Nonlocal Games, Compression Theorems, and the Arithmetical Hierarchy

Abstract

We investigate the connection between the complexity of nonlocal games and the arithmetical hierarchy, a classification of languages according to the complexity of arithmetical formulas defining them. It was recently shown by Ji, Natarajan, Vidick, Wright and Yuen that deciding whether the (finite-dimensional) quantum value of a nonlocal game is 1 or at most 1 2 is complete for the class Σ 1 (i.e., RE). A result of Slofstra implies that deciding whether the commuting operator value of a nonlocal game is equal to 1 is complete for the class Π 1 (i.e., coRE).We prove that deciding whether the quantum value of a two-player nonlocal game is exactly equal to 1 is complete for Π 2 ; this class is in the second level of the arithmetical hierarchy and corresponds to formulas of the form "∀x ∃y φ(x, y)". This shows that exactly computing the quantum value is strictly harder than approximating it, and also strictly harder than computing the commuting operator value (either exactly or approximately).We explain how results about the complexity of nonlocal games all follow in a unified manner from a technique known as compression. At the core of our Π 2 -completeness result is a new "gapless" compression theorem that holds for both quantum and commuting operator strategies. Our compression theorem yields as a byproduct an alternative proof of Slofstra's result that the set of quantum correlations is not closed. We also show how a "gap-preserving" compression theorem for commuting operator strategies would imply that approximating the commuting operator value is complete for Π 1 .

Quantum free games

“…Using a connection to the nonuniform complexity of the halting problem we show that any MIP * protocol for RE requires Ω(log n) bits of communication. It follows that our results in item 3 are optimal up to an O(log * n) factor, and that the gapless compression theorems of [MNY21] are asymptotically optimal. We conjecture that these bounds can be saturated in the gapped case as well.…”

“…3. By the same techniques, we obtain a zero-gap AM * protocol for a Π 2 complete language with constant-size questions and almost logarithmically (O(log n • log * n)) large answers, improving on the headline result of Mousavi, Nezhadi and Yuen [MNY21].…”

“…In this section, we define basic notions concerning nonlocal games, strategies, and protocols. For a more detailed treatment, we refer the reader to [MNY21] and [JNV + 20].…”

“…It is tempting then to ask: can we do infinite randomness expansion [CY13] using only 2 provers by using this self-test? The naïve approach does not work because, in the 'question sampling game' of [MNY21] and the 'introspection game' of [JNV + 20], the probability that the provers are asked the 'introspect' questions which cause them to generate randomness (as opposed to being asked questions that test their consistency with each other) decreases by a constant factor every time one applies question reduction, and we need to apply question reduction approximately log * (n) times in order to make the questions constant sized. Can this obstacle be gotten around?…”

Synergies Between Operations Research and Quantum Information Science

“…Imagine the gratification in finally successfully solving or analyzing a thorny SDP and subsequently receiving a message from your experimental-physicist friend that nature agrees with your findings -a nice pat on the back from the universe. Nonlocal games have fundamental connections to models of (quantum) computation [MNY21], and a nonlocal games perspective has recently enabled a landmark result resolving two longstanding problems: Tsirelson's problem in quantum mechanics and Connes' embedding problem in operator algebras [JNV + 21, PV16].…”