The Power of Unentanglement

TL;DR

This paper advances understanding of the power and limitations of unentangled quantum proofs, providing protocols for satisfiability verification, conditions for protocol amplification, and results on the nonexistence of perfect disentanglers.

Contribution

It introduces a new protocol for satisfiability verification with unentangled witnesses, links protocol amplification to a quantum information conjecture, and proves the nonexistence of perfect disentanglers.

Findings

A protocol for 3SAT verification with ~O(√n) unentangled witnesses.

QMA(2) protocols can be amplified assuming a weak Additivity Conjecture.

No perfect disentanglers exist for simulating multiple Merlins.

Abstract

The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA(k)=QMA(2) for k>=2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. First, we give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given ~O(sqrt(n)) unentangled quantum witnesses with O(log n) qubits each. Our protocol relies on the existence of very short PCPs. Second, we show that assuming a weak version of the Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k)=QMA(2) for all k>=2. Third, we prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one.