The Power of Unentangled Quantum Proofs with Non-negative Amplitudes

Quantum entanglement is a fundamental property of quantum mechanics and it serves as a basic resource in quantum computation and information. Despite its importance, the power and limitations of quantum entanglement are far from being fully understood. Here, we study entanglement via the lens of computational complexity. This is done by studying quantum generalizations of the class NP with multiple unentangled quantum proofs, the so-called QMA(2) and its variants. The complexity of QMA(2) is known to be closely connected to a variety of problems such as deciding if a state is entangled and several classical optimization problems. However, determining the complexity of QMA(2) is a longstanding open problem, and only the trivial complexity bounds QMA subset of QMA(2) subset of NEXP are known. In this work, we study the power of unentangled quantum proofs with non-negative amplitudes, a class which we denote QMA(+)(2). In this setting, we are able to design proof verification protocols for (increasingly) hard problems both using logarithmic size quantum proofs and having a constant probability gap in distinguishing yes from no instances. In particular, we design global protocols for small set expansion (SSE), unique games (UG), and PCP verification. As a consequence, we obtain NP subset of QMA(log)(+) (2) with a constant gap. By virtue of the new constant gap, we are able to "scale up" this result to QMA(+)(2), obtaining the full characterization QMA(+)(2) = NEXP by establishing stronger explicitness properties of the PCP for NEXP. We believe that our protocols are interesting examples of proof verification and property testing in their own right. Moreover, each of our protocols has a single isolated property testing task relying on non-negative amplitudes which if generalized would allow transferring our results to QMA(2). One key novelty of these protocols is the manipulation of quantum proofs in a global and coherent way yielding constant gaps. Previous protocols (only available for general amplitudes) are either local having vanishingly small gaps or treating the quantum proofs as classical probability distributions requiring polynomially many proofs. In both cases, these known protocols do not imply non-trivial bounds on QMA(2).