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CritPt / Challenge_68_main / The quantum f-divergence is defined by …
Problem
Answer published by the source. Consult the official source to check your work against its answer.
code template
Code
def answer():
r"""
Return the value of $\frac{d}{dt} D^{\mathrm{std}}_f(\rho_t \|\sigma)$ at $t = 0.5$.
Inputs
----------
None
Outputs
----------
first_derivative: float, the first derivative $\frac{d}{dt} D^{\mathrm{std}}_f(\rho_t \|\sigma)$ at $t = 0.5$
"""
# ------------------ FILL IN YOUR RESULTS BELOW ------------------
first_derivative = ...
# ---------------------------------------------------------------
return first_derivativeproblem description
# Problem setup:
The quantum -divergence is defined by \begin{align*}
D^{\mathrm{std}}_f(\rho \|\sigma) = \int_0^\infty \mathrm{tr}\bigl[ (\rho - \sigma) \frac{1}{L_\rho + s R_\sigma}(\rho-\sigma) \bigr] d\mu(s),
\end{align*}
where a positive measure on such that . We study the derivatives of the -divergence. Let .
# Main problem:
Calculate at . Please be as accurate as possible
Plain-text mathematical notation (without MathML)
# Problem setup:
The quantum f-divergence is defined by \begin{align*}
D^{\mathrm{std}}_f(\rho \|\sigma) = \int_0^\infty \mathrm{tr}\bigl[ (\rho - \sigma) \frac{1}{L_\rho + s R_\sigma}(\rho-\sigma) \bigr] d\mu(s),
\end{align*}
where μ a positive measure on (0,∞) such that ∫₀^(∞)(1)/(1+s)dμ(s)<∞. We study the derivatives of the f-divergence. Let ρ_(t)=σ+t(ρ−σ).
# Main problem:
Calculate (d)/(dt)D_(f)^(std)(ρ_(t)‖σ) at t=0.5. Please be as accurate as possibleOriginal LaTeX notation
# Problem setup:
The quantum $f$-divergence is defined by \begin{align*}
D^{\mathrm{std}}_f(\rho \|\sigma) = \int_0^\infty \mathrm{tr}\bigl[ (\rho - \sigma) \frac{1}{L_\rho + s R_\sigma}(\rho-\sigma) \bigr] d\mu(s),
\end{align*}
where $\mu$ a positive measure on $(0,\infty)$ such that $\int_0^\infty \frac{1}{1+s} d\mu(s) < \infty$. We study the derivatives of the $f$-divergence. Let $\rho_t = \sigma + t(\rho - \sigma)$.
# Main problem:
Calculate $\frac{d}{dt} D^{\mathrm{std}}_f(\rho_t \|\sigma)$ at $t = 0.5$. Please be as accurate as possibleDiscussion
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