2.1.1 Tensor Product

2.1.1 Tensor Product#

Prompts

  • What is a tensor product of two Hilbert spaces, and how do basis states combine? For a two-particle system where each particle has \(d\) basis states, how many basis states does the combined system have?

  • How do operators acting on individual particles extend to the full multi-particle Hilbert space? Why do operators on different particles always commute?

  • What is a Pauli string, and why do Pauli strings form a complete basis for all multi-qubit Hermitian operators?

  • How does the dimension of the \(N\)-particle Hilbert space grow with \(N\)? What does this exponential growth imply for classical simulation of quantum systems?

Lecture Notes#

Overview#

Chapter 1 described a single qubit. To describe multiple quantum particles—atoms, photons, electrons in a solid—we need a systematic way to combine Hilbert spaces. The tensor product provides this: it constructs the full many-body state space from single-particle spaces, and its dimension grows exponentially with particle number. In this lesson we set up the many-body framework (states, operators, Pauli strings) that will be used throughout the rest of the course.

Many-Body State Space#

Tensor Product of Hilbert Spaces#

Tensor Product

When systems A and B have Hilbert spaces \(\mathcal{H}_A\) and \(\mathcal{H}_B\), the composite system lives in

\[\mathcal{H}_{AB} = \mathcal{H}_A \otimes \mathcal{H}_B\]

If \(\{\vert a_i\rangle\}\) is a basis for A and \(\{\vert b_j\rangle\}\) a basis for B, then \(\{\vert a_i\rangle \otimes \vert b_j\rangle\}\) is a complete orthonormal basis for \(\mathcal{H}_{AB}\):

(25)#\[\langle a_i b_j \vert a_{i'} b_{j'} \rangle = \langle a_i \vert a_{i'}\rangle \langle b_j \vert b_{j'}\rangle = \delta_{ii'}\delta_{jj'}\]

Two Qubits#

Each qubit has basis \(\{\vert 0\rangle, \vert 1\rangle\}\). The two-qubit space \(\mathcal{H} = \mathbb{C}^2 \otimes \mathbb{C}^2\) has dimension \(2 \times 2 = 4\), with basis:

\[\vert 00\rangle,\quad \vert 01\rangle,\quad \vert 10\rangle,\quad \vert 11\rangle\]

where \(\vert ij\rangle \equiv \vert i\rangle_A \otimes \vert j\rangle_B\). Orthonormality: \(\langle i_1 j_1 \vert i_2 j_2\rangle = \delta_{i_1 i_2}\delta_{j_1 j_2}\).

Example: Tensor Product of Two Single-Qubit States

Let

\[ \vert\psi\rangle_A = \alpha\vert0\rangle + \beta\vert1\rangle, \qquad \vert\phi\rangle_B = \gamma\vert0\rangle + \delta\vert1\rangle. \]

Algebraic approach (distribute over superposition):

\[\begin{split} \begin{split} \vert\psi\rangle_A\otimes\vert\phi\rangle_B &=(\alpha\vert0\rangle+\beta\vert1\rangle)\otimes(\gamma\vert0\rangle+\delta\vert1\rangle)\\ &=\alpha\gamma\vert00\rangle+\alpha\delta\vert01\rangle+\beta\gamma\vert10\rangle+\beta\delta\vert11\rangle. \end{split} \end{split}\]

Vector approach (Kronecker product intuition):

\[\begin{split} \vert\psi\rangle_A=\begin{pmatrix}\alpha\\\beta\end{pmatrix}, \qquad \vert\phi\rangle_B=\begin{pmatrix}\gamma\\\delta\end{pmatrix}, \end{split}\]
\[\begin{split} \vert\psi\rangle_A\otimes\vert\phi\rangle_B =\begin{pmatrix}\alpha\,\vert\phi\rangle_B\\\beta\,\vert\phi\rangle_B\end{pmatrix} =\begin{pmatrix}\alpha\begin{pmatrix}\gamma\\\delta\end{pmatrix}\\\beta\begin{pmatrix}\gamma\\\delta\end{pmatrix}\end{pmatrix} =\begin{pmatrix}\alpha\gamma\\\alpha\delta\\\beta\gamma\\\beta\delta\end{pmatrix}, \end{split}\]

which is exactly the coefficient vector of \(\alpha\gamma\vert00\rangle+\alpha\delta\vert01\rangle+\beta\gamma\vert10\rangle+\beta\delta\vert11\rangle\).

General Case: \(N\) Particles#

For \(N\) particles, each with \(d\) single-particle states \(\vert\alpha\rangle\) (\(\alpha = 1, \ldots, d\)), the composite space is \(\mathcal{H}^{\otimes N} = \mathcal{H}_1 \otimes \cdots \otimes \mathcal{H}_N\), with basis:

\[\vert\alpha_1 \alpha_2 \cdots \alpha_N\rangle = \vert\alpha_1\rangle \otimes \vert\alpha_2\rangle \otimes \cdots \otimes \vert\alpha_N\rangle\]

Here \(\alpha\) is a generic single-particle quantum number: it can label momentum \(\boldsymbol{k}\), position \(\boldsymbol{r}\), spin \(s\), an energy level, or any other set of quantum numbers that specifies a single-particle mode.

Hilbert Space Dimension

(26)#\[\dim(\mathcal{H}^{\otimes N}) = d^N\]

This grows exponentially with \(N\). For \(N = 300\) qubits: \(2^{300} \approx 10^{90}\) dimensions—more than the number of atoms in the observable universe.

A general state expands as:

(27)#\[\vert\Psi\rangle = \sum_{\alpha_1, \ldots, \alpha_N} \Psi_{\alpha_1 \cdots \alpha_N}\;\vert\alpha_1 \cdots \alpha_N\rangle\]

with expansion coefficients satisfying \(\sum \vert \Psi_{\alpha_1 \cdots \alpha_N}\vert^2 = 1\).

Many-Body Operators#

Single-Body Operators#

An operator \(\hat{A}\) acting on particle \(i\) alone extends to the full space by tensoring with identity \(\hat{I}\) on all other particles:

(28)#\[\hat{A}_i = \hat{I}_1 \otimes \cdots \otimes \hat{A} \otimes \cdots \otimes \hat{I}_N \quad (\hat{A}\text{ in position }i)\]

Its action on basis states:

\[\hat{A}_i \vert\alpha_1 \cdots \alpha_N\rangle = \vert\alpha_1\rangle \cdots (\hat{A}\vert\alpha_i\rangle) \cdots \vert\alpha_N\rangle\]

Example: \(\hat X_1 = \hat X\otimes\hat I\) on Two Qubits

Use the ordered basis \(\{\vert00\rangle,\vert01\rangle,\vert10\rangle,\vert11\rangle\}\).

Algebraic approach (operator action on kets):

\[\begin{split} \begin{split} \hat X_1\vert00\rangle&=(\hat X\vert0\rangle)\otimes(\hat I\vert0\rangle)=\vert10\rangle,\\ \hat X_1\vert01\rangle&=(\hat X\vert0\rangle)\otimes(\hat I\vert1\rangle)=\vert11\rangle,\\ \hat X_1\vert10\rangle&=(\hat X\vert1\rangle)\otimes(\hat I\vert0\rangle)=\vert00\rangle,\\ \hat X_1\vert11\rangle&=(\hat X\vert1\rangle)\otimes(\hat I\vert1\rangle)=\vert01\rangle. \end{split} \end{split}\]

So \(\hat X_1\) flips the first qubit and leaves the second unchanged.

Matrix approach (Kronecker product):

\[\begin{split} \hat X\otimes\hat I= \begin{pmatrix}0&1\\1&0\end{pmatrix} \otimes \begin{pmatrix}1&0\\0&1\end{pmatrix} = \begin{pmatrix} 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0 \end{pmatrix}. \end{split}\]

Question. Show similarly that

\[\begin{split} \hat X_2=\hat I\otimes\hat X= \begin{pmatrix} 0&1&0&0\\ 1&0&0&0\\ 0&0&0&1\\ 0&0&1&0 \end{pmatrix}, \end{split}\]

which flips the second qubit while leaving the first unchanged.

Operators on Different Particles Commute

(29)#\[[\hat{A}_i, \hat{B}_j] = 0 \quad \text{for } i \neq j\]

This is because they act on different tensor factors.

Two-Body Operators#

An operator coupling particles \(i\) and \(j\) acts nontrivially on both tensor factors:

\[\hat{V}_{ij} = \hat{I}_1 \otimes \cdots \otimes \hat{V} \otimes \cdots \otimes \hat{I}_N \quad (\hat{V}\text{ acts on positions }i,j)\]

Example: Constructing \(\hat{X}_1\hat{X}_2\)

Problem. For two qubits, compute \(\hat{X}_1\hat{X}_2\) in matrix form and verify it matches \(\hat{X}\otimes\hat{X}\).

Solution. First write

\[\hat{X}_1 = \hat{X}\otimes\hat{I}, \qquad \hat{X}_2 = \hat{I}\otimes\hat{X}.\]

Using the computational basis \(\{\vert00\rangle,\vert01\rangle,\vert10\rangle,\vert11\rangle\}\),

\[\begin{split}\hat{X}_1= \begin{pmatrix} 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0 \end{pmatrix}, \qquad \hat{X}_2= \begin{pmatrix} 0&1&0&0\\ 1&0&0&0\\ 0&0&0&1\\ 0&0&1&0 \end{pmatrix}.\end{split}\]

Now multiply the two matrices:

\[\begin{split}\hat{X}_1\hat{X}_2= \begin{pmatrix} 0&0&0&1\\ 0&0&1&0\\ 0&1&0&0\\ 1&0&0&0 \end{pmatrix}.\end{split}\]

If we construct \(\hat{X}\otimes\hat{X}\) directly,

\[\begin{split}\hat{X}\otimes\hat{X} = \begin{pmatrix} 0&0&0&1\\ 0&0&1&0\\ 0&1&0&0\\ 1&0&0&0 \end{pmatrix},\end{split}\]

which is exactly the same matrix. This works because tensor products and matrix products satisfy the mixed-product rule $\( (\hat{A}\otimes\hat{B})(\hat{C}\otimes\hat{D})=(\hat{A}\hat{C})\otimes(\hat{B}\hat{D}), \)\( so here \)(\hat{X}\otimes\hat{I})(\hat{I}\otimes\hat{X})=\hat{X}\otimes\hat{X}$.

General Operator Expansion#

Any operator on the \(N\)-particle space can be expanded in the basis of ket-bra products:

\[\hat{O} = \sum_{\{\alpha\},\{\beta\}} O_{\alpha_1\cdots\alpha_N,\,\beta_1\cdots\beta_N}\;\vert\alpha_1\cdots\alpha_N\rangle\langle\beta_1\cdots\beta_N\vert\]

For multi-qubit systems, a more natural basis is given by Pauli strings.

Pauli Strings#

Pauli String Expansion

Any Hermitian operator on \(N\) qubits can be written as:

(30)#\[\hat{O} = \sum_{\{s_i\}} c_{\{s_i\}} \bigotimes_{i=1}^N \hat{\sigma}_i^{s_i}, \qquad \hat{\sigma}_i^{s_i} \in \{\hat{I},\, \hat{X},\, \hat{Y},\, \hat{Z}\}\]

with \(c_{\{s_i\}} \in \mathbb{R}\). There are \(4^N\) Pauli strings; they form an orthogonal basis for all \(2^N \times 2^N\) Hermitian matrices.

Each Pauli string \(\bigotimes_i \hat{\sigma}_i^{s_i}\) is itself a tensor product of single-qubit operators. The decomposition is unique because the \(4^N\) Pauli strings are orthogonal under the trace inner product \(\langle \hat{A}, \hat{B}\rangle = \mathrm{Tr}(\hat{A}^\dagger \hat{B})\).

Example: Pauli String Decomposition

Problem. Decompose the projector \(\hat{P} = \vert 00\rangle\langle 00\vert\) into Pauli strings.

Solution. Use \(\vert 0\rangle\langle 0\vert = \tfrac{1}{2}(\hat{I} + \hat{Z})\):

\[\vert 00\rangle\langle 00\vert = \vert 0\rangle\langle 0\vert \otimes \vert 0\rangle\langle 0\vert = \tfrac{1}{4}(\hat{I} + \hat{Z}) \otimes (\hat{I} + \hat{Z}) = \tfrac{1}{4}(\hat{I}\hat{I} + \hat{I}\hat{Z} + \hat{Z}\hat{I} + \hat{Z}\hat{Z})\]

Four Pauli strings, each with real coefficient \(\tfrac{1}{4}\).

Discussion: Exponential Complexity

The number of Pauli strings grows as \(4^N\), which is exponential in \(N\). This means specifying a general \(N\)-qubit Hamiltonian requires exponentially many parameters. In practice, physical Hamiltonians are local (each term couples only a few nearby qubits), so only polynomially many Pauli strings have nonzero coefficients. What are the implications for classical simulation of quantum systems?

Summary#

  • Tensor product \(\mathcal{H}^{\otimes N}\) combines single-particle spaces; dimension grows as \(d^N\).

  • Basis states \(\vert\alpha_1 \cdots \alpha_N\rangle\) are products of single-particle states; they form a complete orthonormal set.

  • Single-body operators extend via \(\hat{I}\) on other particles; operators on different particles commute.

  • Two-body operators couple pairs of particles, creating correlations.

  • Pauli strings \(\bigotimes_i \hat{\sigma}_i^{s_i}\) form an orthogonal basis for all multi-qubit Hermitian operators, with real coefficients.

See Also

Homework#

1. System A has dimension \(d_A = 3\) (basis \(\vert 1\rangle, \vert 2\rangle, \vert 3\rangle\)) and system B has \(d_B = 2\) (basis \(\vert 1\rangle, \vert 2\rangle\)). What is \(\dim(\mathcal{H}_A \otimes \mathcal{H}_B)\)? List all basis states.

2. Verify that \(\{\vert 00\rangle, \vert 01\rangle, \vert 10\rangle, \vert 11\rangle\}\) form an orthonormal basis using the tensor product inner product \(\langle \alpha_1 \alpha_2 \vert \beta_1 \beta_2\rangle = \langle\alpha_1\vert\beta_1\rangle\langle\alpha_2\vert\beta_2\rangle\).

3. Write the \(4 \times 4\) matrix for \(\hat{Z} \otimes \hat{I}\) in the ordered basis \(\{\vert 00\rangle, \vert 01\rangle, \vert 10\rangle, \vert 11\rangle\}\).

4. For a product state \(\vert\psi\rangle = \vert\psi_A\rangle \otimes \vert\psi_B\rangle\), prove that \(\langle\psi\vert \hat{A} \otimes \hat{B}\vert\psi\rangle = \langle\psi_A\vert\hat{A}\vert\psi_A\rangle\langle\psi_B\vert\hat{B}\vert\psi_B\rangle\).

5. Prove that \([\hat{A}_i, \hat{B}_j] = 0\) for \(i \neq j\), where \(\hat{A}_i = \hat{I} \otimes \cdots \otimes \hat{A} \otimes \cdots \otimes \hat{I}\).

6. The Heisenberg interaction is \(\hat{H} = J(\hat{X}\otimes\hat{X} + \hat{Y}\otimes\hat{Y} + \hat{Z}\otimes\hat{Z})\). (a) Write \(\hat{H}\) as a \(4\times 4\) matrix. (b) How many Pauli strings appear?

7. Decompose \(\vert 00\rangle\langle 00\vert\) into Pauli strings \(\hat{\sigma}_1^{s_1} \otimes \hat{\sigma}_2^{s_2}\) with \(\hat{\sigma}^s \in \{\hat{I}, \hat{X}, \hat{Y}, \hat{Z}\}\). (Hint: \(\vert 0\rangle\langle 0\vert = \tfrac{1}{2}(\hat{I} + \hat{Z})\).)

8. For \(N\) qubits, how many independent real parameters specify (a) a general normalized state in \((\mathbb{C}^2)^{\otimes N}\), and (b) a normalized product state \(\vert\psi_1\rangle \otimes \cdots \otimes \vert\psi_N\rangle\)? What does the difference represent?