Equality, Revisited

In 1979 Yao published a paper that started the field of communication complexity and asked, in particular, what was the randomised complexity of the Equality function (EQ) in the Simultaneous Message Passing (SMP) model (for the question to be non-trivial, one must consider the setting of private randomness). The tight lower bound Ω( √ n) was given only in 1996 by Newman and Szegedy. In this work we develop a new lower bound method for analysing the complexity of EQ in SMP. Our technique achieves the following: • It leads to the tight lower bounds of Ω( √ n) for both EQ and its negation NE in the non-deterministic version of quantum-classical SMP, where Merlin is also quantum – this is the strongest known version of SMP where the complexity of both EQ and NE remain high (previously known techniques seem to be insufficient for this). • It provides a unified view of the communication complexity of EQ and NE , allowing to obtain tight characterisation in all previously studied and a few newly introduced versions of SMP, including all possible combination of either quantum or randomised Alice, Bob and Merlin in the non-deterministic case. Arguably, it also simplifies the previously known lower bound proofs. Our characterisation also leads to tight trade-offs between the message lengths needed for players Alice, Bob, Merlin, not just the maximum message length among them, and highlights that NE is easier than EQ in the presence of classical proofs, whereas the problems have (roughly) the same complexity when a quantum proof is present. We also construct new protocols for EQ and NE that achieve optimal trade-offs in the “asymmetric” scenarios when the (qu)bits from Merlin are either cheaper or more expensive than those from the trusted parties. Along the way, we give tight analysis of a new primitive, where a honest classical and an untrusted quantum parties help a third party to obtain an approximate copy of a quantum state. ∗Division of Mathematical Sciences, Nanyang Technological University, Singapore. †Institute of Mathematics, Academy of Sciences, Žitna 25, Praha 1, Czech Republic. Partially funded by the grant P202/12/G061 of GA ČR and by RVO: 67985840. Most of this work was done while DG was visiting the Centre for Quantum Technologies at the National University of Singapore, and was partially funded by the Singapore Ministry of Education and the NRF. ‡Division of Mathematical Sciences, Nanyang Technological University, Singapore & Centre for Quantum Technologies, National University of Singapore, Singapore. This work is funded by the Singapore Ministry of Education (partly through the Academic Research Fund Tier 3 MOE2012-T3-1-009) and by the Singapore National Research Foundation.