Building Bell State Circuits

1. The Recipe for a Bell State

The simplest way to generate an entangled state is by creating a Bell state. The most common one, called $|{\Phi }^{+}⟩$, can be created with a two-step "recipe" that uses gates we're already familiar with.

The circuit diagram for this process is surprisingly simple. Let's watch a short video that walks through this exact circuit.

As the video explained, the process involves two main operations:

1. Create Superposition: Apply a Hadamard gate to the first qubit.
2. Create Entanglement: Apply a CNOT gate, using the first qubit as the control and the second as the target.

Let's trace the state of the system step-by-step, just as the video did. We start with both qubits initialized to the $|0⟩$ state.

Step 0: Initial State
Our initial two-qubit state is $|00⟩$. As a tensor product, this is $|0⟩\otimes |0⟩$.

Step 1: Apply Hadamard to Qubit 0
We apply the H-gate to the first qubit ($q_0$) and do nothing (apply the Identity gate, $I$) to the second qubit ($q_1$). The total operation is $H\otimes I$.

Since $H|0⟩=\frac{1}{\sqrt{2}}(|0⟩+|1⟩)$, our state becomes:

At this point, the qubits are in a superposition, but they are not yet entangled. We could still describe the state of each qubit individually.

Step 2: Apply CNOT(0, 1)
Now, we apply the CNOT gate, with $q_0$ as the control and $q_1$ as the target. The CNOT gate flips the target qubit if the control is $|1⟩$. We apply this to our superposition state $|{\psi }_1⟩$ term by term:

• For the $|00⟩$ part: The control $q_0$ is $|0⟩$, so the target $q_1$ is unchanged. The state remains $|00⟩$.
• For the $|10⟩$ part: The control $q_0$ is $|1⟩$, so the target $q_1$ flips from $|0⟩$ to $|1⟩$. The state becomes $|11⟩$.

Combining these results, the final state is:

This final state, $\frac{1}{\sqrt{2}}(|00⟩+|11⟩)$, is our target Bell state, often denoted as $|{\Phi }^{+}⟩$ or $|{\beta }_{00}⟩$.

Notice something profound about this state: it's a superposition of two outcomes, $|00⟩$ and $|11⟩$. If we measure the qubits, we will only ever get the result 00 or 11, each with 50% probability. We will never get 01 or 10. The measurement outcomes are perfectly correlated. This state can no longer be factored into a tensor product of two separate qubit states—this is the signature of entanglement.

2. Building the Bell State Circuit in Qiskit

Now that we understand the theory, let's put it into practice. Your goal is to be able to build and implement quantum algorithms, and this simple two-gate circuit is a fundamental building block.

The following reading provides a step-by-step guide to constructing this exact circuit using Qiskit.

Following the guide, the core Qiskit code to build the Bell state circuit is:

from qiskit import QuantumCircuit




# Create a quantum circuit with 2 qubits and 2 classical bits
qc = QuantumCircuit(2, 2)




# 1. Apply a Hadamard gate to qubit 0, putting it in superposition.
qc.h(0)




# 2. Apply a CNOT gate with qubit 0 as control and qubit 1 as target.
qc.cx(0, 1)




# We'll add measurement gates to see the result.
qc.measure([0, 1], [0, 1])




# Let's see the circuit we built!
qc.draw(output='mpl')

This code generates the following circuit:

3. The Family of Bell States

The $|{\Phi }^{+}⟩$ state we created is just one of four possible Bell states. They are all maximally entangled but have different correlations due to phase and bit-flip variations. The four Bell states are:

• $|{\Phi }^{+}⟩=\frac{1}{\sqrt{2}}(|00⟩+|11⟩)$
• $|{\Phi }^{-}⟩=\frac{1}{\sqrt{2}}(|00⟩-|11⟩)$
• $|{\Psi }^{+}⟩=\frac{1}{\sqrt{2}}(|01⟩+|10⟩)$
• $|{\Psi }^{-}⟩=\frac{1}{\sqrt{2}}(|01⟩-|10⟩)$

You can create any of these states by slightly modifying our original circuit. The key is to change the initial state before applying the Hadamard and CNOT gates. This is where your knowledge of the Pauli gates (especially the X-gate) comes in handy.

From your EE background, you can think of this as changing the input signals to a fixed logic circuit to produce different, well-defined outputs.

To create the other Bell states, you can prepare the initial qubits as follows before the H-gate and CNOT:

• To get $|{\Psi }^{+}⟩$, start with $|10⟩$. Apply qc.x(0).
• To get $|{\Phi }^{-}⟩$, start with $|01⟩$. Apply qc.x(1).
• To get $|{\Psi }^{-}⟩$, start with $|11⟩$. Apply qc.x(0) and qc.x(1).

Conclusion

In this lesson, we successfully constructed our first multi-qubit circuit with a specific purpose: to generate entanglement. This is a significant milestone in our course.

Key Takeaways:

• A Bell state is a two-qubit state that is maximally entangled.
• The standard Bell state $|{\Phi }^{+}⟩=\frac{1}{\sqrt{2}}(|00⟩+|11⟩)$ is created by applying a Hadamard gate to the first qubit followed by a CNOT gate.
• This state exhibits perfect correlation: measuring the qubits will always yield 00 or 11, never 01 or 10.
• We can build this circuit practically in Qiskit using qc.h() and qc.cx().
• The other three Bell states can be generated by preparing different initial states using X-gates before the main H-CNOT sequence.

Preview of the Next Lesson:
We've created this "entangled" state and noted its strange correlations. But what does entanglement truly mean? How is it different from two classical objects that are simply correlated (like a pair of gloves)? In our next lesson, we will dive deep into the concept of quantum entanglement, exploring the features that make it so counter-intuitive and powerful, and learn why it baffled even Einstein.