{"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":"83217e10-1925-5c4c-8117-a0460804bb70","task_key":"train--Challenge~5f67~5fmain","task_revision_id":"1","upstream_id":"Challenge_67_main","short_description":"We study the fundamental limit of a quantum channel called the private channel.…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\nd = sp.symbols('d', integer=True)\\n\\ndef answer(d):\\n    r\\\"\\\"\\\"\\n    Return the expression of the quantum capacity of the channel in Sympy format.\\n\\n    Inputs\\n    ----------\\n    d: sympy.Symbol, dimension paramter, $d$\\n\\n    Outputs\\n    ----------\\n    quantum_capacity: sympy.Expr, the quantum capacity of the channel.\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    quantum_capacity = ... # a SymPy expression of inputs\\n    # ---------------------------------------------------------------\\n\\n    return quantum_capacity\",\"problem_description\":\"\\n\\n# Problem setup:\\nWe study the fundamental limit of a quantum channel called the private channel. To be more specific, the Choi operator of the channel is given by the private state, with the input space given by the private system and shield system. Our channel is determined by the following Werner state: $$ q |\\\\psi_+\\\\rangle \\\\langle \\\\psi_+|^{a_0b_0} \\\\otimes \\\\frac{1}{d_{\\\\text{sym}}} P_{\\\\mathrm{sym}}^{A_0B_0} + (1-q) |\\\\psi_-\\\\rangle \\\\langle \\\\psi_-|^{a_0b_0} \\\\otimes \\\\frac{1}{d_{\\\\mathrm{asym}}} P_{\\\\mathrm{asym}}^{A_0B_0},\\\\ q\\\\in [0,1],$$\\nwhere $a_0,b_0$ have dimension 2 and $A_0,B_0$ have dimension $d$. $P_{\\\\mathrm{sym}}$ is the projection onto the symmetric subspace.\\n\\n\\n# Main problem:\\n\\nWhat is the quantum capacity of this channel when $q = \\\\frac{d+1}{2d}$?\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}