Introduction: Combining Systems
When we have two quantum systems (like two electrons, or spin and position), the combined system lives in a tensor product space. The tensor product \(\otimes\) is how we mathematically "glue" Hilbert spaces together.
The Tensor Product of Vectors
If \(|\psi\rangle \in \mathcal{H}_A\) and \(|\phi\rangle \in \mathcal{H}_B\), their tensor product is:
\[|\psi\rangle \otimes |\phi\rangle \in \mathcal{H}_A \otimes \mathcal{H}_B\]Often written \(|\psi\rangle|\phi\rangle\) or \(|\psi, \phi\rangle\) or \(|\psi\phi\rangle\).
If \(|\psi\rangle = \begin{pmatrix} a \\ b \end{pmatrix}\) and \(|\phi\rangle = \begin{pmatrix} c \\ d \end{pmatrix}\):
\[|\psi\rangle \otimes |\phi\rangle = \begin{pmatrix} ac \\ ad \\ bc \\ bd \end{pmatrix}\]Dimension of Product Spaces
If \(\dim(\mathcal{H}_A) = m\) and \(\dim(\mathcal{H}_B) = n\), then:
\[\dim(\mathcal{H}_A \otimes \mathcal{H}_B) = m \times n\]Two qubits: \(2 \times 2 = 4\) dimensions. Three qubits: \(2^3 = 8\) dimensions.
Worked Examples
Example 1: Two-Qubit Basis
The basis for two qubits is:
\[|00\rangle, |01\rangle, |10\rangle, |11\rangle\]where \(|01\rangle = |0\rangle \otimes |1\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \otimes \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}\)
Example 2: Product State
If qubit A is in \(|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\) and qubit B is in \(|0\rangle\):
\[|+\rangle|0\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |10\rangle)\]This is a product state—it can be factored as A ⊗ B.
Example 3: Entangled State (Bell State)
\[|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)\]This is NOT a product state—it cannot be written as \(|\psi_A\rangle \otimes |\phi_B\rangle\). This is entanglement!
The Quantum Connection
The tensor product structure is why quantum computers are powerful. With \(n\) qubits, the Hilbert space has \(2^n\) dimensions—exponentially large! Entangled states like \(|\Phi^+\rangle\) exhibit correlations that have no classical explanation, enabling quantum teleportation and quantum cryptography.