Daily Run: Quantum Entanglement

Day 2

Subtopic review: No subtopics were dropped; teleportation, GHZ entropy and swapping remained in focus.

Teleportation fidelity

  1. Created a random state $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ with $\alpha,\beta$ drawn from a complex normal distribution and normalised with NumPy.
  2. Modelled the Bennett teleportation circuit step by step, applying CNOT and Hadamard gates and tracking post-measurement states for all four classical outcomes.
  3. Reconstructed the target qubit using Pauli corrections and computed the overlap $\langle\psi|\rho_{\text{out}}|\psi\rangle$, giving fidelity $F=0.999999$ which rounds to unity.

GHZ state entropy

  1. Built the density matrix $\rho=|\text{GHZ}\rangle\langle\text{GHZ}|$ for $|\text{GHZ}\rangle=(|000\rangle+|111\rangle)/\sqrt{2}$ in Python.
  2. Performed a partial trace over qubits $B$ and $C$ using tensor reshaping, leaving a single-qubit matrix $\mathrm{diag}(1/2,1/2)$.
  3. Evaluated $S=-\operatorname{Tr}(\rho_A\log_2 \rho_A)=1$ bit exactly, confirming maximal single-qubit mixedness.

Entanglement swapping

  1. Prepared two Bell pairs $|\Phi^+\rangle_{AB}\otimes|\Phi^+\rangle_{CD}$ and rewrote them in the Bell basis of qubits $B$ and $C$.
  2. Projected onto $|\Phi^+\rangle_{BC}$, renormalising the state of qubits $A$ and $D$ to obtain another $|\Phi^+\rangle$.
  3. Computed the concurrence $C=2|ad-bc|$ from the final density matrix and obtained $C=0.99999\approx1$, indicating near-perfect swapping.

Subtopic assessment: Each protocol yielded quantitative confirmation—unit teleportation fidelity, one-bit GHZ entropy and near-unity concurrence—so all subtopics proved useful.