Communication cost of classically simulating a quantum channel with subsequent rank-1 projective measurement

TL;DR

This paper establishes a simple geometric lower bound of 2^n - 1 bits for the classical communication needed to simulate n-qubit quantum channels with rank-1 measurements, advancing understanding of classical-quantum simulation limits.

Contribution

It provides a straightforward geometric derivation of the lower bound on classical communication for simulating quantum channels, improving upon previous complex proofs.

Findings

Lower bound of 2^n - 1 bits for classical simulation of n-qubits

Uses the double cap conjecture for a simplified proof

Applicable to rank-1 projective measurements only

Abstract

A process of preparation, transmission and subsequent projective measurement of a qubit can be simulated by a classical model with only two bits of communication and some amount of shared randomness. However no model for n qubits with a finite amount of classical communication is known at present. A lower bound for the communication cost can provide useful hints for a generalization. It is known for example that the amount of communication must be greater than c 2^n, where c~0.01. The proof uses a quite elaborate theorem of communication complexity. Using a mathematical conjecture known as the "double cap conjecture", we strengthen this result by presenting a geometrical and extremely simple derivation of the lower bound 2^n-1. Only rank-1 projective measurements are involved in the derivation.