Math Problems Vault

Problem 17.2.1

Let $U$ be a unitary $2 \times 2$ matrix. Can the matrix be reconstructed from $\operatorname {tr}(U)$ , $\operatorname {tr}(U^{2})$ , $\operatorname {tr}(U^{3})$ ?

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Problem 17.2.2

Consider the Hadamard matrix

U = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}

which is a unitary and hermitian matrix with eigenvalues $+1$ and $-1$ . We multiply each column with a phase factor $e^{i\phi_{j}}$ and obtain the matrix

V(\phi _{1},\phi _{2}) = \frac{1}{\sqrt{2}}\begin{pmatrix} e^{i\phi _{1}} & e^{i\phi _{2}} \\ e^{i\phi _{1}} & -e^{i\phi _{2}} \end{pmatrix}.

Is the matrix still unitary? If not find the conditions on $\phi_{1}$ and $\phi_{2}$ such that $V(\phi_{1},\phi_{2})$ is unitary.

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Problem 17.2.3

Can one find a $\theta$ such that

\begin{pmatrix} \cos (\theta )e^{-i\theta } & i\sin (\theta )e^{-i\theta } \\ i\sin (\theta )e^{-i\theta } & \cos (\theta )e^{-i\theta } \end{pmatrix} = \frac{1}{2}\begin{pmatrix} 1+i & 1-i \\ 1-i & 1+i \end{pmatrix} ?

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Problem 17.2.4

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(i)

Is the $2 \times 2$ matrix

U = \begin{pmatrix} u_{1}+iu_{2} & u_{3}+iu_{4} \\ -u_{3}+iu_{4} & u_{1}-iu_{2} \end{pmatrix}

with $u_{1} = \cos (\alpha )$ , $u_{2} = \sin (\alpha )\cos (\beta )$ and

u_{3} = \sin (\alpha )\sin (\beta )\cos (\gamma ), \qquad u_{4} = \sin (\alpha )\sin (\beta )\sin (\gamma )

unitary?

(ii)

Is the $3 \times 3$ matrix

U = \frac{1}{\sqrt{3}}\begin{pmatrix} 1 & 1 & 1 \\ i & i & -i \\ i & -i & -i \end{pmatrix}

unitary?

(iii)

Show that the $3 \times 3$ matrix (Fourier matrix)

V = \frac{1}{\sqrt{3}}\begin{pmatrix} 1 & 1 & 1 \\ 1 & \mathrm{exp}\left(i2\pi /3\right) & \mathrm{exp}\left(i4\pi /3\right) \\ 1 & \mathrm{exp}\left(i4\pi /3\right) & \mathrm{exp}\left(i2\pi /3\right) \end{pmatrix}

is unitary.

(iv)

Are the $4 \times 4$ matrices

A = \frac{1}{2}\begin{pmatrix} 1 & 1 & 1 & 1 \\ -1 & 1 & -1 & 1 \\ -1 & -1 & 1 & 1 \\ 1 & -1 & -1 & 1 \end{pmatrix}, \qquad B = \frac{1}{2}\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ -1 & 1 & 1 & -1 \end{pmatrix}

unitary?

(v)

Show that the $4 \times 4$ matrices

X = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 0 & 0 & -i \\ 0 & 1 & i & 0 \\ 0 & -i & -1 & 0 \\ i & 0 & 0 & -1 \end{pmatrix}, \qquad Y = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -1 & 0 & 1 \\ 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{pmatrix}

are unitary.

(vi)

Let

\sigma := \frac{1}{2}(1-\sqrt{5}), \qquad \tau := \frac{1}{2}(1+\sqrt{5})

with $\tau$ the golden mean number. Are the $4 \times 4$ matrices

U_{1} = -\frac{1}{2}\begin{pmatrix} 1 & -\tau & -\sigma & 0 \\ \tau & 1 & 0 & \sigma \\ \sigma & 0 & 1 & -\tau \\ 0 & -\sigma & \tau & 1 \end{pmatrix}, \qquad U_{2} = -\frac{1}{2}\begin{pmatrix} 1 & -\tau & -\sigma & 0 \\ \tau & 1 & 0 & -\sigma \\ \sigma & 0 & 1 & \tau \\ 0 & \sigma & -\tau & 1 \end{pmatrix}

unitary? Prove or disprove.

(vii)

Show that the $2n \times 2n$ matrix

V = \frac{1}{\sqrt{2}}\begin{pmatrix} I_{n} & iI_{n} \\ I_{n} & -iI_{n} \end{pmatrix}

is unitary.

Problem 17.2.5

(i)(ii)(iii)

Let $A$ be an $n \times n$ hermitian matrix with $A^{2} = I_{n}$ .

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(i)

Show that $U = \frac{1}{\sqrt{2}}(I_{n}+iA)$ is unitary.

(ii)

Show that $U = \frac{1}{\sqrt{2}}(I_{n}-iA)$ is unitary.

(iii)

Let $\phi \in [0,2\pi )$ . Show that the condition on $\phi$ such that

V(\phi ) = \frac{1}{\sqrt{2}}(e^{i\phi }I_{n}+e^{-i\phi }A)

is unitary is given by $\phi = \pi /2$ and $\phi = 3\pi /2$ .

Problem 17.2.6

Let $I_{n}$ be the $n \times n$ identity matrix and $J_{n}$ be the $n \times n$ backward identity matrix i.e. the entries are $1$ at the counter diagonal and $0$ otherwise. Are the matrices

U_{1} = \frac{1}{\sqrt{2}}(I_{n}+iJ_{n}), \qquad U_{2} = \frac{1}{\sqrt{2}}(I_{n}-iJ_{n}), \qquad U_{3} = \frac{1}{\sqrt{2}}((1+i)I_{n}+(1-i)J_{n})

unitary?

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Problem 17.2.7

Let $U$ be a unitary $4 \times 4$ matrix. Assume that each column (vector) can be written as the Kronecker product of two vectors in $\mathbb {C}^{2}$ . Can we conclude that the eigenvectors of such a unitary matrix can also be written a Kronecker product of two vectors in $\mathbb {C}^{2}$ ?

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Problem 17.2.8

Find all $2 \times 2$ hermitian and unitary matrices $A$ , $B$ such that $AB = e^{i\pi }BA$ .

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Problem 17.2.9

(i)(ii)

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(i)

Find the conditions on $u_{11}, u_{12}, u_{21}, u_{22}, u_{23}, u_{32}, u_{33} \in \mathbb {C}$ such that the $3 \times 3$ matrix

U = \begin{pmatrix} u_{11} & u_{12} & 0 \\ u_{21} & u_{22} & u_{23} \\ 0 & u_{32} & u_{33} \end{pmatrix}

is unitary.

(ii)

Let $1 \geq r_{jk} \geq 0$ , $\phi_{jk} \in \mathbb {R}$ ( $j,k = 1,2,3$ ). Find the conditions on $r_{jk}$ and $\phi_{jk}$ such that

\begin{pmatrix} r_{11}e^{i\phi _{11}} & r_{12}e^{i\phi _{12}} & r_{13}e^{i\phi _{13}} \\ r_{21}e^{i\phi _{21}} & r_{22}e^{i\phi _{22}} & r_{23}e^{i\phi _{23}} \\ r_{31}e^{i\phi _{31}} & r_{32}e^{i\phi _{32}} & r_{33}e^{i\phi _{33}} \end{pmatrix}

is a unitary matrix.

Problem 17.2.10

Find all $(n+1) \times (n+1)$ matrices $A$ such that $A^{*}UA = U$ , where $U$ is the unitary matrix

U = \begin{pmatrix} 0 & 0 & i \\ 0_{n-1} & I_{n-1} & 0_{n-1} \\ -i & 0 & 0 \end{pmatrix}

and $\operatorname {det}\left(A\right) = 1$ . Consider first the case $n = 2$ . For $n = 2$ we have

A^{*} \begin{pmatrix} 0 & 0 & i \\ 0 & 1 & 0 \\ -i & 0 & 0 \end{pmatrix} A = \begin{pmatrix} 0 & 0 & i \\ 0 & 1 & 0 \\ -i & 0 & 0 \end{pmatrix}

and find nine equations.

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Problem 17.2.11

Consider $n \times n$ unitary matrices. A scalar product of two $n \times n$ matrices $U$ , $V$ can be defined as

\left\langle U, V \right\rangle := \frac{1}{n}\operatorname {tr}(UV^{*}).

Find two $2 \times 2$ unitary matrices $U$ , $V$ such that $\left\langle U, V \right\rangle = \frac{1}{2}$ .

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Problem 17.2.12

Let $U$ be a unitary matrix. Then the determinant of $U$ must be of the form $e^{i\phi }$ . Find the determinant of $U+U^{*}$ .

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Problem 17.2.13

(i)(ii)

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(i)

Consider the unitary matrix

Y = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \equiv i\sigma _{2}.

Find all $2 \times 2$ matrices $S$ such that $YSY^{-1} = S^{T}$ , $S = S^{*}$ .

(ii)

Can one find a $2 \times 2$ unitary matrix $U$ such that

\begin{pmatrix} \cos (\theta ) & e^{i\phi }\sin (\theta ) \\ e^{-i\phi }\sin (\theta ) & -\cos (\theta ) \end{pmatrix} = U \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} U^{*} ?

Problem 17.2.14

Let $U$ be an $n \times n$ unitary matrix and $A$ an arbitrary $n \times n$ matrix. Then we know that $Ue^{A}U^{-1} = e^{UAU^{-1}}$ . Calculate $Ue^{A}U$ with $U \neq U^{-1}$ .

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Problem 17.2.15

Consider the unit vectors in $\mathbb {C}^{3}$

\overrightarrow {z} = \begin{pmatrix} z_{1} \\ z_{2} \\ z_{3} \end{pmatrix}, \qquad \overrightarrow {w} = \begin{pmatrix} w_{1} \\ w_{2} \\ w_{3} \end{pmatrix}

i.e. $\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2} = 1$ , $\left|w_{1}\right|^{2}+\left|w_{2}\right|^{2}+\left|w_{3}\right|^{2} = 1$ . Assume that (complex unit cone) $z_{1}w_{1}+z_{2}w_{2}+z_{3}w_{3} = 0$ . Show that $U \in SU(3)$ can be written as

U = \begin{pmatrix} \overline{z_{1}} & \overline{z_{2}} & \overline{z_{3}} \\ w_{1} & w_{2} & w_{3} \\ u_{1} & u_{2} & u_{3} \end{pmatrix}

where $u_{j} := \sum_{k,\ell =1}^{3} \epsilon_{jk\ell }z_{k}\overline{w_{\ell }}$ .

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Problem 17.2.16

(i)(ii)(iii)

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(i)

Let $\phi_{1}, \phi_{2} \in \mathbb {R}$ . Show that

U(\phi _{1},\phi _{2}) = \frac{1}{\sqrt{2}}\begin{pmatrix} e^{i\phi _{1}} & -e^{-i\phi _{2}} \\ e^{i\phi _{2}} & e^{-i\phi _{1}} \end{pmatrix}

is unitary. Is $U(\phi_{1},\phi_{2})$ an element of $SU(2)$ ? Find the eigenvalues and eigenvectors of $U(\phi_{1},\phi_{2})$ . Do the eigenvalues and eigenvectors depend on the $\phi$ 's? Show that

V(\phi _{1},\phi _{2}) = \frac{1}{\sqrt{2}}\begin{pmatrix} e^{i\phi _{1}} & e^{-i\phi _{2}} \\ e^{i\phi _{2}} & -e^{-i\phi _{1}} \end{pmatrix}

is unitary. Is $V(\phi_{1},\phi_{2})$ an element of $SU(2)$ ? Find the eigenvalues and eigenvectors of $V(\phi_{1},\phi_{2})$ . Do the eigenvalues and eigenvectors depend on the $\phi$ 's?

(ii)

Let $\phi_{1}, \phi_{2}, \phi_{3}, \phi_{4} \in \mathbb {R}$ . Show that

U(\phi _{1},\phi _{2},\phi _{3},\phi _{4}) = \frac{1}{\sqrt{2}}\begin{pmatrix} e^{i\phi _{1}} & 0 & 0 & -e^{-i\phi _{2}} \\ 0 & e^{i\phi _{3}} & -e^{-i\phi _{4}} & 0 \\ 0 & e^{i\phi _{4}} & e^{-i\phi _{3}} & 0 \\ e^{i\phi _{2}} & 0 & 0 & e^{-i\phi _{1}} \end{pmatrix}

is unitary. Is $U(\phi_{1},\phi_{2},\phi_{3},\phi_{4})$ an element of $SU(4)$ ? Find the eigenvalues and eigenvectors of $U(\phi_{1},\phi_{2},\phi_{3},\phi_{4})$ . Do the eigenvalues and eigenvectors depend on the $\phi$ 's?

(iii)

Let $\phi_{1}, \phi_{2} \in \mathbb {R}$ . Show that

U(\phi _{1},\phi _{2}) = \frac{1}{\sqrt{2}}\begin{pmatrix} e^{i\phi _{1}} & 0 & -e^{-i\phi _{2}} \\ 0 & \sqrt{2} & 0 \\ e^{i\phi _{2}} & 0 & e^{-i\phi _{1}} \end{pmatrix}

is unitary. Is $U(\phi_{1},\phi_{2})$ an element of $SU(2)$ ? Find the eigenvalues and eigenvectors of $U(\phi_{1},\phi_{2})$ . Do the eigenvalues and eigenvectors depend on the $\phi$ 's?

Problem 17.2.17

(i)(ii)(iii)(iv)

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(i)

What can be said about the eigenvalues of a matrix which is unitary and hermitian?

(ii)

What can be said about the eigenvalues of a matrix which is unitary and skew-hermitian?

(iii)

What can be said about the eigenvalues of a matrix which is unitary and $U^{T} = U$ ?

(iv)

What can be said about the eigenvalues of a matrix which is unitary and $U^{3} = U$ ?

Problem 17.2.18

Let $n$ be a prime number and $j,k = 1,\ldots ,n$ . Show that the matrices

(U_{1})_{jk} = \frac{1}{\sqrt{n}}\mathrm{exp}\left((2\pi i/n)(j+k-1)^{2}\right)

(U_{r})_{jk} = \frac{1}{\sqrt{n}}\mathrm{exp}\left((2\pi i/n)r(j+k-1)^{2}\right)

(U_{n-1})_{jk} = \frac{1}{\sqrt{n}}\mathrm{exp}\left(((2\pi i/n)(n-1)(j+k-1)^{2}\right)

(U_{n})_{jk} = \frac{1}{\sqrt{n}}\mathrm{exp}\left((2\pi i/n)jk\right)

(U_{n+1})_{jk} = \delta _{jk}

are unitary.

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Problem 17.2.19

(i)(ii)(iii)

In the Hilbert space $\mathbb {C}^{4}$ consider the Bell states

\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix}, \quad \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \end{pmatrix}, \quad \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ 1 \\ -1 \\ 0 \end{pmatrix}, \quad \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \\ 0 \\ -1 \end{pmatrix}.

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(i)

Let $\omega := e^{2\pi i/4}$ . Apply the Fourier transformation

U_{F} = \frac{1}{2}\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & \omega & \omega ^{2} & \omega ^{3} \\ 1 & \omega ^{2} & 1 & \omega ^{2} \\ 1 & \omega ^{3} & \omega ^{2} & \omega \end{pmatrix}

to the Bell states and study the entanglement of these states.

(ii)

Apply the Haar wavelet transform

U_{H} = \frac{1}{2}\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ \sqrt{2} & -\sqrt{2} & 0 & 0 \\ 0 & 0 & \sqrt{2} & -\sqrt{2} \end{pmatrix}

to the Bell states and study the entanglement of these states.

(iii)

Apply the Walsh-Hadamard transform

U_{W} = \frac{1}{2}\begin{pmatrix} 1 & 1 & 1 & 1 \\ -1 & -1 & 1 & 1 \\ -1 & 1 & 1 & -1 \\ 1 & -1 & 1 & -1 \end{pmatrix}

to the Bell states and study the entanglement of these states.

Extend to the Hilbert space $\mathbb {C}^{2^{n}}$ with the first Bell state given by

\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 0 & \cdots & 0 & 1 \end{pmatrix}^{T}.

Problem 17.2.20

Let $U$ be an $n \times n$ unitary matrix. Then $U+U^{*}$ is a hermitian matrix. Can any hermitian matrix represented in this form? For example for the $n \times n$ zero matrix we have $U = \operatorname {diag}(i,i,\ldots ,i)$ .

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Problem 17.2.21

(i)(ii)

Let $\left\{ \left|a_{0}\right\rangle , \left|a_{1}\right\rangle , \ldots , \left|a_{n-1}\right\rangle \right\}$ be an orthonormal basis in the Hilbert space $\mathbb {C}^{n}$ . The discrete Fourier transform

\left|b_{j}\right\rangle = \frac{1}{\sqrt{n}}\sum _{k=0}^{n-1} \omega ^{jk}\left|a_{k}\right\rangle , \qquad j = 0,1,\ldots ,n-1

where $\omega := \mathrm{exp}\left(2\pi i/n\right)$ is the primitive $n$ -th root of unity.

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(i)

Apply the discrete Fourier transform to the standard basis in $\mathbb {C}^{4}$

\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix}, \quad \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}.

(ii)

Apply the discrete Fourier transform to the Bell basis in $\mathbb {C}^{4}$

\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix}, \quad \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 0 \\ 0 \\ -1 \end{pmatrix}, \quad \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \end{pmatrix}, \quad \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ 1 \\ -1 \\ 0 \end{pmatrix}.

Problem 17.2.22

Consider the $3 \times 3$ unitary matrices

U_{1}(\phi _{1},\phi _{2},\phi _{3}) = \begin{pmatrix} e^{i\phi _{1}} & 0 & 0 \\ 0 & e^{i\phi _{2}} & 0 \\ 0 & 0 & e^{i\phi _{3}} \end{pmatrix}, \qquad U_{2}(\phi _{4},\phi _{5},\phi _{6}) = \begin{pmatrix} 0 & 0 & e^{i\phi _{4}} \\ 0 & e^{i\phi _{5}} & 0 \\ e^{i\phi _{6}} & 0 & 0 \end{pmatrix}.

What is the condition on $\phi_{1}, \ldots , \phi_{6}$ such that

[U_{1}(\phi _{1},\phi _{2},\phi _{3}), U_{2}(\phi _{1},\phi _{2},\phi _{3})] = 0_{3} \; ?

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Problem 17.2.23

Let $U$ be a unitary matrix with $U = U^{T}$ . Show that $U$ can be written as $U = V^{T}V$ , where $V$ is unitary. Note that $(V^{T}V)^{T} = V^{T}V$ .

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Problem 17.2.24

Let $n \geq 2$ . Consider the $n \times n$ matrix (counting from $(0,0)$ )

(a_{jk}) = \frac{1}{\sqrt{n}}(e^{i\pi (j-k)})

where $j,k = 0,1,\ldots ,n-1$ . Is the matrix unitary? Study first the cases $n = 2$ and $n = 3$ .

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Problem 17.2.25

Consider the rotation matrix

R(\alpha ) = \begin{pmatrix} \cos (\alpha ) & -\sin (\alpha ) \\ \sin (\alpha ) & \cos (\alpha ) \end{pmatrix}.

Find all $2 \times 2$ matrices

M(\beta ) = \begin{pmatrix} f_{11}(\beta ) & f_{12}(\beta ) \\ f_{21}(\beta ) & f_{22}(\beta ) \end{pmatrix}

such that $R(\alpha )M(\beta )R^{-1}(\alpha ) = M(\beta +\alpha )$ .

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Problem 17.2.26

(i)(ii)

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(i)

Consider an $n \times n$ unitary matrix $U = (u_{jk})$ with $j,k = 1,\ldots ,n$ . Show that

S = (s_{jk}) = (u_{jk}\overline{u}_{jk})

is a double stochastic matrix.

(ii)

Given a double stochastic $n \times n$ matrix $S$ . Can we construct the unitary matrix $U$ which generates the double stochastic matrix as described in (i).

Problem 17.2.27

(i)(ii)

Let $\sigma_{1}, \sigma_{2}, \sigma_{3}$ be the Pauli spin matrices and $\alpha_{j} \in \mathbb {R}$ for $j = 1,2,3$ .

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(i)

Find the $2 \times 2$ matrices

U_{k} := (I_{2}-i\alpha _{k}\sigma _{k})(I_{2}+i\alpha _{k}\sigma _{k})^{-1}

where $k = 1,2,3$ . Are the matrices unitary?

(ii)

Find the $4 \times 4$ matrices

V_{k} := (I_{4}-i\alpha _{k}\sigma _{k}\otimes \sigma _{k})(I_{4}+i\alpha _{k}\sigma _{k}\otimes \sigma _{k})^{-1}

where $k = 1,2,3$ . Are the matrices unitary?

Problem 17.2.28

Let $U$ be a unitary matrix. Calculate $\mathrm{exp}\left(\epsilon (U+U^{*})\right)$ , where $\epsilon \in \mathbb {R}$ . Note that

(U+U^{*})^{2} = U^{2}+(U^{*})^{2}+2I_{n}, \qquad (U+U^{*})^{3} = U^{3}+(U^{*})^{3}+3U+3U^{*}.

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Problem 17.2.29

Let $U$ , $V$ be $n \times n$ unitary matrices. Is the $2n \times 2n$ matrix

\begin{pmatrix} 0_{n} & U \\ V & 0_{n} \end{pmatrix}

unitary?

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Problem 17.2.30

Let $\phi \in \mathbb {R}$ and $U$ be an $2 \times 2$ unitary matrix. Show that the $4 \times 4$ matrix

V = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \otimes U + \begin{pmatrix} e^{i\phi } & 0 \\ 0 & 0 \end{pmatrix} \otimes I_{2}

is unitary. Note that $V = e^{i\phi }I_{2} \oplus U$ .

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