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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.

problem description

# Problem setup: Optimize the Holevo information over cq states χ=∑xpx|x⟩⟨x|⊗ρx\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} {px}\{p_x\} is a probability distribution over xx, and γx\gamma_x are all real numbers in [0,1][0,1]. Assume that cosθ≠1\cos\theta\neq1. # Main problem: Write the maximal value in terms of an optimization maxx∈[0,1]f(x)\max_{x\in[0,1]} f(x), where f(x)f(x) depends only on xx and nothing else. Find the function form of f(x)f(x) 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.

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 f

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Source and history

upstream_id / Original problem ID
Challenge_22_main
task_id
76ca5c11-31d0-519d-ae55-4443e5a9ff0b
task_key
train--Challenge~5f22~5fmain
task_revision_id
4
config / split
/ train

Official source

initial import