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CritPt / Challenge_69_main / Suppose the quantum amplitude damping channel acting on a qubit system is…
Problem
Answer published by the source. Consult the official source to check your work against its answer.
problem description
# Problem setup:
Suppose the quantum amplitude damping channel acting on a qubit system is defined by
\begin{align*}
\mathcal A_{\gamma}\begin{pmatrix}
\rho_{00} & \rho_{01} \\
\rho_{10} & \rho_{11}
\end{pmatrix} = \begin{pmatrix}
\rho_{00}+ \gamma \rho_{11} & \sqrt{1-\gamma}\rho_{01} \\
\sqrt{1-\gamma} \rho_{10} & (1-\gamma)\rho_{11}
\end{pmatrix}.
\end{align*}
The quantum relative entropy is defined by
\begin{equation*}
D(\rho||\sigma)= \begin{cases}
\mathrm{tr}\Big(\rho\big(\log(\rho)-\log(\sigma)\big)\Big) &\text{ if } \text{supp}(\rho)\subseteq \text{supp}(\sigma), \\
\infty &\text{ else}.
\end{cases}
\end{equation*}
# Main problem:
Denote
\begin{align*}
f(\gamma):=\sup_{\rho \neq \sigma}\frac{D(\mathcal{A}_{\gamma}(\rho) \|\mathcal{A}_{\gamma}(\sigma))}{D(\rho \|\sigma)},\quad \gamma \in [0,1],
\end{align*}
where the supremum is taken over all density operators and . Calculate
Plain-text mathematical notation (without MathML)
# Problem setup:
Suppose the quantum amplitude damping channel acting on a qubit system is defined by
\begin{align*}
\mathcal A_{\gamma}\begin{pmatrix}
\rho_{00} & \rho_{01} \\
\rho_{10} & \rho_{11}
\end{pmatrix} = \begin{pmatrix}
\rho_{00}+ \gamma \rho_{11} & \sqrt{1-\gamma}\rho_{01} \\
\sqrt{1-\gamma} \rho_{10} & (1-\gamma)\rho_{11}
\end{pmatrix}.
\end{align*}
The quantum relative entropy is defined by
\begin{equation*}
D(\rho||\sigma)= \begin{cases}
\mathrm{tr}\Big(\rho\big(\log(\rho)-\log(\sigma)\big)\Big) &\text{ if } \text{supp}(\rho)\subseteq \text{supp}(\sigma), \\
\infty &\text{ else}.
\end{cases}
\end{equation*}
# Main problem:
Denote
\begin{align*}
f(\gamma):=\sup_{\rho \neq \sigma}\frac{D(\mathcal{A}_{\gamma}(\rho) \|\mathcal{A}_{\gamma}(\sigma))}{D(\rho \|\sigma)},\quad \gamma \in [0,1],
\end{align*}
where the supremum is taken over all density operators ρ and σ. Calculate f((1)/(8))+f((1)/(4))+f((1)/(2)).
Original LaTeX notation
# Problem setup:
Suppose the quantum amplitude damping channel acting on a qubit system is defined by
\begin{align*}
\mathcal A_{\gamma}\begin{pmatrix}
\rho_{00} & \rho_{01} \\
\rho_{10} & \rho_{11}
\end{pmatrix} = \begin{pmatrix}
\rho_{00}+ \gamma \rho_{11} & \sqrt{1-\gamma}\rho_{01} \\
\sqrt{1-\gamma} \rho_{10} & (1-\gamma)\rho_{11}
\end{pmatrix}.
\end{align*}
The quantum relative entropy is defined by
\begin{equation*}
D(\rho||\sigma)= \begin{cases}
\mathrm{tr}\Big(\rho\big(\log(\rho)-\log(\sigma)\big)\Big) &\text{ if } \text{supp}(\rho)\subseteq \text{supp}(\sigma), \\
\infty &\text{ else}.
\end{cases}
\end{equation*}
# Main problem:
Denote
\begin{align*}
f(\gamma):=\sup_{\rho \neq \sigma}\frac{D(\mathcal{A}_{\gamma}(\rho) \|\mathcal{A}_{\gamma}(\sigma))}{D(\rho \|\sigma)},\quad \gamma \in [0,1],
\end{align*}
where the supremum is taken over all density operators $\rho$ and $\sigma$. Calculate $$f(\frac{1}{8}) + f(\frac{1}{4}) + f(\frac{1}{2}).$$
code template
Code
def answer():
r"""
Return the value of $f(\frac{1}{8}) + f(\frac{1}{4}) + f(\frac{1}{2})$.
Inputs
----------
None
Outputs
----------
sum: float
Value of $f(\frac{1}{8}) + f(\frac{1}{4}) + f(\frac{1}{2})$,
where $f(\gamma):=\sup_{\rho \neq \sigma}\frac{D(\mathcal{A}_{\gamma}(\rho) \|\mathcal{A}_{\gamma}(\sigma))}{D(\rho \|\sigma)},\quad \gamma \in [0,1]$.
"""
# ------------------ FILL IN YOUR RESULTS BELOW ------------------
sum = ...
# ---------------------------------------------------------------
return sumDiscussion
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