{"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":"34748c84-167f-566b-99dd-91ef2966844e","task_key":"train--Challenge~5f60~5fmain","task_revision_id":"1","upstream_id":"Challenge_60_main","short_description":"In this problem, we want to consider the resolution of single-lens imaging of…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\nDelta_k_sq, gamma = sp.symbols('Delta_k_sq gamma')\\n\\ndef answer(Delta_k_sq, gamma):\\n    r\\\"\\\"\\\"\\n    Return the expression of the quantum Fisher information in Sympy format.\\n\\n    Inputs\\n    ----------\\n    Delta_k_sq: sympy.Symbol, $\\\\Delta k^2 \\\\equiv \\\\int_{-\\\\infty}^{\\\\infty} d x\\\\left[\\\\frac{\\\\partial \\\\psi(x)}{\\\\partial x}\\\\right]^2$\\n    gamma: sympy.Symbol, $\\\\gamma \\\\equiv \\\\int_{-\\\\infty}^{\\\\infty} d x \\\\frac{\\\\partial \\\\psi(x)}{\\\\partial x} \\\\psi\\\\left(x-u_2+u_1 \\\\right)$\\n\\n    Outputs\\n    ----------\\n    QFI: sympy.Expr, quantum Fisher information of estimating $\\\\theta=\\\\frac{1}{3}u_1+\\\\frac{2}{3}u_2$ per each measured photon,\\n         $u_1$ and $u_2$ are the positions of the two point sources\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    QFI = ...  # a SymPy expression of inputs\\n    # ---------------------------------------------------------------\\n\\n    return QFI\",\"problem_description\":\"\\n\\n# Problem setup:\\nIn this problem, we want to consider the resolution of single-lens imaging of thermal sources. The quantum state radiated by the thermal sources can be expressed as\\n\\n$$\\n\\\\rho_s=\\\\int D \\\\alpha \\\\Phi(\\\\alpha)|\\\\alpha\\\\rangle\\\\langle\\\\alpha|,\\n$$\\n\\n\\n\\n\\nwhere $\\\\Phi(\\\\alpha)$ is the Sudarshan-Glauber representation and $D \\\\alpha$ is an appropriate measure. Define  $\\\\alpha=\\\\left(\\\\alpha_1, \\\\ldots, \\\\alpha_J\\\\right)^{\\\\top}$ as a column vector of complex field amplitudes for $J$ optical spatial modes on the source plane and $|\\\\alpha\\\\rangle$ as a multimode coherent state with amplitude $\\\\alpha$.\\n For thermal sources, it is standard to assume $\\\\Phi$ to be a zero-mean complex Gaussian given by\\n\\n$$\\n\\\\Phi(\\\\alpha)=\\\\frac{1}{\\\\operatorname{det}(\\\\pi \\\\Gamma_s)} \\\\exp \\\\left(-\\\\alpha^{\\\\dagger} \\\\Gamma^{-1} \\\\alpha\\\\right)\\n$$\\n\\nwhere $\\\\alpha^{\\\\dagger}=\\\\left(\\\\alpha_1^*, \\\\ldots, \\\\alpha_J^*\\\\right)$ denotes the complex transpose of $\\\\alpha$.  We consider a simple two point source case\\n\\n\\\\begin{equation}\\n[\\\\Gamma_s]_{uv}=\\\\epsilon_0\\\\delta_{uv}[\\\\delta_{uu_1}+\\\\delta_{uu_2}]\\n\\\\end{equation}\\nwhere $\\\\epsilon_0\\\\to 0$, and $u_1$, $u_2$ are the positions of the two point sources. If the quantum state $\\\\rho_s$ passes through a single lens which has point spread function $\\\\psi(x)$. We want to quantify the performance of a naive direct imaging method and the fundamental limit for resolution.\\n\\n# Main problem:\\n\\nDefine $\\\\theta=\\\\frac{1}{3}u_1+\\\\frac{2}{3}u_2$, calculate the quantum Fisher information of estimating $\\\\theta$ per each measured photon, express the answer using\\n$\\n\\\\Delta k^2 \\\\equiv \\\\int_{-\\\\infty}^{\\\\infty} d x\\\\left[\\\\frac{\\\\partial \\\\psi(x)}{\\\\partial x}\\\\right]^2\\n$,\\n$\\n\\\\gamma \\\\equiv \\\\int_{-\\\\infty}^{\\\\infty} d x \\\\frac{\\\\partial \\\\psi(x)}{\\\\partial x} \\\\psi\\\\left(x-u_2+u_1 \\\\right)\\n$.\\n\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}