Math Problems Vault

Problem 22.2.1

Assume in the following that $\mathcal{B}_{1,1}$ and $\mathcal{B}_{1,2}$ are mutually unbiased bases in the Hilbert space $\mathcal{H}_{1}$ and $\mathcal{B}_{2,1}$ and $\mathcal{B}_{2,2}$ are mutually unbiased bases in the Hilbert space $\mathcal{H}_{2}$ , respectively. Show that

\left\{ \left|j_{1}\right\rangle \otimes \left|k_{1}\right\rangle \right\} , \quad \left\{ \left|j_{2}\right\rangle \otimes \left|k_{2}\right\rangle \right\} , \qquad j_{1}, j_{2} = 1, \ldots , d_{1}, \ \ k_{1}, k_{2} = 1, \ldots , d_{2}

are mutually unbiased bases in the finite dimensional product Hilbert space $\mathcal{H}_{1} \otimes \mathcal{H}_{2}$ with $\dim (\mathcal{H}_{1} \otimes \mathcal{H}_{2}) = d_{1} \cdot d_{2}$ and the scalar product in the product Hilbert space

(\left\langle j_{1}\right| \otimes \left\langle k_{1}\right|)(\left|j_{2}\right\rangle \otimes \left|k_{2}\right\rangle ) = \left\langle j_{1}\middle |j_{2}\right\rangle \left\langle k_{1}\middle |k_{2}\right\rangle .

Apply this result to $\mathcal{H}_{2} = \mathbb {C}^{3}$ with

\mathcal{B}_{2,1} = \left\{ \left|1\right\rangle _{2,1} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \ \left|2\right\rangle _{2,1} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \ \left|3\right\rangle _{2,1} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\right\}

\mathcal{B}_{2,2} = \left\{ \left|1\right\rangle _{2,2} = \frac{1}{\sqrt{3}}\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \ \left|2\right\rangle _{2,2} = \begin{pmatrix} 1/\sqrt{3} \\ -i/2 - 1/(2\sqrt{3}) \\ i/2 - 1/(2\sqrt{3}) \end{pmatrix}, \right\}

\left|3\right\rangle _{2,2} = \begin{pmatrix} 1/\sqrt{3} \\ i/2 - 1/(2\sqrt{3}) \\ i/2 - 1/(2\sqrt{3}) \end{pmatrix}.

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Problem 22.2.2

(i)(ii)

The vector space of all $n \times n$ matrices over $\mathbb {C}$ form a Hilbert space with the scalar product defined by $\left\langle A, B \right\rangle := \operatorname {tr}(AB^{*})$ . This implies a norm $\left\| A\right\|^{2} = \operatorname {tr}(AA^{*})$ .

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(i)

Consider the Lie group $U(n)$ . Find two unitary $2 \times 2$ matrices $U_{1}$ , $U_{2}$ such that $\left\| U_{1} - U_{2}\right\|$ takes a maximum.

(ii)

Are the matrices

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

such a pair?

Problem 22.2.3

Consider the two normalized vectors

\overrightarrow {v}(\alpha , \phi ) = \begin{pmatrix} e^{i\phi }\cos (\alpha ) \\ \sin (\alpha ) \end{pmatrix}, \quad \overrightarrow {u}(\beta , \psi ) = \begin{pmatrix} e^{i\psi }\cos (\beta ) \\ \sin (\beta ) \end{pmatrix}

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

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

Note that

\overrightarrow {v}^{*}(\alpha , \phi )\overrightarrow {u}(\beta , \psi ) = e^{i(\psi -\phi )}\cos (\alpha )\cos (\beta ) + \sin (\alpha )\sin (\beta )

and

\left|\overrightarrow {v}^{*}(\alpha , \phi )\overrightarrow {u}(\beta , \psi )\right|^{2} = \cos ^{2}(\alpha ) + 2\cos (\psi -\phi )\cos (\alpha )\cos (\beta ) + \sin ^{2}(\beta ).

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Problem 22.2.4

Consider the two normalized vectors

\overrightarrow {v}_{1}(\phi _{1}, \theta _{1}) = \begin{pmatrix} \cos (\phi _{1})\sin (\theta _{1}) \\ \sin (\phi _{1})\sin (\theta _{1}) \\ \cos (\theta _{1}) \end{pmatrix}, \quad \overrightarrow {v}_{2}(\phi _{2}, \theta _{2}) = \begin{pmatrix} \cos (\phi _{2})\sin (\theta _{2}) \\ \sin (\phi _{2})\sin (\theta _{2}) \\ \cos (\theta _{2}) \end{pmatrix},

Find the conditions on $\phi_{1}, \theta_{1}, \phi_{2}, \theta_{2}$ such that $\left|\overrightarrow {v}_{1}^{*}(\phi_{1}, \theta_{1})\overrightarrow {v}_{2}(\phi_{2}, \theta_{2})\right| = \frac{1}{\sqrt{3}}$ .

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Problem 22.2.5

Consider the Hilbert space $M(2, \mathbb {C})$ of the $2 \times 2$ matrices over the complex numbers with the scalar product $\left\langle A, B \right\rangle = \operatorname {tr}(AB^{*})$ and $A, B \in M(2, \mathbb {C})$ . Two mutually unbiased bases are given by

\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}

and

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

Construct mutually unbiased bases applying the Kronecker product for the Hilbert space $M(4, \mathbb {C})$ .

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