Classical and quantum partition bound and detector inefficiency

We study randomized and quantum efficiency lower bounds in communication complexity. These arise from the study of zero-communication protocols in which players are allowed to abort. Our scenario is inspired by the physics setup of Bell experiments, where two players share a predefined entangled state but are not allowed to communicate. Each is given a measurement as input, which they perform on their share of the system. The outcomes of the measurements should follow a distribution predicted by quantum mechanics; however, in practice, the detectors may fail to produce an output in some of the runs. The efficiency of the experiment is the probability that the experiment succeeds (neither of the detectors fails). When the players share a quantum state, this gives rise to a new bound on quantum communication complexity (eff*) that subsumes the factorization norm. When players share randomness instead of a quantum state, the efficiency bound (eff), coincides with the partition bound of Jain and Klauck. This is one of the strongest lower bounds known for randomized communication complexity, which subsumes all the known combinatorial and algebraic methods including the rectangle (corruption) bound, the factorization norm, and discrepancy. The lower bound is formulated as a convex optimization problem. In practice, the dual form is more feasible to use, and we show that it amounts to constructing an explicit Bell inequality (for eff) or Tsirelson inequality (for eff*). We give an example of a quantum distribution where the violation can be exponentially bigger than the previously studied class of normalized Bell inequalities. For one-way communication, we show that the quantum one-way partition bound is tight for classical communication with shared entanglement up to arbitrarily small error.

💡 Research Summary

The paper introduces a unified framework for proving lower bounds in communication complexity by modeling “zero‑communication” protocols that are allowed to abort, a situation motivated by detector inefficiencies in Bell‑type experiments. In such experiments two distant parties share a pre‑established resource (either shared randomness or an entangled quantum state) and each receives a measurement setting as input. The parties perform local measurements and produce outcomes; however, detectors may fail, causing the protocol to abort. The probability that both detectors fire is called the efficiency of the experiment. The authors define two quantitative lower‑bound measures: eff, the classical efficiency bound when the shared resource is randomness, and eff*, the quantum efficiency bound when the shared resource is an entangled state.

Classical efficiency (eff).
The authors formulate the problem of minimizing the success probability of any zero‑communication protocol (i.e., the maximal abort probability) as a linear program. The primal variables are the conditional output probabilities (p_{ab|xy}) for each input pair ((x,y)) and output pair ((a,b)). The constraints enforce that the protocol reproduces the target distribution on the subset of runs that do not abort. The dual linear program assigns weights to inputs and yields an inequality of the form (\sum_{x,y} w_{xy} \Pr