We prove the existence of (one-way) communication tasks with a subconstant
versus superconstant asymptotic gap, which we call "doubly infinite," between
their quantum information and communication complexities. We do so by studying
the exclusion game [C. Perry et al., Phys. Rev. Lett. 115, 030504 (2015)] for
which there exist instances where the quantum information complexity tends to
zero as the size of the input n increases. By showing that the quantum
communication complexity of these games scales at least logarithmically in n,
we obtain our result. We further show that the established lower bounds and
gaps still hold even if we allow a small probability of error. However in this
case, the n-qubit quantum message of the zero-error strategy can be
compressed polynomially.Comment: 16 pages, 2 figures. v4: minor errors fixed; close to published
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