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Appendix B Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Norm Calculation of a Vector in \(\mathbb{C}^2\)

Find the norm \(\||\psi\rangle\|\) of the vector \(|\psi\rangle = \begin{pmatrix} 2i \\ 1 - i \end{pmatrix}\).

Hint

The norm is given by \(\||\psi\rangle\| = \sqrt{\langle\psi|\psi\rangle}\), where \(\langle\psi|\psi\rangle = \sum_k |z_k|^2\). Calculate the squared absolute value of each component and add them together. \(|2i|^2 = 4\), \(|1-i|^2 = (1)^2 + (-1)^2 = 2\).

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B-2. Inner Product Calculation in \(\mathbb{C}^2\)

Compute the inner product \(\langle\phi|\psi\rangle\) of the following two vectors.

\[|\psi\rangle = \begin{pmatrix} 1 + i \\ 2 \end{pmatrix}, \quad |\phi\rangle = \begin{pmatrix} 3 \\ i \end{pmatrix}\]
Hint

\(\langle\phi|\psi\rangle = \sum_k \phi_k^* \psi_k\). Note that the complex conjugate is applied to the components of the first argument (the bra side). We have \(\langle\phi| = (3^*,\; i^*) = (3,\; -i)\).

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B-3. Normalization of a 2-Dimensional Vector

Normalize the vector \(|v\rangle = \begin{pmatrix} 1 \\ 1 \\ i \end{pmatrix}\). That is, find the vector \(|u\rangle = \frac{|v\rangle}{\||v\rangle\|}\) satisfying \(\langle u|u\rangle = 1\).

Hint

First compute \(\langle v|v\rangle = |1|^2 + |1|^2 + |i|^2\) to find the norm, then divide each component by that norm.

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B-4. Determining Orthogonality

Determine whether the following two vectors are orthogonal.

\[|a\rangle = \begin{pmatrix} 1 \\ i \end{pmatrix}, \quad |b\rangle = \begin{pmatrix} i \\ 1 \end{pmatrix}\]
Hint

Compute the inner product \(\langle a|b\rangle\) and check whether it equals 0. Note that \(\langle a| = (1^*,\; i^*) = (1,\; -i)\).

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B-5. Calculation of Matrix Elements

Given the orthonormal basis \(|e_1\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\), \(|e_2\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\), an operator \(\hat{A}\) acts as

\[\hat{A}|e_1\rangle = \begin{pmatrix} 2 \\ i \end{pmatrix}, \quad \hat{A}|e_2\rangle = \begin{pmatrix} -i \\ 3 \end{pmatrix}\]

Find the matrix representation \((A_{jk}) = \langle e_j|\hat{A}|e_k\rangle\) of \(\hat{A}\).

Hint

Since \(A_{jk} = \langle e_j|\hat{A}|e_k\rangle\), \(A_{11} = \langle e_1|\hat{A}|e_1\rangle\) is the first component of \(\hat{A}|e_1\rangle\), and \(A_{21} = \langle e_2|\hat{A}|e_1\rangle\) is the second component of \(\hat{A}|e_1\rangle\).

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B-6. Calculating the Hermitian Conjugate

Find the Hermitian conjugate \(A^\dagger\) of the matrix

\[A = \begin{pmatrix} 2 & 1 - i \\ 3i & 4 + 2i \end{pmatrix}\]
Hint

The Hermitian conjugate is the operation of "transposing and taking the complex conjugate of each element." Use \((A^\dagger)_{jk} = A_{kj}^*\).

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B-7. Calculation of a Commutator

Calculate the commutator \([A, B] = AB - BA\) of the following \(2 \times 2\) matrices \(A\) and \(B\).

\[A = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\]
Hint

First compute the matrix products \(AB\) and \(BA\) separately, then take their difference. Use the formula for the product of \(2 \times 2\) matrices: \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} ae+bg & af+bh \\ ce+dg & cf+dh \end{pmatrix}\).

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B-8. Determining Linear Independence

Determine whether the following two vectors in \(\mathbb{C}^2\) are linearly independent.

\[|v_1\rangle = \begin{pmatrix} 1 \\ i \end{pmatrix}, \quad |v_2\rangle = \begin{pmatrix} i \\ -1 \end{pmatrix}\]
Hint

Set up \(c_1|v_1\rangle + c_2|v_2\rangle = \mathbf{0}\) and check whether the only solution is \(c_1 = c_2 = 0\). Alternatively, check whether one vector is a scalar multiple of the other. Does there exist an \(\alpha\) such that \(|v_2\rangle = \alpha |v_1\rangle\)?

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B-9. Calculation of Expansion Coefficients

For the orthonormal basis \(|+\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}\), \(|-\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -1 \end{pmatrix}\), find the expansion coefficients \(c_+ = \langle +|\psi\rangle\) and \(c_- = \langle -|\psi\rangle\) for the vector \(|\psi\rangle = \begin{pmatrix} 3 \\ i \end{pmatrix}\).

Hint

For an orthonormal basis, the expansion coefficients are obtained by \(c_k = \langle e_k|\psi\rangle\) (Eq. (B.19)). Note that \(\langle +| = \frac{1}{\sqrt{2}}(1,\; 1)\) (since this basis has real components, complex conjugation is unnecessary).

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B-10. Verification of the Completeness Relation

Using the basis \(|+\rangle\), \(|-\rangle\) from D9, compute \(|+\rangle\langle +| + |-\rangle\langle -|\) as a \(2 \times 2\) matrix and verify that it equals the identity matrix \(\hat{1}\).

Hint

\(|+\rangle\langle +|\) is the product of the ket \(|+\rangle\) (column vector) and the bra \(\langle +|\) (row vector), resulting in a \(2\times 2\) matrix. We have \(|+\rangle\langle +| = \frac{1}{2}\begin{pmatrix}1\\1\end{pmatrix}(1,\;1) = \frac{1}{2}\begin{pmatrix}1 & 1\\1 & 1\end{pmatrix}\). Similarly, compute \(|-\rangle\langle -|\) and add them together.

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Medium

M-1. Gram–Schmidt Orthogonalization

In \(\mathbb{C}^2\), the following two linearly independent vectors are given:

\[|v_1\rangle = \begin{pmatrix} 1 \\ i \end{pmatrix}, \quad |v_2\rangle = \begin{pmatrix} 1 \\ 1 \end{pmatrix}\]

Apply the Gram–Schmidt orthogonalization procedure (Eqs. (B.22)–(B.24)) to construct an orthonormal basis \(\{|e_1\rangle, |e_2\rangle\}\). Also verify that the resulting \(|e_1\rangle\) and \(|e_2\rangle\) are indeed orthonormal (\(\langle e_1|e_2\rangle = 0\), \(\langle e_1|e_1\rangle = \langle e_2|e_2\rangle = 1\)).

Hint

Step 1: Compute \(|e_1\rangle = |v_1\rangle / \||v_1\rangle\|\). We have \(\||v_1\rangle\| = \sqrt{|1|^2 + |i|^2} = \sqrt{2}\).

Step 2: Compute \(|w_2\rangle = |v_2\rangle - \langle e_1|v_2\rangle |e_1\rangle\) and normalize it. Carefully calculate \(\langle e_1|v_2\rangle\) (remember that the complex conjugate is applied to the components of the first argument).

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M-2. Proof that Eigenvalues of Hermitian Matrices are Real

Let \(\hat{A}\) be a Hermitian matrix (\(\hat{A}^\dagger = \hat{A}\)). Prove that in the eigenvalue equation of \(\hat{A}\)

\[\hat{A}|\lambda\rangle = \lambda|\lambda\rangle \quad (|\lambda\rangle \neq \mathbf{0})\]

the eigenvalue \(\lambda\) is necessarily real.

Hint

Multiply both sides of the eigenvalue equation from the left by \(\langle\lambda|\) to obtain \(\langle\lambda|\hat{A}|\lambda\rangle = \lambda\langle\lambda|\lambda\rangle\). Then, using \(\hat{A}^\dagger = \hat{A}\) and the Hermitian property of the inner product \(\langle\lambda|\hat{A}|\lambda\rangle = \langle\hat{A}^\dagger\lambda|\lambda\rangle = \langle\hat{A}\lambda|\lambda\rangle\), show that \(\langle\lambda|\hat{A}|\lambda\rangle\) is real.

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M-3. Eigenvectors belonging to different eigenvalues of a Hermitian matrix are orthogonal

Let \(|\lambda_1\rangle\) and \(|\lambda_2\rangle\) be eigenvectors belonging to two eigenvalues \(\lambda_1 \neq \lambda_2\) of a Hermitian matrix \(\hat{A}\), respectively. Prove that \(\langle\lambda_1|\lambda_2\rangle = 0\).

Hint

Compare the expression obtained by multiplying both sides of \(\hat{A}|\lambda_2\rangle = \lambda_2|\lambda_2\rangle\) from the left by \(\langle\lambda_1|\) with the expression obtained by taking the Hermitian conjugate of \(\hat{A}|\lambda_1\rangle = \lambda_1|\lambda_1\rangle\) and multiplying from the right by \(|\lambda_2\rangle\). Use the fact that \(\lambda_1, \lambda_2\) are real (the result from S2).

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M-4. Derivation of Commutator Identity

For arbitrary linear operators \(\hat{A}, \hat{B}, \hat{C}\), prove the following identity (the Jacobi identity):

\[[\hat{A}, [\hat{B}, \hat{C}]] + [\hat{B}, [\hat{C}, \hat{A}]] + [\hat{C}, [\hat{A}, \hat{B}]] = 0\]
Hint

Use the definition of the commutator \([\hat{X}, \hat{Y}] = \hat{X}\hat{Y} - \hat{Y}\hat{X}\) to expand each term on the left-hand side. Each term yields 4 terms, such as \([\hat{A}, [\hat{B}, \hat{C}]] = \hat{A}\hat{B}\hat{C} - \hat{A}\hat{C}\hat{B} - \hat{B}\hat{C}\hat{A} + \hat{C}\hat{B}\hat{A}\), so write out all 12 terms and verify that they cancel.

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M-5. Proof of the Schwarz Inequality

For any vectors \(|\psi\rangle\), \(|\phi\rangle\) in an inner product space, prove the Schwarz (Cauchy–Schwarz) inequality

\[|\langle\phi|\psi\rangle|^2 \leq \langle\phi|\phi\rangle \cdot \langle\psi|\psi\rangle\]

Also, state the condition under which equality holds.

Hint

For an arbitrary complex number \(t\), define \(|w\rangle = |\psi\rangle - t|\phi\rangle\) and use \(\langle w|w\rangle \geq 0\). Choosing \(t = \langle\phi|\psi\rangle / \langle\phi|\phi\rangle\) leads to the inequality (the case \(|\phi\rangle = \mathbf{0}\) is trivial).

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Advanced

A-1. Basis Transformation by Unitary Matrices and Transformation Rules for Matrix Representations

In an \(N\)-dimensional inner product space, two orthonormal bases \(\{|e_k\rangle\}_{k=1}^N\) and \(\{|f_k\rangle\}_{k=1}^N\) are given. The basis transformation matrix \(U\) is defined by \(U_{jk} = \langle e_j|f_k\rangle\).

(a) Prove that \(U\) is a unitary matrix (\(U^\dagger U = UU^\dagger = \hat{1}\)) by using the completeness relation.

(b) Let \(A^{(e)}\) be the matrix representation of an operator \(\hat{A}\) in the basis \(\{|e_k\rangle\}\), and \(A^{(f)}\) be the matrix representation in the basis \(\{|f_k\rangle\}\). Show that

\[A^{(e)} = U\, A^{(f)}\, U^\dagger\]

holds.

(c) Using this result, prove that "the trace of an operator \(\mathrm{Tr}(\hat{A}) = \sum_k A_{kk}\) is independent of the choice of basis."

Hint

(a) Compute \((U^\dagger U)_{jk} = \sum_l U_{lj}^* U_{lk} = \sum_l \langle f_j|e_l\rangle\langle e_l|f_k\rangle\) and insert the completeness relation for \(\{|e_l\rangle\}\).

(b) Insert \(\hat{1} = \sum_l |f_l\rangle\langle f_l|\) on both the left and right sides of \(\hat{A}\) in \(A^{(e)}_{jk} = \langle e_j|\hat{A}|e_k\rangle\).

(c) Use the cyclic property of the trace \(\mathrm{Tr}(XYZ) = \mathrm{Tr}(ZXY)\), or directly compute \(\sum_k (UAU^\dagger)_{kk}\).

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A-2. Tensor Product Space and Construction of the Bell Basis

Consider the tensor product \(\mathbb{C}^2 \otimes \mathbb{C}^2\) of two \(\mathbb{C}^2\) spaces. Let the standard basis of each space be \(\{|0\rangle, |1\rangle\}\), and the standard basis of the tensor product space be \(\{|00\rangle, |01\rangle, |10\rangle, |11\rangle\}\) (where \(|jk\rangle \equiv |j\rangle \otimes |k\rangle\)).

(a) Show that the following four vectors (the Bell basis) form an orthonormal basis of \(\mathbb{C}^2 \otimes \mathbb{C}^2\).

\[|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)\]
\[|\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)\]
\[|\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)\]
\[|\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)\]

(b) Show that \(|\Phi^+\rangle\) is "entangled." That is, prove by contradiction that there exist no \(|a\rangle \in \mathbb{C}^2\), \(|b\rangle \in \mathbb{C}^2\) such that \(|\Phi^+\rangle = |a\rangle \otimes |b\rangle\).

(c) Expand an arbitrary state written in the standard basis \(|\psi\rangle = \alpha|00\rangle + \beta|01\rangle + \gamma|10\rangle + \delta|11\rangle\) (\(|\alpha|^2 + |\beta|^2 + |\gamma|^2 + |\delta|^2 = 1\)) in the Bell basis. That is, express the expansion coefficients in terms of \(\alpha, \beta, \gamma, \delta\).

Hint

(a) The inner product in the tensor product space is \(\langle jk|lm\rangle = \delta_{jl}\delta_{km}\). Compute all six inner product combinations \(\langle\Phi^+|\Phi^-\rangle\), \(\langle\Phi^+|\Psi^+\rangle\), ... and confirm they are all 0, and verify that the norm of each vector is 1. Four orthonormal vectors in a four-dimensional space form a basis.

(b) Set \(|a\rangle = a_0|0\rangle + a_1|1\rangle\), \(|b\rangle = b_0|0\rangle + b_1|1\rangle\), expand \(|a\rangle\otimes|b\rangle\), and compare coefficients with those of \(|\Phi^+\rangle\). Derive a contradiction from \(a_0 b_0 = 1/\sqrt{2}\), \(a_0 b_1 = 0\), \(a_1 b_0 = 0\), \(a_1 b_1 = 1/\sqrt{2}\).

(c) Use the completeness relation \(\hat{1} = |\Phi^+\rangle\langle\Phi^+| + |\Phi^-\rangle\langle\Phi^-| + |\Psi^+\rangle\langle\Psi^+| + |\Psi^-\rangle\langle\Psi^-|\) and find each expansion coefficient via inner products.

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