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Noncommutative Nullstellensätze and Perfect Games

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Abstract

The foundations of classical algebraic geometry and real algebraic geometry are the Nullstellensatz and Positivstellensatz. Over the last two decades, the basic analogous theorems for matrix and operator theory (noncommutative variables) have emerged. This paper concerns commuting operator strategies for nonlocal games, recalls NC Nullstellensatz which are helpful, extends these, and applies them to a very broad collection of games. In the process, it brings together results spread over different literature studies, hence rather than being terse, our style is fairly expository. The main results of this paper are two characterizations, based on Nullstellensatz, which apply to games with perfect commuting operator strategies. The first applies to all games and reduces the question of whether or not a game has a perfect commuting operator strategy to a question involving left ideals and sums of squares. Previously, Paulsen and others translated the study of perfect synchronous games to problems entirely involving a \(*\)-algebra. The characterization we present is analogous, but works for all games. The second characterization is based on a new Nullstellensatz we derive in this paper. It applies to a class of games we call torically determined games, special cases of which are XOR and linear system games. For these games, we show the question of whether or not a game has a perfect commuting operator strategy reduces to instances of the subgroup membership problem and, for linear systems games, we further show this subgroup membership characterization is equivalent to the standard characterization of perfect commuting operator strategies in terms of solution groups. Both the general and torically determined games characterizations are amenable to computer algebra techniques, which we also develop. For context, we mention that Positivstellensätze are behind the standard NPA upper bound on the score players can achieve for a game using a commuting operator strategy. This paper develops analogous NC real algebraic geometry which bears on perfect games.

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Notes

  1. The \(*\)-algebra \(\mathscr {U}\) is isomorphic to a group algebra and has appeared before in other contexts. For example, in [22] an algebra closely related to \(\mathscr {U}\) was denoted \(\mathcal {A}(X,A)\).

  2. A full set of relations for \(\mathscr {U}\) is listed in Sect. 3.3.1.

  3. In [9] it was already observed that the characterization of binary linear systems games presented could be generalized to any system of equations mod p for any prime p. This generalization is given explicitly in [13].

  4. The question of whether a game has perfect commuting operator value is undecidable [29], meaning these algorithms (or any algorithms!) cannot always identify games with commuting operator value one, but there are many examples where they do.

  5. To see why, note a strategy is perfect iff players following it always provide winning responses to every possible question, so changing the probability of questions being asked or the score associated with non-winning responses does not affect perfectness of a strategy.

  6. This is the finest vector space topology for which every neighborhood of zero contains a convex balanced absorbing set. Equivalently, it is the coarsest topology for which every seminorm on \({\mathcal {A}}\) is continuous. In this case, every linear functional f on \({\mathcal {A}}\) is continuous since |f| is a seminorm. Hence, for a convex subset \(C\subseteq {\mathcal {A}}\) we have

    $$\begin{aligned} {{\,\textrm{cl}\,}}(C)=\{ c\in {\mathcal {A}}\mid \forall \text { linear } \ell :{\mathcal {A}}\rightarrow {\mathbb {C}}\text { with } \ell (C)\subseteq \mathbb {R}_{\ge 0} \text { we have } \ell (c)\ge 0 \} \end{aligned}$$

    by the Hahn–Banach separation theorem.

  7. This notion of Archimedean should not be confused with the algebra \({\mathcal {A}}\) being Archimedean closed, meaning that for any \(a \in {\mathcal {A}}\) with \(a + \epsilon \in {{\,\textrm{SOS}\,}}_{{\mathcal {A}}}\) for each \(\epsilon \in \mathbb {R}_{>0}\) we also have \(a \in {{\,\textrm{SOS}\,}}_{{\mathcal {A}}}\).

  8. Note we have not shown this definition of \(w(2)_\ell \) is unique, i.e., we have not shown that this replacement defines a homomorphism, but we will not need to for the current proof.

  9. We note this claim can hold true for a larger class of games than synchronous ones. In most of this section the only property of synchronous games which will be used is that they satisfy Eq. (8.3). Thus, most of the techniques of this section apply to a slightly larger class of games than synchronous ones.

Abbreviations

\(\mathbb {C}\langle x\rangle \) :

Free algebra on x + variants

\({\mathfrak {I}}\) :

Capital Gothic letters for two-sided ideals    AND

\({\mathfrak {I}}(generators)_{algebra}\) :

To be explicit in terms of generators/algebra

\({\mathfrak {L}}\), \(\mathfrak {R}\) :

Capital Gothic letters for left/right ideals    AND

\({\mathfrak {L}}(generators)_{algebra}\) :

To be explicit in terms of generators/algebra

\(\mathcal {E}(\alpha )\) :

The set of projectors used by player \(\alpha \) in a nonlocal game

\(E(\alpha )^{i}_{a}\) :

The projector in a nonlocal game strategy corresponding to player \(\alpha \) giving a response a to question i

\(e(\alpha )^i_{a}\) :

Formal variables satisfying the same relations as \(E(\alpha )^{i}_{a}\)

\(X(\alpha )^i_{a}\) :

The signature matrix \(2E(\alpha )^{i}_{a} - 1\)

\(x(\alpha )^i_{a}\) :

The formal variable in \(\mathscr {U}\) corresponding to \(X(\alpha )^i_{a}\)

\(\mathscr {U}\) :

The universal game algebra formed by the \(e(\alpha )^i_{a}\)

\({\mathscr {I}}\) :

Universal game ideal

\({\mathcal {A}}\), etc.:

\(C^*\)-algebras

\({\mathbb E}\) :

Conditional expectation

\({{\,\textrm{Alg}\,}}(blah)\) :

(Sub)algebra generated by blah

\({{\,\textrm{Alg}\,}}^*(blah)\) :

\(*\)-Subalgebra generated by blah

\({\mathcal {A}}^{-1}\) :

Invertible elements in an algebra \({\mathcal {A}}\)

\(\mathcal {Y}(\vec {i})\) :

Set of answers corresponding to “valid responses” to question \(\vec {i}\)

\(\mathcal {N}(\vec {i})\) :

Set of answers corresponding to “invalid responses” to question \(\vec {i}\)

\(\mathcal {Y}\) :

Set of elements \(e(\alpha )^i_a\) corresponding to “valid responses −1”

\(\mathcal {N}\) :

Set of elements \(e(\alpha )^i_a\) corresponding to “invalid responses”

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Acknowledgements

Funding was provided by Javna Agencija za Raziskovalno Dejavnost RS (Grant Nos. J1-2453, N1-0217, P1-0222) and National Science Foundation (CCF-1729369).

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Authors and Affiliations

  1. University of Waterloo, Waterloo, Canada

    Adam Bene Watts

  2. University of California San Diego, La Jolla, CA, USA

    J. William Helton

  3. University of Ljubljana, Ljubljana, Slovenia

    Igor Klep

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  1. Adam Bene Watts
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Bene Watts, A., Helton, J.W. & Klep, I. Noncommutative Nullstellensätze and Perfect Games. Ann. Henri Poincaré 24, 2183–2239 (2023). https://doi.org/10.1007/s00023-022-01262-1

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