7  Your First Bell State

Welcome to your first hands-on QPL tutorial! In this chapter, you’ll create your first quantum program: a Bell state - one of the most fundamental entangled quantum states.

7.1 What is a Bell State?

A Bell state is a maximally entangled 2-qubit quantum state. The most common Bell state is:

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

This state has fascinating properties: - Superposition: Equal probability of measuring \(|00\rangle\) or \(|11\rangle\) - Perfect correlation: If you measure qubit A as 0, qubit B will always be 0 - Maximal entanglement: Entropy \(S = 1.0\) (maximum for 2 qubits)

7.2 Setting Up

First, make sure QPL is installed:

pip install quantum-process-language

Create a new Python file my_first_bell_state.py:

from qpl import QPLProgram, create_question, QuestionType

# Create a quantum program
program = QPLProgram("My First Bell State")

7.3 Creating Quantum Systems

In QPL, you don’t create individual “qubits” - you create quantum systems that can then be entangled into relations:

# Create two quantum systems
qubit_a = program.create_system()
qubit_b = program.create_system()

print(f"Created systems: {qubit_a.system_id} and {qubit_b.system_id}")

Each system gets a unique ID. At this point, they’re independent - not yet entangled.

7.4 Entangling Systems

Here’s where QPL’s philosophy shines. Instead of applying gates like CNOT, you directly express the relationship you want:

# Create a Bell pair (maximally entangled state)
bell_pair = program.entangle(qubit_a, qubit_b)

print(f"Bell state created!")
print(f"State vector shape: {bell_pair.state.shape}")
print(f"Entanglement entropy: {bell_pair.entanglement_entropy:.3f}")

Output:

Bell state created!
State vector shape: (4,)
Entanglement entropy: 1.000

The entropy of 1.0 confirms this is maximally entangled.

7.5 Understanding the State

Let’s look at the actual quantum state:

import numpy as np

print("Bell state vector:")
print(bell_pair.state)
print()
print("In basis notation:")
print(f"|00⟩: {bell_pair.state[0]:.3f}")
print(f"|01⟩: {bell_pair.state[1]:.3f}")
print(f"|10⟩: {bell_pair.state[2]:.3f}")
print(f"|11⟩: {bell_pair.state[3]:.3f}")

Output:

Bell state vector:
[0.707+0j  0.+0j  0.+0j  0.707+0j]

In basis notation:
|00⟩: 0.707
|01⟩: 0.000
|10⟩: 0.000
|11⟩: 0.707

This is exactly \(\frac{|00\rangle + |11\rangle}{\sqrt{2}}\)! (Note: \(0.707 \approx \frac{1}{\sqrt{2}}\))

7.6 Complete Example

Here’s the complete program you can run:

from qpl import QPLProgram, create_question, QuestionType

def main():
    # Create program
    program = QPLProgram("Complete Bell State Example")

    # Create quantum systems
    qubit_a = program.create_system()
    qubit_b = program.create_system()

    # Entangle into Bell state
    bell_pair = program.entangle(qubit_a, qubit_b)

    # Show state information
    print("=" * 50)
    print("BELL STATE CREATED")
    print("=" * 50)
    print(f"State vector: {bell_pair.state}")
    print(f"Entanglement entropy: {bell_pair.entanglement_entropy:.3f}")
    print(f"Systems: {list(bell_pair.systems)}")

if __name__ == "__main__":
    main()

7.7 What You Learned

Congratulations! You’ve just:

✅ Created your first quantum program in QPL ✅ Generated a maximally entangled Bell state ✅ Understood entanglement entropy ✅ Explored the quantum state vector

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Next: Chapter 8: Understanding GHZ States - Scale to 3+ qubits