Entanglement and the Bell state

Entanglement links two qubits so that their results always agree, although each one alone is a fair coin toss. The simplest entangled state is called a Bell state.

It matters because entanglement is a resource that a quantum program uses and an ordinary one cannot. In this lesson you build a Bell state in the lab and in Python, and you test whether a state is entangled.

When two qubits are separate, their joint state is the tensor product of the two states. Multiply every amplitude of the first by every amplitude of the second.

For [a0, a1] and [b0, b1] the result is [a0 x b0, a0 x b1, a1 x b0, a1 x b1]. These are the amplitudes of 00, 01, 10 and 11, in that order.

def tensor(a, b):
    result = []
    for x in a:
        for y in b:
            result.append(x * y)
    return result

# Press Run. Then try tensor([1, 0], [0.6, 0.8]) and see which outcomes can appear.
print(tensor([0.6, 0.8], [1, 0]))

The CNOT gate acts on two qubits. The first is the control. The second is flipped only when the control is 1.

It swaps the amplitudes of 10 and 11 and leaves the other two alone. On a state of certain answers, it works like an if statement.

docker exec jupyter python3 -c "from qsim import *; show(CNOT @ apply(X, zero(2), 0))"

Start from 00. Apply H to the first qubit, which gives 0.707|00> + 0.707|10>. Then apply CNOT.

The 10 term becomes 11, so the state is 0.707|00> + 0.707|11>. That is a Bell state. Only 00 and 11 can appear, each with a chance of 0.5, and 01 and 10 never do.

docker exec jupyter python3 steps/bell.py

docker exec jupyter python3 -c "from qsim import *; print(measure(CNOT @ apply(H, zero(2), 0), 10, 3))"

A state of two qubits is a product state if it is the tensor product of two separate states. In that case the first and last amplitudes, multiplied together, equal the middle two multiplied together.

If those two products differ, the state is entangled: no two separate states can make it.