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Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem description

# Problem setup: The quantum ff-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\mu a positive measure on (0,∞)(0,\infty) such that ∫0∞11+sdμ(s)<∞\int_0^\infty \frac{1}{1+s} d\mu(s) < \infty. We study the derivatives of the ff-divergence. Let ρt=σ+t(ρ−σ)\rho_t = \sigma + t(\rho - \sigma). # Main problem: Calculate ddtDfstd(ρt‖σ)\frac{d}{dt} D^{\mathrm{std}}_f(\rho_t \|\sigma) at t=0.5t = 0.5. 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 possible
Original 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 possible

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_derivative

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Source and history

upstream_id / Original problem ID
Challenge_68_main
task_id
1220bd73-c482-56b8-b22d-35ffda192359
task_key
train--Challenge~5f68~5fmain
task_revision_id
4
config / split
/ train

Official source

initial import