Project · Simulate a Bell state

Build a 2-qubit circuit (in Qiskit, or with raw numpy matrices) that prepares the Bell state $|\Phi^+\rangle = (|00\rangle + |11\rangle)/\sqrt{2}$ from $|00\rangle$, measure it 1{,}000 times in the computational basis, and verify empirically that the outcomes are split roughly 50/50 between $|00\rangle$ and $|11\rangle$ with $|01\rangle$ and $|10\rangle$ never appearing. The point is not just to reproduce the distribution — it is to argue convincingly, in writing, that the result demonstrates *entanglement* rather than mere classical correlation, and to identify the experiment that would distinguish the two.

• A working circuit (Qiskit `QuantumCircuit` or numpy matrix product) that produces $|\Phi^+\rangle$ from $|00\rangle$ using exactly one Hadamard and one CNOT, with the explicit state vector printed before measurement.
• A measurement script that runs 1{,}000 shots and prints the empirical counts for $|00\rangle, |01\rangle, |10\rangle, |11\rangle$.
• A short write-up (1 page) stating: (a) the predicted distribution and why, (b) the empirical distribution, (c) the discrepancy and whether it is within the expected sampling error of $\sim 1/\sqrt{1000} \approx 0.03$.
• An explicit argument — in prose — for why the observed distribution is *not* explainable by a classical pre-shared random bit, including the specific second experiment (e.g. measurement in the X-basis) that would distinguish entanglement from classical correlation.

• The state vector printed before measurement has amplitudes $(1/\sqrt{2}, 0, 0, 1/\sqrt{2})$ on $(|00\rangle, |01\rangle, |10\rangle, |11\rangle)$ to numerical precision.
• $|01\rangle$ and $|10\rangle$ counts are exactly zero across 1{,}000 shots; $|00\rangle$ and $|11\rangle$ counts are each within roughly $\pm 30$ of 500 (i.e. within $\sim 2$ standard deviations of 50/50).
• The write-up explicitly states why a classical 'shared coin flip' model produces the same Z-basis statistics, and identifies the X-basis or CHSH-style measurement that breaks the tie.
• The implementation passes a self-check that $|\Phi^+\rangle$ cannot be factored as a tensor product (e.g. by computing the partial trace and showing it is the maximally mixed state, or by exhibiting the algebraic obstruction shown in task 4).