Lab 3: Entanglement Identity — Equations

1. Identity-Preserving Entanglement#

Let two particles A and B be entangled such that their identity states are preserved across measurement frames:

$$ |\Psi\rangle = \frac{1}{\sqrt{2}} \left( |I_A\rangle |I_B\rangle + |I_B\rangle |I_A\rangle \right) $$

Where ( |I_A\rangle ) and ( |I_B\rangle ) represent identity eigenstates.

Define a phase mapping operator ( \hat{\Phi} ) that preserves identity coherence:

$$ \hat{\Phi} |I\rangle = e^{i\phi_I} |I\rangle $$

This ensures entangled identity states remain phase-coherent under transformation.

Introduce a metric ( \mathcal{E}_I ) to quantify identity entanglement strength:

$$ \mathcal{E}_I = |\langle I_A | I_B \rangle|^2 $$

Maximal identity entanglement occurs when ( \mathcal{E}_I = 1 ).

Define a triadic tensor ( T_{ijk} ) for entangled identity propagation:

$$ T_{ijk} = \langle I_i | \hat{E}_j | I_k \rangle $$

Where ( \hat{E}_j ) is an entanglement operator acting on identity basis states.