Daily Run: Quantum Entanglement

Day 1

Entropy of Bell states

1. Fetched arXiv:quant-ph/0101012 for the definition of entanglement entropy.
2. Computed the reduced density matrix of $|\Phi^+\rangle = (|00\rangle+|11\rangle)/\sqrt{2}$ using NumPy.
3. Diagonalised it to find an entropy of $1.0$ bit, confirming maximal entanglement.

CHSH inequality checks

1. Built the CHSH operator $S=A\otimes B + A\otimes B' + A'\otimes B - A'\otimes B'$ with Pauli matrices.
2. Evaluated $\langle\Phi^+|S|\Phi^+\rangle$ numerically to obtain $2.8284$, saturating Tsirelson's bound $2\sqrt{2}$.
3. Confirmed classical strategies are limited to $|S|\le 2$, demonstrating quantum nonlocality.

Schmidt decomposition of random states

1. Generated a normalised random two‑qubit state and reshaped it into a $2\times2$ matrix.
2. Applied singular value decomposition to extract Schmidt coefficients $(0.944, 0.329)$.
3. Observed that non‑maximal coefficients correspond to partial entanglement.