Tensor Products and Multi-Qubit States

qc/math

Multiple quantum systems combine via the tensor product ($\otimes$), not addition. For two single-qubit states $|\varphi ⟩={\alpha }_0|0⟩+{\alpha }_1|1⟩$ and $|\psi ⟩={\beta }_0|0⟩+{\beta }_1|1⟩$:

$|\varphi ⟩\otimes |\psi ⟩={\sum }_{a,b\in \{0,1\}}{\alpha }_a{\beta }_b|ab⟩$

— bilinear in both arguments. In column-vector form this is the Kronecker product:

$|\varphi ⟩\otimes |\psi ⟩=\left(\begin{array}{c}{\alpha }_0\\ {\alpha }_1\end{array}\right)\otimes \left(\begin{array}{c}{\beta }_0\\ {\beta }_1\end{array}\right)=\left(\begin{array}{c}{\alpha }_0{\beta }_0\\ {\alpha }_0{\beta }_1\\ {\alpha }_1{\beta }_0\\ {\alpha }_1{\beta }_1\end{array}\right)$

The same operation combines operators: $M\otimes N$ acts on the combined space, built the same component-wise way. This is exactly the fact behind Pauli Operators’ “tensor products extend this to any number of qubits” line — an $n$-qubit Pauli string like $XZI$ is literally $X\otimes Z\otimes I$.

Why this makes state spaces explode

Key insight: $n$ single qubits combine into one $2^n$-dimensional space, not $2n$-dimensional — this is the exact mechanism behind why quantum states are classically expensive and why statevector simulation doesn’t scale. Two qubits: 4 amplitudes. Twenty qubits: over a million. This is also structurally why entanglement exists at all: a general $2^n$-dimensional vector doesn’t have to factor as a tensor product of $n$ single-qubit vectors — when it doesn’t, that’s what “entangled” means (see Bell States).

Classical parallel

Multiple classical systems combine via the Cartesian product of their state sets, and the same tensor-product operation combines their probability vectors — the math is identical. The difference isn’t the combination rule, it’s that a quantum joint state can hold superposition and phase relationships a classical joint probability distribution can’t.

from qiskit.quantum_info import Statevector
 
phi = Statevector([1, 0])       # |0>
psi = Statevector([0, 1])       # |1>
combined = phi.tensor(psi)      # |0> ⊗ |1> = |01>

Related

• What is a Qubit
• Pauli Operators
• Hamiltonian Simulation — Why It’s Hard
• Simulators — Statevector vs Shot-Based
• Bell States

Self-Check

• Could you compute $|0⟩\otimes |1⟩$ by hand using the Kronecker product formula?
• Why does $n$ qubits give a $2^n$-dimensional space rather than $2n$?
• Why does “not every vector in the combined space factors as a tensor product” matter for what entanglement means?