Large violation of Bell inequalities with low entanglement

In this paper we obtain violations of general bipartite Bell inequalities of order $\frac{\sqrt{n}}{\log n}$ with $n$ inputs, $n$ outputs and $n$-dimensional Hilbert spaces. Moreover, we construct explicitly, up to a random choice of signs, all the elements involved in such violations: the coefficients of the Bell inequalities, POVMs measurements and quantum states. Analyzing this construction we find that, even though entanglement is necessary to obtain violation of Bell inequalities, the Entropy of entanglement of the underlying state is essentially irrelevant in obtaining large violation. We also indicate why the maximally entangled state is a rather poor candidate in producing large violations with arbitrary coefficients. However, we also show that for Bell inequalities with positive coefficients (in particular, games) the maximally entangled state achieves the largest violation up to a logarithmic factor.Comment: Reference [16] added. Some typos correcte

Quantum multi-prover interactive proofs with communicating provers: LOCC and separable protocols

The recent groundbreaking result of $\mathsf{MIP}^∗ = \mathsf{RE}$ has demonstrated that the study of quantum multi-prover interactive proof ($\mathsf{QMIP}$) systems can shed light on many seemingly disjoint fields of computer science, physics, and mathematics. One particularly interesting variant of $\mathsf{QMIP}$ involves a quantum verifier exchanging quantum messages with multiple quantum provers who are not allowed to share entanglement or communicate quantumly, but are allowed unlimited classical communication between them. We call this class $\mathsf{QMIP}^{\mathsf{locc}}$, for its connection to the local operations and classical communication (LOCC) paradigm. A generalization of this class involves allowing the provers to carry out any joint separable strategy, and we call this class $\mathsf{QMIP}^{\mathsf{sep}}$. These classes are both known to be lower bounded by non-deterministic exponential time ($\mathsf{NEXP}$). In this work, we investigate the true strength of these two complexity classes, both proving the exact characterization $\mathsf{QMIP}^{\mathsf{sep}} = \mathsf{NEXP}$ and showing that $\mathsf{QMIP}^{\mathsf{locc}}$ is upper bounded by the class $\mathsf{RE}$ of recursively enumerable languages. We also discuss potential strategies for improving either bound on $\mathsf{QMIP}^{\mathsf{locc}}$. To improve the lower bound, we suggest using the "introspection" technique from the proof of $\mathsf{NEEXP} \subseteq \mathsf{MIP}^∗$; to improve the upper bound, we identify connections to both a quantum analogue of the interactive compression problem in communication complexity/information theory and to finding a succinct characterization of LOCC channels. Our work contributes to a growing canon of work on quantum proof systems, non-local games, and LOCC protocols, which all continue to shed light on the computational power of quantum resources