Exponential communication gap between weak and strong classical simulations of quantum communication

TL;DR

This paper demonstrates a significant exponential gap in classical communication complexity between weak and strong simulation protocols for quantum communication, with weak protocols requiring constant or polynomial resources and strong protocols needing exponential resources.

Contribution

The authors introduce a bounded-error weak simulation protocol with constant or polynomial communication cost, contrasting with exponential cost for strong simulations, improving previous bounds.

Findings

Weak simulation protocol requires communication independent of qubits number

Strong simulation protocol requires exponential communication in qubits number

The new weak protocol improves upon previous bounds based on Johnson-Lindenstrauss lemma

Abstract

The most trivial way to simulate classically the communication of a quantum state is to transmit the classical description of the quantum state itself. However, this requires an infinite amount of classical communication if the simulation is exact. A more intriguing and potentially less demanding strategy would encode the full information about the quantum state into the probability distribution of the communicated variables, so that this information is never sent in each single shot. This kind of simulation is called weak, as opposed to strong simulations, where the quantum state is communicated in individual shots. In this paper, we introduce a bounded-error weak protocol for simulating the communication of an arbitrary number of qubits and a subsequent two-outcome measurement consisting of an arbitrary pure state projector and its complement. This protocol requires an amount of classical communication independent of the number of qubits and proportional to Delta^{-1}, where Delta is the error and a free parameter of the protocol. Conversely, a bounded-error strong protocol requires an amount of classical communication growing exponentially with the number of qubits for a fixed error. Our result improves a previous protocol, based on the Johnson-Lindenstrauss lemma, with communication cost scaling as Delta^{-2} log Delta^{-1}.