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CritPt / Challenge_22_main / Optimize the Holevo information over cq states …
Problem
Answer published by the source. Consult the official source to check your work against its answer.
code template
Code
import sympy as sp
x = sp.symbols('x')
def answer(x):
r"""
Return the expression of $f(x)$ in Sympy format.
Inputs
----------
x : sympy.Symbol, optimization variable $x\in[0,1]$
Outputs
----------
f : sympy.Expr, the explicit function form of $f(x)$
"""
# ------------------ FILL IN YOUR RESULTS BELOW ------------------
f = ... # a SymPy expression of x
# ---------------------------------------------------------------
return fproblem description
# Problem setup:
Optimize the Holevo information over cq states , where
\begin{equation}
\rho_x=\left(\begin{array}{ccc}
\gamma_x \cos ^2 \theta & 0 & 0 \\
0 & \left(1-\gamma_x\right) \cos ^2 \theta & \sqrt{1-\gamma_x} \cos \theta \sin \theta e^{i \phi_x} \\
0 & \sqrt{1-\gamma_x} \cos \theta \sin \theta e^{-i \phi_x} & \sin ^2 \theta
\end{array}\right),
\end{equation}
is a probability distribution over , and are all real numbers in . Assume that .
# Main problem:
Write the maximal value in terms of an optimization , where depends only on and nothing else. Find the function form of explicitly.
Plain-text mathematical notation (without MathML)
# Problem setup:
Optimize the Holevo information over cq states χ=∑_(x)p_(x)|x⟩⟨x|⊗ρ_(x), where
\begin{equation}
\rho_x=\left(\begin{array}{ccc}
\gamma_x \cos ^2 \theta & 0 & 0 \\
0 & \left(1-\gamma_x\right) \cos ^2 \theta & \sqrt{1-\gamma_x} \cos \theta \sin \theta e^{i \phi_x} \\
0 & \sqrt{1-\gamma_x} \cos \theta \sin \theta e^{-i \phi_x} & \sin ^2 \theta
\end{array}\right),
\end{equation}
{p_(x)} is a probability distribution over x, and γ_(x) are all real numbers in [0,1]. Assume that cosθ≠1.
# Main problem:
Write the maximal value in terms of an optimization max_(x∈[0,1])f(x), where f(x) depends only on x and nothing else. Find the function form of f(x) explicitly.
Original LaTeX notation
# Problem setup:
Optimize the Holevo information over cq states $\chi=\sum_x p_x|x\rangle\langle x|\otimes\rho_x$, where
\begin{equation}
\rho_x=\left(\begin{array}{ccc}
\gamma_x \cos ^2 \theta & 0 & 0 \\
0 & \left(1-\gamma_x\right) \cos ^2 \theta & \sqrt{1-\gamma_x} \cos \theta \sin \theta e^{i \phi_x} \\
0 & \sqrt{1-\gamma_x} \cos \theta \sin \theta e^{-i \phi_x} & \sin ^2 \theta
\end{array}\right),
\end{equation}
$\{p_x\}$ is a probability distribution over $x$, and $\gamma_x$ are all real numbers in $[0,1]$. Assume that $\cos\theta\neq1$.
# Main problem:
Write the maximal value in terms of an optimization $\max_{x\in[0,1]} f(x)$, where $f(x)$ depends only on $x$ and nothing else. Find the function form of $f(x)$ explicitly.
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