Tracially Embeddable Strategies Framework

• Tracially embeddable strategies are a framework representing two-player nonlocal-game strategies within finite tracial von Neumann algebras, using state densities and commutant embeddings.
• They enable approximation of general commuting-operator correlations by transferring correlations to finite tracial settings, facilitating quantitative rounding techniques.
• Applications span synchronous and projection games, where approximate synchrony and tracial rounding yield precise error bounds for robust self-testing and constraint verification.

Searching arXiv for the cited papers to ground the article in the current record. Tracially embeddable strategies form a framework for representing and approximating two-player nonlocal-game strategies inside finite tracial von Neumann algebras. In the formulation introduced for commuting-operator correlations, a tracially-embeddable strategy realizes the shared state as a vector of the form $\sigma\ket{\tau}$ in the standard $L^2(M,\tau)$ representation of a finite tracial von Neumann algebra $(M,\tau)$, with one prover’s observables embedded in $M$ and the other’s in the commutant $M'$; the induced correlation is then written as $\tau(\sigma A^x_a \sigma B^y_b)$ (Lin, 2023). A later development adapts the same tracialization paradigm to projection games, where almost perfect quantum or commuting-operator strategies can be rounded to genuinely tracial strategies with quantitative control $1-O((L\varepsilon)^{1/4})$ (Culf, 16 Mar 2026). Taken together, these results provide a route for lifting arguments that are naturally phrased in finite or tracial operator-algebraic settings to the full commuting-operator model.

1. Definition and operator-algebraic setting

For a two-player nonlocal game with question sets $X,Y$ and answer sets $A,B$, a commuting-operator strategy is specified by a Hilbert space $\mathcal H$, a unit vector $L^2(M,\tau)$0, and POVMs $L^2(M,\tau)$1, $L^2(M,\tau)$2 such that all $L^2(M,\tau)$3 commute with all $L^2(M,\tau)$4; the induced correlation is

$L^2(M,\tau)$5

The tracially-embeddable subclass is defined by requiring a finite tracial von Neumann algebra $L^2(M,\tau)$6 in standard GNS form $L^2(M,\tau)$7, a positive operator $L^2(M,\tau)$8 with $L^2(M,\tau)$9, and embeddings

$(M,\tau)$0

such that, under the identification $(M,\tau)$1, one has $(M,\tau)$2 and

$(M,\tau)$3

(Lin, 2023).

This definition isolates the structural feature needed in many nonlocal-game arguments: the ability to work in a finite tracial von Neumann algebra while still retaining a general state through the density $(M,\tau)$4. The notation

$(M,\tau)$5

is used for the set of correlations arising from tracially-embeddable strategies (Lin, 2023).

A related but more restrictive notion appears in the projection-game setting. There, a tracial correlation is given by a von Neumann algebra $(M,\tau)$6, projective-valued measurements $(M,\tau)$7 and $(M,\tau)$8, and a normal faithful tracial state $(M,\tau)$9 such that

$M$0

Every tracial correlation is commuting-operator, and if $M$1 is finite type I it is also quantum (Culf, 16 Mar 2026). The distinction is substantive: the 2023 framework embeds general commuting-operator strategies into a tracial ambient algebra while retaining a state density $M$2, whereas the 2026 projection-game result rounds directly to a genuinely tracial strategy.

2. Universal approximation by tracially-embeddable strategies

The foundational approximation result states that every commuting-operator correlation can be approximated arbitrarily well by correlations from tracially-embeddable strategies: $M$3 This is Theorem 3.6, labeled “Embedding Rounding” (Lin, 2023).

The proof strategy begins with an arbitrary commuting-operator realization and forms the unital $M$4-algebra generated by one prover’s operators. A state reproducing the correlation is then transferred, for every $M$5, to a finite tracial von Neumann algebra $M$6 together with an embedding $M$7, a positive $M$8 with $M$9, and a normal state $M'$0 so that the target matrix coefficients are approximated within $M'$1 (Lin, 2023). The listed tools are a standard Ozawa-type embedding, or Connes-embedding trick plus Arveson extension, followed by GNS to standard form and Radon–Nikodym in the commutant (Lin, 2023).

A central clarification follows from the theorem’s exact formulation: the framework does not claim that every commuting-operator strategy is already tracially embeddable. What is proved is a density statement in correlation space. This distinction matters because many arguments in rigidity and soundness are stable under small correlation error, whereas exact realizability in a finite tracial algebra is stronger and is not asserted by the theorem.

This approximation theorem is the operator-algebraic basis for the slogan in the paper’s title, “Lifting MIP* tricks to MIP$M'$2” (Lin, 2023). The framework permits techniques that had previously depended on finite-dimensional matrix-algebra manipulations to be reformulated in finite tracial von Neumann algebras and then applied to commuting-operator strategies via approximation.

3. Approximate synchrony and decomposition phenomena

One major application concerns synchronous games. For a synchronous game and a commuting-operator correlation $M'$3, synchronicity is defined by

$M'$4

A correlation is called $M'$5-approximately synchronous if $M'$6 (Lin, 2023).

The main rounding theorem in this regime states that if $M'$7 is $M'$8-approximately synchronous, then there exists a measurable family $M'$9 of exactly synchronous commuting-operator correlations and a probability density $\tau(\sigma A^x_a \sigma B^y_b)$0 on $\tau(\sigma A^x_a \sigma B^y_b)$1 such that

$\tau(\sigma A^x_a \sigma B^y_b)$2

This is Theorem 4.1 (Lin, 2023).

The proof combines five ingredients explicitly listed in the exposition: the universal embedding theorem; an orthogonalization lemma in von Neumann algebras to replace POVMs by PVMs up to $\tau(\sigma A^x_a \sigma B^y_b)$3; construction of a joint spectral decomposition via the projectors $\tau(\sigma A^x_a \sigma B^y_b)$4 à la Connes; exact synchrony on each slice $\tau(\sigma A^x_a \sigma B^y_b)$5; and integration of the $\tau(\sigma A^x_a \sigma B^y_b)$6- and total-variation errors to obtain the final $\tau(\sigma A^x_a \sigma B^y_b)$7 bound (Lin, 2023).

The comparison with Vidick’s finite-dimensional result is also explicit. The finite-dimensional matrix algebra $\tau(\sigma A^x_a \sigma B^y_b)$8 is replaced by a general finite von Neumann algebra $\tau(\sigma A^x_a \sigma B^y_b)$9; the Hilbert–Schmidt norm is replaced by the tracial $1-O((L\varepsilon)^{1/4})$0-norm $1-O((L\varepsilon)^{1/4})$1; and the polar/GNS trick is replaced by Connes’ joint distribution plus Arveson extension and Radon–Nikodym in the commutant (Lin, 2023). The conclusion is that the finite-dimensional approximate-synchrony machinery extends to commuting-operator strategies, with an $1-O((L\varepsilon)^{1/4})$2 quantitative bound.

A plausible implication is that the framework separates two issues that are often conflated in nonlocal-game analysis: exact synchrony versus the existence of a tracial model. The 2023 results show that approximate synchrony can be converted into an average of exactly synchronous commuting-operator correlations only after passing through the tracially-embeddable formalism.

4. Projection games and approximate traciality

Projection games form a broader class than synchronous games. A nonlocal game $1-O((L\varepsilon)^{1/4})$3 with finite question sets $1-O((L\varepsilon)^{1/4})$4, answer alphabets $1-O((L\varepsilon)^{1/4})$5, distribution $1-O((L\varepsilon)^{1/4})$6 on $1-O((L\varepsilon)^{1/4})$7, and predicate $1-O((L\varepsilon)^{1/4})$8 is a projection game if for each $1-O((L\varepsilon)^{1/4})$9 there is a map

$X,Y$0

such that

$X,Y$1

(Culf, 16 Mar 2026). Because of the asymmetry between the players, projection games are in general not synchronous, and the synchronous-game structure theory does not apply directly (Culf, 16 Mar 2026).

The 2026 result introduces a projection-game analogue of tracial rounding. For a non-degenerate projection game, the condition-number is

$X,Y$2

equivalently, $X,Y$3 is the minimum conditional probability that Bob’s question equals $X,Y$4 given Alice’s question $X,Y$5 (Culf, 16 Mar 2026). If

$X,Y$6

is a commuting-operator or quantum correlation with winning probability $X,Y$7, then there exist projective measurements $X,Y$8 in some finite or infinite von Neumann algebra $X,Y$9 with faithful normal tracial state $A,B$0 such that

$A,B$1

satisfies

$A,B$2

with an absolute constant $A,B$3 that is stated in the paper as $A,B$4 (Culf, 16 Mar 2026). If the original correlation was finite-dimensional quantum, then $A,B$5 may be chosen finite type I, so $A,B$6 is quantum tracial (Culf, 16 Mar 2026).

The rounding argument is organized into three steps. First, Bob’s algebra $A,B$7 is generated by the POVMs $A,B$8, and Haagerup’s theory identifies the state $A,B$9 on $\mathcal H$0 with a positive $\mathcal H$1 satisfying $\mathcal H$2; de la Salle–Marrakchi are then used to transfer Alice’s operators from $\mathcal H$3 into $\mathcal H$4 as an $\mathcal H$5-valued POVM $\mathcal H$6 satisfying

$\mathcal H$7

(Culf, 16 Mar 2026).

Second, near-optimal winning probability forces commutator control,

$\mathcal H$8

and similarly for the $\mathcal H$9 up to a factor $L^2(M,\tau)$00 (Culf, 16 Mar 2026). De la Salle’s orthogonalization lemma then rounds “almost projective on average” POVMs to true PVMs. The projection-game structure makes Alice’s measurement a joint refinement of marginal POVMs, and the analysis of this joint PVM is exactly where the factor $L^2(M,\tau)$01 enters: if the $L^2(M,\tau)$02 marginals each satisfy $L^2(M,\tau)$03, then the joint POVM satisfies $L^2(M,\tau)$04, and since $L^2(M,\tau)$05 this contributes the factor $L^2(M,\tau)$06 (Culf, 16 Mar 2026).

Third, the core-algebra lemma of Marrakchi–de la Salle is applied in the semifinite core $L^2(M,\tau)$07. Finite families of projections that are almost central for $L^2(M,\tau)$08 can be conjugated by an $L^2(M,\tau)$09 path to projections in the fixed-point algebra $L^2(M,\tau)$10, which carries a genuine tracial state $L^2(M,\tau)$11; the resulting projections $L^2(M,\tau)$12 satisfy

$L^2(M,\tau)$13

and therefore the winning probability is preserved up to the same order (Culf, 16 Mar 2026).

5. Technical instruments and representative applications

Several technical lemmas recur across the framework. In the 2023 paper, an orthogonalization lemma in finite von Neumann algebras and a Connes-style slicing through spectral projectors of $L^2(M,\tau)$14 are central to deriving exact synchrony from approximate synchrony (Lin, 2023). In the 2026 projection-game paper, the corresponding toolkit is stated explicitly as de la Salle orthogonalization, a joint-measurable approximate-PVM lemma, and a core-algebra Connes lemma (Culf, 16 Mar 2026).

The de la Salle orthogonalization lemma says that if $L^2(M,\tau)$15 is a normal state on $L^2(M,\tau)$16 and $L^2(M,\tau)$17 is a POVM with

$L^2(M,\tau)$18

then there is a PVM $L^2(M,\tau)$19 so that

$L^2(M,\tau)$20

(Culf, 16 Mar 2026). The joint-measurable approximate-PVM lemma propagates almost-projectivity from marginals to a joint POVM with an additive loss $L^2(M,\tau)$21 (Culf, 16 Mar 2026). The core-algebra Connes lemma then converts almost-central projections for $L^2(M,\tau)$22 into commuting PVMs in the centralizer of the dual action, with matrix-element error $L^2(M,\tau)$23 (Culf, 16 Mar 2026).

These tools underpin several applications. One is the soundness of the quantum tensor code test in the full commuting-operator model. For a synchronous commuting-operator strategy succeeding in the test with bias $L^2(M,\tau)$24, there exists a global PVM $L^2(M,\tau)$25 such that

$L^2(M,\tau)$26

and by applying the approximate-synchrony theorem one obtains the same soundness for $L^2(M,\tau)$27-approximately synchronous strategies, with an additional $L^2(M,\tau)$28 penalty (Lin, 2023).

A second application is a state-dependent Gowers–Hatami theorem in finite tracial von Neumann algebras. If $L^2(M,\tau)$29 is a finite group, $L^2(M,\tau)$30 is a finite tracial von Neumann algebra, $L^2(M,\tau)$31 satisfies $L^2(M,\tau)$32, and a map $L^2(M,\tau)$33 obeys

$L^2(M,\tau)$34

then there exist an isometry $L^2(M,\tau)$35 commuting with $L^2(M,\tau)$36 and a genuine unitary representation $L^2(M,\tau)$37 such that

$L^2(M,\tau)$38

(Lin, 2023). The stated role of this theorem is to promote approximate group structure in the state seminorm to exact representations usable in robust self-testing (Lin, 2023).

A third application is a sample-efficient Pauli-basis test in the commuting-operator model. For a symmetric, generating distribution $L^2(M,\tau)$39 on $L^2(M,\tau)$40 of spectral gap $L^2(M,\tau)$41, there is a two-prover synchronous game $L^2(M,\tau)$42 of size $L^2(M,\tau)$43 questions such that any tracially-embeddable strategy with success $L^2(M,\tau)$44 yields local isometries and an auxiliary state satisfying the stated EPR and observable-closeness bounds, and choosing $L^2(M,\tau)$45 on an $L^2(M,\tau)$46-sized generating set of constant spectral gap gives a certified $L^2(M,\tau)$47-EPR test using only $L^2(M,\tau)$48 many random subsets (Lin, 2023).

6. Comparative position, limitations, and open questions

Within the nonlocal-game literature, the framework sits between exact tracial realizability and unconstrained commuting-operator strategies. Its first form proves density of tracially-embeddable correlations inside $L^2(M,\tau)$49 (Lin, 2023); its later projection-game form proves that almost perfect strategies can be rounded all the way to tracial strategies with quantitative loss controlled by $L^2(M,\tau)$50 and $L^2(M,\tau)$51 (Culf, 16 Mar 2026). This suggests a progression from qualitative embeddability to game-structured quantitative tracialization.

The comparison with earlier rounding for constraint-system games is explicit. Paddock’s rounding for the constraint-variable formulation of a general constraint-system game with $L^2(M,\tau)$52 constraints each of size at most $L^2(M,\tau)$53 had error bound $L^2(M,\tau)$54 or worse; by working directly with the projection-game form and using the synchronous-game core lemma of Marrakchi–de la Salle, the 2026 result obtains only an $L^2(M,\tau)$55 blow-up, no dependence on the total number $L^2(M,\tau)$56 of constraints, an exponent $L^2(M,\tau)$57 rather than $L^2(M,\tau)$58, and a natural extension to the full commuting-operator model (Culf, 16 Mar 2026). In particular, for fixed maximum context-size $L^2(M,\tau)$59, including $L^2(M,\tau)$60 in a linear system game or any bounded-arity CSP, one gets an $L^2(M,\tau)$61 error with no hidden constants depending on $L^2(M,\tau)$62 or alphabet size (Culf, 16 Mar 2026).

The main limitations are also spelled out in the sources. For projection games, the current quantitative bound depends on the condition-number $L^2(M,\tau)$63, and one open question is whether this dependence is necessary or whether an $L^2(M,\tau)$64-independent bound can be proved (Culf, 16 Mar 2026). Other listed questions are whether the framework extends to arbitrary games with unique-response predicates, and whether there are natural classes of nonlocal games beyond synchronous and projection where a similar “tracial robustness” phenomenon holds (Culf, 16 Mar 2026).

A final conceptual caution is that the word “tracial” is used in two closely related but non-identical senses. In the 2023 framework, the trace $L^2(M,\tau)$65 coexists with a density $L^2(M,\tau)$66 and the correlation formula is state-dependent, $L^2(M,\tau)$67 (Lin, 2023). In the 2026 projection-game theorem, the rounded strategy is tracial in the stricter sense $L^2(M,\tau)$68 with projectors in a von Neumann algebra carrying a faithful normal tracial state (Culf, 16 Mar 2026). The literature treats both as parts of a common paradigm of embedding or rounding nonlocal-game strategies into tracial von Neumann settings, but the exact operator-algebraic conclusions differ.