A system made of two parts — two particles, or one particle’s position and its spin — needs states that say something about both. If one part is in the state $\ket{a}$ and the other in $\ket{b}$, the whole is in the product state written $\ket{a}\ket{b}$. But every superposition of product states is a state too, and most of them are not products: the parts of such a state have no states of their own. The space of the whole is the tensor product of the two spaces. Its basis is every pair of basis states, one from each part, so dimensions multiply rather than add. An operator on one part acts on its own factor and leaves the other alone, and operators on different parts commute.

A lock with two wheels of ten digits has a hundred settings: every digit of one wheel with every digit of the other. The state space of two systems grows the same way, by multiplying — though a quantum state can also be a superposition of settings, which no lock can be.

For parts with state spaces $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$, the whole lives in $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, spanned by the products of their basis states.

• The product is linear in each factor: $(\alpha\ket{a}+\beta\ket{a'})\ket{b}=\alpha\ket{a}\ket{b}+\beta\ket{a'}\ket{b}$.

Let the two operators act one after the other on a product state, in both orders. Each changes only its own factor, so both orders give the same state. Product states span the whole space, so by linearity the operators commute on every state.

Each operator changes its own factor only.

$\hat{B}$ first, then $\hat{A}$.

$\hat{A}$ first, then $\hat{B}$: the same state.

The two agree on every product state, and product states span the space.

The basis of a combined system is every pair of basis states, one from each part. A part with three states and a part with two give six basis states for the whole.

Write $\ket{00}$ for $\ket{0}\ket{0}$, and so on. A product of two qubit states, $(a\ket{0}+b\ket{1})(c\ket{0}+d\ket{1})$, expands into four terms whose coefficients are , , and .

Superpositions of product states are states too, and most of them cannot be factored. In such a state neither part has a state of its own.

An operator that belongs to the first part acts on the first factor of a product state and leaves the second factor untouched.

Great work! You now know how to:

• build the state space of a composite system as a tensor product
• tell product states from states that do not factor
• show that operators on different parts commute