{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"研究水準の物理問題で、科学的理解と多段階の推論・計算を評価するベンチマークです。公開データには70の課題があり、問題文とコード雛形を組み合わせて解答を構成します。\n\nCritPt evaluates scientific understanding and multi-step reasoning and computation on research-level physics problems. Its public dataset contains 70 challenges with problem descriptions and code templates.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://critpt.com/","indexing_mode":"noindex"},"task_id":"76ca5c11-31d0-519d-ae55-4443e5a9ff0b","task_key":"train--Challenge~5f22~5fmain","task_revision_id":"1","upstream_id":"Challenge_22_main","short_description":"Optimize the Holevo information over cq states $\\chi=\\sum_x p_x|x\\rangle\\langle…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\nx = sp.symbols('x')\\n\\ndef answer(x):\\n    r\\\"\\\"\\\"\\n    Return the expression of $f(x)$ in Sympy format.\\n\\n    Inputs\\n    ----------\\n    x : sympy.Symbol, optimization variable $x\\\\in[0,1]$\\n\\n    Outputs\\n    ----------\\n    f : sympy.Expr, the explicit function form of $f(x)$\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    f = ...  # a SymPy expression of x\\n    # ---------------------------------------------------------------\\n\\n    return f\",\"problem_description\":\"# Problem setup:\\n\\nOptimize the Holevo information over cq states $\\\\chi=\\\\sum_x p_x|x\\\\rangle\\\\langle x|\\\\otimes\\\\rho_x$, where\\n\\\\begin{equation}\\n\\\\rho_x=\\\\left(\\\\begin{array}{ccc}\\n\\\\gamma_x \\\\cos ^2 \\\\theta & 0 & 0 \\\\\\\\\\n0 & \\\\left(1-\\\\gamma_x\\\\right) \\\\cos ^2 \\\\theta & \\\\sqrt{1-\\\\gamma_x} \\\\cos \\\\theta \\\\sin \\\\theta e^{i \\\\phi_x} \\\\\\\\\\n0 & \\\\sqrt{1-\\\\gamma_x} \\\\cos \\\\theta \\\\sin \\\\theta e^{-i \\\\phi_x} & \\\\sin ^2 \\\\theta\\n\\\\end{array}\\\\right),\\n\\\\end{equation}\\n$\\\\{p_x\\\\}$ is a probability distribution over $x$, and $\\\\gamma_x$ are all real numbers in $[0,1]$. Assume that $\\\\cos\\\\theta\\\\neq1$.\\n\\n\\n# Main problem:\\n\\nWrite 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.\\n\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}