Matching upper bounds on symmetric predicates in quantum communication complexity

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 $f: \{0, 1\}^n \to \{0, 1\}$ is a symmetric function, $G: \{0, 1\}^j \times \{0, 1\}^k \to \{0, 1\}$ is any function and Alice (resp. Bob) is given $(X_i)_{i \leq n}$ (resp. $(Y_i)_{i \leq n}$). Recently, Chakraborty et al. [STACS 2022] showed that the quantum communication complexity of $f \circ G$ is $O(Q(f)\mathrm{QCC}_\mathrm{E}(G))$ when the parties are allowed to use shared entanglement, where $Q(f)$ is the query complexity of $f$ and $\mathrm{QCC}_\mathrm{E}(G)$ is the exact communication complexity of $G$. In this paper, we first show that the same statement holds \emph{without shared entanglement}, 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 $\textrm{AND}_2 : \{0, 1\} \times \{0, 1\} \to \{0, 1\}$ 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.