High-Dimensional Quantum Communication Complexity beyond Strategies Based on Bell’s Theorem

Quantum resources can improve communication complexity problems (CCPs) beyond their classical constraints. One quantum approach is to share entanglement and create correlations violating a Bell inequality, which can then assist classical communication. A second approach is to resort solely to the preparation, transmission, and measurement of a single quantum system, in other words, quantum communication. Here, we show the advantages of the latter over the former in high-dimensional Hilbert space. We focus on a family of CCPs, based on facet Bell inequalities, study the advantage of high-dimensional quantum communication, and realize such quantum communication strategies using up to ten-dimensional systems. The experiment demonstrates, for growing dimension, an increasing advantage over quantum strategies based on Bell inequality violation. For sufficiently high dimensions, quantum communication also surpasses the limitations of the postquantum Bell correlations obeying only locality in the macroscopic limit. We find that the advantages are tied to the use of measurements that are not rank-one projective, and provide an experimental semi-device-independent falsification of such measurements in Hilbert space dimension six.

Figure 1

(a) Quantum CCP implementation based on the violation of the CGLMP inequalities. (b) Quantum CCP implementation based on communicating a single $d$-level quantum system.

Figure 2

Experimental setup for implementing the CCPs with quantum communication. $d$-dimensional quantum systems are encoded into the linear transverse momentum of single photons. The experiment is composed of two main parts: one for the state preparation and another for performing measurements on the prepared system. Both parts rely on the programmability of spatial light modulators for preparing the required states and measurements.

Figure 3

Experimental results. ${\mathrm{\Delta }}_d^{\mathrm{Exp}}$ is represented by red points. The yellow points represent ${\mathrm{\Delta }}_d^{\text{Bell}}$. The blue points are the theoretical predictions of ${\mathrm{\Delta }}_d^{\mathrm{QS}}$. The green points represent ${\mathrm{\Delta }}_d^{\mathrm{ML}}$.