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تسجيل مستخدم جديدA Direct Product Theorem for One-Way Quantum Communication
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We prove a direct product theorem for the one-way entanglement-assisted quantum communication complexity of a general relation $fsubseteqmathcal{X}timesmathcal{Y}timesmathcal{Z}$. For any $varepsilon, zeta > 0$ and any $kgeq1$, we show that [ mathrm{Q}^1_{1-(1-varepsilon)^{Omega(zeta^6k/log|mathcal{Z}|)}}(f^k) = Omegaleft(kleft(zeta^5cdotmathrm{Q}^1_{varepsilon + 12zeta}(f) - loglog(1/zeta)right)right),] where $mathrm{Q}^1_{varepsilon}(f)$ represents the one-way entanglement-assisted quantum communication complexity of $f$ with worst-case error $varepsilon$ and $f^k$ denotes $k$ parallel instances of $f$. As far as we are aware, this is the first direct product theorem for quantum communication. Our techniques are inspired by the parallel repetition theorems for the entangled value of two-player non-local games, under product distributions due to Jain, Pereszl{e}nyi and Yao, and under anchored distributions due to Bavarian, Vidick and Yuen, as well as message-compression for quantum protocols due to Jain, Radhakrishnan and Sen. Our techniques also work for entangled non-local games which have input distributions anchored on any one side. In particular, we show that for any game $G = (q, mathcal{X}timesmathcal{Y}, mathcal{A}timesmathcal{B}, mathsf{V})$ where $q$ is a distribution on $mathcal{X}timesmathcal{Y}$ anchored on any one side with anchoring probability $zeta$, then [ omega^*(G^k) = left(1 - (1-omega^*(G))^5right)^{Omegaleft(frac{zeta^2 k}{log(|mathcal{A}|cdot|mathcal{B}|)}right)}] where $omega^*(G)$ represents the entangled value of the game $G$. This is a generalization of the result of Bavarian, Vidick and Yuen, who proved a parallel repetition theorem for games anchored on both sides, and potentially a simplification of their proof.
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