Matching upper bounds on symmetric predicates in quantum communication complexity

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In this paper, we focus on the quantum communication complexity of functions of the form $f \circ G = f(G(X_1, Y_1), \ldots, G(X_n, Y_n))$ where $f: \Bset^n \to \Bset$ is a symmetric function, $G: \Bset^j \times \Bset^k \to \Bset$ is any function and Alice (resp. Bob) is given $(X_i)_{i \in [n]}$ (resp. $(Y_i)_{i \in [n]}$). Recently, Chakraborty et al. [STACS 2022] showed that the quantum communication complexity of $f \circ G$ is $O(Q(f)\QCCEX(G))$ when the parties are allowed to use shared entanglement, where $Q(f)$ is the query complexity of $f$ and $\QCCEX(G)$ is the exact communication complexity of $G$. }{In this paper, we first show that the same statement holds \emph{without both shared entanglement and shared randomness}, which generalizes their result. Based on the improved result, we next show tight upper bounds on $f \circ \mathrm{AND}_2$ for any symmetric function $f$ (where $\textrm{AND}_2 : \Bset \times \Bset \to \Bset$ denotes the 2-bit AND function) in both models: with shared entanglement and without shared entanglement. This matches the well-known lower bound by Razborov~[Izv. Math. 67(1) 145, 2003] when shared entanglement is allowed and improves Razborov's bound when shared entanglement is not allowed.