Build a Bell state with N/M and read the results

Build a two-qubit correlation you can inspect

A Bell state is a useful next step after a single-qubit measurement experiment. For the state ( |00> + |11> ) / sqrt(2), computational-basis measurements yield 00 or 11 with equal ideal probability. Each qubit individually looks random, while the pair has a strong correlation.

The goal here is to connect an N/M program to that distribution and then change the program to test your explanation. The experiment runs in a local noiseless simulator; it does not measure a physical quantum processor.

Run the circuit

Paste the following code into the Playground:

module blog_bell;
@seed(23);
fn main() {
  let q = qreg[2];
  sample 512 {
    H(q[0]);
    CNOT(q[0], q[1]);
    let c0 = measure(q[0]);
    let c1 = measure(q[1]);
  }
  return q;
}

Both qubits start at 0. H(q[0]) creates equal amplitudes for the first qubit. CNOT flips the second qubit when the first is 1. The combined state is therefore supported on 00 and 11. The pre-measurement probability display should assign 0.5 to each of those states, with zero probability for 01 and 10 in this ideal model. A single measurement collapses the state; inspect the histogram and pre-measurement distribution separately from that final collapsed state.

The sampled histogram need not split exactly 256/256. Sampling 512 shots produces an estimate of the distribution, not a new exact definition of it. When comparing runs, preserve the program, seed, shot count, simulator settings and version. If a diagram labels bits in a different order, read its ordering convention rather than assuming the leftmost digit always represents the same register index.

Three changes that reveal what the gates do

1. Remove the CNOT line. The second qubit remains 0, and the two-qubit distribution no longer has the Bell correlation.
2. Restore CNOT and remove the first H. A CNOT with a control initially at 0 leaves the register at 00.
3. Restore the original program and add X(q[1]); after CNOT. The supported outcomes change to 01 and 10, producing anticorrelation in the computational basis.

Write down your expected outcomes before running each change. A prediction that disagrees with the simulator gives you a specific gate or basis convention to investigate.

What the histogram cannot prove

A classical mixture containing equal numbers of 00 and 11 also has this computational-basis histogram. A histogram in one basis alone is therefore insufficient to establish entanglement. The simulator prepares a Bell state because of the state evolution prescribed by the gates; experimental verification requires additional measurements and a suitable analysis.

Entanglement also does not provide faster-than-light communication. An observer who sees only one of these qubits cannot control its random local outcome to send a message. Correlations become useful when measurement records and the experimental setup are compared.

For a more demanding exercise, investigate measurements in another basis and explain why the same classical mixture and the coherent Bell state need not behave identically. Keep that question separate from a complete Bell-inequality experiment, which requires chosen measurement settings and an appropriate statistical treatment.

Next steps and sources

• Bit, qubit and measurement: revisit the distinction between probability and sampling.
• N/M documentation: language syntax and simulation workflow.
• IBM: Quantum circuits: circuit composition and entangling gates.
• IBM Quantum tutorials: further practical experiments, including a CHSH-inequality tutorial.

The expected probabilities in this article concern the stated ideal circuit. Device noise, imperfect preparation and finite-shot uncertainty would need separate treatment in a hardware experiment.