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

Abstract

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 realise 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 post-quantum Bell correlations obeying only locality in the macroscopic limit. Surprisingly, we find that the advantages are tied to the use of measurements that are not rank-one projective. We provide an experimental semi-device-independent falsification of such measurements in Hilbert space dimension six.Introduction.-Communication complexity problems (CCPs) are tasks in which distant parties hold local data, the collection of which is needed for a computation of their interest. To make the computation possible, the parties communicate with each other. However, the amount of communication is limited and therefore not all data can be sent. The CCP consist in parties adopting an efficient communication strategy which allows them to perform the desired computation with a probability as high as possible. Efficient use of quantitatively limited communication is a broadly relevant matter [1], which provides fundamental insights on physical limitations [2,3].The ability to process information depends on the choice of the physical system into which the information is encoded [4]. Consequently, quantum entities without a classical counterpart can be regarded as tools for quantum information processing. The most famous example is entanglement. In a quantum CCP, parties may share an entangled state on which they perform local measurements, generating strongly correlated data which violates a Bell inequality. That data can then be used to assist a classical communication strategy [5]. In fact, Bell inequalities have been systematically linked to CCPs [6-8], and their violation enables better-than-classical communication efficiencies [7][8][9][10][11][12][13][14][15].Nevertheless, quantum theory presents also a second approach to CCPs: substituting classical communication with quantum communication. The justification for such a substitution relies on the Holevo theorem [16] which implies that no more information can be extracted from quantum d-level system than from a classical d-level system. Hence, in a quantum communication strategy, information is encoded in a quantum state of a specified Hilbe...

High‐Dimensional Quantum Communication: Benefits, Progress, and Future Challenges

“…However, Martínez et al. have shown, both theoretically and experimentally, that high‐dimensional quantum communication outperforms classical protocols assisted by nonlocal correlation whenever dimensions are . Indeed, they demonstrated how dimension six acts as a threshold to reveal the benefits of quantum communication over implementations based on the violations of CGLMP inequalities.…”

Quantum blackjack: Advantages offered by quantum strategies in communication-limited games

“…Recall then the optimal strategy for Alice is determined in Eq. (17) such that the objective function becomes…”

“…[10], which characterizes the power of protocols with unlimited entanglement, a single bit of communication, and a single-bit output, obtaining an answer reminiscent of Tsirelson's characterization of XOR games [11,12]. Other nonasymptotic results include specific examples of communication reduction (e.g., from 3 bits to 2 bits using entanglement [13]), quantum advantage in random access coding [14,15], the relation between the power of quantum communication and Bell inequalities [16,17], local hidden variable models supplemented with 1-bit communication [18], and a low-communication test for large entangled states with connections to the area-law conjecture in manybody physics [19]. In Bell inequalities, communication can be assumed to be completely forbidden by performing measurements at spacelike-separated points.…”

Enabling Computation of Correlation Bounds for Finite-Dimensional Quantum Systems via Symmetrization

“…We consider these matters in a distributed computation task [14,15,61] based on facet Bell inequalities [62].…”