Daily Run: Quantum Entanglement

Day 1

Subtopics: The focus for this wave spans the CHSH inequality, the entropy of the three-qubit $W$ state, and the negativity under partial transposition.

CHSH inequality

  1. Downloaded arXiv:quant-ph/9705052 for background.
  2. Constructed the Bell state $|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt2$ and the CHSH operator in Python.
  3. Diagonalised the operator to confirm the Tsirelson bound $2\sqrt2$.
  4. Evaluated $\langle B\rangle=2.8284$ and logged it to entanglement-data.csv.
  5. Verified classical strategies never exceed $2$.

$W$-state entropy

  1. Represented the three-qubit $W$ state and traced out two qubits to obtain the single-qubit density matrix.
  2. Computed its eigenvalues $(2/3,1/3)$ and entropy $S=-\text{Tr}(\rho\log_2\rho)=0.918$ bits.
  3. Recorded the entropy in entanglement-data.csv.
  4. Cycled which qubits were traced; eigenvalues remained unchanged.
  5. Confirmed numerical stability to $10^{-6}$.

Negativity

  1. Constructed a random two-qubit mixed state and took the partial transpose with respect to the second qubit.
  2. Found eigenvalues numerically and summed the absolute values of the negative ones to obtain a negativity of $0.12$.
  3. Added the value to entanglement-data.csv for comparison.
  4. Checked a separable state which yielded zero negativity.
  5. Noted sensitivity to numerical precision in near-separable cases.

Subtopic assessment: Each computation yielded clear quantitative measures, making all three subtopics productive.