Tutorial 1: Bell State¶

Time: 10 minutes
Level: Beginner
Concepts: Superposition, entanglement, measurement

Try it interactively

This tutorial is also available as a Jupyter notebook you can run locally:
Open 02_bell_states.ipynb

What You'll Build¶

A Bell State — the simplest example of quantum entanglement. Two qubits become perfectly correlated: measuring one instantly determines the other.

Background¶

The four Bell States are:

\[|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)\]

\[|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)\]

\[|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)\]

\[|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)\]

We'll create \(|\Phi^+\rangle\), the most common Bell State.

Step 1: Create the Circuit¶

import quantsdk as qs

circuit = qs.Circuit(2, name="bell-state")

We need 2 qubits, both starting in \(|0\rangle\).

Step 2: Apply Hadamard¶

circuit.h(0)  # Put qubit 0 into superposition

After this, qubit 0 is in state \(\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\), while qubit 1 is still \(|0\rangle\).

Combined state: \(\frac{1}{\sqrt{2}}(|00\rangle + |10\rangle)\)

Step 3: Apply CNOT¶

circuit.cx(0, 1)  # Entangle: CNOT with control=0, target=1

The CNOT flips qubit 1 when qubit 0 is \(|1\rangle\):

\(|00\rangle \rightarrow |00\rangle\) (control is 0, no flip)
\(|10\rangle \rightarrow |11\rangle\) (control is 1, flip target)

Result: \(\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) = |\Phi^+\rangle\)

Step 4: Measure¶

circuit.measure_all()

Step 5: Run and Verify¶

result = qs.run(circuit, shots=1000)
print(result.counts)
# {'00': ~500, '11': ~500}

You should see roughly equal counts of 00 and 11, with no 01 or 10 — that's entanglement!

Complete Code¶

import quantsdk as qs

# Create Bell State |Phi+>
circuit = qs.Circuit(2, name="bell-state")
circuit.h(0).cx(0, 1).measure_all()

# Run
result = qs.run(circuit, shots=1000)

# Analyze
print(f"Counts: {result.counts}")
print(f"Probabilities: {result.probabilities}")
print(f"Most likely: {result.most_likely}")

# Visualize
result.plot_histogram()

Other Bell States¶

Create all four Bell States by adding gates before measurement:

# |Phi+> = (|00> + |11>) / sqrt(2)
phi_plus = qs.Circuit(2).h(0).cx(0, 1).measure_all()

# |Phi-> = (|00> - |11>) / sqrt(2)
phi_minus = qs.Circuit(2).h(0).cx(0, 1).z(0).measure_all()

# |Psi+> = (|01> + |10>) / sqrt(2)
psi_plus = qs.Circuit(2).h(0).cx(0, 1).x(1).measure_all()

# |Psi-> = (|01> - |10>) / sqrt(2)
psi_minus = qs.Circuit(2).h(0).cx(0, 1).x(1).z(0).measure_all()

Key Takeaways¶

Hadamard creates superposition — one qubit in two states at once
CNOT creates entanglement — correlating two qubits
Measurement collapses the state — you only see 00 or 11, never 01 or 10
Entanglement is a resource used in teleportation, cryptography, and error correction

Next Tutorial¶

Quantum Teleportation — use entanglement to "teleport" quantum information.