8. Multi-Qubit: The Bell States

The Bell States

You have analyzed the concept of entanglement (the "what"). Now you need the toolkit (the "how"). The Bell States are not just random entangled vectors. They are the four specific, maximally entangled states that form a complete orthonormal basis for the two-qubit Hilbert space. Just as $|00\rangle, |01\rangle, |10\rangle, |11\rangle$ form the standard "Computational Basis," the four Bell states form the "Bell Basis."

1. The Four Bell States

These are the "North, South, East, and West" of the entangled world.

1. $|\Phi^+\rangle$ (Phi-Plus): The standard Bell state. $$|\Phi^+\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}}$$ (Correlation: Same results. If you measure 0, they measure 0.)
2. $|\Phi^-\rangle$ (Phi-Minus): The phase-flipped version. $$|\Phi^-\rangle = \frac{|00\rangle - |11\rangle}{\sqrt{2}}$$ (Correlation: Same results, but carries a phase difference).
3. $|\Psi^+\rangle$ (Psi-Plus): The "parity" flip. $$|\Psi^+\rangle = \frac{|01\rangle + |10\rangle}{\sqrt{2}}$$ (Correlation: Opposite results. If you measure 0, they measure 1.)
4. $|\Psi^-\rangle$ (Psi-Minus): The "singlet" state (crucial in physics). $$|\Psi^-\rangle = \frac{|01\rangle - |10\rangle}{\sqrt{2}}$$ (Correlation: Opposite results, with a phase difference).

2. The Circuit: How to Make Them

You don't find these in nature; you build them. The recipe is universal for all quantum computers.

The Ingredients:

1. Two qubits.
2. One Hadamard (H) gate.
3. One CNOT gate.

The Process (to create $|\Phi^+\rangle$):

1. Start: Initialize both qubits to $|00\rangle$ (Qubit A is left, Qubit B is right).
2. Superposition: Apply $H$ to Qubit A. $$|00\rangle \xrightarrow{H \otimes I} \frac{(|0\rangle + |1\rangle)}{\sqrt{2}} \otimes |0\rangle = \frac{|00\rangle + |10\rangle}{\sqrt{2}}$$
3. Entanglement: Apply CNOT with Qubit A as Control and Qubit B as Target.
   • $|00\rangle \to |00\rangle$ (Control is 0, do nothing).
   • $|10\rangle \to |11\rangle$ (Control is 1, flip target).
   • Result: $\frac{|00\rangle + |11\rangle}{\sqrt{2}}$

3. Why This Matters: Change of Basis

Because these four states form a valid Basis, you can measure a two-qubit system in the Bell Basis. This is the secret sauce behind Quantum Teleportation and Superdense Coding.

In Teleportation, you don't measure "0" or "1"; you measure "Which of the 4 Bell states are these two qubits in?" The answer tells you how to reconstruct the state on the other side of the universe.

Your Task: Deriving the "Singlet" ($|\Psi^-\rangle$)

To master quantum circuits, you must be able to trace the state vector step-by-step. You want to generate the state $|\Psi^-\rangle = \frac{|01\rangle - |10\rangle}{\sqrt{2}}$.

The Setup:

• Input State: $|11\rangle$ (Qubit A is 1, Qubit B is 1).
• Gate Sequence: Apply $H$ to Qubit A, then apply CNOT (Control A, Target B).

Derive the final state:

1. After H on A: Apply the Hadamard to the first $|1\rangle$. Recall that $H|1\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}}$. What is the combined 2-qubit state vector at this midpoint?
2. After CNOT: Take the result from step 1 and apply the CNOT logic. (Remember: flip the second bit only if the first bit is 1). Does your result match the definition of $|\Psi^-\rangle$ above?