Quantum Correlation Self-Testing

Updated 9 November 2025

Correlation self-testing is a framework that identifies the unique set of quantum correlations by relying solely on optimal performance in specialized information tasks.
It employs adaptive CHSH games to ensure that only quantum systems can reach a win rate of approximately 0.8536, effectively ruling out classical and generalized probabilistic models.
Experimental implementations using entangled states and Bell measurements provide a robust, device-independent method for operationally axiomatizing quantum theory.

Correlation self-testing of quantum theory refers to the identification and certification of the unique set of input–output correlations that can be realized within quantum theory, solely by observing the optimal performance in specific information-processing tasks, typically in the context of Bell-type or more general multipartite games. Unlike standard self-testing, which aims to certify the underlying quantum state and measurements up to local isometries, correlation self-testing targets the global correlation set, enabling the potential experimental exclusion of alternative physical theories (such as generalized probabilistic theories, or GPTs) that produce strictly larger or smaller sets of correlations. The paradigm is now central in efforts to axiomatize quantum theory from operational criteria and to experimentally rule out candidate post-quantum or subquantum models.

1. Conceptual Framework: From Self-Testing to Correlation Self-Testing

Standard self-testing demonstrates, under minimal assumptions and in a device-independent framework, that certain observed correlations can only be explained by a unique (up to local isometries) quantum state and measurement strategy. This is typically established by achieving extremal or near-extremal violations of well-chosen Bell inequalities (e.g., maximal CHSH violation self-tests the singlet and Pauli observables).

Correlation self-testing generalizes this to the level of physical theories: it seeks information-processing tasks such that only correlations achievable in quantum theory can reach optimal performance, ruling out both more-nonlocal (e.g., PR box) and less-nonlocal (e.g., classical, restricted GPT) theories. A correlation self-test is thus a pair (causal structure, task functional) $\left(C, f\right)$ such that

Any alternative theory $\mathsf{T}$ achieving the quantum optimum in $f$ must produce exactly the quantum set of correlations for all possible $C'$ .

Formally, given a set of candidate theories $\{T\}$ and their respective achievable distribution sets $T(C)$ , the task $f$ is a correlation self-test for quantum theory in $\{T\}$ if

$\max_{P \in \bigcup_T T(C)} f(P) = \max_{P \in Q(C)} f(P)$

and, if $T$ achieves the quantum optimum in $C$ , then $T(C') = Q(C')$ for all $C'$ . This establishes a full correspondence between optimal performance in the chosen task and quantum correlations in any scenario.

2. The Adaptive CHSH Game: Paradigmatic Task for Correlation Self-Testing

The adaptive CHSH game, introduced in (Weilenmann et al., 2020) and (Weilenmann et al., 2020), provides a concrete scenario in which quantum theory uniquely achieves a maximal figure of merit even when evaluated against large swathes of exotic GPTs.

Game structure:

Three parties: Alice, Bob, Charlie.
Two independent sources: $S_{AB}$ (shared between Alice and Bob) and $S_{B'C}$ (Bob and Charlie).
Alice and Charlie receive random bits $r_A, r_C \in \{0,1\}$ as measurement settings; Bob receives no input but performs a joint measurement on his two subsystems, outputting $b \in \{00, 01, 10, 11\}$ (encoding four distinct CHSH-like clauses).
The team wins if $a \oplus c = r_A r_C \oplus$ [(linear combination of $r_A, r_C$ determined by $b$ )].

The performance is quantified by the overall probability of winning ( $P_\mathrm{win}$ ), which—due to the structure of the game—maps directly onto the mean of four distinct CHSH-like operators.

Quantum optimality:

In standard quantum theory, the maximum CHSH value is $2\sqrt{2}$ , yielding $P_\mathrm{win}^{\mathrm{Q}} = \frac{1}{2} + \frac{1}{2\sqrt{2}} \approx 0.8536$ .
Achievable by entanglement swapping: Bob performs a Bell basis measurement on his subsystems; Alice/Charlie measure in optimal (Pauli $Z/X$ or $\pm 45^\circ$ rotation) bases.
No classical or GPT model whose bipartite composition is either the minimal or maximal tensor product (i.e., only separable states or all non-signalling correlations, with only separable effects) can exceed $P_\mathrm{win} = 0.75$ .
All known polygon GPTs, including those with self-duality or non-trivial convex sets, fail to achieve the quantum-optimal value in this game.

Summary table:

Model Class	Maximum $P_\mathrm{win}$
Quantum ( $\otimes$ standard)	$0.8536$
Classical/local (separable)	$0.75$
Boxworld (all PR boxes allowed)	$0.75$
Polygon theories (n-gons, up to 30)	$<0.8536$

Thus, empirical observation of $P_\mathrm{win} > 0.75$ (and in particular $P_\mathrm{win} \approx 0.8536$ ) in an experiment suffices to rule out all non-quantum theories from the large classes covered, up to the numerically established bounds of these models (Weilenmann et al., 2020, Weilenmann et al., 2020, Sengupta et al., 4 Nov 2025).

3. Analysis in Generalized Probabilistic Theories (GPTs) and the Role of Compositional Consistency

GPTs serve as a flexible framework to paper theories beyond quantum mechanics—encoding a compact convex state space for each system, a dual effect cone, and (crucially) rules for composing joint systems via tensor product constructions.

Minimal tensor product ($\otimes_\min$): joint states are convex combinations of product states, no entanglement.
Maximal tensor product ($\otimes_\max$): joint states are all positive on effects, includes all non-signalling boxes.

For two-dimensional systems (so-called "gbits"), extensive numerical and analytic evidence establishes that:

No model with a bipartite state space between $\otimes_\min$ and $\otimes_\max$ (including self-dualizations and various restricted no-restriction hypotheses) achieves the quantum-optimal adaptive CHSH win rate (Weilenmann et al., 2020, Weilenmann et al., 2020).

An important strengthening of the analysis introduces compositional consistency: for an effect to be physically allowed, it must be possible to incorporate it into a measurement that preserves the validity of states when acting on arbitrary subsystems—a property formalized as minimal $k$ -preservability (Sengupta et al., 4 Nov 2025). This requirement directly singles out Tsirelson's bound; any GPT with effects not minimally 2-preserving admits postquantum CHSH violations exceeding the quantum bound.

4. Practical and Experimental Realization

Realization of such correlation self-tests is within reach of standard photonic or superconducting qubit experiments:

Prepare two entangled Bell pairs ("EPR pairs") and distribute as required.
Bob performs a Bell-basis measurement on his two systems (possible with linear-optics circuits and post-selection).
Alice/Charlie perform fast, randomly chosen projective measurements in complementary bases.
The joint statistics are collected to estimate $P_\mathrm{win}$ .

Observation of $P_\mathrm{win} \gtrsim 0.85$ , with controlled error bars, is sufficient to statistically rule out all GPT models failing to reach the quantum bound—including all convex hulls of local plus finitely many PR boxes with restricted relabelling (Sengupta et al., 4 Nov 2025). This makes the experimental program robustly device-independent.

5. Broader Implications: Toward a Single-Task Axiom and Axiomatization of Quantum Theory

Correlation self-testing via the adaptive CHSH game is a concrete instantiation of an operational axiom: quantum theory is the unique GPT that achieves the maximal possible success in this task (Weilenmann et al., 2020). This establishes a candidate "single-task" principle, analogous in spirit to information causality and local orthogonality.

If this optimality holds in all causal structures and for all conceivable alternative GPTs, it would provide a new route toward a foundational operational axiom for quantum mechanics. Current analysis is comprehensive for 2D systems; open questions remain regarding higher-dimensional systems, multipartite generalizations, and exotic infinite-dimensional or nonlocally tomographic GPTs (Weilenmann et al., 2020, Weilenmann et al., 2020).

6. Connections to General Self-Testing, Operator-Algebraic, and Graph-Theoretic Approaches

Correlation self-testing is distinct from—but systematically connected to—other forms of device-independent self-testing:

Standard self-testing (e.g., via maximal violation of CHSH or Mermin inequalities) certifies state and measurements up to local isometry for a given Bell experiment (Zhang et al., 2018, Coopmans et al., 2019, Panwar et al., 2022).
Operator-algebraic and operator-system approaches formalize uniqueness of correlation as uniqueness of an abstract state (e.g., a functional on a universal $C^*$ -algebra), and often equivalently characterize self-testing as uniqueness among states respecting a given algebraic structure (Paddock et al., 2023, Crann et al., 22 Jun 2025).
Graph-theoretic formulations via exclusivity graphs and the Lovász theta number unify many standard self-tests, showing that uniqueness of the quantum optimum in a semidefinite program corresponds to self-testing (Bharti et al., 2021).

In all these frameworks, when the appropriate uniqueness and optimality conditions are met, self-testing statements extend to a correlation-level certification—sometimes allowing extension to infinite-dimensional or commuting operator models (Paddock et al., 2023, Crann et al., 22 Jun 2025).

7. Open Problems and Future Directions

Key current directions in correlation self-testing include:

Proving the impossibility for any non-quantum (including higher-dimensional and nonlocally tomographic) GPT to achieve the quantum bound in the adaptive CHSH or any similar task.
Characterizing closed-form bounds for correlation self-testing tasks in multipartite and/or higher-dimensional scenarios.
Developing robustness analyses and self-testing criteria based on the geometry (extremality, facet structure, non-exposed points) of the quantum correlation set (Chen et al., 2022, Rai et al., 2021).
Further elucidating the mathematical connections among self-testing, uniqueness of state extensions, and dilation theory in $C^*$ -algebraic settings.

The development of robust, experimentally accessible correlation self-tests is expected to play a central role in both the axiomatization of quantum theory and the certification of quantum devices in future quantum networks.