Math Problems Vault

Problem 22.1.1

(i)(ii)

The four Dirac matrices are given by

\gamma _{1} = \begin{pmatrix} 0 & 0 & 0 & -i \\ 0 & 0 & -i & 0 \\ 0 & i & 0 & 0 \\ i & 0 & 0 & 0 \end{pmatrix}, \quad \gamma _{2} = \begin{pmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix},

\gamma _{3} = \begin{pmatrix} 0 & 0 & -i & 0 \\ 0 & 0 & 0 & i \\ i & 0 & 0 & 0 \\ 0 & -i & 0 & 0 \end{pmatrix}, \quad \gamma _{4} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}

and

\gamma _{5} = \gamma _{1}\gamma _{2}\gamma _{3}\gamma _{4} = \begin{pmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{pmatrix}.

We define the $4 \times 4$ matrices

\sigma _{jk} := \frac{i}{2}[\gamma _{j}, \gamma _{k}], \qquad j < k

where $j = 1, 2, 3$ , $k = 2, 3, 4$ .

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(i)

Calculate $\sigma_{12}$ , $\sigma_{13}$ , $\sigma_{14}$ , $\sigma_{23}$ , $\sigma_{24}$ , $\sigma_{34}$ .

(ii)

Do the 16 matrices

I_{4}, \ \gamma _{1}, \ \gamma _{2}, \ \gamma _{3}, \ \gamma _{4}, \ \gamma _{5}, \ \gamma _{5}\gamma _{1}, \ \gamma _{5}\gamma _{2}, \ \gamma _{5}\gamma _{3}, \ \gamma _{5}\gamma _{4}, \ \sigma _{12}, \ \sigma _{13}, \ \sigma _{14}, \ \sigma _{23}, \ \sigma _{24}, \ \sigma _{34}

form a basis in the Hilbert space $M_{4}(\mathbb {C})$ ? If so is the basis orthogonal?

Problem 22.1.2

Let $P$ be the $n \times n$ primary permutation matrix

P := \begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & 1 \\ 1 & 0 & 0 & \ldots & 0 \end{pmatrix}

and $V$ be the $n \times n$ unitary diagonal matrix ( $\zeta \in \mathbb {C}$ )

V := \begin{pmatrix} 1 & 0 & 0 & & 0 \\ 0 & \zeta & 0 & & 0 \\ 0 & 0 & \zeta ^{2} & & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \ldots & \zeta ^{n-1} \end{pmatrix}

where $\zeta^{n} = 1$ . Then the set of matrices

\left\{ P^{j}V^{k} : j, k = 0, 1, \ldots , n-1\right\}

provide a basis in the Hilbert space for all $n \times n$ matrices with the scalar product

\left\langle A, B \right\rangle := \frac{1}{n}\operatorname {tr}(AB^{*})

for $n \times n$ matrices $A$ and $B$ . Write down the basis for $n = 2$ .

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Problem 22.1.3

Consider the two normalized vectors

\overrightarrow {v}(\alpha ) = \begin{pmatrix} \cos (\alpha ) \\ \sin (\alpha ) \end{pmatrix}, \quad \overrightarrow {u}(\beta ) = \begin{pmatrix} \cos (\beta ) \\ \sin (\beta ) \end{pmatrix}

in $\mathbb {R}^{2}$ . Find the condition on $\alpha , \beta \in \mathbb {R}$ such that

\left|\overrightarrow {v}^{*}(\alpha )\overrightarrow {u}(\beta )\right|^{2} = \frac{1}{2}.

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Problem 22.1.4

Consider the four nonnormal $2 \times 2$ matrices

Q = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}, \quad R = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 0 \\ 1 & -1 \end{pmatrix},

S = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -1 \\ 0 & 0 \end{pmatrix}, \quad T = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix}.

Show that they form an orthonormal basis in the Hilbert space of the $2 \times 2$ matrices with the scalar product $\left\langle X, Y \right\rangle = \operatorname {tr}(XY^{*})$ .

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Problem 22.1.5

Consider the Hilbert space $M_{d}(\mathbb {C})$ of $d \times d$ matrices with scalar product $\left\langle A, B \right\rangle := \operatorname {tr}(AB^{*})$ , $A, B \in M_{d}(\mathbb {C})$ . Consider an orthogonal basis of $d^{2}$ $d \times d$ hermitian matrices $B_{1}, B_{2}, \ldots , B_{d^{2}}$ , i.e.

\left\langle B_{j}, B_{k} \right\rangle = \operatorname {tr}(B_{j}B_{k}) = d\delta _{jk}

since $B_{k}^{*} = B_{k}$ for a hermitian matrix. Let $M$ be a $d \times d$ hermitian matrix. Let

m_{j} = \operatorname {tr}(B_{j}M) \qquad j = 1, \ldots , d^{2}.

Given $m_{j}$ and $B_{j}$ ( $j = 1, \ldots , d^{2}$ ). Find $M$ .

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Problem 22.1.6

(i)(ii)

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(i)

Consider the Hilbert space $\mathcal{H}$ of the $2 \times 2$ matrices over the complex numbers. Show that the rescaled Pauli spin matrices $\mu_{j} = \frac{1}{\sqrt{2}}\sigma_{j}$ , $j = 1, 2, 3$

\mu _{1} = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \mu _{2} = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \mu _{3} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

plus the rescaled $2 \times 2$ identity matrix

\mu _{0} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

form an orthonormal basis in the Hilbert space $\mathcal{H}$ .

(ii)

Find an orthonormal basis given by hermitian matrices in the Hilbert space $\mathcal{H}$ of $4 \times 4$ matrices over $\mathbb {C}$ .

Problem 22.1.7

Let $A$ , $B$ be hermitian matrices. Find the scalar product

\left\langle A \otimes B, B \otimes A \right\rangle .

Is $\left\langle A \otimes B, B \otimes A \right\rangle \geq 0$ . Prove or disprove.

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Problem 22.1.8

Consider the Hilbert space of the $n \times n$ matrices over $\mathbb {C}$ . Let $A$ , $B$ be two $n \times n$ matrices over $\mathbb {C}$ . Assume that $\left\langle A, B \right\rangle = 0$ . Calculate

\left\langle A \otimes A, B \otimes B \right\rangle , \qquad \left\langle A \otimes B, A \otimes B \right\rangle , \qquad \left\langle A \otimes B, B \otimes A \right\rangle .

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Problem 22.1.9

(i)(ii)

Consider the Hilbert space $\mathcal{H} = \mathbb {C}^{4}$ and the hermitian matrices

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

A = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \qquad B = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}.

We consider the $16 \times 16$ matrix

K = \frac{1}{2}(H \otimes I_{4} + I_{4} \otimes H) - \lambda A \otimes A - \mu B \otimes B

acting in the Hilbert space $\mathbb {C}^{16}$ , where $\lambda , \mu \geq 0$ .

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(i)

Consider the permutation matrix

R = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{pmatrix}.

Thus the inverse of $R$ exists. Calculate $(R \otimes R)^{-1}K(R \otimes R)$ and the commutator $[R \otimes R, K]$ . Discuss.

(ii)

Consider the permutation matrices

S = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \qquad T = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}.

Find the commutators $[S \otimes S, K]$ and $[T \otimes T, K]$ . Discuss.

Problem 22.1.10

Consider the Hilbert space of $2 \times 2$ matrices. Let $\alpha , \beta \in \mathbb {R}$ and

A(\alpha ) = \frac{1}{\sqrt{2}}\begin{pmatrix} \cos (\alpha ) & \sin (\alpha ) \\ \sin (\alpha ) & -\cos (\alpha ) \end{pmatrix}, \qquad B(\beta ) = \frac{1}{\sqrt{2}}\begin{pmatrix} \cos (\beta ) & \sin (\beta ) \\ \sin (\beta ) & -\cos (\beta ) \end{pmatrix}.

Find the conditions on $\alpha$ and $\beta$ such that $\left\langle A(\alpha ), B(\beta ) \right\rangle = \frac{1}{2}$ . The matrices $A(\alpha )$ , $B(\beta )$ contain $\frac{1}{\sqrt{2}}\sigma_{3}$ with $\alpha = 0$ and $\frac{1}{\sqrt{2}}\sigma_{1}$ with $\alpha = \pi /2$ .

Study the case

\left\langle A(\alpha ) \otimes A(\alpha )\right\rangle = \left\langle B(\beta ) \otimes B(\beta )\right\rangle = \frac{1}{4}.

Extend the matrix $A(\alpha )$ to

A(\alpha , \phi ) = \frac{1}{\sqrt{2}}\begin{pmatrix} \cos (\alpha ) & e^{i\phi }\sin (\alpha ) \\ e^{-i\phi }\sin (\alpha ) & -\cos (\alpha ) \end{pmatrix}.

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Problem 22.1.11

Consider the Pauli spin matrices $\sigma_{3}$ , $\sigma_{1}$ , $\sigma_{2}$ . Show that the normalized eigenvectors of $\sigma_{3}$ , $\sigma_{1}$ , $\sigma_{2}$ provide a set of mutually unbiased bases.

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Problem 22.1.12

Consider the three $4 \times 4$ matrices $\sigma_{3} \otimes \sigma_{3}$ , $\sigma_{1} \otimes \sigma_{1}$ , $\sigma_{2} \otimes \sigma_{2}$ . Find mutually unbiased bases for these matrices applying the result from the previous problem.

Extend the results from problem 11 and problem 12 the case in the Hilbert space $\mathbb {C}^{8}$ with $\sigma_{3} \otimes \sigma_{3} \otimes \sigma_{3}$ , $\sigma_{1} \otimes \sigma_{1} \otimes \sigma_{1}$ , $\sigma_{2} \otimes \sigma_{2} \otimes \sigma_{2}$ . Extend to $n$ Kronecker products of the Pauli spin matrices.

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Problem 22.1.13

(i)(ii)(iii)

Let $d \geq 2$ . Consider the Hilbert space $\mathbb {C}^{d}$ and $\overrightarrow {e}_{0}, \overrightarrow {e}_{1}, \ldots , \overrightarrow {e}_{d-1}$ be the standard basis

\overrightarrow {e}_{0} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \quad \overrightarrow {e}_{1} = \begin{pmatrix} 0 \\ 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \quad \ldots , \quad \overrightarrow {e}_{d-1} = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{pmatrix}.

Let $\omega_{d} := e^{i2\pi /d}$ . Then we can form new orthonormal bases via

\overrightarrow {v}_{m;b} \equiv \left|m;b\right\rangle = \frac{1}{\sqrt{d}}\sum _{n=0}^{d-1} \omega ^{bn(n-1)/2-nm}\overrightarrow {e}_{n}, \qquad b, m = 0, 1, \ldots , d-1

where the $d$ labelled by the $b$ are the bases and $m$ labels the state within a basis. Thus we find mutually unbiased bases.

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(i)

Consider $d = 2$ and $b = 0$ . Find the basis $\left|0;0\right\rangle$ , $\left|1;0\right\rangle$ .

(ii)

Consider $d = 3$ and $b = 0$ . Find the basis $\left|1;0\right\rangle$ , $\left|2;0\right\rangle$ , $\left|3;0\right\rangle$ .

(iii)

Consider $d = 4$ and $b = 0$ . Find the basis $\left|0;0\right\rangle$ , $\left|1;0\right\rangle$ , $\left|2;0\right\rangle$ , $\left|3;0\right\rangle$ . Find out whether the states can be written as Kronecker product of two vectors in $\mathbb {C}^{2}$ , i.e. whether the states are entangled or not?