What Is Entanglement?

The leap from one qubit to two introduces one of the most striking features of quantum mechanics: entanglement. Understanding entanglement starts with a question — can a two-qubit state be pulled apart?

Product states and separability

A general two-qubit state lives in a four-dimensional space spanned by $|00\rangle$ , $|01\rangle$ , $|10\rangle$ , and $|11\rangle$ :

$|\psi\rangle = \alpha_{00}|00\rangle + \alpha_{01}|01\rangle + \alpha_{10}|10\rangle + \alpha_{11}|11\rangle.$

A state is called separable (or a product state) if it can be written as the tensor product of two independent single-qubit states:

= \big(a|0\rangle + b|1\rangle\big)\otimes\big(c|0\rangle + d|1\rangle\big). $$ Expanding that product gives $ac|00\rangle + ad|01\rangle + bc|10\rangle + bd|11\rangle$, so the four amplitudes must satisfy a very specific relationship: $\alpha_{00}\,\alpha_{11} = \alpha_{01}\,\alpha_{10}$. This is the separability condition. If no factoring exists — if the amplitudes fail that test — the state is **entangled**. ## A concrete example Consider the equal superposition of $|00\rangle$ and $|11\rangle$: $$ |\Phi^+\rangle = \tfrac{1}{\sqrt{2}}\big(|00\rangle + |11\rangle\big). $$ Here $\alpha_{00} = \alpha_{11} = \tfrac{1}{\sqrt{2}}$ and $\alpha_{01} = \alpha_{10} = 0$. The separability condition demands $\alpha_{00}\,\alpha_{11} = \alpha_{01}\,\alpha_{10}$, but the left side is $\tfrac{1}{2}$ while the right side is $0$. They disagree, so no product state can reproduce $|\Phi^+\rangle$. Now contrast that with $\tfrac{1}{\sqrt{2}}(|00\rangle + |01\rangle)$. Factor out qubit 0: $$ \tfrac{1}{\sqrt{2}}(|00\rangle + |01\rangle) = |0\rangle \otimes \tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle). $$ Qubit 0 is simply $|0\rangle$ and qubit 1 is in an independent superposition — the two qubits do not "know" about each other, so this state is separable. <Callout type="tip"> Entanglement is not the same as correlation. Classical coins can be correlated (one lands heads iff the other lands tails), but that correlation was set in advance. Entanglement means the joint state genuinely cannot be decomposed into independent parts — there is no "hidden" pre-assigned value that explains the correlations. </Callout> ## Why it matters When you measure one qubit of an entangled pair, the outcome of the other is no longer independent — even if the two qubits are separated by any distance. This is the basis of superdense coding, quantum teleportation, and every quantum-key-distribution protocol. The rest of this module builds entangled states in the circuit model and shows how they behave under measurement. ## Try it The state $\tfrac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$ is one of the four Bell states. Is it entangled or separable? Work through the separability condition and return `true` if it is entangled, `false` if it can be factored. <RunnableExample />