 8/8 |
IIT and entanglement measure different things. Entanglement is about quantum correlations. Phi is about irreducible causal structure in classical measurement statistics. A Bell state has perfect quantum correlations, but when you convert its measurement statistics into a transition probability matrix, every measurement outcome is causally independent.
But the emergence metrics reveal hidden structure
The Bell state has the highest emergence index (0.50) and the lowest effective rank (2/4). Only 2 of 4 measurement outcomes are independent — the other two are determined by the first two. This IS the entanglement, captured not as Phi but as dimensional compression.
The GHZ state has emergence = 0.33 with rank = 4/8. The W state has emergence = 0.07 with rank = 7/8. The emergence index tracks entanglement structure better than Phi does.
The product state (no entanglement) has the highest EI (2.08 bits) and full rank (8/8). No emergence. Every measurement outcome carries independent information. This is the quantum version of the rainforest vs. monoculture finding: the most "complex" state (product, all qubits independent) has the highest determinism and zero emergence. The most "integrated" state (GHZ) has the lowest determinism and highest emergence.
This result suggests that emergence index, not Phi, is the right IIT-derived measure for quantum systems. It recovers the entanglement hierarchy (Product < W < GHZ < Bell) when Phi does not. This is a concrete prediction that could be tested on real quantum hardware.
States: Bell (2q), GHZ (3q), Product (3q), W (3q), Random depth-5 (3q) Engine: ruvector-consciousness v2.1.0, exact TPM: Measurement statistics P(outcome_j | input_i) in computational basis
cargo run --release -p quantum-consciousnessTimestamp: 2026-03-31 UTC