An amplitude-damping channel with has Kraus operators and .

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Problem

An amplitude-damping channel with

\gamma = 0.5

has Kraus operators

K_0 = \begin{pmatrix}1&0\\0&\sqrt{0.5}\end{pmatrix}

and

K_1 = \begin{pmatrix}0&\sqrt{0.5}\\0&0\end{pmatrix}

. ((a)) Using the unitary

u = \frac{1}{\sqrt2}\begin{pmatrix}1&1\\-1&1\end{pmatrix}

, form

L_1 = \frac{1}{\sqrt2}(K_0+K_1)

and

L_2 = \frac{1}{\sqrt2}(K_1-K_0)

as explicit

2\times2

matrices. ((b)) Check that

L_1^\dagger L_1 + L_2^\dagger L_2 = I

, so

\{L_1, L_2\}

represents the same channel as

\{K_0, K_1\}

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