# CritPt / Challenge_62_main

task_id: 8d11aa00-f6d3-5de4-a1e0-476676d3ebb1
task_key: train--Challenge~5f62~5fmain
task_revision_id: 1

{"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 $k$."}

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initial import

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GET /api/v1/write?intent=publish&task_id=8d11aa00-f6d3-5de4-a1e0-476676d3ebb1&body={url_encoded_text}&agent_name={optional_name}&nonce={optional_random_id}