Tensor Product

When two quantum systems are brought together, their joint state lives in a larger Hilbert space built by the tensor product. It explains why quantum computers with $n$ qubits have $2^n$-dimensional state spaces, and why entanglement cannot be described by looking at subsystems individually.

Tensor product of vectors

For qubits $|a\rangle, |b\rangle \in \mathbb{C}^2$, their tensor product $|a\rangle \otimes |b\rangle$ is a vector in $\mathbb{C}^4$ formed by multiplying every component of $|a\rangle$ by the entire vector $|b\rangle$:

$|a\rangle \otimes |b\rangle = \begin{pmatrix} a_0 \\ a_1 \end{pmatrix} \otimes \begin{pmatrix} b_0 \\ b_1 \end{pmatrix} = \begin{pmatrix} a_0 b_0 \\ a_0 b_1 \\ a_1 b_0 \\ a_1 b_1 \end{pmatrix}$

The two-qubit computational basis is $\{|00\rangle, |01\rangle, |10\rangle, |11\rangle\}$, the four standard basis vectors of $\mathbb{C}^4$.

Tensor product of matrices (Kronecker product)

For matrices $A \in \mathbb{C}^{m\times m}$ and $B \in \mathbb{C}^{n\times n}$, the Kronecker product $A \otimes B$ is an $mn \times mn$ matrix where each entry $a_{ij}$ of $A$ is replaced by the block $a_{ij} B$:

$A \otimes B = \begin{pmatrix} a_{00}B & a_{01}B \\ a_{10}B & a_{11}B \end{pmatrix}$

This is how gates on multi-qubit systems are constructed. The Hadamard gate acting only on qubit 1 of a two-qubit system, leaving qubit 2 alone, is:

$H \otimes I = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \end{pmatrix}$

Separable vs entangled states

A state is separable if $|\psi\rangle = |\alpha\rangle \otimes |\beta\rangle$; measuring qubit 1 gives no information about qubit 2. The Bell state $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ is entangled: attempting $a_0 b_0 = \frac{1}{\sqrt{2}}$, $a_0 b_1 = 0$, $a_1 b_0 = 0$, $a_1 b_1 = \frac{1}{\sqrt{2}}$ yields inconsistent equations; no factored form exists.

Exponential growth of Hilbert space

With $n$ qubits the Hilbert space has dimension $2^n$:

Qubits	Dimension	Classical memory (64-bit)
10	1,024	8 KB
30	~1 billion	~8 GB
50	~$10^{15}$	~8 petabytes

Classical simulation of a general $n$-qubit state requires exponential memory, the root of quantum computational advantage.

Code example

import numpy as np

# Single-qubit basis states
ket0 = np.array([1, 0])
ket1 = np.array([0, 1])

# Two-qubit state |01> = |0> tensor |1>
ket01 = np.kron(ket0, ket1)
print("|01> =", ket01)  # [0, 1, 0, 0]

# Bell state |Phi+> = (|00> + |11>) / sqrt(2)
bell = (np.kron(ket0, ket0) + np.kron(ket1, ket1)) / np.sqrt(2)
print("|Phi+> =", bell)

# Gate on first qubit only: H tensor I
H = np.array([[1, 1], [1, -1]]) / np.sqrt(2)
I = np.eye(2)
H_on_first = np.kron(H, I)
print("\nH tensor I:\n", H_on_first)

# Apply to |00>: result is |+>|0> = [1,0,1,0]/sqrt(2)
ket00 = np.kron(ket0, ket0)
print("\n(H tensor I)|00> =", H_on_first @ ket00)

See also

• Qubit
• Entanglement
• Quantum Gate
• Superposition