{"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":"8d11aa00-f6d3-5de4-a1e0-476676d3ebb1","task_key":"train--Challenge~5f62~5fmain","task_revision_id":"1","upstream_id":"Challenge_62_main","short_description":"Quantum superposition of a single particle can be demonstrated by the…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\nphi = sp.symbols('phi')\\nk = sp.symbols('k', integer=True)\\n\\ndef answer(phi, k, k_value):\\n    r\\\"\\\"\\\"\\n    Return the violation as a function of $\\\\phi$ for the case $k=1$, and the expressions of $T$ and $\\\\phi_{\\\\max}$ in Sympy format for any given k_value.\\n\\n    Inputs\\n    ----------\\n    phi: sympy.Symbol, phase angle $\\\\phi$\\n    k: sympy.Symbol, $k$\\n    k_value: int\\n        A specific value of $k$ used to determine which interval expression to return, but is *not* substituted into the final result.\\n\\n    Outputs\\n    ----------\\n    violation: sympy.Expr, the violation $\\\\delta$ as a function of $\\\\phi$ for the case $k=1 (N=2k+1=3)$.\\n    T: sympy.Interval, the interval $T \\\\subseteq [0,\\\\pi]$ of phases for which a quantum violation occurs.\\n        Explicitly set the parameters `left_open` and `right_open`.\\n    phi_max: sympy.Expr, the phase $\\\\phi_{\\\\max}$ at which the maximal quantum violation occurs.\\n\\n    The returned expressions of T and phi_max should preserve full symbolic dependence on k, while `k_value` is used internally solely to to select which interval expression to return.\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    violation = ...\\n    T = ...\\n    phi_max = ...\\n    # ---------------------------------------------------------------\\n\\n    return violation, T, phi_max\",\"problem_description\":\"\\n\\n# Problem setup:\\nQuantum superposition of a single particle can be demonstrated by the interference pattern observed in a double-slit experiment. Although it has been shown that no genuine triple-slit interference exists for a single quantum particle, an $N$-slit experiment can still reveal stronger nonclassical patterns, detectable through violations of the following inequality (Note: A classical theory that allows $N-1$-slit inteference cannot violate such an inequality.):\\n\\n$$\\np(0|0,\\\\cdots,0) + \\\\sum_{i=1}^N p(1|0,\\\\cdots,1_i,\\\\cdots,0) \\\\leq N,\\n$$\\n\\nwhere $p(b|a_1,\\\\cdots,a_n)$ represents the statistics of getting outcome $b$ when each path encodes $a_i$.\\n\\nTo see the violation, a quantum particle is prepared among $N$ paths, such that $|\\\\Psi_N\\\\rangle = \\\\frac{1}{\\\\sqrt{N}} \\\\sum^N_{i=1} |e_i\\\\rangle$  with $|e_i\\\\rangle=|0\\\\rangle_{A_1}\\\\cdots|1\\\\rangle_{A_i}\\\\cdots|0⟩_{A_N}$. Each party encodes a local phase on their path $i$ given input $a_i$, by applying the unitary $U_i(\\\\phi_{a_i}) = e^{i\\\\phi_{a_i}}|1_i\\\\rangle\\\\langle1_i|$. The resulting encoded state is given by\\n\\n$$\\\\sigma_{a_1,\\\\cdots,a_n}= \\\\bigotimes_i U_i(\\\\phi_{a_i}) |\\\\Psi_N\\\\rangle\\\\langle\\\\Psi_N|\\\\bigotimes_i U_i^{\\\\dagger}(\\\\phi_{a_i}),$$\\n\\nwhich is measured using a fixed measurement $\\\\{\\\\Pi_b\\\\}_b$, and the statistics are given by\\n\\n$$p(b|a_1,\\\\cdots,a_n)=Tr[\\\\Pi_b \\\\sigma_{a_1,\\\\cdots,a_n}].$$\\n\\nWe consider a specific encoding strategy for an odd number of paths, $N = 2k + 1$, as follows:\\nSet $\\\\phi_{1_i} = \\\\phi$ for $i \\\\leq k$; $\\\\phi_{1_i} = \\\\pi$ for $i = k + 1$; $\\\\phi_{1_i} = -\\\\phi$ for $i > k + 1$; and $\\\\phi_{i_0} = 0$ for all $i$.\\n\\n\\n\\n\\n# Main problem:\\n\\n(1) For the case $k=1 (N=2k+1=3)$, express the violation as a function of $\\\\phi$, i.e., the violation is defined as\\n\\n$$\\n\\\\delta= p(0|0,\\\\cdots,0) + \\\\sum_{i=1}^N p(1|0,\\\\cdots,1_i,\\\\cdots,0)- N.\\n$$\\n\\n(2) Determine the range of $\\\\phi$ for which a quantum violation occurs, denoted as $T$, where $T \\\\subseteq [0, \\\\pi]$ is a function of $k$.\\n\\n(3) Determine the value of $\\\\phi$ at which the maximal quantum violation occurs, denoted as $\\\\phi_{\\\\max} $, as a function of $k$.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}