{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"CritPt 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","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[]}},"task_id":"70e5fa51-5e86-5e0e-a05d-b2d58522e47b","task_key":"train--Challenge~5f69~5fmain","task_revision_id":"1","upstream_id":"Challenge_69_main","short_description":"Suppose the quantum amplitude damping channel acting on a qubit system is…","config":"","split":"train","body":"{\"code_template\":\"def answer():\\n    r\\\"\\\"\\\"\\n    Return the value of $f(\\\\frac{1}{8}) + f(\\\\frac{1}{4}) + f(\\\\frac{1}{2})$.\\n\\n    Inputs\\n    ----------\\n    None\\n\\n    Outputs\\n    ----------\\n    sum: float\\n        Value of $f(\\\\frac{1}{8}) + f(\\\\frac{1}{4}) + f(\\\\frac{1}{2})$,\\n        where $f(\\\\gamma):=\\\\sup_{\\\\rho \\\\neq \\\\sigma}\\\\frac{D(\\\\mathcal{A}_{\\\\gamma}(\\\\rho) \\\\|\\\\mathcal{A}_{\\\\gamma}(\\\\sigma))}{D(\\\\rho \\\\|\\\\sigma)},\\\\quad \\\\gamma \\\\in [0,1]$.\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    sum = ...\\n    # ---------------------------------------------------------------\\n\\n    return sum\",\"problem_description\":\"\\n\\n# Problem setup:\\nSuppose the quantum amplitude damping channel acting on a qubit system is defined by\\n\\\\begin{align*}\\n    \\\\mathcal A_{\\\\gamma}\\\\begin{pmatrix}\\n        \\\\rho_{00} & \\\\rho_{01} \\\\\\\\\\n        \\\\rho_{10} & \\\\rho_{11}\\n    \\\\end{pmatrix} = \\\\begin{pmatrix}\\n        \\\\rho_{00}+ \\\\gamma \\\\rho_{11} & \\\\sqrt{1-\\\\gamma}\\\\rho_{01} \\\\\\\\\\n        \\\\sqrt{1-\\\\gamma} \\\\rho_{10} & (1-\\\\gamma)\\\\rho_{11}\\n    \\\\end{pmatrix}.\\n\\\\end{align*}\\nThe quantum relative entropy is defined by\\n\\\\begin{equation*}\\nD(\\\\rho||\\\\sigma)= \\\\begin{cases}\\n \\\\mathrm{tr}\\\\Big(\\\\rho\\\\big(\\\\log(\\\\rho)-\\\\log(\\\\sigma)\\\\big)\\\\Big) &\\\\text{ if } \\\\text{supp}(\\\\rho)\\\\subseteq \\\\text{supp}(\\\\sigma), \\\\\\\\\\n\\\\infty &\\\\text{ else}.\\n\\\\end{cases}\\n\\\\end{equation*}\\n\\n\\n# Main problem:\\n\\nDenote\\n\\\\begin{align*}\\n    f(\\\\gamma):=\\\\sup_{\\\\rho \\\\neq \\\\sigma}\\\\frac{D(\\\\mathcal{A}_{\\\\gamma}(\\\\rho) \\\\|\\\\mathcal{A}_{\\\\gamma}(\\\\sigma))}{D(\\\\rho \\\\|\\\\sigma)},\\\\quad \\\\gamma \\\\in [0,1],\\n\\\\end{align*}\\nwhere the supremum is taken over all density operators $\\\\rho$ and $\\\\sigma$. Calculate $$f(\\\\frac{1}{8}) + f(\\\\frac{1}{4}) + f(\\\\frac{1}{2}).$$\\n\\n\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}