---
title: Tracially Embeddable Strategies Framework
url: https://www.emergentmind.com/topics/tracially-embeddable-strategies-framework
type: topic
---

# Tracially Embeddable Strategies Framework

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 $\tau(\sigma A^x_a \sigma B^y_b)$ [2304.01940]. 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 $1-O((L\varepsilon)^{1/4})$ [2603.14746]. 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 $\ket\psi\in\mathcal H$, and POVMs $\{A^x_a\}\subset B(\mathcal H)$, $\{B^y_b\}\subset B(\mathcal H)$ such that all $A^x_a$ commute with all $B^y_b$; the induced correlation is
\[
C_{x,y,a,b}=\bra\psi A^x_a B^y_b \ket\psi.
\]
The tracially-embeddable subclass is defined by requiring a finite tracial von Neumann algebra $(M,\tau)$ in standard GNS form $\bigl(\pi_\tau,L^2(M,\tau),\ket\tau\bigr)$, a positive operator $\sigma\in M^+$ with $\|\sigma\|_2^2=\tau(\sigma^2)=1$, and embeddings
\[
A^x_a\in M,\qquad B^y_b\in M'=\{M\}'\subset B\bigl(L^2(M,\tau)\bigr),
\]
such that, under the identification $L^2(M,\tau)\cong\mathcal H$, one has $\ket\psi=\sigma\ket\tau$ and
\[
\bra\psi A^x_a B^y_b\ket\psi=\tau\bigl(\sigma A^x_a \sigma B^y_b\bigr)
\]
[2304.01940].

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 $\sigma$. The notation
\[
\Corr^{\rm Tr}_{qc}(X,A;Y,B)\subset \Corr_{qc}(X,A;Y,B)
\]
is used for the set of correlations arising from tracially-embeddable strategies [2304.01940].

A related but more restrictive notion appears in the projection-game setting. There, a tracial correlation is given by a von Neumann algebra $M$, projective-valued measurements $\{A^i_a\}\subset M$ and $\{B^j_b\}\subset M$, and a normal faithful tracial state $\tau$ such that
\[
c(a,b\mid i,j)=\tau(A^i_a B^j_b).
\]
Every tracial correlation is commuting-operator, and if $M$ is finite type I it is also quantum [2603.14746]. The distinction is substantive: the 2023 framework embeds general commuting-operator strategies into a tracial ambient algebra while retaining a state density $\sigma$, 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:
\[
\overline{\Corr^{\rm Tr}_{qc}(X,A;Y,B)}=\Corr_{qc}(X,A;Y,B).
\]
This is Theorem 3.6, labeled “Embedding Rounding” [2304.01940].

The proof strategy begins with an arbitrary commuting-operator realization and forms the unital $*$-algebra generated by one prover’s operators. A state reproducing the correlation is then transferred, for every $\varepsilon>0$, to a finite tracial von Neumann algebra $(M,\tau)$ together with an embedding $\pi$, a positive $\sigma\in M$ with $\|\sigma\|_2=1$, and a normal state $\widetilde\phi(a)=\tau(\sigma a \sigma)$ so that the target matrix coefficients are approximated within $\varepsilon$ [2304.01940]. 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 [2304.01940].

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$^{co}$” [2304.01940]. 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 $C\in\Corr_{qc}(X,A;X,A)$, synchronicity is defined by
\[
\delta(C):=\mathbb E_{x\sim\mu}\sum_{a\neq b} C_{x,x,a,b}.
\]
A correlation is called $\delta$-approximately synchronous if $\delta(C)\le \delta$ [2304.01940].

The main rounding theorem in this regime states that if $C$ is $\delta$-approximately synchronous, then there exists a measurable family $\{C^\lambda\}_\lambda\subset\Corr_{qc}^s$ of exactly synchronous commuting-operator correlations and a probability density $P(\lambda)$ on $[0,\infty)$ such that
\[
\mathbb E_{x,y\sim\mu}\sum_{a,b}\left|C_{x,y,a,b}-\int_0^\infty P(\lambda)C^\lambda_{x,y,a,b}\,d\lambda\right|
\le O\!\bigl(\delta^{1/8}\bigr).
\]
This is Theorem 4.1 [2304.01940].

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 $O(\delta^{1/4})$; construction of a joint spectral decomposition via the projectors $\chi_{\ge\sqrt\lambda}(\sigma)$ à la Connes; exact synchrony on each slice $\lambda$; and integration of the $\ell_2$- and total-variation errors to obtain the final $O(\delta^{1/8})$ bound [2304.01940].

The comparison with Vidick’s finite-dimensional result is also explicit. The finite-dimensional matrix algebra $M_d(\mathbb C)$ is replaced by a general finite von Neumann algebra $(M,\tau)$; the Hilbert–Schmidt norm is replaced by the tracial $L^2$-norm $\|x\|_2=\tau(x^*x)^{1/2}$; and the polar/GNS trick is replaced by Connes’ joint distribution plus Arveson extension and Radon–Nikodym in the commutant [2304.01940]. The conclusion is that the finite-dimensional approximate-synchrony machinery extends to commuting-operator strategies, with an $O(\delta^{1/8})$ 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 $G$ with finite question sets $I,J$, answer alphabets $A_i,B_j$, distribution $\mu$ on $I\times J$, and predicate $V(a,b\mid i,j)\in\{0,1\}$ is a projection game if for each $(i,j)$ there is a map
\[
p_{i,j}:A_i\to B_j
\]
such that
\[
V(a,b\mid i,j)=1 \iff b=p_{i,j}(a)
\]
[2603.14746]. Because of the asymmetry between the players, projection games are in general not synchronous, and the synchronous-game structure theory does not apply directly [2603.14746].

The 2026 result introduces a projection-game analogue of tracial rounding. For a non-degenerate projection game, the condition-number is
\[
L:=\max_{(i,j):\mu(i,j)>0}\frac{\mu(i)}{\mu(i,j)},
\]
equivalently, $1/L$ is the minimum conditional probability that Bob’s question equals $j$ given Alice’s question $i$ [2603.14746]. If
\[
c(a,b\mid i,j)=\langle\psi|P^i_a Q^j_b|\psi\rangle
\]
is a commuting-operator or quantum correlation with winning probability $w_G(c)=1-\varepsilon$, then there exist projective measurements $\{p^i_a\},\{q^j_b\}$ in some finite or infinite von Neumann algebra $M$ with faithful normal tracial state $\tau$ such that
\[
c'(a,b\mid i,j):=\tau(p^i_a q^j_b)
\]
satisfies
\[
w_G(c')\ge 1-C\cdot (L\varepsilon)^{1/4},
\]
with an absolute constant $C$ that is stated in the paper as $148$ [2603.14746]. If the original correlation was finite-dimensional quantum, then $M$ may be chosen finite type I, so $c'$ is quantum tracial [2603.14746].

The rounding argument is organized into three steps. First, Bob’s algebra $M$ is generated by the POVMs $\{Q^j_b\}$, and Haagerup’s theory identifies the state $\omega(x)=\langle\psi|x|\psi\rangle$ on $M$ with a positive $h\in L^1(M)$ satisfying $\tau(h)=1$; de la Salle–Marrakchi are then used to transfer Alice’s operators from $M'$ into $M$ as an $M$-valued POVM $\{\bar P^i_a\}$ satisfying
\[
\langle\psi|P^i_aQ^j_b|\psi\rangle
=
\tau(\bar P^i_a h^{1/2} Q^j_b h^{1/2})
\]
[2603.14746].

Second, near-optimal winning probability forces commutator control,
\[
\sum_{i,j,b}\mu(i,j)\|[Q^j_b,h^{1/2}]\|_2^2\le O(\varepsilon),
\]
and similarly for the $\bar P^i_a$ up to a factor $L$ [2603.14746]. 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 $L$ enters: if the $n$ marginals each satisfy $\sum_a \tau(P_i^2)\ge 1-\delta$, then the joint POVM satisfies $\sum_{\vec i}\tau(J_{\vec i}^2)\ge 1-n\delta$, and since $n\le L$ this contributes the factor $L$ [2603.14746].

Third, the core-algebra lemma of Marrakchi–de la Salle is applied in the semifinite core $c(M)$. Finite families of projections that are almost central for $h^{1/2}$ can be conjugated by an $h^{it}$ path to projections in the fixed-point algebra $c(M)^\theta$, which carries a genuine tracial state $\tau_{\rm core}$; the resulting projections $r^i_a,r^j_b$ satisfy
\[
\bigl|\tau(p^i_a h^{1/2} q^j_b h^{1/2})-\tau_{\rm core}(r^i_a r^j_b)\bigr|
\le O((L\varepsilon)^{1/4}),
\]
and therefore the winning probability is preserved up to the same order [2603.14746].

## 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 $\sigma$ are central to deriving exact synchrony from approximate synchrony [2304.01940]. 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 [2603.14746].

The de la Salle orthogonalization lemma says that if $\rho$ is a normal state on $M$ and $\{A_i\}$ is a POVM with
\[
\sum_i \rho(A_i^2)\ge 1-\delta,
\]
then there is a PVM $\{p_i\}\subset M$ so that
\[
\sum_i \rho(|A_i-p_i|^2)\le 9\delta
\]
[2603.14746]. The joint-measurable approximate-PVM lemma propagates almost-projectivity from marginals to a joint POVM with an additive loss $n\delta$ [2603.14746]. The core-algebra Connes lemma then converts almost-central projections for $h^{1/2}$ into commuting PVMs in the centralizer of the dual action, with matrix-element error $O(\delta^{1/4})$ [2603.14746].

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 $\varepsilon$, there exists a global PVM $\{G_c\}_{c\in\Sigma^m}\subset M$ such that
\[
\mathbb E_{u\sim \Sigma^m}\sum_{c:\,c(u)\neq a}\tau\!\bigl(\sigma A^u_a G_c \sigma\bigr)
\le \poly(m,t,d)\,\poly(\varepsilon)+\mathrm{negl}(n),
\]
and by applying the approximate-synchrony theorem one obtains the same soundness for $\delta$-approximately synchronous strategies, with an additional $O(\delta^{1/8})$ penalty [2304.01940].

A second application is a state-dependent Gowers–Hatami theorem in finite tracial von Neumann algebras. If $G$ is a finite group, $(M,\tau)$ is a finite tracial von Neumann algebra, $\rho\in M^+$ satisfies $\tau(\rho)=1$, and a map $\phi:G\to \mathcal U(M)$ obeys
\[
\phi(g^{-1})=\phi(g)^*
\quad\text{and}\quad
\mathbb E_{g,h\in G}\|\phi(g)\phi(h)-\phi(gh)\|_\rho^2\le \eta,
\]
then there exist an isometry $V:L^2(M,\tau)\to \mathbb C^{|G|}\otimes L^2(M,\tau)$ commuting with $M'$ and a genuine unitary representation $\pi':G\to \mathcal U(\mathbb C^{|G|})$ such that
\[
\mathbb E_{g\in G}\bigl\|\phi(g)-V^*(\pi'(g)\otimes I)V\bigr\|_\rho^2\le \eta
\]
[2304.01940]. The stated role of this theorem is to promote approximate group structure in the state seminorm to exact representations usable in robust self-testing [2304.01940].

A third application is a sample-efficient Pauli-basis test in the commuting-operator model. For a symmetric, generating distribution $\mu$ on $\{0,1\}^n$ of spectral gap $\kappa(\mu)$, there is a two-prover synchronous game $\Gamma_\mu$ of size $O(n)$ questions such that any tracially-embeddable strategy with success $1-\varepsilon$ yields local isometries and an auxiliary state satisfying the stated EPR and observable-closeness bounds, and choosing $\mu$ on an $O(n)$-sized generating set of constant spectral gap gives a certified $n$-EPR test using only $O(n)$ many random subsets [2304.01940].

## 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 $\Corr_{qc}$ [2304.01940]; its later projection-game form proves that almost perfect strategies can be rounded all the way to tracial strategies with quantitative loss controlled by $L$ and $\varepsilon$ [2603.14746]. 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 $m$ constraints each of size at most $k$ had error bound $O(\poly(m)\cdot \exp(O(k))\cdot \varepsilon^{1/8})$ 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 $L=\max_i |V_i|$ blow-up, no dependence on the total number $m$ of constraints, an exponent $1/4$ rather than $1/8$, and a natural extension to the full commuting-operator model [2603.14746]. In particular, for fixed maximum context-size $L$, including $k=2$ in a linear system game or any bounded-arity CSP, one gets an $O(\varepsilon^{1/4})$ error with no hidden constants depending on $m$ or alphabet size [2603.14746].

The main limitations are also spelled out in the sources. For projection games, the current quantitative bound depends on the condition-number $L$, and one open question is whether this dependence is necessary or whether an $L$-independent bound can be proved [2603.14746]. 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 [2603.14746].

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 $\tau$ coexists with a density $\sigma$ and the correlation formula is state-dependent, $\tau(\sigma A \sigma B)$ [2304.01940]. In the 2026 projection-game theorem, the rounded strategy is tracial in the stricter sense $\tau(AB)$ with projectors in a von Neumann algebra carrying a faithful normal tracial state [2603.14746]. 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.

Source: https://www.emergentmind.com/topics/tracially-embeddable-strategies-framework