QMA(2) with Limited Shared Entanglement

Abstract

In $\mathsf{QMA}(2)$, two computationally unbounded provers submit unentangled quantum witnesses to a polynomial-time quantum verifier. We ask whether allowing limited shared entanglement between the two provers changes the power of $\mathsf{QMA}(2)$. To this end, we formalize $\mathsf{QMA}(2;h)$, a variant in which the provers initially share $h$ EPR pairs, as suggested by Aaronson, Beigi, Drucker, Fefferman, and Shor (ToC~2009). We show that polynomially many shared EPR pairs give the provers no cheating advantage: $\mathsf{QMA}(2)\subseteq \mathsf{QMA}(2;h)$ for $h=\mathrm{poly}(n)$. For logarithmically many pairs, the power of $\mathsf{QMA}(2)$ remains unchanged, strengthening this inclusion to $\mathsf{QMA}(2)=\mathsf{QMA}(2;h)$ for $h=O(\log n)$. Finally, we introduce a variant that also allows classical communication between the provers when preparing the witnesses. This setting is particularly revealing for logarithmic-size witnesses: assuming $\mathsf{NP}⊈\mathsf{BQP}$, we show that superlogarithmically many shared EPR pairs strictly reduce the model's computational power.

arXiv comment27 pages (including appendices), 1 table, 0 figures. v3: updated arXiv metadata