Interactive proofs with approximately commuting provers (1510.00102v1)

Abstract: The class $\MIP^*$ of promise problems that can be decided through an interactive proof system with multiple entangled provers provides a complexity-theoretic framework for the exploration of the nonlocal properties of entanglement. Little is known about the power of this class. The only proposed approach for establishing upper bounds is based on a hierarchy of semidefinite programs introduced independently by Pironio et al. and Doherty et al. This hierarchy converges to a value that is only known to coincide with the provers' maximum success probability in a given proof system under a plausible but difficult mathematical conjecture, Connes' embedding conjecture. No bounds on the rate of convergence are known. We introduce a rounding scheme for the hierarchy, establishing that any solution to its $N$-th level can be mapped to a strategy for the provers in which measurement operators associated with distinct provers have pairwise commutator bounded by $O(\ell^2/\sqrt{N})$ in operator norm, where $\ell$ is the number of possible answers per prover. Our rounding scheme motivates the introduction of a variant of $\MIP^*$, called $\MIP_\delta^*$, in which the soundness property is required to hold as long as the commutator of operations performed by distinct provers has norm at most $\delta$. Our rounding scheme implies the upper bound $\MIP_\delta^* \subseteq \DTIME(\exp(\exp(\poly)/\delta^2))$. In terms of lower bounds we establish that $\MIP^*_{2^{-\poly}}$, with completeness $1$ and soundness $1-2^{-\poly}$, contains $\NEXP$. The relationship of $\MIP_\delta^*$ to $\MIPstar$ has connections with the mathematical literature on approximate commutation. Our rounding scheme gives an elementary proof that the Strong Kirchberg Conjecture implies that $\MIPstar$ is computable. We discuss applications to device-independent cryptography.

• The paper introduces a novel rounding scheme that transforms solutions from the semidefinite programming hierarchy into strategies with approximately commuting provers.
• It establishes an upper bound for the MIPδ* class by demonstrating that interactive proofs with nearly commuting provers can capture complexity classes such as NEXP.
• The work links approximate commutation to key conjectures and offers practical insights for device-independent cryptography and robust quantum protocols.

Interactive Proofs with Approximately Commuting Provers: A Summary

This paper investigates the complexity class $MIP^*$, which encompasses promise problems that can be decided using an interactive proof system with multiple entangled quantum provers. The exploration centers on the nonlocal properties of entanglement and the computational boundaries of $MIP^*$. Earlier attempts at characterizing $MIP^*$ rely heavily on semidefinite programming hierarchies, yet these methods depend on unresolved mathematical conjectures, like Connes' embedding conjecture, to align the field-theoretic and maximum success probabilities of the provers with quantum strategies.

Significantly, this work introduces a novel rounding scheme for approximating solutions within the semidefinite program hierarchy. The authors propose that any solution to the $N$-th level of this hierarchy can be transformed into a strategy where measurement operators between distinct provers have commutators bounded by $O(\ell^2/\sqrt{N})$ in operator norm, a significant refinement over prior convergence guarantees. This new scheme supports the introduction of a variant of the multiprover quantum interactive proof system, termed $MIP_\delta^*$, wherein the provers are permitted to carry out operations in a shared Hilbert space as long as their operations approximately commute.

The implications are compelling: the authors establish an upper bound for the $MIP_\delta^*$ class of $DTIME(\exp(\exp(\poly)/\delta^2))$. They further demonstrate that $MIP^*_{2^{-\poly}}$, with completeness 1 and soundness $1-2^{-\poly}$, contains the class $NEXP$. The relationship between $MIP_\delta^*$ and $MIP^*$ correlates with broader mathematical inquiries into approximate commutation. Also, the paper offers an elementary proof that the Strong Kirchberg Conjecture implies computational bounds for $MIP$.

For practical applications, it underscores possible advancements in device-independent cryptography by enabling protocols robust to a degree of provers' misalignment. The focus on approximate commutation opens further avenues for understanding the interplay between quantum noise and system performance, suggesting an alternative approach to quantum nonlocality's stringent conditions.

In summary, this paper situates itself at the crossroads of quantum computing, complexity theory, and quantum information science by proposing a modification of the class $MIP^*$ that both reflects the computational power of entangled systems and aligns better with experimental realities. Practical and theoretical advancements as established by $MIP^*_\delta$ pave the way for examining uncharted territories in quantum proofs and provide a scaffold for evaluating quantum systems when exact commuting conditions are unattainable.