ruvector/examples/quantum-consciousness/RESEARCH.md

Quantum Circuit Consciousness: IIT Phi and Entanglement

Motivation

Integrated Information Theory (IIT) and quantum entanglement both formalize the idea that a system is "more than the sum of its parts." This example explores whether IIT's Phi measure, applied to quantum circuit measurement statistics, captures the same structure as standard entanglement measures.

Quantum States Analyzed

1. Bell State (2 qubits)

The maximally entangled two-qubit state:

|Psi> = (|00> + |11>) / sqrt(2)

Prepared by H(0) followed by CNOT(0,1). Measurement in the computational basis yields 00 or 11 with equal probability, never 01 or 10. This maximal correlation should produce HIGH Phi.

2. GHZ State (3 qubits)

The Greenberger-Horne-Zeilinger state:

|GHZ> = (|000> + |111>) / sqrt(2)

Genuinely multipartite entangled: tracing out any single qubit destroys all entanglement. Expected to show HIGH Phi and high emergence (the 3-party correlations cannot be reduced to 2-party).

3. Product State (3 qubits)

|Psi> = |0> x |0> x |0>

Completely separable, no entanglement. The TPM is the identity matrix (each input maps to itself). Expected Phi = 0 since the system decomposes perfectly into independent parts.

4. W State (3 qubits)

|W> = (|001> + |010> + |100>) / sqrt(3)

Bipartite entanglement that survives partial trace: tracing out any one qubit still leaves the other two entangled. Different entanglement structure from GHZ. Expected: Phi between product and GHZ.

5. Random Circuit (3 qubits, depth 5)

Random single-qubit rotations interleaved with CNOT gates. The resulting entanglement depends on the specific random gates chosen. Serves as a control to show that Phi varies continuously with circuit structure.

TPM Construction for Quantum Circuits

The key mapping from quantum mechanics to IIT:

TPM[i][j] = |<j|U|i>|^2

where U is the circuit unitary, and i, j are computational basis states. This gives the probability of measuring outcome j when the input is the basis state i. The resulting TPM is doubly stochastic for unitary circuits (both rows and columns sum to 1).

Expected Phi Hierarchy

Standard entanglement measures predict:

Product (0) < W < Bell <= GHZ

IIT's Phi may not follow this ordering exactly because:

1. Phi measures integrated information across the minimum information partition (MIP), not entanglement per se
2. The Bell state is 2-qubit while GHZ/W are 3-qubit, so the partition spaces differ
3. W state has different entanglement structure (robust to qubit loss) which may be valued differently by Phi

Entanglement Measures for Comparison

• Concurrence (2 qubits): measures entanglement of formation
• Tangle (3 qubits): measures genuine 3-party entanglement
• Entanglement entropy: von Neumann entropy of reduced density matrix

The GHZ state has maximal tangle but zero concurrence for any pair. The W state has zero tangle but nonzero concurrence for every pair.

Causal Emergence in Quantum Systems

Causal emergence asks: is there a macro-level description of the quantum system that is more informative than the qubit-level description? For entangled states, the answer may be yes -- the entangled subsystem behaves as a single effective degree of freedom.

Limitations

1. Classical TPM: we use |<j|U|i>|^2, discarding quantum phases. IIT on the full quantum state (quantum IIT) is an active research area.
2. Measurement basis dependence: Phi depends on the choice of computational basis. A different measurement basis could yield different Phi values.
3. Small systems: 3 qubits = 8x8 TPM, well within exact Phi computation limits but far from interesting quantum advantage regimes.

References

• Tononi, G. (2008). Consciousness as Integrated Information.
• Zanardi, P. et al. (2018). Quantum Integrated Information Theory.
• Greenberger, D.M. et al. (1989). Going Beyond Bell's Theorem.
• Dur, W. et al. (2000). Three qubits can be entangled in two inequivalent ways.
• Hoel, E.P. (2017). When the Map Is Better Than the Territory.