The set of quantum correlations is not closed

TL;DR

This paper constructs a linear system non-local game demonstrating that the set of quantum correlations is not closed, providing new insights into quantum non-locality and the Tsirelson problem.

Contribution

It introduces a novel non-local game that separates finite-dimensional and infinite-dimensional quantum strategies, and proves the undecidability of certain quantum strategy questions.

Findings

The set of tensor-product quantum correlations is not closed.

A new counterexample to the Tsirelson problem is provided.

Decidability of perfect strategies for linear system games is proven to be undecidable.

Abstract

We construct a linear system non-local game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert space, or even with any tensor-product strategy. In particular, this shows that the set of (tensor-product) quantum correlations is not closed. The constructed non-local game provides another counterexample to the "middle" Tsirelson problem, with a shorter proof than our previous paper (though at the loss of the universal embedding theorem). We also show that it is undecidable to determine if a linear system game can be played perfectly with a finite-dimensional strategy, or a limit of finite-dimensional quantum strategies.

Equations

$$p(a,b|x,y)=\sum _{i=1}^k{\lambda }_i{p}_a^{ix}{q}_b^{iy}\text{ for all }(a,b,x,y)\in {\mathcal{O}}_A\times {\mathcal{O}}_B\times {\mathcal{I}}_A\times {\mathcal{I}}_B.$$

$$p(a,b|x,y)=⟨\psi |M_a^x\otimes N_b^y|\psi ⟩\text{ for all }(a,b,x,y)\in {\mathcal{O}}_A\times {\mathcal{O}}_B\times {\mathcal{I}}_A\times {\mathcal{I}}_B.$$

$$C_c\subseteq C_q\subseteq C_{qs}\subseteq C_{qa}.$$

$$p(a,b|x,y)=\omega (M_a^x\otimes N_b^y)$$

$$p(a,b|x,y)=⟨\psi |{\varphi }_A(M_a^x)\otimes {\varphi }_B(N_b^y)|\psi ⟩$$

$$\{n\in \mathbb{N}:{\mathcal{G}}_n\text{ has a perfect strategy in }{C}_q\},$$

$$\Vert \varphi (r)-\text{mathbbm}1\Vert \le ϵ$$

$$\Vert \psi (x)-\varphi (x)\Vert \le \delta$$

$${\ell }^{fa}(g):=\underset{ϵ\to 0^{+}}{\lim }\underset{\varphi }{\sup }\Vert \varphi (w)-\text{mathbbm}1\Vert ,$$

$$\Vert \gamma (\Psi (w))-\text{mathbbm}1\Vert =\Vert \varphi (w)\oplus {\varphi }^{\prime }(w)-\text{mathbbm}1\Vert \ge \frac{1}{\sqrt{N}}\Vert \varphi (w)-\text{mathbbm}1\Vert$$

$$⟨S_0:R_0{⟩}_{{\mathbb{Z}}_2}:=⟨S_0\cup \{J\}:R_0\cup \left\{[J,s]=e:s\in S_0\right\}\cup \{J^2=e\}⟩,$$

$$p(\underline{a},b|i,j)=⟨\psi |M_{\underline{a}}^i\otimes N_b^j|\psi ⟩,1\le i\le m,1\le j\le n,\underline{a}\in S_i,b\in {\mathbb{Z}}_2.$$

$$p_{ij}:=\sum _{\underline{a},b:a_j=b}p(\underline{a},b|i,j).$$

$$Y_{ij}=\sum _{\underline{a}\in S_i}(-1{)}^{a_j}{M}_{\underline{a}}^i$$

$$\begin{array}{rl}⟨\psi |Y_{ij}\otimes X_j|\psi ⟩& =⟨\psi |\sum _{\underline{a}\in S_i}(-1{)}^{a_j}{M}_{\underline{a}}^i\otimes \sum _{b\in {\mathbb{Z}}_2}(-1{)}^b{N}_b^j|\psi ⟩\\ & =\sum _{\underline{a}\in S_i,b\in {\mathbb{Z}}_2}(-1{)}^{a_j+b}p(\underline{a},b|i,j)\\ & =2\left[\sum _{\underline{a},b:a_j=b}p(\underline{a},b|i,j)\right]-1=2p_{ij}-1,\end{array}$$

$$\displaystyle\Gamma(A,b):=\Big{\langle}x_{1},\ldots,x_{n}:\;$$

$$\prod _{j=1}^n{x}_j^{A_{ij}}=J^{b_i}\text{ for all }1\le i\le m,\text{ and }$$

$$\displaystyle x_{j}x_{k}=x_{k}x_{j}\text{ if }j,k\in V_{i}\text{ for some }1\leq i\leq m\;\Big{\rangle}_{\mathbb{Z}_{2}}$$

$$⟨\psi |Y_{ij}\otimes X_j|\psi ⟩\ge 1-ϵ\text{ for all }1\le i\le n,j\in V_i.$$

$$\Vert D-X\Vert \le \left(1+\frac{1}{\sqrt{2}}\right)\Vert X^2-\text{mathbbm}1\Vert .$$

$$|{\alpha }^2-1{|}^2$$

$$=(a^2-1{)}^2+2b^2+2a^2{b}^2+b^4.$$

$$\Vert {\left(\mathrm{Re}X\right)}^2-\text{mathbbm}1\Vert =\sqrt{\frac{1}{d}\sum _j{\left[{\left(\mathrm{Re}{X}_{jj}\right)}^2-1\right]}^2}\le \sqrt{\frac{1}{d}\sum _j|X_{jj}^2-1{|}^2}=\Vert X^2-\text{mathbbm}1\Vert ,$$

$$\Vert \mathrm{Re}X-X\Vert =\Vert \mathrm{Im}X\Vert =\sqrt{\frac{1}{d}\sum _j|\mathrm{Im}{X}_{jj}{|}^2}\le \sqrt{\frac{1}{2d}\sum _j|X_{jj}^2-1{|}^2}=\frac{1}{\sqrt{2}}\Vert X^2-\text{mathbbm}1\Vert .$$

$$|a^2-1|=|1+a||1-a|=(1+|a|)|\mathrm{sgn}a-a|\ge |\mathrm{sgn}a-a|$$

$$\Vert Z-Y\Vert \le \left(1+\frac{1}{2\sqrt{2}}\right)\Vert X_{n}Y-Y{X}_n\Vert .$$

$$\Vert Z_0^2-\text{mathbbm}1\Vert$$

$$\le \frac{1}{4}\Vert Y{X}_{n}Y{X}_n-\text{mathbbm}1\Vert +\frac{1}{4}\Vert X_{n}Y{X}_{n}Y-\text{mathbbm}1\Vert =\frac{1}{2}\Vert X_{n}Y-Y{X}_n\Vert .$$

$$\Vert Z-Z_0\Vert \le \left(1+\frac{1}{\sqrt{2}}\right)\Vert Z_0^2-\text{mathbbm}1\Vert \le \left(\frac{1}{2}+\frac{1}{2\sqrt{2}}\right)\Vert X_{n}Y-Y{X}_n\Vert .$$

$$\Vert Y-Z_0\Vert =\frac{1}{2}\Vert Y-X_{n}Y{X}_n\Vert =\frac{1}{2}\Vert X_{n}Y-Y{X}_n\Vert ,$$

$$⟨x_1,\dots ,x_k:x_i^2=e,[x_i,x_j]=e\text{ for all }i\ne j⟩.$$

$$\Vert \psi (x_i)-\varphi (x_i)\Vert \le Cϵ$$

Full text

1. Introduction

A two-player non-local game $\mathcal{G}$ consists of finite question sets $\mathcal{I}_{A}$ and $\mathcal{I}_{B}$, finite output sets $\mathcal{O}_{A}$ and $\mathcal{O}_{B}$, and a function $V:\mathcal{O}_{A}\times\mathcal{O}_{B}\times\mathcal{I}_{A}\times\mathcal{I}_{B}\rightarrow\{0,1\}$. During the game, the two players, commonly called Alice and Bob, are given inputs $x\in\mathcal{I}_{A}$ and $y\in\mathcal{I}_{B}$ respectively, and return outputs $a\in\mathcal{O}_{A}$ and $b\in\mathcal{O}_{B}$ respectively. The players win if $V(a,b|x,y)=1$, and lose if $V(a,b|x,y)=0$. The players know the rules of the game, and can decide ahead of time on their strategy. However, once the game is in progress, they are unable to communicate, meaning they do not know each others inputs or subsequent choices. This can make it impossible for the players to win with certainty.

Imagine that the game is played repeatedly. To an outside observer, Alice and Bob’s actions during the game are described by the probability $p(a,b|x,y)$ that Alice and Bob output $a\in\mathcal{O}_{A}$ and $b\in\mathcal{O}_{B}$ on inputs $x\in\mathcal{I}_{A}$ and $y\in\mathcal{I}_{B}$. The collection $\{p(a,b|x,y)\}\subset\mathbb{R}^{\mathcal{O}_{A}\times\mathcal{O}_{B}\times\mathcal{I}_{A}\times\mathcal{I}_{B}}$ is called a correlation matrix (or a behaviour). Which correlation matrices can be achieved depends on the physical model. For instance, a correlation matrix $\{p(a,b|x,y)\}$ is said to be classical if it can be achieved using classical shared randomness. Formally, this means that there must be some integer $k\geq 1$, a probability distribution $\{\lambda_{i}\}$ on $\{1,\ldots,k\}$, probability distributions $\{p^{ix}_{a}\}$ on $\mathcal{O}_{A}$ for each $1\leq i\leq k$ and $x\in\mathcal{I}_{A}$, and probability distributions $\{q^{iy}_{b}\}$ on $\mathcal{O}_{B}$ for each $1\leq i\leq k$ and $y\in\mathcal{I}_{B}$, such that

[TABLE]

The set of classical correlation matrices is denoted by $C_{c}(\mathcal{O}_{A},\mathcal{O}_{B},\mathcal{I}_{A},\mathcal{I}_{B})$, although we typically write $C_{c}$ when the output and input sets are clear.

In quantum information, we are interested in what correlations can be achieved with a shared quantum state. Accordingly, a correlation matrix is said to be quantum if there are finite-dimensional Hilbert spaces $H_{A}$ and $H_{B}$, a quantum state $\ket{\psi}\in H_{A}\otimes H_{B}$, projective measurements111A projective measurement on a Hilbert space $H$ is a collection $\{P_{x}\}_{x\in X}$ of self-adjoint operators on $H$, such that $P_{x}^{2}=P_{x}$ for all $x\in X$, and $\sum_{x\in X}P_{x}=\mathbbm{1}$. The set $X$ is interpreted as the set of measurement outcomes. $\{M^{x}_{a}\}_{a\in\mathcal{O}_{A}}$ on $H_{A}$ for every $x\in\mathcal{I}_{A}$, and projective measurements $\{N^{y}_{b}\}_{b\in\mathcal{O}_{B}}$ on $H_{B}$ for every $y\in\mathcal{I}_{B}$, such that

[TABLE]

The set of quantum correlation matrices is denoted by $C_{q}\cong C_{q}(\mathcal{O}_{A},\mathcal{O}_{B},\mathcal{I}_{A},\mathcal{I}_{B})$. There are two natural variations on this definition. We can drop the requirement that $H_{A}$ and $H_{B}$ be finite-dimensional, in which case we get another set of correlations often denoted by $C_{qs}$. We can also look at correlations which can be realized as limits of finite-dimensional quantum correlations; the corresponding correlation set is the closure of $C_{q}$, and is typically denoted by $C_{qa}$. It is well-known that $C_{qs}\subseteq C_{qa}$, and consequently $C_{qa}$ is also the closure of $C_{qs}$ [SW08].

Since $C_{qs}\subseteq C_{qa}$, we get a hierarchy of correlation sets

[TABLE]

All the sets involved are convex, and $C_{c}$ and $C_{qa}$ are both closed. Bell’s celebrated theorem [Bel64] states that $C_{c}\neq C_{q}$, and furthermore that the two sets can be separated by a hyperplane. It has been a longstanding open problem to determine the relationship between the quantum correlation sets, and in particular to determine whether $C_{q}$ and $C_{qs}$ are closed (see, i.e., [Tsi06, WCD08, Fri12, BLP17]). Part of the interest in this latter question comes from the resource theory of non-local games: $C_{q}\neq C_{qa}$ if and only if there is a non-local game which can be played optimally (with respect to some payoff function) using a limit of finite-dimensional quantum strategies, but cannot be played optimally using any fixed dimension. Numerical evidence has suggested that even very simple non-local games might have this property [PV10, LW]. For variants of non-local games (for instance, with quantum questions, or infinite output sets), there are several examples of games with this property [LTW13, MV14, RV15].

The purpose of this paper is to show that there are indeed non-local games (with finite classical input and output sets) that cannot be played optimally using any fixed dimension. A perfect strategy for a non-local game $\mathcal{G}$ is a correlation matrix $\{p(a,b|x,y)\}$ such that Alice and Bob win with probability one on every pair of inputs $x$ and $y$. Formally, this means that for all $(a,b,x,y)\in\mathcal{O}_{A}\times\mathcal{O}_{B}\times\mathcal{I}_{A}\times\mathcal{I}_{B}$, if $V(a,b|x,y)=0$, then $p(a,b|x,y)=0$.

Theorem 1.1**.**

There is a non-local game with a perfect strategy in $C_{qa}$, but no perfect strategy in $C_{qs}$.

In particular, neither $C_{q}$ or $C_{qs}$ are closed. The proof is constructive, with the game in question having input sets of size $184$ and $235$, and output sets of size $8$ and $2$.

The set $C_{q}$ is related to the cone of completely positive-semidefinite (cpsd) matrices defined in [LP15]. An $n\times n$ matrix $M$ is said to be cpsd if there are non-negative operators $P_{1},\ldots,P_{n}$ on some finite-dimensional Hilbert space with $M_{ij}=\operatorname{tr}(P_{i}P_{j})$ for all $1\leq i,j\leq n$. By a theorem of Sikora and Varvitsiotis [SV16], the set $C_{q}$ is an affine slice of the cone of cpsd matrices, so the cone of cpsd matrices is not closed as a consequence of Theorem 1.1.

The fact that $C_{qs}\neq C_{qa}$ also has an interesting reformulation. Let $G_{i}$ be the $n$-fold free product $\mathbb{Z}_{m}*\cdots*\mathbb{Z}_{m}$, where $n=|\mathcal{I}_{i}|$ and $m=|\mathcal{O}_{i}|$, for $i=A,B$. Let $M^{x}_{a}$ denote the $a$th spectral projector of the $x$th factor of $G_{A}$ in the full group $C^{*}$-algebra $C^{*}(G_{A})$ of $G_{A}$, and define $M^{y}_{b}$ similarly for $C^{*}(G_{B})$. For each $i=A,B$, find a faithful representation $\nu_{i}$ of $C^{*}(G_{i})$ on some Hilbert space $H_{i}$. The minimal (or spatial) tensor product $C^{*}(G_{A})\otimes_{s}C^{*}(G_{B})$ is the norm-closure of the image $\nu_{A}(C^{*}(G_{A}))\otimes\nu_{B}(C^{*}(G_{B}))$ in the $C^{*}$-algebra $\mathcal{B}(H_{A}\otimes H_{B})$. A correlation matrix $\{p(a,b|x,y)\}$ belongs to $C_{qa}$ if and only if there is a state $\omega$ on the $C^{*}$-algebra $C^{*}(G_{A})\otimes_{s}C^{*}(G_{B})$ with

[TABLE]

for all $(a,b,x,y)\in\mathcal{O}_{A}\times\mathcal{O}_{B}\times\mathcal{I}_{A}\times\mathcal{I}_{B}$ [SW08, Fri12]. On the other hand, the correlation matrix belongs to $C_{qs}$ if and only if there are representations $\phi_{i}$ of $G_{i}$ on $H_{i}$, $i=A,B$, and a vector state $\ket{\psi}\in H_{A}\otimes H_{B}$, with

[TABLE]

for all $(a,b,x,y)\in\mathcal{O}_{A}\times\mathcal{O}_{B}\times\mathcal{I}_{A}\times\mathcal{I}_{B}$. Since $C_{qs}\neq C_{qa}$, there can be states on the minimal tensor product $C^{*}(G_{A})\otimes_{s}C^{*}(G_{B})$ which do not come from vector states on some tensor-product $\phi_{A}\otimes\phi_{B}$ of representations $\phi_{A}$ and $\phi_{B}$.

There is another candidate set of quantum correlations, the commuting-operator correlations $C_{qc}$, which contains $C_{qa}$. Determining whether $C_{qc}$ is known to be equal to $C_{t}$ for any $t\in{q,qs,qa}$ is known as Tsirelson’s problem [Tsi06, DP16]. In a previous paper [Slo16], we showed that $C_{qs}\neq C_{qc}$. By showing that $C_{qs}\neq C_{qa}$, we provide another proof of this fact. The proof that $C_{qs}\neq C_{qc}$ in [Slo16] uses a universal embedding theorem, which states that every finitely-presented group embeds in the solution group of a linear system game. In this paper, we follow a similar line, proving a restricted embedding theorem for a subclass of finitely-presented groups which we call linear-plus-conjugacy groups. For the proof of this restricted embedding theorem, we use a completely different method from [Slo16], with the result that the proof is much shorter. However, it remains an open problem to prove the universal embedding theorem via the new approach.

An easy consequence of the universal embedding theorem is that it is undecidable to determine if a linear system game has a perfect strategy in $C_{qc}$. In this paper we prove a stronger result by applying our restricted embedding theorem to Kharlampovich’s example [Kha82] of a finitely presented solvable group with an undecidable word problem.

Theorem 1.2**.**

There is a (recursive) family of linear system games such that

1. (a)

it is undecidable to determine if a game in the family has a perfect strategy in $C_{qa}$, and 2. (b)

every game in the family has a perfect strategy in $C_{qc}$ if and only if it has a perfect strategy in $C_{qa}$.

Kharlampovich’s construction has been extended by Kharlampovich, Myasnikov, and Sapir to show that the word problem for finitely-presented residually-finite groups can be as hard as any computable function [KMS17].222The word problem for finitely-presented residually-finite groups is always decidable, so this is the best possible lower bound. Using this extension, we can show:

Theorem 1.3**.**

Let $f:\mathbb{N}\rightarrow\mathbb{N}$ be a computable function. Then there is a family of linear system games $\mathcal{G}_{n}$, $n\in\mathbb{N}$, such that

1. (a)

the games $\mathcal{G}_{n}$ have input and output sets of size $\exp(O(n))$, and the function $n\mapsto\mathcal{G}_{n}$ is computable in $\exp(O(n))$-time; 2. (b)

for any algorithm accepting the language

[TABLE]

the maximum running time over inputs $n\leq N$ is at least $f(N)$ when $N$ is sufficiently large; 3. (c)

$\mathcal{G}_{n}$* has a perfect strategy in $C_{qc}$ if and only if it has a perfect strategy in $C_{q}$.*

Theorem 1.3 has the following corollary.

Corollary 1.4**.**

It is undecidable to determine if a linear system game has a perfect strategy in $C_{q}$.

1.1. Acknowledgements

I thank Jason Crann, Richard Cleve, Tobias Fritz, Li Liu, Martino Lupini, Narutaka Ozawa, Vern Paulsen, Mark Sapir, Jamie Sikora, and Thomas Vidick for helpful comments and conversations.

2. Group theory preliminaries

2.1. Group presentations

Given a set $S$, let $\mathcal{F}(S)$ denote the free group generated by $S$. If $H$ is a group, then homomorphisms $\mathcal{F}(S)\rightarrow H$ can be identified with functions $S\rightarrow H$, and we use these two types of objects interchangeably. If $R$ is a subset of $\mathcal{F}(S)$, then the quotient of $\mathcal{F}(S)$ by the normal subgroup generated by $R$ is denoted by $\langle S:R\rangle$. If $G=\langle S:R\rangle$ and $R^{\prime}\subset\mathcal{F}(S\cup S^{\prime})$, then we write $\langle G,S^{\prime}:R^{\prime}\rangle$ to mean $\langle S\cup S^{\prime}:R\cup R^{\prime}\rangle$.

A group $G$ is said to be finitely presentable if $G=\langle S:R\rangle$ for some finite sets $S$ and $R$. A finitely presented group is a tuple $(G,S,R)$, where $G=\langle S:R\rangle$. In other words, a finitely presented group is a finitely presentable group along with a choice of finite presentation.

2.2. Approximate representations

Let $\left\lVert\cdot\right\rVert$ be the normalized Hilbert-Schmidt norm, i.e. if $T$ is an endomorphism of a finite-dimensional Hilbert space $H$, then $\left\lVert T\right\rVert=\sqrt{\operatorname{tr}(T^{*}T)}/\sqrt{\dim H}$.

Definition 2.1**.**

Let $G=\langle S:R\rangle$ be a finitely presented group. A finite-dimensional $\epsilon$-approximate representation (or $\epsilon$-representation for short) is a homomorphism $\phi:\mathcal{F}(S)\rightarrow\mathcal{U}(H)$ from $\mathcal{F}(S)$ to the unitary group $\mathcal{U}(H)$ of some finite-dimensional Hilbert space $H$, such that

[TABLE]

for all $r\in R$.

Note that the normalized Hilbert-Schmidt norm is invariant under conjugation by unitaries, so the set of $\epsilon$-representations is independent of the cyclic order of the relations $r\in R$. That means that, for instance, we can write the relation $x=y$ without worrying about whether we mean $xy^{-1}=e$ or $y^{-1}x=e$.

There are several different notions of approximate representations in the literature. The notion we are using comes from the study of stable relations of $C^{*}$-algebras (see, for instance, Section 4.1 of [Lor97]). For the purposes of this paper, we could also use the closely related notion of approximate homomorphisms as in [CL15, Section II]. However, Definition 2.1 is very convenient for working with examples, as we frequently do in this paper. The main disadvantage of this definition is that it depends on the choice of presentation. We can work around this using the following easy lemma.

Lemma 2.2**.**

Let $\psi:G\rightarrow H$ be a homomorphism, where $G=\langle S:R\rangle$ and $H=\langle S^{\prime}:R^{\prime}\rangle$ are finitely presented groups. If $\Psi:\mathcal{F}(S)\rightarrow\mathcal{F}(S^{\prime})$ is a lift of $\psi$, then there is a constant $C>0$ such that if $\phi$ is an $\epsilon$-representation of $H$, then $\phi\circ\Psi$ is a $C\epsilon$-representation of $G$.

We record two other simple lemmas for later use.

Lemma 2.3**.**

Let $G=\langle S:R\rangle$, and let $M$ be the length of the longest relation in $R$. If $\phi$ is an $\epsilon$-representation of $G$, and $\psi$ is an approximate representation of $G$ with

[TABLE]

for all $x\in S$, then $\psi$ is an $(M\delta+\epsilon)$-representation.

Given approximate representations $\phi:\mathcal{F}(S)\rightarrow\mathcal{U}(H)$ and $\psi:\mathcal{F}(S)\rightarrow\mathcal{U}(H^{\prime})$ of $G=\langle S:R\rangle$, we can form new approximate representations $\phi\oplus\psi:\mathcal{F}(S)\rightarrow\mathcal{U}(H\oplus H^{\prime})$ and $\phi\otimes\psi:\mathcal{F}(S)\rightarrow\mathcal{U}(H\otimes H^{\prime})$.

Lemma 2.4**.**

Suppose $\phi$ and $\psi$ are $\epsilon$- and $\epsilon^{\prime}$-representations of $G$ respectively. Then $\phi\oplus\psi$ is a $\max(\epsilon,\epsilon^{\prime})$-representation, and $\phi\otimes\psi$ is an $(\epsilon+\epsilon^{\prime})$-representation.

A group $G$ is said to be residually finite-dimensional if every non-trivial element of $G$ is non-trivial in some finite-dimensional representation. More generally, the set of elements which are trivial in finite-dimensional representations forms a normal subgroup of $G$. We let $G^{fin}$ denote the quotient of $G$ by this normal subgroup (alternatively, $G^{fin}$ is the image of $G$ in its profinite completion). Any homomorphism $\phi:G\rightarrow H$ descends to a homomorphism $G^{fin}\rightarrow H^{fin}$.

Definition 2.5**.**

A homomorphism $\phi:G\rightarrow H$ is a $fin$-embedding if the induced map $G^{fin}\rightarrow H^{fin}$ is injective, and a $fin^{*}$-embedding if $\phi$ is both injective and a $fin$-embedding.

Equivalently, $\phi$ is a $fin$-embedding if $\phi(g)$ is non-trivial in finite-dimensional representations whenever $g\in G$ is non-trivial in finite-dimensional representations.

We can similarly look at elements which are non-trivial in approximate representations:

Definition 2.6**.**

Let $G$ be a finitely presentable group. An element $g\in G$ is non-trivial in (finite-dimensional) approximate representations if there is a finite presentation $G=\langle S:R\rangle$, a representative $w\in\mathcal{F}(S)$ for $g$, and some constant $\delta>0$ such that, for all $\epsilon>0$, there is an $\epsilon$-representation $\phi$ of $G$ with $\left\lVert\phi(w)-\mathbbm{1}\right\rVert>\delta$.

Alternatively, if $g\in G=\langle S:R\rangle$, let

[TABLE]

where $w$ is a representative for $g$, and the supremum is across $\epsilon$-representations $\phi$ of $G$. It is easy to see that the right-hand side is independent of the choice of representative $w$. By Lemma 2.2, if $\psi:G\rightarrow H$ is a homomorphism, then $\ell^{fa}(g)\geq\ell^{fa}(\psi(g))$. Consequently, $\ell^{fa}(g)$ is independent of the chosen presentation $\langle S:R\rangle$, and $g$ is non-trivial in approximate representations if and only if $\ell^{fa}(g)>0$. This makes it apparent that the choice of presentation $\langle S:R\rangle$ and representative $w$ in Definition 2.6 is arbitrary.

Standard amplification arguments show that the constant $\delta$ in Definition 2.6 is also somewhat arbitrary; in fact, $\ell^{fa}(g)$ never takes values in $(0,\sqrt{2})$. The same amplification arguments can be used to show that a finitely-presented group $G$ is hyperlinear if and only if every non-trivial element of $G$ is non-trivial in approximate representations, and this can be used as the definition of hyperlinearity for finitely-presented groups. We refer Section II.2 of [CL15] for the standard definition of hyperlinearity, along with the amplification arguments needed to prove the equivalence.

Clearly $\ell^{fa}(g)\geq 0$ for all $g\in G$, and it is easy to see that $\ell^{fa}(gh)\leq\ell^{fa}(g)+\ell^{fa}(h)$ and $\ell^{fa}(hgh^{-1})=\ell^{fa}(g)$ for all $g,h\in G$. Thus the set of elements of $G$ which are trivial in approximate representations (i.e. for which $\ell^{fa}(g)=0$) forms a normal subgroup of $G$. Let $G^{fa}$ be the quotient of $G$ by this normal subgroup. Because $\ell^{fa}$ is decreasing via homomorphisms, any homomorphism $\phi:G\rightarrow H$ between finitely presentable groups descends to a homomorphism $G^{fa}\rightarrow H^{fa}$.

Definition 2.7**.**

A homomorphism $\phi:G\rightarrow H$ is an $fa$-embedding if the induced map $G^{fa}\rightarrow H^{fa}$ is injective, and an $fa^{*}$-embedding if $\phi$ is injective, a $fin$-embedding, and an $fa$-embedding.

Equivalently, $\phi$ is an $fa$-embedding if $\phi(g)$ is non-trivial in approximate representations whenever $g\in G$ is non-trivial in approximate representations.

If $\phi$ and $\psi$ are approximate representations, then we say that $\phi$ is a direct summand of $\psi$ if $\psi=\phi\oplus\phi^{\prime}$ for some other approximate representation $\phi^{\prime}$. We use the following simple trick to construct $fa^{*}$-embeddings.

Lemma 2.8**.**

Let $G=\langle S:R\rangle$ and $H=\langle S^{\prime}:R^{\prime}\rangle$ be two finitely presented groups, and let $\Psi:\mathcal{F}(S)\rightarrow\mathcal{F}(S^{\prime})$ be a lift of a homomorphism $\psi:G\rightarrow H$.

1. (a)

Suppose that for every representation (resp. finite-dimensional representation) $\phi$ of $G$, there is a representation (resp. finite-dimensional representation) $\gamma$ of $H$ such that $\phi$ is a direct summand of $\gamma\circ\psi$. Then $\psi$ is injective (resp. a $fin$-embedding). 2. (b)

Suppose that there is an integer $N>0$ and a real number $C>0$ such that for every $d$-dimensional $\epsilon$-representation $\phi$ of $G$, where $\epsilon>0$, there is an $Nd$-dimensional $C\epsilon$-representation $\gamma$ of $H$ such that $\phi$ is a direct summand of $\gamma\circ\Psi$. Then $\psi$ is an $fa$-embedding.

Proof.

Part (a) is clear, so we prove (b). Suppose $\phi$ is an $\epsilon$-representation of $G$, where $\epsilon>0$. If $\gamma\circ\Psi=\phi\oplus\phi^{\prime}$, where $\phi$ is $d$-dimensional and $\phi^{\prime}$ is $(N-1)d$-dimensional, then

[TABLE]

for all $w\in\mathcal{F}(S)$. So $\ell^{fa}(\psi(g))\geq\ell^{fa}(g)/\sqrt{N}$, and $\psi$ is an $fa$-embedding. ∎

In our applications it will be possible to check parts (a) and (b) of Lemma 2.8 simultaneously, in which case $\psi$ will be an $fa^{*}$-embedding.

2.3. Groups over $\mathbb{Z}_{2}$

For convenience, we use the following definition from from [Slo16]: A group over $\mathbb{Z}_{2}$ is a pair $(G,J)$, where $J$ is a central element of $G$ of order two. Note that $J$ is allowed to be the identity element. Typically we drop the pair notation, and just use the symbol $J$ (or $J_{G}$ where necessary) to refer to the special element of a group $G$ over $\mathbb{Z}_{2}$, in the same way that we use $e$ to refer to the identity element. If $G$ and $H$ are groups over $\mathbb{Z}_{2}$, then a morphism $G\rightarrow H$ over $\mathbb{Z}_{2}$ is a group homomorphism $G\rightarrow H$ which sends $J_{G}\mapsto J_{H}$.

If a group $G$ over $\mathbb{Z}_{2}$ is finitely presentable, then it has a finite presentation $\langle S:R\rangle$ where $J\in S$, and $R$ includes the relations $J^{2}=e$ and $[J,s]=e$ for every $s\in S\setminus\{J\}$. We use presentations of this form often enough that it is helpful to have some notation for them. Suppose that $S_{0}$ is a set of indeterminates, and $R_{0}\subset\mathcal{F}(S_{0}\cup\{J\})$. Then we set

[TABLE]

and call $\langle S_{0}:R_{0}\rangle_{\mathbb{Z}_{2}}$ a presentation over $\mathbb{Z}_{2}$. As with ordinary presentations, if $G=\langle S:R\rangle$ or $\langle S:R\rangle_{\mathbb{Z}_{2}}$, then $\langle G,S^{\prime}:R^{\prime}\rangle_{\mathbb{Z}_{2}}:=\langle S\cup S^{\prime}:R\cup R^{\prime}\rangle_{\mathbb{Z}_{2}}$.

3. Linear system games and solution groups

Let $Ax=b$ be an $m\times n$ linear system over $\mathbb{Z}_{2}$. To the system $Ax=b$, we can associate a non-local game, called a linear system game, as follows. For each $1\leq i\leq m$, let $V_{i}=\{j:A_{ij}\neq 0\}$ be the set of indices of variables appearing in the $i$th equation. Let $S_{i}\subset\mathbb{Z}_{2}^{V_{i}}$ be the set of assignments to variables $x_{j}$, $j\in V_{j}$ satisfying the $i$th equation, i.e. $\underline{a}\in\mathbb{Z}_{2}^{V_{i}}$ belongs to $S_{i}$ if and only if $\sum_{j\in V_{j}}a_{j}=b_{i}$. Then Alice receives an equation as input, represented by an integer $1\leq i\leq m$, and must output an element $\underline{a}\in S_{i}$. Bob receives a variable, represented by an integer $1\leq j\leq n$, and must output an assignment $b$ for $x_{j}$. The players win if either $j\not\in V_{i}$, or $j\in V_{i}$ and $a_{j}=b$, i.e. Alice’s and Bob’s outputs are consistent.

A quantum strategy (presented in terms of measurements) for a linear system game consists of

1. (1)

a pair of Hilbert spaces $H_{A}$ and $H_{B}$, 2. (2)

a projective measurement $\{N^{j}_{b}\}_{b\in\mathbb{Z}_{2}}$ on $H_{B}$ for every integer $1\leq j\leq n$, 3. (3)

a projective measurement $\{M^{i}_{\underline{a}}\}_{\underline{a}\in S_{i}}$ on $H_{A}$ for every integer $1\leq i\leq m$, and 4. (4)

a quantum state $\ket{\psi}\in H_{A}\otimes H_{B}$.

The strategy is finite-dimensional if $H_{A}$ and $H_{B}$ are finite-dimensional. The associated quantum correlation matrix $\{p(\underline{a},b|i,j)\}$ is defined by

[TABLE]

As in the introduction, we also use the term strategy to refer to the correlation matrix $\{p(\underline{a},b|i,j)\}$. If $j\in V_{i}$, then the probability that Alice and Bob win on inputs $i$ and $j$ is

[TABLE]

A strategy is perfect if and only if $p_{ij}=1$ for all $1\leq i\leq m$ and $j\in V_{i}$.

For linear system games, it is often convenient to work with strategies presented in terms of $\pm 1$-valued observables—self-adjoint operators which square to the identity—rather than measurement operators. A quantum strategy (presented in terms of observables) consists of

1. (a)

a pair of Hilbert spaces $H_{A}$ and $H_{B}$; 2. (b)

a collection of self-adjoint operators $X_{j}$, $1\leq j\leq n$, on $H_{B}$ such that $X_{j}^{2}=\mathbbm{1}$ for every $1\leq j\leq n$; 3. (c)

a collection of self-adjoint operators $Y_{ij}$, $1\leq i\leq m$, $j\in V_{i}$ on $H_{A}$ such that

1. (i)

$Y_{ij}^{2}=\mathbbm{1}$ for every $1\leq i\leq m$ and $j\in V_{i}$, 2. (ii)

$\prod_{j\in V_{i}}Y_{ij}=(-\mathbbm{1})^{b_{i}}$ for every $1\leq i\leq m$, and 3. (iii)

$Y_{ij}Y_{il}=Y_{il}Y_{ij}$ for every $1\leq i\leq m$ and $j,l\in V_{i}$;

and 4. (d)

a quantum state $\ket{\psi}\in H_{A}\otimes H_{B}$.

Given a quantum strategy presented in terms of measurements, we can get a quantum strategy presented in terms of observables by setting $X_{j}=N^{j}_{0}-N^{j}_{1}$ for every $1\leq j\leq n$, and

[TABLE]

for $1\leq i\leq m$ and $j\in V_{i}$. Conversely, given a quantum strategy in terms of observables, we can recover the measurement presentation using the spectral decomposition of the observables. So the two notions of strategy are equivalent. Note that if $j\in V_{i}$, then

[TABLE]

where $p_{ij}$ is, again, the probability that Alice and Bob win on inputs $i$ and $j$. The quantity $2p_{ij}-1$ is called the winning bias on inputs $i$ and $j$.

To every linear system, we can also associate a finitely presented group over $\mathbb{Z}_{2}$, as follows.

Definition 3.1**.**

Let $Ax=b$ be an $m\times n$ linear system. The solution group of this system is the group

[TABLE]

We say that a group over $\mathbb{Z}_{2}$ is a solution group if it has a presentation over $\mathbb{Z}_{2}$ of this form.

Solution groups and linear system games are related as follows.

Theorem 3.2** ([CM14], see also [CLS16]).**

Let $\mathcal{G}$ be the linear system game associated to a system $Ax=b$. Then the following are equivalent:

1. (a)

$\mathcal{G}$* has a perfect strategy in $C_{qs}$.* 2. (b)

$\mathcal{G}$* has a perfect strategy in $C_{q}$.* 3. (c)

$J_{\Gamma}$* is non-trivial in some finite-dimensional representation of $\Gamma=\Gamma(A,b)$.*

Although we haven’t defined the set of commuting-operator correlations $C_{qc}$, we can work with $C_{qc}$ through the following result.

Theorem 3.3** ([CLS16]).**

The linear system game associated to a system $Ax=b$ has a perfect strategy in $C_{qc}$ if and only if $J_{\Gamma}$ is non-trivial in $\Gamma=\Gamma(A,b)$.

The main point of this section is to prove an analog of one direction of Theorem 3.2 for approximate representations.

Proposition 3.4**.**

Let $\Gamma=\Gamma(A,b)$ be a solution group. If $J_{\Gamma}$ is non-trivial in finite-dimensional approximate representations of $\Gamma$ then the linear system game associated to $Ax=b$ has a perfect strategy in $C_{qa}$.

The proof of Proposition 3.4 is a straightforward application of a number of easy stability lemmas. We start by pinning down what we want to prove.

Lemma 3.5**.**

The linear system game associated to $Ax=b$ has a perfect strategy in $C_{qa}$ if and only if, for all $\epsilon>0$, there is a finite-dimensional quantum strategy (presented in terms of observables) $\{Y_{ij}\}$, $X_{j}$, $\ket{\psi}$ such that

[TABLE]

Proof.

Since $C_{qa}$ is the closure of $C_{q}$, the linear system game associated to $Ax=b$ has a perfect strategy in $C_{qa}$ if and only if, for every $\epsilon>0$, there is a finite-dimensional quantum strategy such that the winning probability $p_{ij}\geq 1-\epsilon/2$ for every $1\leq i\leq m$ and $j\in V_{i}$. But $p_{ij}\geq 1-\epsilon/2$ if and only if the winning bias $2p_{ij}-1\geq 1-\epsilon$, so the lemma follows from equation (3.1). ∎

Next, we come to the stability lemmas, which will allow us to turn approximate representations of the solution group $\Gamma$ into quantum strategies. The following lemmas are all likely well-known to experts (see, for instance, [Gle10, FK10]); we include the proofs for completeness.

Lemma 3.6**.**

For any diagonal matrix $X$, there is a diagonal matrix $D$ with $D^{2}=\mathbbm{1}$ and

[TABLE]

Proof.

Suppose $X$ is a $d\times d$ matrix, and let $D_{ii}=\operatorname{sgn}\operatorname{Re}X_{ii}$ for all $1\leq i\leq d$, where $\operatorname{sgn}x=1$ if $x\geq 0$ and $-1$ if $x<0$. To show that the desired inequality holds, consider a complex number $\alpha=a+bi$. Then

[TABLE]

In particular, this implies that $|\alpha^{2}-1|^{2}$ is greater than or equal to $(a^{2}-1)^{2}$ and $2b^{2}$. Consequently,

[TABLE]

and

[TABLE]

By considering the cases $a\geq 0$ and $a<0$ separately, we see that

[TABLE]

for all $a\in\mathbb{R}$. Thus, as above, $\left\lVert D-\operatorname{Re}X\right\rVert\leq\left\lVert(\operatorname{Re}X)^{2}-\mathbbm{1}\right\rVert$, and the lemma follows. ∎

Lemma 3.7**.**

Suppose $X_{1},\ldots,X_{n}$ are commuting unitary matrices, with $X_{i}^{2}=\mathbbm{1}$ for all $1\leq i\leq n$, and $Y$ is a unitary matrix such that $Y^{2}=\mathbbm{1}$ and $Y$ commutes with $X_{i}$ for all $1\leq i\leq n-1$. Then there is a unitary matrix $Z$ such that $Z^{2}=\mathbbm{1}$, $Z$ commutes with $X_{i}$ for all $1\leq i\leq n$, and

[TABLE]

Proof.

Let $Z_{0}=\frac{1}{2}(Y+X_{n}YX_{n})$. Clearly $Z_{0}$ commutes with $X_{i}$ for all $1\leq i\leq n-1$. Since $X_{n}^{2}=\mathbbm{1}$, we also have that $X_{n}Z_{0}=\frac{1}{2}(X_{n}Y+YX_{n})=Z_{0}X_{n}$. Since $Y^{2}=\mathbbm{1}=(X_{n}YX_{n})^{2}$ as well, we have that

[TABLE]

Since $X_{n}$ and $Y$ are self-adjoint, $Z_{0}$ is self-adjoint, so we can simultaneously diagonalize $X_{1},\ldots,X_{n}$ and $Z_{0}$. Hence by Lemma 3.6, there is a matrix $Z$ such that $Z^{2}=\mathbbm{1}$, $Z$ commutes with $X_{i}$ for all $1\leq i\leq n$, and

[TABLE]

Finally,

[TABLE]

so the lemma follows. ∎

Lemma 3.8**.**

Consider $\mathbb{Z}_{2}^{k}$ as a finitely-presented group with presentation

[TABLE]

Then there is a constant $C>0$, depending on $k$, such that if $\phi$ is an $\epsilon$-representation of $\mathbb{Z}_{2}^{k}$ on a Hilbert space $H$, then there is a representation $\psi$ of $\mathbb{Z}_{2}^{k}$ on $H$ with

[TABLE]

for all $1\leq i\leq k$.

Proof.

Suppose $\psi$ is an $\epsilon$-representation of $\mathbb{Z}_{2}^{k}$ such that the following properties hold for some $1\leq l\leq k-1$:

1. (a)

$\psi(x_{i})^{2}=\mathbbm{1}$ for all $1\leq i\leq k$, and 2. (b)

$\psi(x_{i})$ commutes with $\psi(x_{j})$ for all $1\leq i\leq l-1$ and $1\leq j\leq k$.

In particular, property (b) requires that $\psi(x_{1}),\ldots,\psi(x_{l})$ pairwise commute. Then by Lemma 3.7, for each $l<j\leq k$ there is a unitary matrix $X_{j}$ such that $X_{j}^{2}=\mathbbm{1}$, $X_{j}$ commutes with $\psi(x_{i})$ for all $1\leq i\leq l$, and

[TABLE]

where $C_{0}=1+\frac{1}{2\sqrt{2}}$. Define an approximate representation $\psi^{\prime}$ of $G$ by $\psi^{\prime}(x_{i})=\psi(x_{i})$ if $i\leq l$ and $\psi^{\prime}(x_{i})=X_{i}$ if $i>l$. Then $\psi^{\prime}(x_{i})^{2}=\mathbbm{1}$ for all $1\leq i\leq k$, and $\psi^{\prime}(x_{i})$ commutes with $\psi^{\prime}(x_{j})$ for all $1\leq i\leq l$ and $1\leq j\leq k$. In other words, $\psi^{\prime}$ satisfies properties (a) and (b) with $l$ replaced by $l+1$. Finally, $\left\lVert\psi^{\prime}(x_{i})-\psi(x_{i})\right\rVert\leq C_{0}\epsilon$ for all $1\leq i\leq k$, so $\psi^{\prime}$ is a $(4C_{0}+1)\epsilon$-representation by Lemma 2.3.

Now suppose that $\phi$ is any $\epsilon$-representation of $\mathbb{Z}_{2}^{k}$. By Lemma 3.6, there is an approximate representation $\psi_{1}$ of $\mathbb{Z}_{2}^{k}$ with $\psi_{1}(x_{i})^{2}=\mathbbm{1}$ and $\left\lVert\psi_{1}(x_{i})-\phi(x_{i})\right\rVert\leq C_{1}\epsilon$ for all $1\leq i\leq k$, where $C_{1}=(1+\frac{1}{\sqrt{2}})$. By Lemma 2.3, $\psi_{1}$ is a $(4C_{1}+1)\epsilon$-representation. Clearly, $\psi_{1}$ satisfies conditions (a) and (b) with $l=1$. Using the argument in the previous paragraph, we can then iteratively define approximate representations $\psi_{2},\ldots,\psi_{k-1}$, where $\psi_{j}$ satisfies conditions (a) and (b) with $l=j$ for all $1\leq j\leq k-1$. Let $\epsilon_{l}=(4C_{0}+1)^{l-1}(4C_{1}+1)\epsilon$, so $\psi_{1}$ is an $\epsilon_{1}$-representation. It is not hard to check that $\psi_{l}$ is an $\epsilon_{l}$-representation, and furthermore that

[TABLE]

for all $1\leq i\leq k$. Since $\psi_{k-1}$ is an exact representation, we can take

[TABLE]

∎

Lemma 3.9**.**

Suppose $G=\langle S_{0}:R_{0}\rangle_{\mathbb{Z}_{2}}$, where $R_{0}$ includes the relations $s^{2}=e$ for all $s\in S_{0}$. If $J_{G}$ is non-trivial in finite-dimensional approximate representations of $G$, then for every $\epsilon>0$ there is an $\epsilon$-representation $\phi$ of $G$ such that $\phi(J)=-\mathbbm{1}$, and $\phi(s)^{2}=\mathbbm{1}$ for all $s\in S_{0}$.

Proof.

Suppose $A$ is an $m\times n$ matrix, and let $S=S_{0}\cup\{J\}$. If $J$ is non-trivial in approximate representations, then there is a $\delta>0$ such that for all $\epsilon>0$, there is an $\epsilon$-representation $\phi$ with $\left\lVert\phi(J)-\mathbbm{1}\right\rVert>\delta$.

By Lemmas 2.3, 3.6, and 3.7, there are constants $C,C^{\prime}>0$ such that if $\phi$ is an $\epsilon$-representation, then there is a $C^{\prime}\epsilon$-representation $\psi$ such that

1. (1)

$\psi(x)^{2}=\mathbbm{1}$ for all $x\in S$, 2. (2)

$\psi(s)$ and $\psi(J)$ commute for all $s\in S_{0}$, and 3. (3)

$\left\lVert\psi(J)-\phi(J)\right\rVert\leq C\epsilon$.

(We can take $C=(1+\frac{1}{\sqrt{2}})$, while $C^{\prime}$ will depend on the length of the longest defining relation of $G$.) If $\left\lVert\phi(J)-\mathbbm{1}\right\rVert>\delta$, and $\epsilon<\delta/(2C)$, then

[TABLE]

so $\left\lVert\psi(J)-\mathbbm{1}\right\rVert\geq\frac{\delta}{2}$. Thus we conclude that for all $\epsilon>0$, there is an $\epsilon$-representation $\psi$ satisfying conditions (1) and (2), and with $\left\lVert\psi(J)-\mathbbm{1}\right\rVert>\frac{\delta}{2}$.

Suppose $\psi$ is an $\epsilon$-representation satisfying conditions (1) and (2), and with $\left\lVert\psi(J)-\mathbbm{1}\right\rVert>\frac{\delta}{2}$. Choose a basis with $\psi(J)=\mathbbm{1}_{d_{0}}\oplus(-\mathbbm{1}_{d_{1}})$. Since $\psi(s)$ commutes with $\psi(J)$ for all $s\in S_{0}$, we must have $\psi=\psi_{0}\oplus\psi_{1}$, where $\psi_{a}$ is an approximate representation of dimension $d_{a}$, and $\psi_{a}(J)=(-\mathbbm{1})^{a}$, $a=0,1$. Since $\psi(s)^{2}=\mathbbm{1}$, we also have $\psi_{a}(s)^{2}=\mathbbm{1}$ for all $s\in S_{0}$, $a=0,1$. To finish the proof, we just need to show that $\psi_{1}$ is a $C^{\prime\prime}\epsilon$-representation for some constant $C^{\prime\prime}$ independent of $\psi$. If $w\in\mathcal{F}(S)$, then

[TABLE]

If $w=J$, then $\left\lVert\psi_{0}(w)-\mathbbm{1}\right\rVert=0$ and $\left\lVert\psi_{1}(w)-\mathbbm{1}\right\rVert=\left\lVert-2\mathbbm{1}\right\rVert=4$, so we conclude that

[TABLE]

so $d_{1}/(d_{0}+d_{1})>\delta^{2}/16$. On the other hand, if $w=r$ is one of the defining relations of $G$, then

[TABLE]

Thus $\psi_{1}$ is a $4\epsilon/\delta$-representation with $\psi_{1}(J)=-\mathbbm{1}$ and $\psi_{1}(s)^{2}=\mathbbm{1}$ for all $s\in S_{0}$. Since $\delta$ is a constant, the lemma follows. ∎

Proof of Proposition 3.4.

Suppose $J$ is non-trivial in finite-dimensional approximate representations of $\Gamma$. Given $\epsilon>0$, let $\phi$ be an $\epsilon$-representation of $\Gamma$ with $\phi(J)=-\mathbbm{1}$ and $\phi(x_{j})^{2}=\mathbbm{1}$ for all $1\leq j\leq n$, as in Lemma 3.9. Suppose $\phi$ has dimension $d$, and let $\ket{v}$ be the maximally entangled state on $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$. For each $1\leq j\leq n$, set $X_{j}=\phi(x_{j})$. For each $1\leq i\leq m$, let $j_{i}$ be the maximal element of $V_{i}$, and set $W_{i}:=V_{i}\setminus\{j_{i}\}$. The restriction of $\phi$ to the subgroup $\langle x_{j}:j\in W_{i}\rangle$ is an $\epsilon$-representation of $\mathbb{Z}_{2}^{W_{i}}$, and by Lemma 3.8, there is a representation $\psi_{i}$ of $\mathbb{Z}_{2}^{W_{i}}$ with $\left\lVert\psi_{i}(x_{j})-\phi(x_{j})\right\rVert\leq O(\epsilon)$.333For this proof, we use the notation $O(\epsilon)$ to hide constants which are independent of $\epsilon$, $\phi$, and so on. The constants can still depend on the linear system $Ax=b$, however. Set $Y_{ij}:=\psi_{i}(x_{j})^{T}$ (the transpose of $\psi_{i}(x_{j})$ in a Schmidt basis for $\ket{v}$) for all $j\in W_{i}$, and set $Y_{i{j_{i}}}:=(-1)^{b_{i}}\prod_{j\in W_{i}}Y_{ij}$.

Suppose $j\in W_{i}$ for some $1\leq i\leq m$. Since $Y_{ij}$ and $X_{j}$ are self-adjoint, we have that

[TABLE]

so $\frac{1}{d}\operatorname{tr}(Y_{ij}^{T}X_{j})\geq 1-O(\epsilon^{2})$. For the remaining variable in $V_{i}$, we have that

[TABLE]

where the last equality uses the fact that $\phi(x_{j_{i}})^{2}=\mathbbm{1}$. Because the $Y_{ij}$’s commute for all $j\in W_{i}$, $Y_{ij_{i}}$ is also self-adjoint, so once again we conclude that

[TABLE]

or in other words that $\frac{1}{d}\operatorname{tr}(Y_{ij_{i}}^{T}X_{j})\geq 1-O(\epsilon^{2})$.

Now clearly $\{Y_{ij}\}$, $\{X_{j}\}$, $\ket{v}$ is a strategy for the linear system game associated to $Ax=b$. If $A$ and $B$ are any two $d\times d$ matrices, it follows from the definition of maximally entangled states that

[TABLE]

We conclude that $\bra{v}Y_{ij}\otimes X_{j}\ket{v}=\frac{1}{d}\operatorname{tr}(Y_{ij}^{T}X_{j})\geq 1-O(\epsilon^{2})$ for all $j\in V_{i}$, $1\leq i\leq m$. The proposition follows from Lemma 3.5. ∎

4. Linear-plus-conjugacy groups

The goal of the next two sections is to show that there is a solution group $\Gamma$ such that $J_{\Gamma}$ is trivial in finite-dimensional representations, but non-trivial in approximate representations. In this section, we start by showing that it suffices to construct more general types of group with these properties.

Given an $m\times n$ linear system $Ax=b$, we once again let $V_{i}=V_{i}(A):=\{1\leq j\leq n:A_{ij}\neq 0\}$.

Definition 4.1**.**

Suppose $Ax=b$ is an $m\times n$ linear system over $\mathbb{Z}_{2}$, and $\mathcal{C}\subseteq[n]\times[n]\times[n]$, where $[n]=\{1,\ldots,n\}$. Let

[TABLE]

Lacking a better term, we say that a group over $\mathbb{Z}_{2}$ is a linear-plus-conjugacy group if it has a presentation over $\mathbb{Z}_{2}$ of this form.

The conjugacy part of the name comes from the fact that since $x_{i}$ is an involution, the relation $x_{i}x_{j}x_{i}=x_{k}$ is equivalent to the relation $x_{i}x_{j}x_{i}^{-1}=x_{k}$, so $\Gamma(A,b,\mathcal{C})$ can be thought of as a solution group with additional conjugacy relations. In the context of linear-plus-conjugacy and related groups, we use the term conjugacy relations as a convenient shorthand for relations of the form $xyx=z$. We also use the term linear relation $x_{1}\cdots x_{n}=e$ to refer to the set of relations

[TABLE]

Finally, observe that there are two ways to make generators $x_{i}$ and $x_{j}$ commute in a linear-plus-conjugacy group: we can add a conjugacy relation $x_{i}x_{j}x_{i}=x_{j}$, or add an additional generator $x_{n+1}$ and a linear relation $x_{i}x_{j}x_{n+1}=e$. We pick and choose from these two methods based on what is convenient.

The main point of this section is to prove:

Proposition 4.2**.**

Let $G$ be a linear-plus-conjugacy group. Then there is an $fa^{*}$-embedding $G\rightarrow\Gamma$ over $\mathbb{Z}_{2}$, where $\Gamma$ is a solution group.

We prove Proposition 4.2 by first showing that linear-plus-conjugacy groups can be embedded in linear-plus-conjugacy groups of a certain form.

Definition 4.3**.**

A linear-plus-conjugacy group is nice if it has a presentation of the form $\Gamma(A,b,\mathcal{C})$, where $A$ is an $m\times n$ matrix over $\mathbb{Z}_{2}$, $b\in Z_{2}^{m}$, and $\mathcal{C}\subseteq[n]\times[n]\times[n]$ is such that if $(i,j,k)\in\mathcal{C}$, then $j,k\in V_{l}$ for some $1\leq l\leq m$.

This means that if $x_{i}x_{j}x_{i}=x_{k}$ is a defining relation of a nice linear-plus-conjugacy group, then $x_{j}x_{k}=x_{k}x_{j}$ will also be a defining relation.

Lemma 4.4**.**

Let $G$ be a linear-plus-conjugacy group. Then there is an $fa^{*}$-embedding $G\rightarrow K$ over $\mathbb{Z}_{2}$, where $K$ is a nice linear-plus-conjugacy group.

Proof.

Suppose $G=\Gamma(A,b,\mathcal{C})$, where $A$ is an $m\times n$ matrix. Let

[TABLE]

Since the generators are involutions, note that the relations imply that $fw_{j}=w_{j}f$, $y_{j}z_{j}=z_{j}y_{j}$, and $fz_{j}f=y_{j}$ for all $1\leq j\leq n$. If $(i,j,k)\in\mathcal{C}$, then

[TABLE]

[TABLE]

in $K$. Thus there is a homomorphism $\psi:G\rightarrow K$ sending $x_{i}\mapsto x_{i}$.

Suppose $\phi$ is an $\epsilon$-representation of $G$, where $\epsilon>0$. Define an approximate representation $\gamma$ of $K$ by

[TABLE]

[TABLE]

[TABLE]

It is straightforward to check that $\gamma$ is an $\epsilon$-representation of $K$. If $\Psi$ is the lift of $\psi$ sending $x_{i}\mapsto x_{i}$, then $\gamma\circ\Psi=\phi\oplus\phi$. When $\phi$ is an exact representation of dimension $d$ (possibly infinite), the same construction gives an exact representation $\gamma$ of dimension $2d$. By Lemma 2.8, $\psi$ is an $fa^{*}$-embedding.

Finally, we observe that $K$ is a nice linear-plus-conjugacy group. Indeed, since the relation $x_{i}=y_{i}z_{i}$ forces $y_{i}$ and $z_{i}$ to commute, this relation is equivalent to the relations

[TABLE]

which means that we can make $x_{i}=y_{i}z_{i}$, and similarly $x_{i}=fw_{i}$, part of the “linear” relations. By adding ancilla variables $g_{jk}$, the commuting relations $y_{j}z_{k}=z_{k}y_{j}$ can also be replaced with equivalent linear relations $g_{jk}y_{j}z_{k}=e$. The conjugacy relations $fy_{j}f=z_{j}$ and $w_{i}y_{j}w_{i}=z_{k}$ will then satisfy the requirements of Definition 4.3. ∎

Proof of Proposition 4.2.

By Lemma 4.4, we can assume that $G$ is a nice linear-plus-conjugacy group. Let $G=\Gamma(A,b,\mathcal{C})$ be a presentation satisfying the conditions of Definition 4.3. Augment the linear system $Ax=b$ by adding additional variables $y_{Ij}$ for each $I\in\mathcal{C}$ and $1\leq j\leq 7$, and additional relations

[TABLE]

for every $I=(i,j,k)\in\mathcal{C}$. Let $\Gamma$ be solution group of this augmented linear system, so

[TABLE]

where $R$ consists of the new relations (now written in multiplicative form)

[TABLE]

for every $I=(i,j,k)\in\mathcal{C}$, as well as the corresponding commutation relations. In $\Gamma$, we have that

[TABLE]

for every $I=(i,j,k)\in\mathcal{C}$. So once again we get a homomorphism $\psi:G\rightarrow\Gamma$ sending $x_{i}\mapsto x_{i}$.

Suppose $\phi$ is an $\epsilon$-representation of $G$. Define an approximate representation $\gamma$ of $\Gamma$ by

[TABLE]

for all $I=(i,j,k)\in\mathcal{C}$. It is straightforward to show that $\gamma$ is a $C\epsilon$-representation of $\Gamma$, where $C$ is a positive constant $\leq 15$. For instance, consider the relation $y_{I5}^{2}=e$. To show that $\gamma(y_{I5})^{2}\approx\mathbbm{1}$, we need to show that $\phi(x_{j}x_{i}x_{j})^{2}\approx\mathbbm{1}$. Write $X\approx_{\epsilon}Y$ to mean that $\left\lVert X-Y\right\rVert\leq\epsilon$. Since $\phi(x_{i})^{2}\approx_{\epsilon}\mathbbm{1}$ and $\phi(x_{j})^{2}\approx\mathbbm{1}$, we have $\phi(x_{i}x_{k}x_{i})^{2}\approx_{3\epsilon}\mathbbm{1}$. We can conclude from this that $\gamma(y_{I5})^{2}\approx_{3\epsilon}\mathbbm{1}$ (we can do slightly better by averaging over the blocks of $\gamma(y_{I5})$, but we ignore this to simplify the analysis). We can similarly show that $\gamma(y_{Ij})^{2}\approx_{3\epsilon}\mathbbm{1}$ for all $1\leq j\leq 7$, and that the linear relations in Equation (4.1) hold to within $3\epsilon$.

This leaves the commuting relations. Consider the relation $y_{I3}y_{I4}y_{I5}=e$. We want to show that $\gamma(y_{I3})$, $\gamma(y_{I4})$, and $\gamma(y_{I5})$ approximately commute. But since $\gamma(y_{I3})\gamma(y_{I4})\gamma(y_{I5})\approx_{3\epsilon}\mathbbm{1}$ and $\gamma(y_{Ij})^{2}\approx_{3\epsilon}\mathbbm{1}$, we conclude that

[TABLE]

or in other words, $\gamma(y_{I4})\gamma(y_{I5})\approx_{15\epsilon}\gamma(y_{I5})\gamma(y_{I4})$. The other commuting relations follow similarly.

Let $\Psi$ be the lift of $\psi$ sending $x_{i}\mapsto x_{i}$. Then $\gamma\circ\Psi=\phi\oplus\phi$. Once again, the same construction applies when $\psi$ is an exact representation, so $\psi$ is an $fa^{*}$-embedding by Lemma 2.8. ∎

Note that if $j=k$ in a relation $x_{i}x_{j}x_{i}=x_{k}$, then the system in Equation (4.1) is precisely the Mermin-Peres magic square [Mer90, Per90]. The magic square has previously been used by Ji to show that linear system games can require a (finite but) arbitrarily high amount of entanglement to play perfectly [Ji13].

The proof of Proposition 4.2 has several interesting features:

Remark 4.5**.**

Let $G=\Gamma(A,b,\mathcal{C})$ be an $m\times n$ linear-plus-conjugacy group, and let $\Gamma^{\prime}=\Gamma^{\prime}(A^{\prime},b^{\prime})$ be the solution group constructed in the proof of Proposition 4.2. Then, accounting for Lemma 4.4, the system $A^{\prime}x=b^{\prime}$ has $11n+8c+1$ variables and $8n+m+7c$ equations, where $c=|\mathcal{C}|$ is the number of conjugacy relations. A presentation for $\Gamma^{\prime}$ can be constructed in polynomial time in $m$, $n$, and $c$.

The proofs of Lemma 4.4 and Proposition 4.2 show that there is a constant $C>0$, and a lift $\Psi$ of the homomorphism $G\rightarrow\Gamma^{\prime}$ to the defining free groups, such that for any $d$-dimensional $\epsilon$-representation $\phi$ of $G$, there is a $4d$-dimensional $C\epsilon$-representation $\psi$ of $\Gamma^{\prime}$ with $\psi\circ\Psi=\phi^{\oplus 4}$. Taking into account the fact that we have to change the presentation of the group $K$ in the proof of Lemma 4.4, we can take the constant $C\leq 75$. The lift $\Psi$ can be chosen to send the generators of $G$ to generators of $\Gamma^{\prime}$ (although not every generator of $\Gamma^{\prime}$ will lie in the image of $\Psi$).

It is important for our argument that the $fa^{*}$-embedding in Proposition 4.2 is over $\mathbb{Z}_{2}$. However, we can go a little further in what type of groups can be embedded if we drop this requirement.

Definition 4.6**.**

Suppose $A$ is an $m\times n$ matrix over $\mathbb{Z}_{2}$, and $\mathcal{C}\subseteq[n]\times[n]\times[n]$. Let

[TABLE]

We say that a group $G$ is a homogeneous-linear-plus-conjugacy group if it has a presentation of this form.

Since $\operatorname{\Gamma_{0}}(A,\mathcal{C})$ is not presented over $\mathbb{Z}_{2}$, a homogeneous-linear-plus-conjugacy group is not a linear-plus-conjugacy group. However, the two types of groups are closely related, as $\operatorname{\Gamma_{0}}(A,\mathcal{C})\times\mathbb{Z}_{2}=\Gamma(A,0,\mathcal{C})$.

Definition 4.7**.**

Suppose $A$ is an $m\times n$ matrix over $\mathbb{Z}_{2}$, $\mathcal{C}_{0}\subseteq[n]\times[n]\times[n]$, $\mathcal{C}_{1}\subseteq[\ell]\times[n]\times[n]$, and $L$ is an $\ell\times\ell$ lower-triangular matrix with non-negative integer entries. Let

[TABLE]

We refer to the generators $x_{i}$ in this presentation as involutary generators, and to the generators $y_{j}$ as non-involutary generators. We say that a group $G$ is an extended homogeneous-linear-plus-conjugacy group if it has a presentation of this form.

Proposition 4.8**.**

Let $G=\operatorname{E\Gamma_{0}}(A,\mathcal{C}_{0},\mathcal{C}_{1},L)$ as in Definition 4.7, where $A$ is an $m\times n$ matrix. Then there is an $m\times n^{\prime}$ matrix $A^{\prime}$ and a set $\mathcal{C}^{\prime}\subset[n^{\prime}]\times[n^{\prime}]\times[n^{\prime}]$, where $n\leq n^{\prime}$, such that there is an $fa^{*}$-embedding $\psi:G\rightarrow\operatorname{\Gamma_{0}}(A^{\prime},\mathcal{C}^{\prime})$ with $\psi(x_{i})=x_{i}$ for all $1\leq i\leq n$.

Proof.

Suppose $G$ has $\ell$ non-involutary generators, and let

[TABLE]

We claim that the natural morphism $\psi:G\rightarrow G^{\prime}$ is an $fa^{*}$-embedding. Indeed, let $\Psi:\mathcal{F}(S)\rightarrow\mathcal{F}(S\cup\{z,w\})$ be the natural inclusion, where $S=\{x_{1},\ldots,x_{n},y_{1},\ldots,y_{\ell}\}$. Given an $\epsilon$-representation $\phi$ of $G$, define an approximate representation $\gamma$ of $G^{\prime}$ by

[TABLE]

[TABLE]

Because $L$ is lower-triangular, $G^{\prime}$ has no defining relations of the form $y_{1}y_{i}y_{1}^{-1}=y_{i}^{L_{1i}}$. Suppose $L_{i1}>0$, so that $\phi(y_{i})\phi(y_{1})\phi(y_{i})^{*}\approx_{\epsilon}\phi(y_{1})^{L_{i1}}$, where once again $X\approx_{\epsilon}Y$ means that $\left\lVert X-Y\right\rVert\leq\epsilon$. Then $\phi(y_{i})\phi(y_{1})^{*}\phi(y_{i})^{*}\approx_{\epsilon}\phi(y_{1})^{-L_{i1}}$, so $\psi(y_{i})\psi(y_{1})\psi(y_{i})^{*}\approx_{\epsilon}\psi(y_{1})^{L_{i1}}$. It is easy to see that the remaining defining relations of $G^{\prime}$ hold to within $\epsilon$, so $\psi$ is an $\epsilon$-representation of $G^{\prime}$. Since $\phi$ is a direct summand of $\gamma\circ\Psi$, we can apply Lemma 2.8 with $N=2$ and $C=1$ to see that $\psi$ is an $fa$-embedding. The same construction for exact representations shows that $\psi$ is an $fa^{*}$-embedding.

Next, observe that $G^{\prime}$ is an extended homogeneous-linear-plus-conjugacy group with $\ell-1$ non-involutary generators. Indeed, suppose $(1,j,k)\in\mathcal{C}_{1}$. Then the defining relation $y_{1}x_{j}y_{1}^{-1}=x_{k}$ is equivalent to the relation $zwx_{j}wz=x_{k}$. By adding an ancilla variable $Z_{jk}$ with $Z_{jk}^{2}=e$, we can replace this relation with the two conjugacy relations $wx_{j}w=Z_{jk}$ and $zZ_{jk}z=x_{k}$. Similarly, suppose $L_{i1}>0$. Then the relation $y_{i}y_{1}y_{i}=y_{1}^{L_{i1}}$ is equivalent to the relation $y_{i}wy_{i}^{-1}=w(zw)^{L_{i1}-1}$. Once again, we can replace this relation with a sequence of conjugacy relations by adding ancilla variables. For instance, if $L_{i1}=3$, then we would add ancilla variables $W_{i0}$ and $W_{i1}$ with $W_{i0}^{2}=W_{i1}^{2}=e$, and conjugacy relations $zwz=W_{i0}$, $wW_{i0}w=W_{i1}$, and $y_{i}wy_{i}^{-1}=W_{i1}$. After making these replacements, the only relation containing $y_{1}$ is $y_{1}=zw$, so we can remove $y_{1}$ from the set of generators. The commuting relations added in $G^{\prime}$ are equivalent to $y_{i}zy_{i}^{-1}=z$ for all $2\leq i\leq\ell$, so $G^{\prime}$ is an extended homogeneous-linear-plus-conjugacy group. The additional variables (including the ancilla) are involutary generators, so $G^{\prime}$ has $\ell-1$ non-involutary generators.

Iterating this construction, we get a sequence of $fa^{*}$-embeddings terminating in a homogeneous-linear-plus-conjugacy group, as desired. ∎

The reason the above argument does not apply for groups over $\mathbb{Z}_{2}$ is that, if we set $\gamma(J)=\phi(J)\oplus\mathbbm{1}$, then $\gamma(J)$ would not commute with $\gamma(z)$ and $\gamma(w)$, while if we set $\gamma(J)=\phi(J)\oplus\phi(J)$, then any linear relations containing $J$ would not be satisfied.

Remark 4.9**.**

The above proof shows that, in Proposition 4.8, we can take

[TABLE]

and

[TABLE]

where $\ell$ is the number of non-involutary generators, $\operatorname{sum}(L)$ is the sum of the entries of $L$, and $\#(L)$ is the number of non-zero entries of $L$. The matrix $A^{\prime}$ and set $\mathcal{C}^{\prime}$ can be constructed in polynomial time in $m$, $n$, $\ell$, $|\mathcal{C}_{0}|$, $|\mathcal{C}_{1}|$, and $\operatorname{sum}(L)$.

5. Proof of Theorem 1.1

The point of this section is to prove the following proposition, and hence finish the proof of Theorem 1.1.

Proposition 5.1**.**

There is a solution group $\Gamma$ for which $J$ is trivial in finite-dimensional representations, but non-trivial in finite-dimensional approximate representations.

For the proof of Proposition 5.1, it is convenient to work with sofic groups. We do not need to know the definition of soficity, just that the class of sofic groups has the following properties:

1. (1)

Amenable groups are sofic. 2. (2)

Sofic groups are hyperlinear. 3. (3)

If $H$ is an amenable subgroup of a sofic group $G$, and $\alpha:H\rightarrow G$ is injective homomorphism, then the HNN extension of $G$ by $\alpha$ is sofic.

An expository treatment of sofic groups can be found in [CL15]. In particular, the last “closure property” can be found in [CL15, Section II.4].

We need one more general-purpose lemma before proceeding to the proof.

Lemma 5.2**.**

Suppose $G=\langle S:R\rangle$ is a finitely-presented group, where $R$ contains the relation $a^{2}=e$ for some $a\in S$. Let

[TABLE]

where $J,t\not\in S$. If $a$ is non-trivial in approximate representations of $G$, then $J$ is non-trivial in approximate representations of $\widehat{G}$.

Note that $\widehat{G}$ is the “$\mathbb{Z}_{2}$-HNN extension” of $G\times\mathbb{Z}_{2}$, where $J$ is the generator of the $\mathbb{Z}_{2}$ factor, by the order-two automorphism sending $a\mapsto Ja$ and $J\mapsto J$.

Proof.

For the purposes of this proof, if $X$ is a linear operator on a finite-dimensional Hilbert space $H$, let $\widetilde{\operatorname{tr}}(X):=\operatorname{tr}(X)/\dim H$. Suppose $\phi$ is an $\epsilon$-representation of $G$ with $\phi(a)^{2}=1$ and $\operatorname{tr}(\phi(a))\geq 0$. Because the eigenvalues of $\phi(a)$ belong to $\{\pm 1\}$, we can choose a basis so that $\phi(a)=\mathbbm{1}_{d_{0}}\oplus(-\mathbbm{1})_{d_{0}}\oplus\mathbbm{1}_{d_{1}}$, where $d_{1}=\operatorname{tr}(\phi(a))$. Define an approximate representation $\psi$ of $\widehat{G}$ by

[TABLE]

Clearly $\left\lVert\psi(r)-\mathbbm{1}\right\rVert=\left\lVert\phi(r)-\mathbbm{1}\right\rVert\leq\epsilon$ for all relations $r\in R$, $\psi([J,s])=\mathbbm{1}$ for all $s\in S\cup\{t\}$, and $\psi(t)^{2}=\psi(J)^{2}=\mathbbm{1}$. For the remaining relation,

[TABLE]

So $\psi$ will be a $\max\left(\epsilon,2\sqrt{\widetilde{\operatorname{tr}}(\phi(a))}\right)$-representation with $\left\lVert\psi(J)-\mathbbm{1}\right\rVert=2$.

To make $\widetilde{\operatorname{tr}}(\phi(a))$ small, we can use the tensor-power trick as in Section II.2 of [CL15]. Suppose $a$ is non-trivial in approximate representations of $G$. By Lemmas 2.3 and 3.6, there is a constant $\delta>0$, such that for all $\epsilon>0$, there is an $\epsilon$-representation $\phi$ of $G$ with $\left\lVert\phi(a)-\mathbbm{1}\right\rVert>\delta$ and $\phi(a)^{2}=\mathbbm{1}$. Given $\epsilon>0$, find an integer $k$ such that

[TABLE]

and let $\phi$ be an $\frac{\epsilon}{k}$-representation with $\left\lVert\phi(a)-\mathbbm{1}\right\rVert>\delta$ and $\phi(a)^{2}=\mathbbm{1}$. Suppose $\phi$ has dimension $d$, and let $\gamma$ be the direct sum of $\phi$ with $d$ copies of the trivial representation. Then $\gamma$ is an $\frac{\epsilon}{k}$-representation of $G$ by Lemma 2.4, and furthermore $\gamma(a)^{2}=\mathbbm{1}$, $\operatorname{tr}(\gamma(a))=d+\operatorname{tr}(\phi(a))\geq 0$, and

[TABLE]

Since $\gamma(a)$ is self-adjoint,

[TABLE]

so we conclude that

[TABLE]

Since $\widetilde{\operatorname{tr}}(X^{\otimes k})=\widetilde{\operatorname{tr}}(X)^{k}$, Lemma 2.4 implies that $\gamma^{\otimes k}$ is an $\epsilon$-representation of $\widehat{G}$ with

[TABLE]

Applying the argument of the first paragraph to $\gamma^{\otimes k}$, we get an $\epsilon$-representation $\psi$ of $G$ with $\left\lVert\psi(J)-\mathbbm{1}\right\rVert=2$. This shows that $J$ is non-trivial in approximate representations of $\widehat{G}$. ∎

We are now ready to prove Proposition 5.1. Note that any hyperlinear but non-residually-finite group has an element which is trivial in finite-dimensional representations, but non-trivial in approximate representations. To prove Proposition 5.1, we show that

[TABLE]

is an extended homogeneous-linear-plus-conjugacy group which is hyperlinear but non-residually finite. Indeed, to see that $K$ has a presentation as in Definition 4.7, we can introduce a third variable $c$ with $c^{2}=e$ and $c=ab$. Then $K$ is equivalent to the extended homogeneous-linear-plus-conjugacy group with three involutary generators $a,b,c$, one linear relation $abc=e$ (along with the corresponding commuting relations), two non-involutary generators $x$ and $y$, and three conjugacy relations $yay^{-1}=a$, $yby^{-1}=c$, and $xyx^{-1}=y^{2}$. For the remainder of this section, $K$ will refer to this group.

Lemma 5.3**.**

$K$* is sofic, and the element $a\in K$ is non-trivial.*

Proof.

$K_{1}:=\langle y,a,b:a^{2}=b^{2}=e,ab=ba,yay^{-1}=a,yby^{-1}=ab\rangle$ is isomorphic to $\mathbb{Z}\ltimes(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$, and in particular is solvable (hence amenable). The group $K$ is the HNN extension of $K_{1}$ by the injective endomorphism of $\langle y\rangle\cong\mathbb{Z}$ sending $y\mapsto y^{2}$. Hence $K$ is sofic by properties (1) and (3) of sofic groups above. In addition, the natural morphism $K_{1}\rightarrow K$ is injective. Since $a$ is clearly non-trivial in $K_{1}$, we conclude that $a$ is non-trivial in $K$. ∎

The following lemma comes from discussions with Tobias Fritz.

Lemma 5.4**.**

The element $a\in K$ is trivial in all finite-dimensional representations of $K$.

Proof.

By a theorem of Mal’cev [Mal65], it suffices to show that $a$ is trivial in finite representations, rather than finite-dimensional representations. So let $\phi:G\rightarrow H$ be a homomorphism from $G$ to a finite group $H$. Now the order $k$ of $\phi(x)$ is finite, so $\phi(y)=\phi(x)^{k}\phi(y)\phi(x)^{-k}=\phi(y)^{2^{k}}$. It follows that the order $m=|\phi(y)|$ of $\phi(y)$ divides $2^{k}-1$, and in particular is odd. Since $\phi(y)\phi(b)\phi(y)^{-1}=\phi(ab)$ and $\phi(y)\phi(ab)\phi(y)^{-1}=\phi(b)$, we conclude that $\phi(b)=\phi(y)^{m}\phi(b)\phi(y)^{-m}=\phi(ab)$. Consequently $\phi(a)=\mathbbm{1}$ as desired. ∎

Proof of Proposition 5.1.

By Proposition 4.8, there is an $fa$-embedding of $K$ to a homogeneous-linear-plus-conjugacy group $G=\operatorname{\Gamma_{0}}(A,\mathcal{C})$, in which $a\in K$ is mapped to a generator $x_{i}$ of $G$. Let

[TABLE]

The relation $tx_{i}t=Jx_{i}$ can be replaced with the relations $tx_{i}t=Z$ and $Zx_{i}=J$, where $Z$ is an ancilla variable with $Z^{2}=e$. With this presentation, $\widehat{G}$ is a linear-plus-conjugacy group. By Proposition 4.2, there is an $fa$-embedding over $\mathbb{Z}_{2}$ of $\widehat{G}$ to a solution group $\Gamma$.

By Lemma 5.3, $a$ is non-trivial in approximate representations of $K$, and hence $x_{i}$ is non-trivial in approximate representations of $G$. By Lemma 5.2, $J_{\widehat{G}}$ is non-trivial in approximate representations of $\widehat{G}$, and we conclude that $J_{\Gamma}$ is non-trivial in approximate representations of $\Gamma$.

Finally, there is a morphism from $K$ to $\widehat{G}$ which sends $a$ to $x_{i}$, so $x_{i}$ will be trivial in all finite-dimensional representations of $\widehat{G}$ by Lemma 5.4. But since $J_{\widehat{G}}=[t,x_{i}]$, this means that $J_{\widehat{G}}$ (and hence $J_{\Gamma}$) is trivial in all finite-dimensional representations of $\widehat{G}$. ∎

Proof of Theorem 1.1.

Let $\Gamma$ be the solution group from Proposition 5.1, and let $\mathcal{G}$ be the associated game. Since $J$ is trivial in finite-dimensional representations, Theorem 3.2 implies that $\mathcal{G}$ does not have a perfect strategy in $C_{qs}$. But since $J$ is non-trivial in approximate representations, Proposition 3.4 implies that $\mathcal{G}$ has a perfect strategy in $C_{qa}$. ∎

Remark 5.5**.**

By Remarks 4.5 and 4.9, the linear system constructed in the proof of Theorem 1.1 will have $235$ variables and $184$ equations.

6. Proofs of Theorems 1.2 and 1.3

To prove Theorem 1.2, we want to find a hyperlinear group with an undecidable word problem, which $fa$-embeds in a solution group. For Theorem 1.3, we want to find a family of residually finite groups with arbitrarily hard (albeit computable) word problems, which $fin$-embed in solution groups. Fortunately, such groups are provided by Kharlampovich [Kha82] and Kharlampovich, Myasnikov, and Sapir [KMS17]. Since the presentations are rather complicated, we do not repeat them here. Instead, we summarize some points of the construction from [KMS17] in the following theorem.

It is helpful to use the following notation: given $S_{0}\subseteq S_{1}$, let $\mathcal{N}(S_{0},S_{1})$ denote the normal subgroup generated by $S_{0}$ in the free group $\mathcal{F}(S_{1})$. Note that if $S_{1}\subseteq S$, then $\mathcal{N}(S_{0},S_{1})$ is a (not necessarily normal) subgroup of $\mathcal{F}(S)$ in a natural way. Also, if $x,y$ are group elements, recall that $[x,y]=xyx^{-1}y^{-1}$, and $x^{y}=yxy^{-1}$.444This is the reverse of the convention in [KMS17], where $[x,y]=x^{-1}y^{-1}xy$ and $x^{y}=y^{-1}xy$.

Theorem 6.1** ([KMS17], see also [Kha82]).**

Let $X\subseteq\mathbb{N}$ be recursively enumerable. Then there is a finitely-presented solvable group $K_{X}=\langle S:R\rangle$ with the following properties:

1. (1)

The set $S$ is divided into three subsets $L_{i}$, $i=0,1,2$. 2. (2)

The relations in $R$ come in three types:

1. (a)

$R$* contains the relations $x^{2}=e$ for all $x\in L_{0}\cup L_{1}$.* 2. (b)

$R$* also contains commuting relations of the form $xy=yx$, for certain pairs $x,y\in S$.* 3. (c)

Every other relation $r\in R$ belongs to some normal subgroup $\mathcal{N}(S_{0},S_{1})$, where $S_{1}\subseteq S$ and $S_{0}\subseteq(L_{0}\cup L_{1})\cap S_{1}$ are such that the image of $\mathcal{N}(S_{0},S_{1})$ in $K_{X}$ is abelian. 3. (3)

The image of $\mathcal{N}(L_{0},S)$ in $K_{X}$ is abelian. 4. (4)

There are elements $z_{0},z_{1}\in L_{0}$, $A_{1},A_{2}\in L_{1}$, and $a,a^{\prime}\in L_{2}$, such that $n\in X$ if and only if

[TABLE]

in $K_{X}$, where $w(m)$ is defined by

[TABLE] 5. (5)

If $X$ is recursive, then $K_{X}$ is residually finite.

Note that there is some overlap between relations of type (2b) and (2c). Indeed, if $[x,y]=e$ is a relation, then the image of $\mathcal{N}(\{x\},\{x,y\})$ in $K_{X}$ is equal to $\langle x\rangle$, and in particular is abelian. Since $[x,y]$ belongs to $\mathcal{N}(\{x\},\{x,y\})$, any relation $[x,y]=e$ of type (2b) with $x\in L_{0}\cup L_{1}$ can also be regarded as a relation of type (2c).

To see that property (4) of the theorem holds from the description in [KMS17], it is helpful to note that, by properties (1), (2a), and (3) of the theorem, $w(m)$ is an involution for all $m\geq 1$.

Lemma 6.2**.**

Suppose $K=\langle S:R\rangle$ is a finitely-presented group satisfying properties (1) and (2) of Theorem 6.1. Then $K$ is an extended homogeneous-linear-plus-conjugacy group (as in Definition 4.7).

Furthermore, if $S_{0}\subseteq S_{1}\subseteq S$ are two subsets such that $S_{0}\subseteq L_{0}\cup L_{1}$, and the image of $\mathcal{N}(S_{0},S_{1})$ in $K$ is abelian, then for every $w\in\mathcal{N}(S_{0},S_{1})$, there is a presentation of $K$ as an extended homogeneous-linear-plus-conjugacy group in which $w$ is equal in $K$ to one of the involutary generators $x_{i}$.

Proof.

The generating set of $K$ is split into involutary generators $L_{0}\cup L_{1}$ and non-involutary generators $L_{2}$. Since the order on non-involutary generators matters in Definition 4.7, choose an arbitrary enumeration $y_{1},\ldots,y_{k}$ of $L_{2}$. According to property (2) of Theorem 6.1, the defining relations for $K$ (aside from the involutary relations on $L_{0}\cup L_{1}$) fall into two types: (2b) and (2c). Both types of relations can be rewritten as linear and conjugacy relations of the types allowed in Definition 4.7. Indeed, commuting relations (relations of type (2b)) can be regarded as conjugacy relations (note that for relations $y_{i}y_{j}=y_{j}y_{i}$, we can choose either $y_{i}y_{j}y_{i}^{-1}=y_{j}$ or $y_{j}y_{i}y_{j}^{-1}=y_{i}$ depending on whether $i>j$ or $i<j$).

This leaves relations of type (2c). For this, we first prove the second part of the lemma. Suppose that the image of $\mathcal{N}(S_{0},S_{1})$ is abelian in $K$, where $S_{0}\subset L_{0}\cup L_{1}$. We claim that for any non-trivial element $w\in\mathcal{N}(S_{0},S_{1})$, there is a finite set of generators $S_{w}$ and relations $R_{w}\subset\mathcal{F}(L_{0}\cup L_{1}\cup L_{2}\cup S_{w})$ such that

1. (i)

$R_{w}$ consists of linear and conjugacy relations as in Definition 4.7, 2. (ii)

the relations

[TABLE]

imply that $w$ is equal to an element of $L_{0}\cup S_{w}$, and 3. (iii)

the added generators $S_{w}$ and relations $R_{w}$ do not change the group, i.e. the inclusion

[TABLE]

is an isomorphism.

To prove the claim, we use induction on the length of $w$ in $\mathcal{F}(S_{1})$. The claim is trivially true if $w\in S_{0}^{\pm}$. Suppose $w=zxz^{-1}$, where $x\in\mathcal{N}(S_{0})$ has length less than $w$, and $z\in S_{1}$. By induction, there is a set of ancilla variables $S_{x}$ and relations $R_{x}$ satisfying properties (i)-(iii) for $x$. In particular, the relations $\widetilde{R}_{x}$ imply that $x$ is equal to some $X\in S_{0}\cup S_{x}$. Then we can set $S_{w}:=S_{x}\cup\{W\}$, where $W$ is a new indeterminate, and $R_{w}:=R_{x}\cup\{W=zXz\}$ or $R_{x}\cup\{W=zXz^{-1}\}$ depending on whether $z\in L_{0}\cup L_{1}$ or $z\in L_{2}$. If $w=z^{-1}xz$, then we do the same thing, but using $zWz^{-1}=X$ in place of $W=zXz^{-1}$. Finally, suppose that $w=x_{1}\cdots x_{n}$, where each $x_{i}\in\mathcal{N}(S_{0},S_{1})$ has smaller length than $w$. By induction, there are sets $S_{x_{i}}$ and relations $R_{x_{i}}$ implying that $x_{i}$ is equal to some $X_{i}\in L_{0}\cup S_{x_{i}}$. We then set $S_{w}:=\bigcup S_{x_{i}}\cup\{W\}$, where $W$ is again a new indeterminate, and

[TABLE]

Since the image of $\mathcal{N}(S_{0},S_{1})$ in $K$ is abelian, adding the relations $R_{w}$ does not change $K$. This proves the claim.

Now suppose that $K$ has a defining relation in $\mathcal{N}(S_{0},S_{1})$. If $r=zxz^{-1}$ for some $x\in\mathcal{N}(S_{0},S_{1})$ and $z\in S_{1}^{\pm}$, then $r$ can be replaced with the simpler relation $x$. So we can assume without loss of generality that $r=x_{1}\cdots x_{n}$, where each $x_{i}\in\mathcal{N}(S_{0},S_{1})$. By the claim, we can add ancilla variables and relations such that each $x_{i}$ is equal to an involutary generator $X_{i}$ in $K$, and the relation $r$ can be replaced with the linear relation $X_{1}\cdots X_{n}=e$. We conclude that $K$ is an extended homogeneous-linear-plus-conjugacy group. The claim also immediately implies the second part of the lemma. ∎

We now come to the main result of this section.

Proposition 6.3**.**

Let $X\subseteq\mathbb{N}$ be a recursively enumerable set. Then there is a family of solution groups $\Gamma_{n}=\Gamma\left(A^{(n)},b^{(n)}\right)$, $n\geq 1$, such that

1. (a)

$A^{(n)}x=b^{(n)}$* is an $\exp(O(n))\times\exp(O(n))$ linear system;* 2. (b)

the function $n\mapsto\left(A^{(n)},b^{(n)}\right)$ is computable in $\exp(O(n))$-time; 3. (c)

$J_{\Gamma_{n}}$* is non-trivial in $\Gamma_{n}$ if and only if $n\in X$;* 4. (d)

if $J_{\Gamma_{n}}$ is non-trivial in $\Gamma_{n}$, then $J_{\Gamma_{n}}$ is non-trivial in approximate representations; and 5. (e)

if $X$ is recursive and $J_{\Gamma}$ is non-trivial in $\Gamma_{n}$, then $J_{\Gamma_{n}}$ is non-trivial in finite-dimensional representations.

Before giving the proof, we need the following exact version of Lemma 5.2.

Lemma 6.4**.**

Suppose $G=\langle S:R\rangle$ is a finitely-presented group, where $R$ contains the relation $a^{2}=e$ for some $a\in S$. Let

[TABLE]

where $J,t\not\in S$. If $a$ is non-trivial in finite-dimensional representations of $G$, then $J$ is non-trivial in finite-dimensional representations of $\widehat{G}$.

Proof.

Suppose $a$ is non-trivial in finite-dimensional representations of $G$. A theorem of Baumslag states that the free product of two residually finite groups amalgamated over a finite subgroup is residually finite [Bau63]. Let $\widetilde{G}:=G\times\mathbb{Z}_{2}$, where the generator of the $\mathbb{Z}_{2}$ factor is denoted by $J$, and let $H=\langle t,a:t^{2}=a^{2}=e,tat=aJ\rangle_{\mathbb{Z}_{2}}\cong\mathbb{Z}_{2}\ltimes\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. Then $\widehat{G}$ is isomorphic to amalgamated free product of $\widetilde{G}$ and $H$ over $\langle a,J\rangle\cong\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, a finite group. While $\widetilde{G}$ is not necessarily residually finite, the group $\widetilde{G}^{fin}$ is residually finite by definition, and there is natural map from $\widehat{G}$ to the amalgamated free product of $\widetilde{G}^{fin}$ and $H$ over $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. The image of $J_{\widetilde{G}}$ is non-trivial in $\widetilde{G}^{fin}$, and hence in the amalgamated product of $\widetilde{G}^{fin}$ and $H$. So $J$ is non-trivial in finite-dimensional representations of $\widehat{G}$ by Baumslag’s result. ∎

Proof of Proposition 6.3.

Given a recursively enumerable subset $X\subseteq\mathbb{N}$, let $K_{X}=\langle S:R\rangle$ be the associated group from Theorem 6.1. Using the notation from property (4) of Theorem 6.1, let $c(n)=[A_{2},[A_{1},w(2^{n})]][A_{2},[A_{1},z_{0}]]^{-1}$, so that $c(n)=e$ in $K_{X}$ if and only if $n\in X$. Since $c(n)$ belongs to $\mathcal{N}(L_{0},S)$, Lemma 6.2 and property (3) of Theorem 6.1 implies that $K_{X}$ has a presentation $\langle S_{n}:R_{n}\rangle$ as an extended homogeneous-linear-plus-conjugacy group, in which $c(n)$ is equal to some involutary generator in $S_{n}$. Since the presentation $\langle S:R\rangle$ is fixed, the size of $\langle S_{n}:R_{n}\rangle$ depends only on the number of ancilla generators and relations needed to set $c(n)$ equal to one of the involutary generators. Inspection of the argument from Lemma 6.2 reveals that we need to add $4m$ ancilla generators and relations to set $w(m)$ to an involutary generator. Thus $S_{n}$ and $R_{n}$ will have size $O(2^{n})$, and the function $n\mapsto(S_{n},R_{n})$ can be computed in $O(2^{n})$-time.

By Proposition 4.8, there is an $fa^{*}$-embedding from $\langle S_{n}:R_{n}\rangle$ to a homogeneous-linear-plus-conjugacy group $G_{n}$, in which $c(n)$ is mapped to some generator $x_{i}$. As in the proof of Proposition 5.1, let

[TABLE]

Then $\widehat{G}_{n}$ is a linear-plus-conjugacy group, and by Proposition 4.2, there is an $fa^{*}$-embedding of $\widehat{G}_{n}$ in a solution group $\Gamma_{n}=\Gamma(A^{(n)},b^{(n)})$. By Remarks 4.5 and 4.9, $A^{(n)}$ and $b^{(n)}$ can be constructed in time polynomial in $|S_{n}|$ and $|R_{n}|$, so $A^{(n)}$ and $b^{(n)}$ satisfy parts (a) and (b) of the proposition.

Suppose $c(n)$ is non-trivial. Since $K_{X}$ is solvable, it is hyperlinear, so $c(n)$ is non-trivial in approximate representations. By Lemma 5.2, $J_{\Gamma_{n}}$ will be non-trivial in approximate representations. If $X$ is recursive, then $K_{X}$ will be residually finite by property (5) of Theorem 6.1, and hence $J_{\Gamma_{n}}$ will be non-trivial in finite-dimensional representations by Lemma 6.4 (this uses the fact that $fa^{*}$-embeddings are also $fin$-embeddings). On the other hand, if $c(n)$ is trivial then $J_{\Gamma_{n}}$ will be trivial. Hence parts (c)-(e) of the proposition follow from property (4) of Theorem 6.1. ∎

Proof of Theorem 1.2.

Let $X\subseteq\mathbb{N}$ be a recursively enumerable but non-recursive set, and take the family $\{\mathcal{G}_{n}:n\in\mathbb{N}\}$ of games associated to the solution groups $\{\Gamma_{n}:n\in\mathbb{N}\}$ constructed in Proposition 6.3. By Theorem 3.3 and part (c) of Proposition 6.3, $\mathcal{G}_{n}$ will have a perfect strategy in $C_{qc}$ if and only if $n\in X$. By Proposition 3.4 and part (d) of Proposition 6.3, $\mathcal{G}_{n}$ will have a perfect strategy in $C_{qc}$ if and only if it has a perfect strategy in $C_{qa}$. Because the function $n\mapsto\mathcal{G}_{n}$ is computable by part (b) of Proposition 6.3, it is undecidable to determine if the games in this family have perfect srategies in $C_{qa}$. ∎

Proof of Theorem 1.3.

Given a computable function $f(n)$, let $X\subseteq\mathbb{N}$ be a recursive subset such that for any Turing machine accepting $X$, the running time over inputs $n\leq N$ is at least $f(N)$ when $N$ is sufficiently large.555Often when talking about the running time, we look at the maximum running time over inputs of size $\leq N$, rather than value $\leq N$. However, thinking of the running time in terms of the values of the inputs does not change the fact that such sets $X$ exist. Once again, we can take the family of games $\{\mathcal{G}_{n}:n\in\mathbb{N}\}$ associated to the solution groups $\{\Gamma_{n}:n\in\mathbb{N}\}$ from Proposition 6.3. Then part (a) of Theorem 1.3 follows from parts (a) and (b) of Proposition 6.3, while parts (b) and (c) of Theorem 1.3 follow from parts (c) and (e) of Proposition 6.3, as well as Theorems 3.2 and 3.3. ∎

Proof of Corollary 1.4.

Suppose there is an algorithm to decide if a linear system game has a perfect strategy in $C_{q}$. Let $g(n)$ be the running time of this algorithm on games coming from linear systems with at most $n$ rows and columns. Note that $g(n)$ is an increasing function. Let $f(n)$ be any computable function such that

[TABLE]

for all $n\geq 1$. Let $\mathcal{G}_{n}$ be the family of games associated to $f(n)$ as in Theorem 1.3. Then there is a constant $C$ such that $\mathcal{G}_{n}$ has size $\leq 2^{Cn}$ for all $n\geq 1$, and the function $n\mapsto\mathcal{G}_{n}$ is computable in time $2^{Cn}$. Plugging $\mathcal{G}_{n}$ into the algorithm to decide whether a linear system game has a perfect strategy in $C_{q}$, we get an algorithm for the language

[TABLE]

with running time at most $g(2^{CN})+2^{CN}$ on inputs $n\leq N$. But by part (b) of Theorem 1.3, when $N$ is sufficiently large the maximum running time on inputs $n\leq N$ for any algorithm for $X$ must be at least $f(N)$. Since $N^{2}$ will eventually be larger than $CN$, we get a contradiction. Thus there is no algorithm to decide if a linear system game has a perfect strategy in $C_{q}$. ∎