Bell States

Overview

Bell states are four special two-qubit states that are:

• entangled,
• symmetric and easy to recognize,
• and maximally informative examples of what entanglement does to measurement outcomes.

This page defines the four Bell states and analyzes their correlations under measurement, keeping the discussion rule-based and precise.

Intuition

A Bell state is a joint pure state where each individual qubit looks random on its own, but the pair has strong structure.

For example, in one Bell state you measure two qubits and always get matching bits (00 or 11). In another, you always get opposite bits (01 or 10). Importantly:

• this is not because each qubit had a definite classical value all along,
• but because the joint state assigns amplitudes to joint outcomes.

Entanglement lives in the joint description.

Formal Description

The four Bell states are:

$$|\Phi^+\rangle = \tfrac{1}{\sqrt{2}}(|00\rangle+|11\rangle),\quad |\Phi^-\rangle = \tfrac{1}{\sqrt{2}}(|00\rangle-|11\rangle),$$ $$|\Psi^+\rangle = \tfrac{1}{\sqrt{2}}(|01\rangle+|10\rangle),\quad |\Psi^-\rangle = \tfrac{1}{\sqrt{2}}(|01\rangle-|10\rangle).$$

Each state is normalized because it has two basis terms, each with amplitude magnitude $1/\sqrt{2}$, so the total probability sums to 1.

Correlations in the computational basis

If you measure both qubits in the computational basis:

• For $|\Phi^\pm\rangle$, outcomes are $00$ or $11$, each with probability $1/2$.
• For $|\Psi^\pm\rangle$, outcomes are $01$ or $10$, each with probability $1/2$.

Notice that the plus/minus sign does not change these computational-basis probabilities. The sign is a relative phase and becomes visible when you measure in other bases.

“Maximally entangled” (operational meaning)

One operational way to understand “maximally entangled” is:

• each single qubit, viewed alone, has no definite value (it behaves like a fair random bit for many measurements),
• but the pair shows perfect correlations/anti-correlations that cannot be produced by assigning each qubit its own pure state.

We will make the “viewed alone” idea precise in the next page (reduced states).

Worked Example

Take

$$|\Phi^+\rangle = \tfrac{1}{\sqrt{2}}(|00\rangle+|11\rangle).$$

Measure both qubits in the computational basis.

The amplitudes are:

• amplitude of $|00\rangle$ is $1/\sqrt{2}$,
• amplitude of $|11\rangle$ is $1/\sqrt{2}$,
• amplitudes of $|01\rangle$ and $|10\rangle$ are 0.

So Born’s rule gives:

$$P(00)=\left|\tfrac{1}{\sqrt{2}}\right|^2=\tfrac{1}{2},\quad P(11)=\tfrac{1}{2},\quad P(01)=0,\quad P(10)=0.$$

Now look at correlations:

• The first qubit is 0 exactly when the second is 0.
• The first qubit is 1 exactly when the second is 1.

So the outcomes always match, even though each outcome (00 or 11) is individually random.

Turtle Tip

Common Pitfalls

Quick Check

What’s Next

Bell states highlight a key idea: even when the joint state is pure, each subsystem can behave as if it were “random.” Next we’ll formalize what it means to describe a subsystem on its own using reduced states and the idea of a partial trace, without diving into heavy density-matrix machinery.