Project · Simulate a Bell state

project

hard

module project

Ship something real. Submit your work when you're done.

Brief

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.

Deliverables

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.

How we grade it

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).

Hints

Two equivalent ways to do this. Qiskit: qc = QuantumCircuit(2, 2); qc.h(0); qc.cx(0, 1); qc.measure([0,1], [0,1]), then run on AerSimulator() (or the new StatevectorSampler / SamplerV2 API depending on your Qiskit version) with shots=1000. Pure numpy: build

H \otimes I

\mathrm{CNOT}

, multiply onto

|00\rangle = e_0

, then sample from the squared-amplitude distribution using np.random.choice. Both are fine — the rubric scores the analysis, not the framework.

Be careful with Qiskit's bit ordering. By default, Qiskit prints two-qubit basis states as

|q_1 q_0\rangle

(little-endian). The Bell state shows up as

00

and

11

in either ordering, but if you generalise to other Bell states the labels will look swapped from the textbook unless you flip your interpretation.

The "could this be classical correlation?" question is the key one. A pre-shared random bit (flip a coin, both write down the result) reproduces the 50/50 Z-basis distribution exactly. The thing it cannot reproduce is the correlation in every basis. Mention this even if you don't implement the X-basis check.

Sampling error on a 50/50 binomial with

N = 1000

has standard deviation

\sqrt{N \cdot 0.5 \cdot 0.5} \approx 15.8

. Expect each of

and in the range most runs.

Expected output

A run looks something like:

Statevector before measurement: [0.7071+0.j 0. +0.j 0. +0.j 0.7071+0.j] |00> |01> |10> |11> Measurement counts (1000 shots): |00>: 498 |01>: 0 |10>: 0 |11>: 502 Predicted P(|00>) = P(|11>) = 0.5, P(|01>) = P(|10>) = 0.0 Empirical fractions: 0.498 | 0.000 | 0.000 | 0.502 Discrepancy: 0.002 (within 1-sigma sampling error of 0.0158) Entanglement check: tried to factor as |a> tensor |b>; system is inconsistent.

…followed by the prose argument for entanglement vs. classical correlation, naming the X-basis measurement as the experiment that would falsify the classical-correlation hypothesis.

Stretch goals

Implement the X-basis measurement explicitly: apply

H \otimes H

before measuring, and verify that you still see only

|00\rangle

and

|11\rangle

outcomes (with

|\Phi^+\rangle

). A classical pre-shared-bit model would predict 50/50 across all four outcomes here — show this is the experimental signature of entanglement.

Generalise to all four Bell states: produce

by varying the input to the circuit () and confirm each one's signature distribution.