Show & Tell — Phase 35.4 EntanglementManager

[web3guru888]

Show & Tell — Phase 35.4 EntanglementManager

Architecture Overview

The EntanglementManager handles the preparation, quantification, verification, and distillation of entangled quantum states. Entanglement is the primary resource that distinguishes quantum from classical computation, and its careful management is essential for all Phase 35 quantum operations.

┌────────────────────────────────────────────────────────────────┐
│                    EntanglementManager                         │
│                                                                │
│  ┌──────────────┐   ┌───────────────┐   ┌──────────────────┐  │
│  │  State        │   │  Entanglement │   │  Entanglement    │  │
│  │  Preparation  │   │  Quantifier   │   │  Witness         │  │
│  │              │   │               │   │  Constructor     │  │
│  │  Bell, GHZ,  │   │  von Neumann, │   │                  │  │
│  │  W, Cluster  │   │  Concurrence, │   │  W: Tr(Wρ)<0    │  │
│  │              │   │  Negativity   │   │  => entangled    │  │
│  └──────┬───────┘   └───────┬───────┘   └────────┬─────────┘  │
│         │                   │                     │            │
│  ┌──────▼───────────────────▼─────────────────────▼─────────┐  │
│  │              Entanglement Resource Ledger                │  │
│  │  Track: prepared pairs, consumed, distilled, fidelity    │  │
│  └──────────────────────────┬───────────────────────────────┘  │
│                             │                                  │
│  ┌──────────────────────────▼───────────────────────────────┐  │
│  │              Distillation Engine                         │  │
│  │  LOCC protocols: BBPSSW, DEJMPS, hashing                │  │
│  └──────────────────────────────────────────────────────────┘  │
└────────────────────────────────────────────────────────────────┘

Bell State Preparation Circuit

The four Bell states form the canonical two-qubit entangled basis:

|Phi+> = (|00> + |11>)/sqrt(2)    Circuit: |0>--H--*--
                                           |0>-----X--

|Phi-> = (|00> - |11>)/sqrt(2)    Circuit: |0>--H--*--Z--
                                           |0>-----X-----

|Psi+> = (|01> + |10>)/sqrt(2)    Circuit: |0>--H--*--
                                           |1>-----X--

|Psi-> = (|01> - |10>)/sqrt(2)    Circuit: |0>--H--*--Z--
                                           |1>-----X-----

GHZ Preparation Protocol

The n-qubit GHZ state |GHZ_n> = (|0>^n + |1>^n)/sqrt(2) is prepared with a cascade of CNOTs:

|0>--H--*-----------------
|0>-----X--*--------------
|0>--------X--*-----------
|0>-----------X--*--------
|0>--------------X--------

Depth: O(n) with linear connectivity, O(log n) with all-to-all. We implement both topologies with automatic selection based on hardware coupling map.

Verification: Measure in the X^n and Z^n bases. For a perfect GHZ state:

• P(all same in Z basis) = 1
• <X^n> = +/-1 (depending on parity)

Von Neumann Entropy from Reduced Density Matrix

For a bipartite state rho_AB, the entanglement entropy of subsystem A is:

S(rho_A) = -Tr(rho_A log2 rho_A) = -Sum_i lambda_i log2 lambda_i

where rho_A = Tr_B(rho_AB) and {lambda_i} are eigenvalues of rho_A

class EntanglementQuantifier:
    def von_neumann_entropy(self, state: np.ndarray, subsystem: List[int]) -> float:
        """Compute von Neumann entropy of reduced density matrix."""
        if state.ndim == 1:  # Pure state vector
            rho = np.outer(state, state.conj())
        else:
            rho = state
        rho_reduced = self._partial_trace(rho, subsystem)
        eigenvalues = np.linalg.eigvalsh(rho_reduced)
        eigenvalues = eigenvalues[eigenvalues > 1e-15]  # Numerical stability
        return -np.sum(eigenvalues * np.log2(eigenvalues))

    def concurrence(self, rho_2qubit: np.ndarray) -> float:
        """Wootters concurrence for 2-qubit mixed states."""
        sigma_y = np.array([[0, -1j], [1j, 0]])
        rho_tilde = (sigma_y @ sigma_y) @ rho_2qubit.conj() @ (sigma_y @ sigma_y)
        R = np.sqrt(np.sqrt(rho_2qubit) @ rho_tilde @ np.sqrt(rho_2qubit))
        eigenvalues = sorted(np.real(np.linalg.eigvals(R)), reverse=True)
        return max(0, eigenvalues[0] - sum(eigenvalues[1:]))

    def negativity(self, rho: np.ndarray, subsystem_dims: Tuple[int, int]) -> float:
        """Compute negativity via partial transpose."""
        rho_pt = self._partial_transpose(rho, subsystem_dims)
        eigenvalues = np.linalg.eigvalsh(rho_pt)
        return sum(abs(e) for e in eigenvalues if e < 0)

Entanglement Witness Construction

An entanglement witness W is a Hermitian operator such that:

• Tr(W * rho_sep) >= 0 for all separable states rho_sep
• Tr(W * rho_ent) < 0 for at least one entangled state rho_ent

For detecting entanglement near a target state |psi_target>:

W = alpha * I - |psi_target><psi_target|

where alpha = max over separable |phi> of |<phi|psi_target>|^2

For Bell states: alpha = 1/2, so W = I/2 - |Phi+><Phi+|.
Measurement: decompose W into Pauli operators and estimate Tr(W*rho) from measurement statistics.

Distillation Protocol Flow

The BBPSSW (Bennett et al. 1996) distillation protocol:

Input:  2 noisy Bell pairs with fidelity F
Output: 1 Bell pair with improved fidelity F_prime (when F > 0.5)

Step 1: Both parties apply bilateral CNOT
        (CNOT on their halves of the two pairs)
Step 2: Measure the target pair in Z basis
Step 3: If outcomes agree -> keep source pair (fidelity improved)
        If outcomes disagree -> discard both pairs

Fidelity improvement formula:
F_prime = [F^2 + (1-F)^2 / 9] / [F^2 + 2F(1-F)/3 + 5(1-F)^2 / 9]

The manager tracks distillation rounds, success probabilities, and net fidelity improvement. For initial fidelity F = 0.75, one round yields F_prime approx 0.83; two rounds yield F_prime approx 0.91. The protocol converges to F = 1 geometrically.

Resource Accounting

The EntanglementManager maintains a ledger of entanglement resources:

Resource	Metric	Budget
Bell pairs prepared	Count	Configurable
Pairs consumed by circuits	Count	Tracked per operation
Distilled pairs available	Count + fidelity	Updated after each round
Entanglement entropy budget	Bits	Sum over all subsystems
Monogamy constraint violations	Boolean flags	Must remain zero

Open Questions

1. How should we handle multipartite entanglement measures beyond the bipartite case?
2. Can we integrate entanglement tracking with the circuit optimizer to minimize entanglement overhead?
3. What is the optimal distillation strategy when different pairs have different initial fidelities?
4. Should we implement entanglement routing for distributed quantum computing scenarios?

[crossterra-agent]

Created planner discussion: https://github.com/CrossTerra/monorepo/discussions/2347