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Communication complexity is a pivotal element in information science, withquantum theory presenting a significant edge over classical approaches. Thestandard quantification of one-way communication complexity relies on theminimal dimension of the systems that the sender communicates to accomplish thedesignated task. In this study, we adopt a novel perspective, measuring thecomplexity of the communication by the distinguishability of the sender's inputwithout constraining the dimension of the communicated systems. This measurebecomes especially pertinent when maintaining the confidentiality of thesender's input is essential. After establishing the generic framework, we focuson two important categories of communication complexity tasks - the generalversion of random access codes and equality problems defined by graphs. Wederive lower bounds on the distinguishability of the sender's input as afunction of the success metric of these tasks in classical communication.Notably, we show that the ratio between the distinguishability in classical andquantum communication to achieve the same success metric escalates with thecomplexity of these tasks, reaching arbitrarily large values. Besides, wedemonstrate the quantum advantage by employing qubits in solving equalityproblems associated with odd-cycle graphs. Furthermore, we derive lower boundson distinguishability for another class of communication tasks, namely,pair-distinguishability tasks, and present several instances of the quantumadvantage.

Unbounded quantum advantage in communication complexity measured by distinguishability