A system of two parts lives in the tensor product of their spaces: basis states $\ket{a}\ket{b}$, one from each part, so dimensions multiply. A product state $\ket{\alpha}\ket{\beta}$ gives each part a state of its own. Most superpositions are not products — the singlet of two spins, $(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow})/\sqrt{2}$, is the famous example — and such states are called entangled: the whole has a definite state and neither part does. For mixed states the same line is drawn: independent parts have $\hat{\rho}_{A}\otimes\hat{\rho}_{B}$, mixtures of such products are separable, with correlations no stronger than those of a list of shared instructions, and every other state is entangled.

Send one of a pair of gloves to each of two cities. Each city finds a glove of unknown hand, but the two always differ: correlated, as a separable mixture is. Entangled parts are correlated too, and a later chapter shows their correlations can be stronger than any such shared list allows.

For two parts $A$ and $B$, with $\ket{\uparrow\downarrow}$ written for $\ket{\uparrow}\ket{\downarrow}$:

• $\dim(\mathcal{H}_{A}\otimes\mathcal{H}_{B})=d_{A}d_{B}$; operators on different parts commute. (The chapter on adding angular momenta treats tensor products at length.)

A product has coefficients $a_{i}b_{j}$, and those satisfy the condition at once. Conversely, the condition says the $2\times2$ matrix of coefficients has zero determinant, so its rows are proportional, and a matrix with proportional rows is of the form — a product.

Multiply a product out.

Every product satisfies the condition.

The condition says the matrix of coefficients is singular.

A singular $2\times2$ matrix that is not zero has proportional rows.

So the state factors.

The basis of a two-part system is every pair of basis states, one from each part. The dimensions multiply.

A pure state gives each part a state of its own exactly when it is a product. For two qubits that is when $c_{00}c_{11}=c_{01}c_{10}$.

The singlet of two spins $\tfrac12$ is the difference of the two mixed products, divided by $\sqrt{2}$. Along any common axis its two spins come out opposite.

A mixture of product states is separable. Its parts can be correlated, but only the way two sealed envelopes with matching slips inside are.

Great work! You now know how to:

• describe the state space of two parts and its product states
• test a pure state of two qubits for being a product
• tell entangled states from separable ones