Bounding the asymptotic quantum value of all multipartite compiled non-local games

TL;DR

The paper proves that Kalai et al.’s multipartite compiler is asymptotically quantum-sound: as cryptographic security becomes perfect, the best achievable quantum score in the compiled game is bounded above by the commuting-operator value of the original multipartite non-local game. The authors develop a robust operator-algebraic framework, including a chain rule for Radon–Nikodym derivatives of CP maps on C*-algebras, and show that sequential operational-no-signalling correlations converge to commuting-operator strategies in the limit. They construct universal C*-algebras for sequential PVMs/POVMs and connect IND-CPA security to algebraic constraints on strategies via block encodings, enabling a transposition from cryptographic limitations to quantum-information structure. The results extend known bipartite quantum-soundness to all multipartite scenarios, with potential implications for cryptographic assumptions, self-testing, and complexity-theoretic constructs arising from compiled non-local games. This work thus provides a principled path to certifying quantum behavior in distributed protocols when physical separation is replaced by cryptographic security flaws and, in the asymptotic regime, ties the quantum capabilities of a single prover to the fundamental commuting-operator framework of the original game.

Abstract

Non-local games are a powerful tool to distinguish between correlations possible in classical and quantum worlds. Kalai et al. (STOC'23) proposed a compiler that converts multipartite non-local games into interactive protocols with a single prover, relying on cryptographic tools to remove the assumption of physical separation of the players. While quantum completeness and classical soundness of the construction have been established for all multipartite games, quantum soundness is known only in the special case of bipartite games. In this paper, we prove that the Kalai et al.'s compiler indeed achieves quantum soundness for all multipartite compiled non-local games, by showing that any correlations that can be generated in the asymptotic case correspond to quantum commuting strategies. Our proof uses techniques from the theory of operator algebras, and relies on a characterisation of sequential operationally no-signalling strategies as quantum commuting operator strategies in the multipartite case, thereby generalising several previous results. On the way, we construct universal C*-algebras of sequential PVMs and prove a new chain rule for Radon-Nikodym derivatives of completely positive maps on C*-algebras which may be of independent interest.

Figures (14)

• Figure 1: Graphical overview of our results and contributions. In all scenarios, the verifier is depicted in grey and the prover(s) are depicted in orange. The horizontal axis represents the spatial dimension, whereas time is flowing vertically, from top to bottom. Although all scenarios are depicted here involving three players, all results in our manuscript directly generalise and are proven in the more general scenario of any finite number of players. Starting at the top-left, counter-clockwise: non-local game; KLVY-compiled game without space-like separations (bottom-left); Sequential game with approximate no-signalling (bottom-right); Algebraic strategies with operational no-signalling (top-right); non-local game with commuting-operator strategy (top-left, again).
• Figure 2: Commutative diagrams of universal C*-algebras capturing PVMs (POVMs) in the two-player case (left) and sequential PVMs (POVMs) in the three-player case (right).
• Figure 3: On the left, a graphical representation of a $3$-players non-local game. On the right, the game obtained through KLVY22Quantum compilation. Time is flowing downwards. The players are represented in orange, and the verifier is in gray.
• Figure 4: Efficient state preparation, efficient measurement, and efficient quantum instrument. Every wire represents a register consisting of a polynomial number of qubits.
• Figure 5: Circuit representation of a general efficient compiled quantum correlation for bipartite games. (1) is the efficient state preparation of $\rho^\lambda$, (2) is the efficient quantum instrument $\Tilde{A}_{\mathtt{a}|\mathtt{x}}^\lambda$ and (3) is the efficient measurement $B_{b|y}^\lambda$.

Theorems & Definitions (90)

• Theorem : Informal version of Theorem \ref{['th:asymptotic-quantum-soundness-k-players']}
• Theorem : Informal version of theorem \ref{['th:chained-rn-k']}
• Theorem : Informal version of Corollary \ref{['cor:k-seq-players']}
• Theorem : Informal version of theorem \ref{['th:3pl-ind-cpa']}
• Definition 1: C$^*$-algebra
• Definition 2: Representation $(\pi, \mathcal{H})$ of $\mathscr{A}$
• Theorem 1: Gelfand-Naimark-Segal (GNS) construction
• Theorem 2: Radon-Nikodym Theorem for positive linear functionals conway
• Lemma 1: Theorem 3.3 of Rag03radon
• Theorem 3: Stinespring dilation theorem