Postulate 5: Composite Systems

The first four postulates describe a single system. The fifth tells us how to build a description of several systems together — two qubits, an atom and a field, a whole register. The rule is the tensor product, and from it springs entanglement, the resource that makes quantum information distinctive.

The composite-system postulate

Postulate 5. The state space of a composite physical system is the tensor product of the state spaces of the component systems. If system $A$ has space $\mathcal{H}_A$ and system $B$ has space $\mathcal{H}_B$ , the joint space is
$\mathcal{H}_{AB} = \mathcal{H}_A \otimes \mathcal{H}_B.$
If $A$ is prepared in $|\psi\rangle_A$ and $B$ in $|\phi\rangle_B$ independently, the joint state is the product state $|\psi\rangle_A \otimes |\phi\rangle_B$ .

Dimensions multiply

A key consequence: dimensions multiply, not add. If $\dim\mathcal{H}_A = d_A$ and $\dim\mathcal{H}_B = d_B$ , then $\dim\mathcal{H}_{AB} = d_A d_B$ . For $n$ qubits the joint space has dimension $2^n$ . This exponential growth — $2^{300}$ exceeds the number of atoms in the observable universe by the time $n = 300$ — is why quantum systems are so expensive to simulate classically and so powerful as computational devices.

A basis for the joint space is the set of tensor products of basis vectors. For two qubits:

\{\,|0\rangle\otimes|0\rangle,\; |0\rangle\otimes|1\rangle,\; |1\rangle\otimes|0\rangle,\; |1\rangle\otimes|1\rangle\,\} = \{\,|00\rangle, |01\rangle, |10\rangle, |11\rangle\,\}.

Tensor-product mechanics

The tensor product is bilinear and distributes over addition. Concretely, with column vectors,

\begin{pmatrix} a_0 \\ a_1 \end{pmatrix} \otimes \begin{pmatrix} b_0 \\ b_1 \end{pmatrix} = \begin{pmatrix} a_0 b_0 \\ a_0 b_1 \\ a_1 b_0 \\ a_1 b_1 \end{pmatrix}.

For example $|+\rangle\otimes|0\rangle = \tfrac{1}{\sqrt2}(|0\rangle + |1\rangle)\otimes|0\rangle = \tfrac{1}{\sqrt2}(|00\rangle + |10\rangle)$ . Inner products factor across the tensor product: $\big(\langle\psi_1|\otimes\langle\phi_1|\big)\big(|\psi_2\rangle\otimes|\phi_2\rangle\big) = \langle\psi_1|\psi_2\rangle\,\langle\phi_1|\phi_2\rangle$ , which is what lets us compute joint probabilities.

Entangled states: not every state is a product

The decisive feature of the tensor product is that most joint states are not product states. Consider the Bell state

|\Phi^+\rangle = \frac{1}{\sqrt2}\big(|00\rangle + |11\rangle\big).

Suppose it factored as $(\alpha|0\rangle + \beta|1\rangle)\otimes(\gamma|0\rangle + \delta|1\rangle)$ . Expanding gives amplitudes $\alpha\gamma$ for $|00\rangle$ , $\alpha\delta$ for $|01\rangle$ , $\beta\gamma$ for $|10\rangle$ , $\beta\delta$ for $|11\rangle$ . Matching $|01\rangle$ and $|10\rangle$ to zero forces $\alpha\delta = 0$ and $\beta\gamma = 0$ , but then either $\alpha\gamma$ or $\beta\delta$ also vanishes — contradicting that both $|00\rangle$ and $|11\rangle$ have amplitude $1/\sqrt2$ . No factorization exists. Such states are entangled: the subsystems have no independent state of their own.

Operators on composite systems

Operators also tensor. A gate $\hat{U}_A$ acting only on $A$ becomes $\hat{U}_A \otimes \hat{I}_B$ on the joint space, leaving $B$ alone. Local operations can never create entanglement from a product state — entanglement requires an interaction term, a genuinely two-body gate such as CNOT, that is not of the form $\hat{U}_A\otimes\hat{U}_B$ . This is exactly why the controlled gates of the simulator are the ingredients that build Bell states out of product states.