Doubly infinite separation of quantum information and communication

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.

Figure 1

Complexities of ${\mathrm{EXC}}_{n,m,\gamma }$ with (a) $m\in \omega \left(\sqrt{nlogn}\right), m\in \tilde{o}\left(n\right), \gamma =0$; (b) $m\in \omega \left(\sqrt{nlogn}\right), m\in \tilde{o}\left(n\right), \gamma ={(n+1)}^{-m}$. Solid arrows denote established gaps (pointing towards the larger complexity), while the dashed ones denote unknown gaps. “$\infty \infty$” means doubly infinite.

Figure 2

Illustrations of infinite gaps in the (a) linear scale and (b) logarithmic scale. “$\infty$” means singly infinite; “$\infty \infty$” means doubly infinite.