Quantum states cannot be transmitted efficiently classically

TL;DR

This paper proves that classical protocols require exponential communication to simulate quantum measurement outcomes, establishing fundamental limits and demonstrating clear quantum-classical separations in information processing tasks.

Contribution

It establishes an optimal exponential lower bound on classical communication for simulating quantum measurements, revealing fundamental quantum advantages.

Findings

Classical simulation of quantum measurement outcomes requires exponential communication.

Quantum sampling can be achieved with a single quantum query, but classical methods need exponentially many.

Certain nonlocal quantum tasks can be performed with Bell pairs, but classical solutions demand exponential communication.

Abstract

We show that any classical two-way communication protocol with shared randomness that can approximately simulate the result of applying an arbitrary measurement (held by one party) to a quantum state of $n$ qubits (held by another), up to constant accuracy, must transmit at least $\Omega(2^n)$ bits. This lower bound is optimal and matches the complexity of a simple protocol based on discretisation using an $\epsilon$-net. The proof is based on a lower bound on the classical communication complexity of a distributed variant of the Fourier sampling problem. We obtain two optimal quantum-classical separations as easy corollaries. First, a sampling problem which can be solved with one quantum query to the input, but which requires $\Omega(N)$ classical queries for an input of size $N$. Second, a nonlocal task which can be solved using $n$ Bell pairs, but for which any approximate classical solution must communicate $\Omega(2^n)$ bits.