Bell states

The Bell states are four specific 2-qubit states that are all maximally entangled — concurrence exactly $1$, no room for more correlation between the two qubits. They’re named after the physicist John Bell, who showed in the 1960s that the weird statistical correlations these states produce cannot be reproduced by any classical “hidden variable” model. Whatever is going on in the quantum world, it’s not just classical randomness with extra steps.

Learning the Bell states is a rite of passage. You’ll see them in essentially every quantum protocol from here on.

Meet the four

Here they are:

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

$|\Phi^-\rangle = \frac{|00\rangle - |11\rangle}{\sqrt{2}}$

$|\Psi^+\rangle = \frac{|01\rangle + |10\rangle}{\sqrt{2}}$

$|\Psi^-\rangle = \frac{|01\rangle - |10\rangle}{\sqrt{2}}$

The $\Phi$ states have the two qubits agreeing (both 0 or both 1). The $\Psi$ states have them disagreeing (one 0 and one 1). The $+$ and $-$ signs control the relative phase between the two terms — and they matter, because they tell different Bell states apart.

All four are maximally entangled (concurrence $= 1$). All four are orthogonal to each other — they form a basis of the 2-qubit state space, called the Bell basis. Any 2-qubit state can be written as a combination of Bell states instead of computational basis states, and in some problems the Bell basis is much more natural.

Building $|\Phi^+\rangle$ from scratch

Let’s build the first Bell state step by step. The widget below walks through the construction: start in $|00\rangle$, apply $H$ to qubit 0, then apply $CNOT_{0 \to 1}$. After both steps, you should be looking at $(|00\rangle + |11\rangle)/\sqrt{2}$ with concurrence $1$.

Complete all three steps. Then click “Measure both” — over and over, or use “×50” to zoom through 50 measurements at once. Watch the result column on the right.

Every single measurement gives either $(0, 0)$ or $(1, 1)$. Never $(0, 1)$. Never $(1, 0)$. The two qubits always agree — perfectly, on every single trial.

If you got into the habit (from classical statistics) of saying “well, there’s a 50/50 chance of each of the four outcomes,” you’d be wrong. There’s a 50/50 chance between $(0, 0)$ and $(1, 1)$, and a 0% chance of anything else. The two outcomes are correlated in a way that persists no matter how many times you repeat the measurement.

Critically, this isn’t because the qubits were secretly $(0, 0)$ or $(1, 1)$ all along. Before you measured, each qubit individually had no definite value — it was in a superposition, 50/50. The joint state, though, had perfect correlation already baked in. That’s the key distinction and the hardest one to internalize.

Making the other three Bell states

Once you have a way to build $|\Phi^+\rangle$, the other three follow by tweaking the input. Start from a different 2-qubit computational basis state before the H-then-CNOT ritual, and you get a different Bell state:

You can verify this by hand (or with the widget below). Reset, pick a preset like $|01\rangle$, apply $H_0$, apply $CNOT_{0 \to 1}$. You should land on $\Psi^+$ — the state $(|01\rangle + |10\rangle)/\sqrt{2}$, which is the “disagree” version.

The EPR paradox, briefly

In 1935, Einstein, Podolsky, and Rosen wrote a paper arguing that quantum mechanics must be incomplete, because it predicts the very kind of correlations you just saw. Their reasoning went something like this:

1. Prepare a Bell pair $|\Phi^+\rangle$.
2. Send one qubit to Alice in Tokyo, the other to Bob in Berlin.
3. Alice measures her qubit. Instantly, Bob’s qubit takes on the matching value.
4. But nothing physical can travel faster than light! Therefore, Bob’s qubit must have “already been” in that value all along. Quantum mechanics just doesn’t tell us which, because it’s incomplete.

This intuition is natural and was shared by most physicists for decades. And it’s wrong. In 1964, John Bell proved a theorem (now called Bell’s inequality) that says: if Bob’s qubit was “already” in a definite value determined by a local classical mechanism (a “hidden variable”), then the measurement correlations between Alice and Bob would satisfy a specific inequality. Quantum mechanics predicts a violation of that inequality. Experiments have confirmed the violation over and over.

So one of Einstein’s unstated assumptions had to give. Either signals can travel faster than light (which would break special relativity) or Bob’s qubit genuinely did not have a definite value until Alice’s measurement happened. Almost all physicists accept the second conclusion.

We’ll look at Bell’s inequality properly in Module 5. For now, the takeaway is: Bell states exhibit correlations that no classical theory can reproduce, and that’s not a loose philosophical statement — it’s a mathematical theorem backed by experiment.

Why Bell states matter in practice

Bell pairs are the currency of most quantum protocols:

• Quantum teleportation uses a Bell pair to “send” a qubit’s quantum state from Alice to Bob using only classical messages.
• Superdense coding uses a Bell pair to send two classical bits per qubit.
• Quantum key distribution uses Bell-pair correlations to detect eavesdroppers — any attempt to measure in transit disturbs the correlations in a detectable way.
• Device-independent quantum cryptography uses Bell inequality violations as a proof that your hardware is actually quantum.

All of these protocols rest on the same key fact: the correlations between Bell-paired qubits are perfect and unfakeable by classical means.

What’s next — and what you’ve done

You have now completed Module 4. Take a moment to appreciate this: you can write down, build, and reason about entangled 2-qubit states. You know the four Bell states by name and how to prepare each one. You know the algebraic test for entanglement. You’ve seen with your own measurements that Bell correlations are perfect.

Module 5 will close the loop on entanglement’s strangeness: we’ll look at Bell’s inequality and see, with a concrete game, why the correlations you just produced cannot be explained by any classical theory. Module 6 takes entanglement and builds quantum teleportation and superdense coding on top of it. Module 7 starts the algorithms — Deutsch-Jozsa, Grover, Shor — all of which use entangled superpositions as their raw material.

From here, it’s all payoff.