Quantum Computing Bell States
Bell states are the four simplest and most famous entangled states of two qubits. John Bell studied them in 1964, and experiments on them later earned a Nobel Prize. Nearly every entanglement demonstration, communication protocol, and hardware test starts with a Bell state.
The Four Bell States
| Name | Description | Measured Results |
|---|---|---|
| Phi plus | Equal mix of 00 and 11 | Always matching |
| Phi minus | 00 and 11 with opposite sign | Always matching |
| Psi plus | Equal mix of 01 and 10 | Always opposite |
| Psi minus | 01 and 10 with opposite sign | Always opposite |
Each state gives random results for a single qubit, yet the pair follows a strict rule. Matching states always agree. Opposite states always disagree.
Building Phi Plus
Two gates build the first Bell state from two qubits starting at 0.
q0: |0> ---[ H ]-----*-----
|
q1: |0> -----------[ X ]----
Start: 00
After H: (00 + 10) / root 2
After CNOT: (00 + 11) / root 2 <-- Bell state
Step by Step
- The Hadamard gate places the first qubit in an equal blend.
- The CNOT gate flips the second qubit only when the first qubit holds 1.
- The two qubits now share one joint state with no separate description.
Building the Other Three
Start with the Phi plus circuit and add single-qubit gates before it.
| Target State | Extra Gate Before the Circuit |
|---|---|
| Phi plus | None |
| Phi minus | X on qubit 0 |
| Psi plus | X on qubit 1 |
| Psi minus | X on both qubits |
Check each result with a simulator. Phi states return 00 and 11. Psi states return 01 and 10.
Measuring a Bell State
Run 10 shots of Phi plus: Shot 1: 11 Shot 2: 00 Shot 3: 00 Shot 4: 11 Shot 5: 11 Shot 6: 00 Shot 7: 11 Shot 8: 00 Shot 9: 00 Shot 10: 11 Each qubit looks random. The pair never disagrees.
Distinguishing the Four States
Reverse the building circuit to tell Bell states apart. Apply CNOT first, then Hadamard, then measure both qubits. Each Bell state maps to a unique two-bit code. The method forms the heart of teleportation and dense coding.
| Bell State | Reading After Reverse Circuit |
|---|---|
| Phi plus | 00 |
| Phi minus | 10 |
| Psi plus | 01 |
| Psi minus | 11 |
Why Bell States Matter
- They serve as the standard test for entanglement on new hardware.
- They provide shared randomness for secure communication.
- They supply the resource that teleportation consumes.
- They demonstrate that nature rejects local hidden variable explanations, as Bell proved.
Layman Example: Two Magic Dice
Picture two dice thrown in different cities. Each die shows a random number when you look at it. When the cities compare notes, the dice match every time. Neither die knew its value in advance, and no message traveled between them. Bell states create exactly this kind of perfect agreement.
Key Takeaways
Four states, two gates, and one rule explain the entire family. Practice building Phi plus until it feels automatic. Every advanced entanglement protocol builds on that small circuit.
Common Beginner Mistakes
Beginners often assume that each qubit in a Bell pair secretly holds a fixed value from the start. Experiments prove otherwise. The matching results appear only when the two measurements meet in a comparison. Another frequent mistake swaps the gate order. Applying the CNOT gate before the Hadamard gate creates no entanglement, because the control qubit still holds a plain 0. Qubit labels cause confusion as well, since many toolkits print results with the first qubit on the right. Run small tests, write down each prediction, and compare it with the simulator output. Those habits catch errors quickly and build lasting confidence.
Practice Questions
- Which two gates create Phi plus?
- Which outcomes does Phi plus return?
- What rule do Psi states follow?
The answers are Hadamard and CNOT, 00 and 11, and the two results always disagree.