Tensor Product Exercises

These exercises practice finite-dimensional tensor products for distinguishable subsystems. They use the two-qubit basis order

$$\lvert00\rangle,\quad \lvert01\rangle,\quad \lvert10\rangle,\quad \lvert11\rangle,$$

with subsystem $A$ written first and subsystem $B$ second. Identity factors are written explicitly when an operator acts only on one subsystem.

For identical particles, do not use these exercises as the final word: after forming tensor products one must impose exchange symmetry. See the identical-particle and Fock-space pages for that layer.

Conventions

For qubits, use

$$I= \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}, \qquad X= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix},$$

and

$$Y= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, \qquad Z= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}.$$

Thus

$$Z\lvert0\rangle=\lvert0\rangle, \qquad Z\lvert1\rangle=-\lvert1\rangle.$$

Exercises

1. Let $\mathcal H_A$ have basis $\{\lvert0\rangle,\lvert1\rangle,\lvert2\rangle\}$ and let $\mathcal H_B$ have basis $\{\lvert g\rangle,\lvert e\rangle\}$. Construct a product basis of $\mathcal H_A\otimes\mathcal H_B$ and give the dimension.

Solution

Pair every basis vector of $A$ with every basis vector of $B$. With $A$ listed first, a natural order is

$$\lvert0g\rangle,\quad \lvert0e\rangle,\quad \lvert1g\rangle,\quad \lvert1e\rangle,\quad \lvert2g\rangle,\quad \lvert2e\rangle.$$

The dimension is

$$\dim(\mathcal H_A\otimes\mathcal H_B) = 3\cdot2 = 6.$$

2. Expand the product state

$$\bigl(\alpha\lvert0\rangle+\beta\lvert1\rangle\bigr) \otimes \bigl(\gamma\lvert0\rangle+\delta\lvert1\rangle\bigr)$$

in the standard two-qubit basis order.

Solution

Distribute the tensor product:

$$\begin{aligned} \lvert\psi\rangle &= \alpha\gamma\lvert00\rangle +\alpha\delta\lvert01\rangle \\ &\quad +\beta\gamma\lvert10\rangle +\beta\delta\lvert11\rangle. \end{aligned}$$

In the basis order $\lvert00\rangle,\lvert01\rangle,\lvert10\rangle,\lvert11\rangle$, the coordinate column is

$$\begin{pmatrix} \alpha\gamma\\ \alpha\delta\\ \beta\gamma\\ \beta\delta \end{pmatrix}.$$

The coefficient array factors as $c_{ij}=a_i b_j$, so it has rank one when written as a $2\times2$ coefficient matrix.

3. Decide whether

$$\lvert\psi\rangle = \frac{1}{\sqrt{10}} \bigl( \lvert00\rangle +2\lvert01\rangle +\lvert10\rangle +2\lvert11\rangle \bigr)$$

is a product state. If it is, factor it.

Solution

The coefficient matrix in the $A,B$ order is

$$C = \frac{1}{\sqrt{10}} \begin{pmatrix} 1&2\\ 1&2 \end{pmatrix}.$$

The two rows are proportional, so the matrix has rank one. Therefore the state is a product state. One factorization is

$$\lvert\psi\rangle = \frac{\lvert0\rangle+\lvert1\rangle}{\sqrt2} \otimes \frac{\lvert0\rangle+2\lvert1\rangle}{\sqrt5}.$$

Multiplying the two normalized factors gives the four amplitudes $1/\sqrt{10}$, $2/\sqrt{10}$, $1/\sqrt{10}$, and $2/\sqrt{10}$.

4. Compute the matrices of $Z\otimes I$, $I\otimes X$, and $Z\otimes Z$ in the standard two-qubit basis order.

Solution

The first operator acts with $Z$ on the first qubit:

$$Z\otimes I = \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{pmatrix}.$$

The second operator flips the second qubit:

$$I\otimes X = \begin{pmatrix} 0&1&0&0\\ 1&0&0&0\\ 0&0&0&1\\ 0&0&1&0 \end{pmatrix}.$$

The product operator multiplies the $Z$ eigenvalues:

$$Z\otimes Z = \begin{pmatrix} 1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&1 \end{pmatrix}.$$

5. Prove that local operators on different subsystems commute:

$$[A\otimes I_B,\ I_A\otimes B]=0.$$

Solution

Use the product rule for tensor-product operators:

$$(A\otimes I_B)(I_A\otimes B) = A\otimes B.$$

In the opposite order,

$$(I_A\otimes B)(A\otimes I_B) = A\otimes B.$$

The two products are the same, so their difference is zero:

$$[A\otimes I_B,\ I_A\otimes B]=0.$$

6. Compute the commutator

$$[X\otimes I,\ Z\otimes Z].$$

Solution

Because the second factor is the identity in the first operator,

$$[X\otimes I,\ Z\otimes Z] = [X,Z]\otimes Z.$$

The Pauli commutator is

$$[X,Z] = -2iY.$$

Therefore

$$[X\otimes I,\ Z\otimes Z] = -2i\,Y\otimes Z.$$

This shows a local operator need not commute with an interaction operator.

7. Diagonalize the two-qubit Hamiltonian

$$H = \omega_A Z\otimes I +\omega_B I\otimes Z +J Z\otimes Z.$$

Solution

All three terms are diagonal in the standard product basis, so the product basis vectors are eigenvectors. Let $z_0=+1$ and $z_1=-1$. For $\lvert ab\rangle$,

$$E_{ab} = \omega_A z_a +\omega_B z_b +J z_a z_b.$$

Thus

$$\begin{aligned} E_{00}&=\omega_A+\omega_B+J,\\ E_{01}&=\omega_A-\omega_B-J,\\ E_{10}&=-\omega_A+\omega_B-J,\\ E_{11}&=-\omega_A-\omega_B+J. \end{aligned}$$

The coupling shifts energies according to whether the two $Z$ eigenvalues are aligned or anti-aligned.

8. The controlled-phase operator has matrix

$$U_{\mathrm{CZ}} = \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&-1 \end{pmatrix}.$$

Show that it cannot be written as a single product $A\otimes B$ with diagonal one-qubit matrices $A$ and $B$.

Solution

Suppose

$$A= \begin{pmatrix} a_0&0\\ 0&a_1 \end{pmatrix}, \qquad B= \begin{pmatrix} b_0&0\\ 0&b_1 \end{pmatrix}.$$

Then

$$A\otimes B = \operatorname{diag} (a_0b_0,\ a_0b_1,\ a_1b_0,\ a_1b_1).$$

Matching the first three diagonal entries of $U_{\mathrm{CZ}}$ requires

$$a_0b_0=1,\qquad a_0b_1=1,\qquad a_1b_0=1.$$

These imply $b_1=b_0$ and $a_1=a_0$, hence $a_1b_1=1$. But the fourth diagonal entry of $U_{\mathrm{CZ}}$ is $-1$. This contradiction shows that controlled phase is not a single product operator.

Cross-Links

• Tensor Products of Hilbert Spaces
• Product Bases
• Operators on Composite Systems
• Product States
• Entangled States
• Bell States
• Common Composite States
• Formula Sheet
• Mathematical Toolkit: Tensor Products

References

• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.