# Constant-sized robust self-tests for states and measurements of unbounded
dimension

Laura Mančinska  Department of Mathematical Sciences, Universitetsparken 5,
2100 København, Denmark<EMAIL_ADDRESS>,  Jitendra Prakash  Department
of Mathematical Sciences, Universitetsparken 5, 2100 København, Denmark
<EMAIL_ADDRESS>and  Christopher Schafhauser  Department of Mathematics,
University of Nebraska – Lincoln, USA<EMAIL_ADDRESS>
###### Abstract.

We consider correlations, $p_{n,x}$, arising from measuring a maximally
entangled state using $n$ measurements with two outcomes each, constructed
from $n$ projections that add up to $xI$. We show that the correlations
$p_{n,x}$ robustly self-test the underlying states and measurements. To
achieve this, we lift the group-theoretic Gowers-Hatami based approach for
proving robust self-tests to a more natural algebraic framework. A key step is
to obtain an analogue of the Gowers-Hatami theorem allowing to perturb an
“approximate” representation of the relevant algebra to an exact one.

For $n=4$, the correlations $p_{n,x}$ self-test the maximally entangled state
of every odd dimension as well as 2-outcome projective measurements of
arbitrarily high rank. The only other family of constant-sized self-tests for
strategies of unbounded dimension is due to Fu (QIP 2020) who presents such
self-tests for an infinite family of maximally entangled states with _even_
local dimension. Therefore, we are the first to exhibit a constant-sized self-
test for measurements of unbounded dimension as well as all maximally
entangled states with odd local dimension.

###### Key words and phrases:

Quantum information, operator algebras, self-testing

###### 2020 Mathematics Subject Classification:

Primary 81P40; Secondary 47C15

###### Contents

  1. 1 Introduction
    1. 1.1 Organisation of the paper
  2. 2 Preliminaries
  3. 3 Synchronous quantum correlations
  4. 4 Robust self-testing: definition and basic properties
  5. 5 Correlations from projections adding up to scalar times identity
  6. 6 Robust self-testing of projections adding up to scalar times identity
  7. 7 Implications
  8. A A short introduction to C*-algebras
  9. B A recipe to generate four projections adding up to scalars times identity

##  1\. Introduction

One of the key tasks in the development of reliable quantum technologies is
the certification of quantum devices. This ensures that the devices are
performing according to their specification. One of the ways such
certification may be carried out is by using _self-testing_ methods which
enable us to infer the quantum-mechanical description of a device merely from
classical observations (measurement statistics). We can then treat these
devices as black boxes as we need not trust the inner workings of the system,
a scenario which one refers to as _device-independence_. Beginning with
[MY04], in which the term was first coined, self-testing has found many
applications, such as device-independent quantum cryptography [MY98, MY04],
delegated quantum computation [CGJV19], entanglement detection [BvCA18a,
BvCA18b], investigating the structure of the quantum correlation set [CS17,
GKW+18], and quantum complexity theory [FJVY19, NV18, NW19]. Self-testing is
also one of the ingredients behind the recent breakthrough result establishing
that $\text{MIP}^{*}=\text{RE}$ [JNV+20]; which further implies a negative
answer to the celebrated Connes’ embedding problem [Con76] from the theory of
von Neumann algebras.

In this work, we prove that certain quantum strategies constructed from
projections which sum up to some particular scalar times identity can be
robustly self-tested from the quantum correlations that they induce.
Specifically, we begin with $d\times d$ projections
$\widetilde{P}_{1},\dots,\widetilde{P}_{n}$ ($n\geq 3$) such that
$\widetilde{P}_{1}+\dots+\widetilde{P}_{n}=xI_{d}$ for some specific scalar
$x\in\mathbb{R}$. The scalar $x$ and the projections
$\widetilde{P}_{1},\dots,\widetilde{P}_{n}$ have the property that whenever
$P_{1},\dots,P_{n}$ are any other projections such that
$P_{1}+\dots+P_{n}=xI$, then $P_{i}=I\otimes\widetilde{P}_{i}$ for all $1\leq
i\leq n$ in some basis. We then define a quantum strategy
$\widetilde{\mathscr{S}}$ and its induced quantum correlation $\widetilde{p}$
corresponding to these projections. We show that if one observes the quantum
correlation $\widetilde{p}$ induced from an arbitrary quantum strategy
$\mathscr{S}$, then $\mathscr{S}$ is related to $\widetilde{\mathscr{S}}$ via
a local isometry. For the robust case, we prove that if we observe a quantum
correlation $p$ such that $\|p-\widetilde{p}\|\leq\epsilon$, then its inducing
strategy $\mathscr{S}$ is “approximately” related to $\widetilde{\mathscr{S}}$
via a local isometry.

Often times self-testing results are proven using ad-hoc techniques, which
means that to obtain new results one essentially has to start from scratch.
One notable exception is an approach that uses perturbative representation
theory of groups to establish robust self-testing [Vid18, CS17, CMMN20]. The
basic idea is that in the ideal case one deduces algebraic relations which the
measurement operators must satisfy on the quantum state. One then associates a
suitable finite group such that one gets a representation of the group. This
transfers to the well-studied field of the representation theory of finite
groups and the problem reduces to identifying suitable irreducible
representations of the group. In the robust case, one gets approximate
versions of the algebraic relations which yield “approximate” representations
of the group (with respect to state-dependent distance). A key tool used in
the approximate case is the Gowers–Hatami theorem [GK17, Gow17, Vid18] which
relates an “approximate” representation of a group to a representation of that
group in some suitable sense. A caveat to this approach is that, in general,
it is not clear what group should be associated to the algebraic relations
identified and if this is at all possible. In fact, it would be more natural
to associate an _algebra_ rather than a group to these algebraic relations.
Moreover, there are known families of non-local games whose optimal strategies
do not have an apparent underlying group yet an underlying algebra can easily
be identified (e.g. binary constraint system games [CM14] beyond linear ones,
synchronous games [PSS+16, HMPS19], graph homomorphism [MR16] and isomorphism
games [AMR+19]).

One of our main contributions is that we showcase how the above general method
for self-testing from this _group_ framework can be lifted to an _algebraic_
framework. Instead of seeking an appropriate group to associate with the
algebraic relations, we simply work with the algebra generated by those
relations. To accomplish this, a major step is to obtain some sort of analogue
of Gowers–Hatami theorem for algebras. We show how this can be done for a
particular algebras, but the approach can be easily generalized to algebras
arising from other synchronous correlations whenever a version of Gowers-
Hatami Theorem holds; we expect this to be the case for algebras with well-
understood representation theory such as finite dimensional algebras. The
analogue we prove in this paper has a non-constructive
($\epsilon$,$\delta$)-dependence which is the most pressing question left open
by the current work.

In comparison to measurements, self-testing of quantum states is relatively
well understood. For example, we know that any (pure) bipartite entangled
state in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ can be self-tested from a
correlation with $3$ inputs and $d$ outputs [CGS17]. For applications, it
would be efficient to have small-sized correlations that robustly self-test
states with large dimensions. The only family of _constant-sized_ correlations
that self-test states of arbitrarily large dimension [Fu19] are constructed by
building upon Slofstra’s group embedding procedure [Slo19] for linear binary
constraint system games. Specifically [Fu19] shows that for each
$d\in\mathscr{D}$, where $\mathscr{D}$ is an infinite subset of the primes
given by

$\displaystyle\mathscr{D}=\\{d:d\text{ is an odd prime, and the smallest
generator of the group }\mathbb{Z}_{d}^{\times}\text{ is }2,3\text{ or }5\\},$

the maximally entangled state $\varphi_{4(d-1)}$ can be robustly self-tested
from correlations with over $100$ questions per party. We obtain the following
corollary which complements the result in [Fu19], but with correlations of
considerably smaller size:

######  Corollary.

For each odd dimension $d\geq 3$, the maximally entangled state $\varphi_{d}$
can be robustly self-tested by quantum correlations with four inputs and two
outputs.

When compared to [Fu19], the strength of our proof is the simplicity and the
small correlation size while the weakness is the non-constructive
$(\epsilon,\delta)$ dependence in the robustness proof.

In their work [Fu19], the author establishes certain algebraic relationships
between the measurements needed to induce the considered constant-sized
correlations but they do not present111Nevertheless, we believe that a
continuation of the arguments presented in [Fu19] should lead to constant-
sized self-tests of measurements. a self-test of measurements according to the
standard definition [ŠB20]. Therefore, our work is the first one to show that
measurements of arbitrarily large dimension can be self-tested from constant-
sized correlations. In addition, to the best of our knowledge, we are the
first to establish self-testing of measurements with operators of rank higher
than one:

######  Corollary.

Given any natural number $k$ there exist four projections of rank $k$ which
can be robustly self-tested by quantum correlations with four inputs and two
outputs.

Most of the known self-tests for _infinite families_ of measurements are for
tensor-products of Pauli matrices (for example, [NV17, Col17]) or Clifford
unitaries [CGJV19]. There are a few results which are different from these,
for instance, [SSKA19, CMMN20]. Our main theorem yields another example of an
infinite family of measurements that goes beyond a tensor-product of Paulis or
Clifford unitaries.

The following example illustrates the concept of robust self-testing and also
outlines the method that we use to obtain our results. Consider the _CHSH
game_ [CHSH69, CHTW04], where two non-communicating players, Alice and Bob,
want to win the game against a referee. The referee picks a pair of bits
$(v,w)\in\\{0,1\\}^{2}$ uniformly and sends $v$ to Alice and $w$ to Bob. Alice
replies with a bit $i$ and Bob replies with a bit $j$. Alice and Bob win the
game if $vw=i+j\pmod{2}$. Classical strategies can only help them win the game
with probability 3/4. On the other hand, quantum strategies provide a better
winning probability of $\omega_{q}(\mathrm{CHSH})\approx 0.85$. Indeed, the
following quantum strategy (which we term _canonical_)

(1.1)
$\displaystyle\widetilde{\mathscr{S}}=\mathopen{}\mathclose{{}\left(\varphi_{2}\in\mathbb{C}^{2}\otimes\mathbb{C}^{2},\mathopen{}\mathclose{{}\left\\{\widetilde{A}_{0}=Z,\widetilde{A}_{1}=X}\right\\},\mathopen{}\mathclose{{}\left\\{\widetilde{B}_{0}=\frac{Z+X}{\sqrt{2}},\widetilde{B}_{1}=\frac{Z-X}{\sqrt{2}}}\right\\}}\right),$

where $\varphi_{2}$ is the maximally entangled state and $X,Z$ are the Pauli
matrices, achieves the maximum quantum winning probability.

Suppose that Alice and Bob use an unknown quantum strategy (in terms of
observables) given by

(1.2)
$\displaystyle\mathscr{S}=\mathopen{}\mathclose{{}\left(\psi\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}},\mathopen{}\mathclose{{}\left\\{A_{0},A_{1}}\right\\},\mathopen{}\mathclose{{}\left\\{B_{0},B_{1}}\right\\}}\right)$

which also yields the maximum winning probability $\omega_{q}(\mathrm{CHSH})$.
What can then be said about the strategy $\mathscr{S}$? It has been shown that
in such a case, there exist isometries
$V_{A}\colon\mathbb{C}^{d_{A}}\to\mathbb{C}^{2}\otimes\mathcal{K}_{A}$ and
$V_{B}\colon\mathbb{C}^{d_{B}}\to\mathbb{C}^{2}\otimes\mathcal{K}_{B}$ for
some Hilbert spaces $\mathcal{K}_{A},\mathcal{K}_{B}$, and a quantum state
$\psi_{\mathrm{junk}}\in\mathcal{K}_{A}\otimes\mathcal{K}_{B}$ such that

(1.3) $\displaystyle(V_{A}\otimes V_{B})\psi$
$\displaystyle=\varphi_{2}\otimes\psi_{\mathrm{junk}},\text{ and}$ (1.4)
$\displaystyle(V_{A}\otimes V_{B})(A_{i}\otimes B_{j})\psi$
$\displaystyle=((\widetilde{A}_{i}\otimes\widetilde{B}_{j})\varphi_{2})\otimes\psi_{\mathrm{junk}},\text{
for all }i,j\in\\{0,1\\}.$

That is, all quantum strategies which yield the maximum quantum winning
probability for the CHSH game are related to the canonical one via local
isometries, which can be viewed as a rigidity property of
$\widetilde{\mathscr{S}}$.

This rigidity result can be extended to the approximate case (which we term
robust self-testing): say the winning probability $\omega_{q}(\mathscr{S})$
obtained by the strategy $\mathscr{S}$ in (1.2) above satisfies
$|\omega_{q}(\mathscr{S})-\omega_{q}(\mathrm{CHSH})|\leq\epsilon,$ for some
$\epsilon\geq 0$. Then, in this case, there exist isometries
$V_{A}\colon\mathbb{C}^{d_{A}}\to\mathbb{C}^{2}\otimes\mathcal{K}_{A}$ and
$V_{B}\colon\mathbb{C}^{d_{B}}\to\mathbb{C}^{2}\otimes\mathcal{K}_{B}$ for
some Hilbert spaces $\mathcal{K}_{A},\mathcal{K}_{B}$, and a quantum state
$\psi_{\mathrm{junk}}\in\mathcal{K}_{A}\otimes\mathcal{K}_{B}$ which satisfy
approximate versions of Equations (1.3) and (1.4):

(1.5) $\displaystyle\|(V_{A}\otimes
V_{B})\psi-\varphi_{2}\otimes\psi_{\mathrm{junk}}\|$ $\displaystyle\leq
f(\epsilon),\text{ and}$ (1.6) $\displaystyle\|(V_{A}\otimes
V_{B})(A_{i}\otimes
B_{j})\psi-((\widetilde{A}_{i}\otimes\widetilde{B}_{j})\varphi_{2})\otimes\psi_{\mathrm{junk}}\|$
$\displaystyle\leq f(\epsilon),\text{ for all }i,j\in\\{0,1\\},$

for some $f(\epsilon)\geq 0$ such that $f(\epsilon)\to 0$ as $\epsilon\to 0$.

There are many proofs for these results [MY04, MYS12, Kan17]. We describe the
approach taken in [Vid18] as this illustrates the connection between self-
testing and representation theory of groups. One starts by noting that the
canonical observables $\widetilde{A}_{0},\widetilde{A}_{1}$ for Alice in (1.1)
generate a finite group $\mathcal{P}$ (called the _Pauli group_ which is
isomorphic to the dihedral group of order eight). In the case when
$\omega_{q}(\mathscr{S})=\omega_{q}(\mathrm{CHSH})$, the function
$f:\mathcal{P}\to\mathscr{U}_{d_{A}}$ (where $\mathscr{U}_{d_{A}}$ is the
group of $d_{A}\times d_{A}$ unitaries) defined by $\widetilde{A}_{0}\mapsto
A_{0}$ and $\widetilde{A}_{1}\mapsto A_{1}$ defines a group representation
with respect to the state $\rho_{A}=\operatorname{Tr}_{B}(\psi\psi^{*})$. On
the other hand, in the approximate case
$|\omega_{q}(\mathscr{S})-\omega_{q}(\mathrm{CHSH})|\leq\epsilon$, one gets an
“approximate” group representation with respect to the state $\rho_{A}$. In
the second step, to get the existence of isometries one invokes the
Gowers–Hatami Theorem [GK17, Vid18].

######  Theorem 1.1 (Gowers–Hatami).

Let $G$ be a finite group, $d\in\mathbb{N}$, $\epsilon>0$,
$\sigma\in\mathbb{M}_{d}$ be a density matrix, and $f:G\to\mathscr{U}_{d}$ be
an $(\epsilon,\sigma)$-representation of $G$. Then, there exists $D\geq d$, an
isometry $V\colon\mathbb{C}^{d}\to\mathbb{C}^{D}$, and a representation
$g\colon G\to\mathscr{U}_{D}$ such that
$\mathopen{}\mathclose{{}\left\|f(a)-V^{*}g(a)V}\right\|_{\sigma}\leq|G|\sqrt{\epsilon}$
for all $a\in G$.

Finally, these two steps are stitched together to get the robust self-testing
statements for the CHSH game.

For the proof of our results, we follow the blueprint mentioned above for the
CHSH game with certain necessary modifications. As noted above, optimal
strategies for the CHSH game may be described in terms of unitary
representations of the Pauli group $\mathcal{P}$. Further, unitary
representations of $\mathcal{P}$ are in bijection with Hilbert space
representations of the group C∗-algebra
$\mathrm{C}^{*}(\mathcal{P})\cong\mathbb{C}^{4}\oplus\mathbb{M}_{2}$. In our
situation, the strategies we consider do not arise from unitary
representations of a group but will still arise from Hilbert space
representations of a certain C∗-algebra $\mathscr{P}_{n,x}$ (which is not
isomorphic to a group C∗-algebra). The C∗-algebra $\mathscr{P}_{n,x}$ is the
universal C∗-algebra generated by projections $r_{1},\dots,r_{n}$ satisfying
the algebraic relation $r_{1}+\dots+r_{n}=x1$. Studying the (approximate)
representation theory of this C∗-algebra will provide an analogue of Theorem
1.1 for $\mathscr{P}_{n,x}$, which leads to the desired self-testing results
in Section 6. Analogues of Theorem 1.1 are ubiquitous in C∗-algebra theory,
but usually approximations are considered in operator norm (c.f. [Lor97]) or
in the 2-norm arising from a tracial state (c.f. [HS18]). In our case, we will
need to work with the 2-norm arising from an “approximately tracial” state
which presents new difficulties (see Theorem 6.5).

###  1.1. Organisation of the paper

We fix some notations and conventions, and some introductory material in
Section 2. In Section 3, we define quantum strategies, their induced quantum
correlations, and the special subset of synchronous quantum correlations. We
prove some of its properties that we will be using later. We give a formal
definition of robust self-testing in Section 4. In Section 5, we collect some
of the basic properties of projections adding up to scalar times identity, and
we define (Definition 5.2) the families of quantum strategies obtained by such
projections and the correlations that they induce. Finally, in Section 6, we
show that the quantum correlations in Definition 5.2 robustly self-test their
canonical quantum strategies, and in Section 7 we discuss some of its
implications. We include two short appendices: Appendix A on some key
definitions from C∗-algebras that we’ll be using in the article, and Appendix
B to outline a method to construct four projections adding to a scalar.

##  2\. Preliminaries

Let $\mathbb{N}$ denote the set of natural numbers
$\mathopen{}\mathclose{{}\left\\{1,2,3,\dots}\right\\}$. Given
$n\in\mathbb{N}$, define
$\mathopen{}\mathclose{{}\left[n}\right]\coloneqq\mathopen{}\mathclose{{}\left\\{1,\dots,n}\right\\}$.
A fraction $\frac{b}{d}$ ($b,d\in\mathbb{N}$) is said to be in _lowest terms_
if $\gcd(b,d)=1$. The cardinality of a set $X$ is denoted $|X|$.

For each $d\in\mathbb{N}$ we let $\mathbb{R}^{d}$ and $\mathbb{C}^{d}$
represent the $d$-dimensional real and complex Euclidean spaces, respectively.
For $d_{A},d_{B}\in\mathbb{N}$ we let $\mathbb{M}_{d_{A},d_{B}}$ denote the
space of all $d_{A}\times d_{B}$ complex matrices. In particular, for
$d\in\mathbb{N}$, we let $\mathbb{M}_{d}\coloneqq\mathbb{M}_{d,d}$ denote the
algebra of all $d\times d$ complex matrices. The $d\times d$ identity matrix
is denoted by $I_{d}$. For $X\in\mathbb{M}_{d_{A},d_{B}}$ we let $X^{T}$
denote the transpose of $X$. For $X\in\mathbb{M}_{d}$ we let
$\operatorname{Tr}(X)$ denote its trace, whereas we shall use
$\mathrm{tr}_{d}(X)\coloneqq\frac{1}{d}\operatorname{Tr}(X)$ to denote its
_normalized_ trace.

Let $\mathcal{H}$ be a (complex) Hilbert space. We assume that the inner
product on $\mathcal{H}$ is linear in the first argument and conjugate-linear
in the second. We denote the algebra of all bounded operators on $\mathcal{H}$
by $\mathfrak{B}(\mathcal{H})$. The identity operator on $\mathcal{H}$ will be
denoted by $I_{\mathcal{H}}$. Given vectors $\xi,\eta\in\mathcal{H}$ and an
$\epsilon\geq 0$, we write $\xi\approx_{\epsilon}\eta$ if
$\|\xi-\eta\|\leq\epsilon$. We will usually identify a $d$-dimensional Hilbert
space $\mathcal{H}$ with $\mathbb{C}^{d}$. The algebra of all operators on
$\mathbb{C}^{d}$ is identified with $\mathbb{M}_{d}$ with respect to the
standard orthonormal basis $\\{e_{i}\\}_{i=1}^{d}$ of $\mathbb{C}^{d}$. Given
$X\in\mathbb{M}_{d}$, a subspace $\mathcal{K}\subseteq\mathbb{C}^{d}$ is
called an _invariant subspace_ of $X$ if $X(\mathcal{K})\subseteq\mathcal{K}$.

A matrix $X\in\mathbb{M}_{d}$ is called _positive_ if
$\mathopen{}\mathclose{{}\left\langle{X\xi},{\xi}}\right\rangle\geq 0$ for all
$\xi\in\mathbb{C}^{d}$. For two Hermitian matrices $X,Y\in\mathbb{M}_{d}$ we
say that $X\leq Y$ if $Y-X$ is positive. A matrix $P\in\mathbb{M}_{d}$ is
called a _projection_ if $P=P^{*}=P^{2}$, where $P^{*}$ is the adjoint of $P$.
A finite set of positive matrices $X_{1},\dots,X_{k}\in\mathbb{M}_{d}$ is
called a _positive operator-valued measure_ (POVM) if
$X_{1}+\dots+X_{k}=I_{d}$. In particular, a finite set of projections
$P_{1},\dots,P_{k}\in\mathbb{M}_{d}$ is called a _projection-valued measure_
(PVM) if $P_{1}+\dots+P_{k}=I_{d}$.

We recall the Schmidt decomposition [Per93, Section 5.3].

######  Lemma 2.1 (Schmidt decomposition).

Let $\psi\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$. There exist
orthonormal sets $\\{\xi_{l}\\}_{l=1}^{r}$ in $\mathbb{C}^{d_{A}}$,
$\\{\eta_{l}\\}_{l=1}^{r}$ in $\mathbb{C}^{d_{B}}$, and strictly positive real
numbers $\\{\alpha_{l}\\}_{l=1}^{r}$ such that
$\psi=\sum_{l=1}^{r}\alpha_{l}\xi_{l}\otimes\eta_{l}$. The numbers
$\\{\alpha_{l}\\}_{l=1}^{r}$, called the _Schmidt coefficients_ of $\psi$, are
unique up to permutation.

Let $\psi=\sum_{l=1}^{r}\alpha_{l}\xi_{l}\otimes\eta_{l}$ be a Schmidt
decomposition of a vector
$\psi\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$. Any orthonormal basis of
$\mathbb{C}^{d_{A}}$ (resp., $\mathbb{C}^{d_{B}}$) containing
$\\{\xi_{l}\\}_{l=1}^{r}$ (resp., $\\{\eta_{l}\\}_{l=1}^{r}$) is called a
_Schmidt basis_ for $\mathbb{C}^{d_{A}}$ (resp., $\mathbb{C}^{d_{B}}$). The
number $r$ is called the _Schmidt rank_ of $\psi$, and we say that $\psi$ is
of _full Schmidt rank_ if $r=d_{A}=d_{B}$. Finally, define the subspaces
$\mathrm{supp}_{A}(\psi)\coloneqq\mathrm{span}\\{\xi_{l}\\}_{l=1}^{r}$ and
$\mathrm{supp}_{B}(\psi)\coloneqq\mathrm{span}\\{\eta_{l}\\}_{l=1}^{r}$.

A matrix $\rho\in\mathbb{M}_{d}$ is called a _density matrix_ if $\rho\geq 0$
and $\operatorname{Tr}(\rho)=1$. If
$\rho\in\mathbb{M}_{d_{A}}\otimes\mathbb{M}_{d_{B}}$ is a density matrix, then
the density matrices

$\displaystyle\rho_{A}\coloneqq\operatorname{Tr}_{B}(\rho)\coloneqq(\mathrm{id}_{A}\otimes\operatorname{Tr})(\rho),\quad\text{
and
}\quad\rho_{B}\coloneqq\operatorname{Tr}_{A}(\rho)\coloneqq(\operatorname{Tr}\otimes\mathrm{id}_{B})(\rho),$

where $\mathrm{id}_{A}$ and $\mathrm{id}_{B}$ are identity maps on
$\mathbb{M}_{d_{A}}$ and $\mathbb{M}_{d_{B}}$, respectively, are called its
_reduced density matrices_.

A density matrix $\rho\in\mathbb{M}_{d}$ induces a semi-inner product on
$\mathbb{M}_{d}$ by

$\displaystyle\mathopen{}\mathclose{{}\left\langle{X},{Y}}\right\rangle_{\rho}\coloneqq\operatorname{Tr}_{\rho}(Y^{*}X)\coloneqq\operatorname{Tr}(Y^{*}X\rho),\qquad\text{
for all }X,Y\in\mathbb{M}_{d}.$

This semi-inner product induces a semi-norm
$\|X\|_{\rho}\coloneqq\mathopen{}\mathclose{{}\left\langle{X},{X}}\right\rangle_{\rho}^{1/2}$
for all $X\in\mathbb{M}_{d}$. This semi-inner product (resp., semi-norm) is an
inner product (resp., a norm) if and only if $\rho$ is invertible.

Usually we will be in the following situation. Let
$\psi\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$ be a unit vector. Let
$\rho=\psi\psi^{*}$ be the corresponding density matrix and let
$\rho_{A},\rho_{B}$ be its reduced density matrices. Then,
$\mathopen{}\mathclose{{}\left\langle{.},{.}}\right\rangle_{\rho_{A}}$ and
$\mathopen{}\mathclose{{}\left\langle{.},{.}}\right\rangle_{\rho_{B}}$ are
semi-inner products on $\mathbb{M}_{d_{A}}$ and $\mathbb{M}_{d_{B}}$,
respectively, and they are inner products if and only if $\psi$ is of full
Schmidt rank. Observe that for any $X\in\mathbb{M}_{d_{A}}$ and
$Y\in\mathbb{M}_{d_{B}}$,

(2.1) $\displaystyle\mathopen{}\mathclose{{}\left\|X}\right\|_{\rho_{A}}^{2}$
$\displaystyle=\operatorname{Tr}(X^{*}X\rho_{A})=\mathopen{}\mathclose{{}\left\langle{\mathopen{}\mathclose{{}\left(X^{*}X\otimes
I_{d_{B}}}\right)\psi},{\psi}}\right\rangle=\|(X\otimes
I_{d_{B}})\psi\|^{2},\text{ and}$ (2.2)
$\displaystyle\mathopen{}\mathclose{{}\left\|Y}\right\|_{\rho_{B}}^{2}$
$\displaystyle=\operatorname{Tr}(Y^{*}Y\rho_{B})=\mathopen{}\mathclose{{}\left\langle{\mathopen{}\mathclose{{}\left(I_{d_{A}}\otimes
Y^{*}Y}\right)\psi},{\psi}}\right\rangle=\|(I_{d_{A}}\otimes Y)\psi\|^{2}.$

The _operator-vector correspondence_ is the linear map
$\mathrm{vec}\colon\mathbb{M}_{d_{A},d_{B}}\rightarrow\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$
given on the canonical matrix units
$\\{E_{i,j}:(i,j)\in[d_{A}]\times[d_{B}]\\}$ for $\mathbb{M}_{d_{A},d_{B}}$ by
$\mathrm{vec}(E_{i,j})\coloneqq e_{i}\otimes e_{j}$ for all
$(i,j)\in[d_{A}]\times[d_{B}]$. In particular, for $\xi\in\mathbb{C}^{d_{A}}$
and $\eta\in\mathbb{C}^{d_{B}}$, we have
$\mathrm{vec}(\xi\eta^{*})=\xi\otimes\overline{\eta}$. The map $\mathrm{vec}$
is an isometry:
$\mathopen{}\mathclose{{}\left\langle{\mathrm{vec}(X)},{\mathrm{vec}(Y)}}\right\rangle=\mathopen{}\mathclose{{}\left\langle{X},{Y}}\right\rangle_{2}\coloneqq\operatorname{Tr}(Y^{*}X)$,
for $X,Y\in\mathbb{M}_{d_{A},d_{B}}$. We shall use the following identity: for
$X\in\mathbb{M}_{d_{A}^{\prime},d_{A}},Y\in\mathbb{M}_{d_{B}^{\prime},d_{B}},D\in\mathbb{M}_{d_{A},d_{B}}$,

(2.3) $\displaystyle(X\otimes Y)\mathrm{vec}(D)$
$\displaystyle=\mathrm{vec}(XDY^{T}).$

For each $d\in\mathbb{N}$, the unit vector
$\varphi_{d}=\frac{1}{\sqrt{d}}\sum_{i=1}^{d}e_{i}\otimes e_{i}$ is called the
(_canonical_) _maximally entangled state_ on
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$. Then the diagonal matrix
$D=\frac{1}{\sqrt{d}}I_{d}$ satisfies $\mathrm{vec}(D)=\varphi_{d}$. For
$X,Y\in\mathbb{M}_{d}$, using Identity (2.3) and the isometry property, we
have

(2.4) $\displaystyle(X\otimes Y)\varphi_{d}$
$\displaystyle=\frac{1}{\sqrt{d}}\mathrm{vec}(XY^{T}),\text{ and}$ (2.5)
$\displaystyle\mathopen{}\mathclose{{}\left\langle{(X\otimes
Y)\varphi_{d}},{\varphi_{d}}}\right\rangle$
$\displaystyle=\frac{1}{d}\operatorname{Tr}(XY^{T}).$

##  3\. Synchronous quantum correlations

Let us briefly describe the framework of non-local games [CHTW04] in which the
discussion of correlations and strategies takes a natural form. A _non-local
game_ $\mathscr{G}$ is played by two spatially separated players, Alice and
Bob, against a referee. There are (non-empty) finite sets $I_{A}$ and $I_{B}$,
called the _question sets_ , and $O_{A}$ and $O_{B}$, called the _answer sets_
, and a _rule function_

$\lambda\colon I_{A}\times I_{B}\times O_{A}\times O_{B}\to\\{0,1\\},$

all of which are known to the players. However, once the game begins, the
players cannot communicate with each other. The game begins with the referee
randomly picking a pair of questions $(v,w)\in I_{A}\times I_{B}$ according to
some probability distribution222The probability distribution on $I_{A}\times
I_{B}$ should be given as part of the game, but as we are only interested in
winning (or perfect) strategies, the probability distribution will not be
relevant for us. For our purposes, it suffices to view $I_{A}\times I_{B}$
with the uniform distribution, so each pair of questions is chosen with equal
probability. and sending $v$ to Alice and $w$ to Bob. Alice returns an answer
$i\in O_{A}$ to the referee, and so does Bob with $j\in O_{B}$. Alice and Bob
win the round of the game if $\lambda(v,w,i,j)=1$, or otherwise they lose. The
goal of Alice and Bob is to maximize their winning probability, and for this,
they can take the help of either classical resources or quantum resources. It
is well-known that for many games, quantum resources exhibit larger winning
probabilities as compared to classical resources [CHTW04]. A classic example
is the _CHSH game_ as described in the Introduction; here
$I_{A}=I_{B}=O_{A}=O_{B}=\\{0,1\\}$, and $\lambda(v,w,i,j)=1$ if and only if
$vw=i+j\pmod{2}$.

As the game is played for many rounds, an outside observer will see a
probability distribution which encodes the strategies employed by the players.
Let $p(i,j|v,w)$ be the probability of Alice responding with $i$ and Bob
responding with $j$ given that they received $v$ and $w$, respectively.
_Classical strategies_ usually employ either a _deterministic_ strategy (given
by a function $I_{A}\times I_{B}\rightarrow O_{A}\times O_{B}$) or by using
_local/shared randomness_. On the other hand, _quantum strategies_ are given
by a _quantum state_ (usually _entangled_) shared by Alice and Bob and sets of
_quantum measurements_. Specifically, we have the following definition.

######  Definition 3.1 (Quantum strategies and quantum correlations).

Given (non-empty) finite sets $I_{A},I_{B},O_{A},O_{B}$ with
$|I_{A}|=n_{A},|I_{B}|=n_{B},|O_{A}|=k_{A}$ and $|O_{B}|=k_{B}$, a _quantum
strategy_ $\mathscr{S}$ is given by a triple:

(3.1)
$\displaystyle\mathscr{S}=\mathopen{}\mathclose{{}\left(\psi\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}},\mathopen{}\mathclose{{}\left\\{E_{v,i}:i\in
O_{A},v\in I_{A}}\right\\},\mathopen{}\mathclose{{}\left\\{F_{w,j}:j\in
O_{B},w\in I_{B}}\right\\}}\right),$

where $\psi$ is a unit vector (also referred to as a _quantum state_), for
each $v\in I_{A}$ the set $\\{E_{v,i}:i\in O_{A}\\}$ is a POVM on
$\mathbb{C}^{d_{A}}$, and for each $w\in I_{B}$, the set $\\{F_{w,j}:j\in
O_{B}\\}$ is a POVM on $\mathbb{C}^{d_{B}}$.

Each quantum strategy $\mathscr{S}$ induces a collection of probability
distributions called a _quantum correlation_ as follows: for each $(v,w)\in
I_{A}\times I_{B}$, define

(3.2) $\displaystyle
p(i,j|v,w)=\mathopen{}\mathclose{{}\left\langle{\mathopen{}\mathclose{{}\left(E_{v,i}\otimes
F_{w,j}}\right)\psi},{\psi}}\right\rangle,\quad\text{ for all }(i,j)\in
O_{A}\times O_{B}.$

Fixing $n_{A},n_{B},k_{A},k_{B}\in\mathbb{N}$, the set of quantum correlations
arising from all choices of quantum strategies is denoted by
$C_{q}(n_{A},n_{B},k_{A},k_{B})$. In particular, we define
$C_{q}(n,k)\coloneqq C_{q}(n,n,k,k)$ for all $n,k\in\mathbb{N}$.

If $p=(p(i,j|v,w))\in C_{q}(n_{A},n_{B},k_{A},k_{B})$ is a quantum
correlation, then the POVM assumptions imply that $p(i,j|v,w)\geq 0$ for all
$i,j,v$ and $w$, and

$\displaystyle\sum_{(i,j)\in O_{A}\times O_{B}}p(i,j|v,w)=1,\quad\text{for all
}(v,w)\in I_{A}\times I_{B}.$

Moreover, quantum correlations satisfy _non-signaling conditions_ ; that is,
they have well-defined _marginals_ $p_{A}(i|v)$ and $p_{B}(j|w)$ in the sense
that

(3.3) $\displaystyle p_{A}(i|v)$ $\displaystyle\coloneqq\sum_{j^{\prime}\in
O_{B}}p(i,j^{\prime}|v,w)=\sum_{j^{\prime}\in
O_{B}}p(i,j^{\prime}|v,w^{\prime}),\text{ and}$ (3.4) $\displaystyle
p_{B}(j|w)$ $\displaystyle\coloneqq\sum_{i^{\prime}\in
O_{A}}p(i^{\prime},j|v,w)=\sum_{i^{\prime}\in
O_{A}}p(i^{\prime},j|v^{\prime},w),$

for all $v,v^{\prime}\in I_{A},w,w^{\prime}\in I_{B},i\in O_{A},j\in O_{B}$.
Indeed, using the definition of a POVM, Equations (3.3) and (3.4) yield

(3.5) $\displaystyle
p_{A}(i|v)=\mathopen{}\mathclose{{}\left\langle{\mathopen{}\mathclose{{}\left(E_{v,i}\otimes
I_{d_{B}}}\right)\psi},{\psi}}\right\rangle\quad\text{and}\quad
p_{B}(j|w)=\mathopen{}\mathclose{{}\left\langle{\mathopen{}\mathclose{{}\left(I_{d_{A}}\otimes
F_{w,j}}\right)\psi},{\psi}}\right\rangle.$

Thus $p_{A}(i|v)$ is independent of $w$, and $p_{B}(j|w)$ is independent of
$v$. (In the context of non-local games, the non-signaling condition is
imposed due to the assumption that the players may not communicate.)

######  Remark 3.2.

The set $C_{q}(n_{A},n_{B},k_{A},k_{B})$ of quantum correlations is convex for
all natural numbers $n_{A},n_{B},k_{A}$ and $k_{B}$. However, in general, it
is not closed. This was first shown by Slofstra who proved that
$C_{q}(184,235,8,2)$ is not closed [Slo19]. Thereafter, non-closure of quantum
correlations sets with smaller values of $n_{A},n_{B},k_{A},k_{B}$ have been
proved [DPP19, MR20, Col20]—in particular, $C_{q}(5,2)$ is not closed as was
first shown in [DPP19].

######  Remark 3.3.

For notational simplicity, we shall simply take

$\displaystyle I_{A}=[n_{A}],I_{B}=[n_{B}],O_{A}=[k_{A}],\text{ and
}O_{B}=[k_{B}]$

in the definition of a quantum correlation.

Introduced in [PSS+16], the subset of synchronous correlations arises
naturally in the context of many non-local games, for example, the _graph
coloring games_ [CHTW04] and _graph homomorphism games_ [MR14]. In this
article, we shall be self-testing some quantum strategies yielding synchronous
quantum correlations, and therefore here is a formal definition.

######  Definition 3.4.

A correlation $(p(i,j|v,w))\in C_{q}(n,k)$ is called _synchronous_ if
$k\coloneqq k_{A}=k_{B}$, $n\coloneqq n_{A}=n_{B}$, and $p(i,j|v,v)=0$ for all
$i,j\in[k]$ and $v\in[n]$ with $i\neq j$. We let $C_{q}^{s}(n,k)$ denote the
subset of all synchronous correlations of $C_{q}(n,k)$, and call it the set of
_synchronous quantum correlations_.

A quantum strategy which induces a synchronous quantum correlation has several
nice properties. For instance, when restricted to $\mathrm{supp}_{A}(\psi)$
(resp., $\mathrm{supp}_{B}(\psi)$), the measurement operators for Alice
(resp., Bob) are projections, Bob’s measurement operators can be expressed in
terms of Alice’s, and $X\mapsto\mathopen{}\mathclose{{}\left\langle{(X\otimes
I)\psi},{\psi}}\right\rangle$ defines a _tracial state_ (see Appendix A) on
the C∗-algebra generated by Alice’s measurements [PSS+16, Theorem 5.5]. The
next two lemmas show that if a correlation is “approximately” synchronous,
then these properties “approximately” hold for its inducing strategy. These
lemmas are not new and appear in various places in the literature in different
guises, but for the sake of completeness, we provide proofs.

Since correlations are collections of probability distributions, it makes
sense to work with the $1$-norm. That is, given two correlations
$p_{1},p_{2}\in C_{q}(n_{A},n_{B},k_{A},k_{B})$, we define

(3.6)
$\displaystyle\mathopen{}\mathclose{{}\left\|p_{1}-p_{2}}\right\|_{1}\coloneqq\sum_{i,j,v,w}\mathopen{}\mathclose{{}\left|p_{1}(i,j|v,w)-p_{2}(i,j|v,w)}\right|.$

######  Lemma 3.5.

Let $p\in C_{q}(n,k)$ be induced by a strategy

$\displaystyle\mathscr{S}=\mathopen{}\mathclose{{}\left(\psi\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}},\mathopen{}\mathclose{{}\left\\{E_{v,i}:i\in[k],v\in[n]}\right\\},\mathopen{}\mathclose{{}\left\\{F_{w,j}:j\in[k],w\in[n]}\right\\}}\right).$

Let $\rho_{A},\rho_{B}$ be the reduced density matrices of the density matrix
$\rho=\psi\psi^{*}$. If $\|p-\widetilde{p}\|_{1}\leq\delta$ for some
synchronous $\widetilde{p}\in C_{q}^{s}(n,k)$ and for some $\delta\geq 0$,
then for all $i\in[k]$ and $v\in[n]$,

  1. (a)

$\|(E_{v,i}\otimes I_{d_{B}})\psi-(I_{d_{A}}\otimes
F_{v,i})\psi\|\leq\sqrt{\delta}$,

  2. (b)

$\|(E_{v,i}\otimes I_{d_{B}})\psi-(E_{v,i}\otimes
F_{v,i})\psi\|\leq\sqrt{\delta}$,

  3. (c)

$\|(I_{d_{A}}\otimes F_{v,i})\psi-(E_{v,i}\otimes
F_{v,i})\psi\|\leq\sqrt{\delta}$,

  4. (d)

$\|E_{v,i}-E_{v,i}^{2}\|_{\rho_{A}}=\|(E_{v,i}\otimes
I_{d_{B}})\psi-(E_{v,i}^{2}\otimes I_{d_{B}})\psi\|\leq 2\sqrt{\delta}$,

  5. (e)

$\|F_{v,i}-F_{v,i}^{2}\|_{\rho_{B}}=\|(I_{d_{A}}\otimes
F_{v,i})\psi-(I_{d_{A}}\otimes F_{v,i}^{2})\psi\|\leq 2\sqrt{\delta}$.

###### Proof.

Using the fact that $0\leq E_{v,i}^{2}\leq E_{v,i}$ and $0\leq F_{v,i}^{2}\leq
F_{v,i}$ for all $v\in[n]$,

$\displaystyle\|(E_{v,i}\otimes I_{d_{B}})\psi-(I_{d_{A}}\otimes
F_{v,i})\psi\|^{2}$
$\displaystyle=\mathopen{}\mathclose{{}\left\langle{(E_{v,i}^{2}\otimes
I_{d_{B}})\psi},{\psi}}\right\rangle+\mathopen{}\mathclose{{}\left\langle{(I_{d_{A}}\otimes
F_{v,i}^{2})\psi},{\psi}}\right\rangle-2\mathopen{}\mathclose{{}\left\langle{(E_{v,i}\otimes
F_{v,i})\psi},{\psi}}\right\rangle$
$\displaystyle\leq\mathopen{}\mathclose{{}\left\langle{(E_{v,i}\otimes
I_{d_{B}})\psi},{\psi}}\right\rangle+\mathopen{}\mathclose{{}\left\langle{(I_{d_{A}}\otimes
F_{v,i})\psi},{\psi}}\right\rangle-2\mathopen{}\mathclose{{}\left\langle{(E_{v,i}\otimes
F_{v,i})\psi},{\psi}}\right\rangle$
$\displaystyle=p_{A}(i|v)+p_{B}(i|v)-2p(i,i|v,v)$
$\displaystyle=\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{k}p(i,j|v,v)+p(j,i|v,v)$
$\displaystyle=\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{k}|p(i,j|v,v)-\widetilde{p}(i,j|v,v)|+|p(j,i|v,v)-\widetilde{p}(j,i|v,v)|$
$\displaystyle\leq\|p-\widetilde{p}\|_{1}\leq\delta,$

where the penultimate line is due to the fact that $\widetilde{p}$ is
synchronous. Thus (a) follows.

Similarly, to see (b), note that

$\displaystyle\|(E_{v,i}\otimes I_{d_{B}})\psi-(E_{v,i}\otimes
F_{v})\psi\|^{2}$
$\displaystyle=\|(E_{v,i}\otimes(I_{d_{B}}-F_{v,i}))\psi\|^{2}$
$\displaystyle=\mathopen{}\mathclose{{}\left\langle{(E_{v,i}^{2}\otimes(I_{d_{B}}-F_{v,i})^{2})\psi},{\psi}}\right\rangle$
$\displaystyle\leq\mathopen{}\mathclose{{}\left\langle{(E_{v,i}\otimes(I_{d_{B}}-F_{v,i}))\psi},{\psi}}\right\rangle$
$\displaystyle=\mathopen{}\mathclose{{}\left\langle{(E_{v,i}\otimes
I_{d_{B}})\psi},{\psi}}\right\rangle-\mathopen{}\mathclose{{}\left\langle{(E_{v,i}\otimes
F_{v,i})\psi},{\psi}}\right\rangle$ $\displaystyle=p_{A}(i|v)-p(i,i|v,v)$
$\displaystyle=\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{k}p(i,j|v,v)=\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{k}|p(i,j|v,v)-\widetilde{p}(i,j|v,v)|$
$\displaystyle\leq\|p-\widetilde{p}\|_{1}\leq\delta.$

Part (c) is similar.

The equalities in parts (d) and (e) follow from Equations (2.1) and (2.2).
Since $\|E_{v,i}\|\leq 1$ and $\|F_{v,i}\|\leq 1$, we deduce from (a) that for
all $v\in[n]$,

$\displaystyle\|(E_{v,i}^{2}\otimes I_{d_{B}})\psi-(E_{v,i}\otimes
F_{v,i})\psi\|$ $\displaystyle\leq\sqrt{\delta}\text{ and}$
$\displaystyle\|(I_{d_{A}}\otimes F_{v,i}^{2})\psi-(E_{v,i}\otimes
F_{v,i})\psi\|$ $\displaystyle\leq\sqrt{\delta}.$

Using these two inequalities with (b) and (c) and applying triangle
inequality, we get the inequalities in part (d) and (e). ∎

We record the special case of $\delta=0$ in Lemma 3.5.

######  Corollary 3.6.

Let $p\in C_{q}^{s}(n,k)$ be a synchronous correlation induced by a strategy

$\displaystyle\mathscr{S}=\mathopen{}\mathclose{{}\left(\psi\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}},\mathopen{}\mathclose{{}\left\\{E_{v,i}:i\in[k],v\in[n]}\right\\},\mathopen{}\mathclose{{}\left\\{F_{w,j}:j\in[k],w\in[n]}\right\\}}\right).$

Then, for all $i\in[k]$ and $v\in[n]$,

  1. (a)

$(E_{v,i}\otimes I_{d_{B}})\psi=(I_{d_{A}}\otimes F_{v,i})\psi$,

  2. (b)

$(E_{v,i}\otimes I_{d_{B}})\psi=(E_{v,i}^{2}\otimes I_{d_{B}})\psi$,

  3. (c)

$(I_{d_{A}}\otimes F_{v,i})\psi=(I_{d_{A}}\otimes F_{v,i}^{2})\psi$.

Moreover, if $\psi$ is of full Schmidt rank $r=d_{A}=d_{B}$, then for all
$v\in[n]$, the matrices $E_{v,i},F_{v,i}$ are projections, and
$F_{v,i}=D^{T}E_{v,i}^{T}(D^{-1})^{T}$, where $D\in\mathbb{M}_{r}$ is such
that $\mathrm{vec}(D)=\psi$.

###### Proof.

Statements (a), (b), and (c) follow immediately from the corresponding
statements in Lemma 3.5. When $\psi$ has full Schmidt rank, the seminorms
$\|\cdot\|_{\rho_{A}}$ and $\|\cdot\|_{\rho_{B}}$ are norms, and so $E_{v,i}$
and $F_{v,i}$ are projections by statements (d) and (e) in Lemma 3.5. Finally,
(a) and Equation (2.3) imply $E_{v,i}D=DF_{v,i}^{T}$. Since $\psi$ has full
Schmidt rank, $D$ is invertible, and the last equation follows. ∎

######  Lemma 3.7 (Approximately tracial state).

Let $p\in C_{q}(n,k)$ be induced by a strategy

$\displaystyle\mathscr{S}=\mathopen{}\mathclose{{}\left(\psi\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}},\mathopen{}\mathclose{{}\left\\{E_{v,i}:i\in[k],v\in[n]}\right\\},\mathopen{}\mathclose{{}\left\\{F_{w,j}:j\in[k],w\in[n]}\right\\}}\right).$

Let $\rho_{A},\rho_{B}$ be the reduced density matrices of the density matrix
$\rho=\psi\psi^{*}$. Let $\mathcal{A}\subseteq\mathbb{M}_{d_{A}}$ (resp.,
$\mathcal{B}\subseteq\mathbb{M}_{d_{B}}$) be the $C^{*}$-algebra generated by
$\\{E_{v,i}\\}_{v,i}$ (resp., $\\{F_{w,j}\\}_{w,j}$). If
$\|p-\widetilde{p}\|_{1}\leq\delta$ for some synchronous $\widetilde{p}\in
C_{q}^{s}(n,k)$ and for some $\delta\geq 0$, then the states
$\operatorname{Tr}_{\rho_{A}}\colon X\mapsto\operatorname{Tr}(X\rho_{A})$ on
$\mathcal{A}$ and $\operatorname{Tr}_{\rho_{B}}\colon
Y\mapsto\operatorname{Tr}(Y\rho_{B})$ on $\mathcal{B}$ satisfy

(3.7) $\displaystyle|\operatorname{Tr}_{\rho_{A}}(WX-XW)|$ $\displaystyle\leq
2\ell\sqrt{\delta}\text{ and}$ (3.8)
$\displaystyle|\operatorname{Tr}_{\rho_{B}}(UY-YU)|$ $\displaystyle\leq
2\ell\sqrt{\delta},$

respectively, where $W$ (resp., $U$) is any word in $\\{E_{v,i}\\}_{v,i}$
(resp., $\\{F_{w,j}\\}_{w,j}$) of length $\ell$, $X\in\mathcal{A}$ with
$\|X\|\leq 1$, and $Y\in\mathcal{B}$ with $\|Y\|\leq 1$.

###### Proof.

Let $W=E_{v,i}W^{\prime}$ where $W^{\prime}$ is of length $\ell-1$, $i\in[k]$
and $v\in[n]$. Then making use of Lemma 3.5(a) a couple of times and that
$\|E_{v,i}\|\leq 1$ and $\|X\|\leq 1$, we have

$\displaystyle\operatorname{Tr}_{\rho_{A}}(WX)$
$\displaystyle=\mathopen{}\mathclose{{}\left\langle{(E_{v,i}W^{\prime}X\otimes
I_{d_{B}})\psi},{\psi}}\right\rangle$
$\displaystyle=\mathopen{}\mathclose{{}\left\langle{(W^{\prime}X\otimes
I_{d_{B}})\psi},{(E_{v,i}\otimes I_{d_{B}})\psi}}\right\rangle$
$\displaystyle\approx_{\sqrt{\delta}}\mathopen{}\mathclose{{}\left\langle{(W^{\prime}X\otimes
I_{d_{B}})\psi},{(I_{d_{A}}\otimes F_{v,i})\psi}}\right\rangle$
$\displaystyle=\mathopen{}\mathclose{{}\left\langle{(W^{\prime}X\otimes
F_{v,i})\psi},{\psi}}\right\rangle$
$\displaystyle=\mathopen{}\mathclose{{}\left\langle{(W^{\prime}X\otimes
I_{d_{B}})(I_{d_{A}}\otimes F_{v,i})\psi},{\psi}}\right\rangle$
$\displaystyle\approx_{\sqrt{\delta}}\mathopen{}\mathclose{{}\left\langle{(W^{\prime}X\otimes
I_{d_{B}})(E_{v,i}\otimes I_{d_{B}})\psi},{\psi}}\right\rangle$
$\displaystyle=\mathopen{}\mathclose{{}\left\langle{(W^{\prime}XE_{v,i}\otimes
I_{d_{B}})\psi},{\psi}}\right\rangle.$

Thus, continuing in this fashion for the whole length of $W^{\prime}$, we will
obtain Expression (3.7). Similarly, Expression (3.8) holds. ∎

######  Remark 3.8.

In Lemma 3.5 and Lemma 3.7, it is enough to assume that $p\in C_{q}(n,k)$
satisfies $\mathopen{}\mathclose{{}\left|p(i,j|v,v)}\right|\leq\delta$ for all
$i\neq j$ and for all $v\in[n]$.

##  4\. Robust self-testing: definition and basic properties

In many cases, given a quantum correlation $\widetilde{p}$, there is a
“unique” quantum strategy $\widetilde{\mathscr{S}}$ which induces the
correlation $\widetilde{p}$, in the sense that, any other strategy
$\mathscr{S}$ which also induces the correlation $\widetilde{p}$ is related to
$\widetilde{\mathscr{S}}$ by a _local isometry_. The way these two strategies
$\mathscr{S}$ and $\widetilde{\mathscr{S}}$ are related is what we term as a
_local dilation_. For our purpose, we work with a more general definition of a
_local $\epsilon$-dilation_ which relates strategy $\mathscr{S}$ to
$\widetilde{\mathscr{S}}$ in the case when the correlation $p$ induced by
$\mathscr{S}$ is $\epsilon$-close to $\widetilde{p}$ (in the 1-norm as in
Equation 3.6). This general definition is useful in practical scenarios where
it is inevitable that errors creep in while conducting experiments.

######  Definition 4.1 (Local $\epsilon$-dilation).

Given $\epsilon\geq 0$ and two finite-dimensional strategies

$\displaystyle\widetilde{\mathscr{S}}$
$\displaystyle=\big{(}\widetilde{\psi}\in\widetilde{\mathcal{H}}_{A}\otimes\widetilde{\mathcal{H}}_{B},\\{\widetilde{E}_{v,i}:v\in[n_{A}],i\in[k_{A}]\\},\\{\widetilde{F}_{w,j}:w\in[n_{B}],j\in[k_{B}]\\}\big{)},$
$\displaystyle\mathscr{S}$
$\displaystyle=\mathopen{}\mathclose{{}\left(\psi\in\mathcal{H}_{A}\otimes\mathcal{H}_{B},\\{E_{v,i}:v\in[n_{A}],i\in[k_{A}]\\},\\{F_{w,j}:w\in[n_{B}],j\in[k_{B}]\\}}\right),$

we say that $\widetilde{\mathscr{S}}$ is a _local $\epsilon$-dilation_ of
$\mathscr{S}$ if there exist isometries
$V_{A}\colon\mathcal{H}_{A}\to\widetilde{\mathcal{H}}_{A}\otimes\mathcal{K}_{A}$
and
$V_{B}\colon\mathcal{H}_{B}\to\widetilde{\mathcal{H}}_{B}\otimes\mathcal{K}_{B}$
for some finite-dimensional Hilbert spaces $\mathcal{K}_{A}$ and
$\mathcal{K}_{B}$, and a quantum state
$\psi_{\mathrm{junk}}\in\mathcal{K}_{A}\otimes\mathcal{K}_{B}$ such that

(4.1) $\displaystyle(V_{A}\otimes V_{B})\psi$
$\displaystyle\approx_{\epsilon}\widetilde{\psi}\otimes\psi_{\mathrm{junk}}\text{
and}$ (4.2) $\displaystyle\mathopen{}\mathclose{{}\left(V_{A}\otimes
V_{B}}\right)(E_{v,i}\otimes F_{w,j})\psi$
$\displaystyle\approx_{\epsilon}\big{(}(\widetilde{E}_{v,i}\otimes\widetilde{F}_{w,j})\widetilde{\psi}\big{)}\otimes\psi_{\mathrm{junk}}$

for all $i$, $j$, $v$, and $w$.

When $\epsilon=0$, we say that $\widetilde{\mathscr{S}}$ is a _local dilation_
of $\mathscr{S}$ (instead of a local $0$-dilation). In this case, the
approximations in Expressions (4.1) and (4.2) are replaced by equalities.

######  Remark 4.2.

We are abusing notation slightly in the above definition. In Estimates (4.1)
and (4.2), the vectors on the left belong to the Hilbert space
$(\widetilde{\mathcal{H}}_{A}\otimes\mathcal{K}_{A})\otimes(\widetilde{\mathcal{H}}_{B}\otimes\mathcal{K}_{B})$
and the vectors on the right belong to the Hilbert space
$(\widetilde{\mathcal{H}}_{A}\otimes\widetilde{\mathcal{H}}_{B})\otimes(\mathcal{K}_{A}\otimes\mathcal{K}_{B})$.
We are identifying these two Hilbert spaces via the unitary which flips the
second and third tensor factors.

######  Remark 4.3.

A note on the terminology that we have chosen: In the literature (for instance
[ŠB20]), the phrase “equivalent up to a local isometry” is often used instead
of “local dilation”. However, we feel that using the word “equivalence” is
somewhat misleading, and that “local dilation” would be an appropriate
replacement, since the term “local dilation” has a directional connotation
(for instance, we can write
$\mathscr{S}\xhookrightarrow{\epsilon}\widetilde{\mathscr{S}}$ to mean that
$\widetilde{\mathscr{S}}$ is a local $\epsilon$-dilation of $\mathscr{S}$)
which is not present if we use the term “equivalence up to local isometry”.

######  Remark 4.4.

Observe that if a strategy $\widetilde{\mathscr{S}}$ induces a correlation
$\widetilde{p}$, and if $\widetilde{\mathscr{S}}$ is a local dilation of any
other strategy $\mathscr{S}$, then $\mathscr{S}$ also induces the same
correlation.

We now recall the definition of self-testing and robust self-testing.

######  Definition 4.5 (Self-testing).

A correlation $\widetilde{p}\in C_{q}(n_{A},n_{B},k_{A},k_{B})$ _self-tests_ a
strategy $\widetilde{\mathscr{S}}$ if for any strategy $\mathscr{S}$ which
also induces $\widetilde{p}$, $\widetilde{\mathscr{S}}$ is a local dilation of
$\mathscr{S}$.

######  Definition 4.6 (Robust self-testing).

A correlation $\widetilde{p}\in C_{q}(n_{A},n_{B},k_{A},k_{B})$ self-tests a
strategy $\widetilde{\mathscr{S}}$ _robustly_ if $\widetilde{p}$ self-tests
$\widetilde{\mathscr{S}}$ and the following condition is satisfied. For each
$\epsilon\geq 0$, there exists $\delta\geq 0$, such that if there is some
$p\in C_{q}(n_{A},n_{B},k_{A},k_{B})$ with
$\mathopen{}\mathclose{{}\left\|p-\widetilde{p}}\right\|_{1}\leq\delta$, and
$p$ is induced by a strategy $\mathscr{S}$, then $\widetilde{\mathscr{S}}$ is
a local $\epsilon$-dilation of $\mathscr{S}$.

For local $\epsilon$-dilations, we establish a transitivity statement which is
helpful in reducing certain results to the case when the quantum state is of
full Schmidt rank. In terms of the notation in Remark 4.3, we establish

$\displaystyle\big{(}\mathscr{S}_{3}\xhookrightarrow{\epsilon_{2}}\mathscr{S}_{2}\xhookrightarrow{\epsilon_{1}}\mathscr{S}_{1}\big{)}\Rightarrow\big{(}\mathscr{S}_{3}\xhookrightarrow{\epsilon_{1}+\epsilon_{2}}\mathscr{S}_{1}\big{)}.$

######  Lemma 4.7 (Transitivity of local $\epsilon$-dilations).

Let $\mathscr{S}_{1},\mathscr{S}_{2}$ and $\mathscr{S}_{3}$ be finite-
dimensional quantum strategies. Let $\epsilon_{1},\epsilon_{2}\geq 0$. If
$\mathscr{S}_{1}$ is a local $\epsilon_{1}$-dilation of $\mathscr{S}_{2}$, and
$\mathscr{S}_{2}$ is a local $\epsilon_{2}$-dilation of $\mathscr{S}_{3}$,
then $\mathscr{S}_{1}$ is a local $(\epsilon_{1}+\epsilon_{2})$-dilation of
$\mathscr{S}_{3}$.

###### Proof.

Let the quantum strategies be given by

$\displaystyle\mathscr{S}_{1}$
$\displaystyle=(\psi_{1}\in\mathcal{H}_{A,1}\otimes\mathcal{H}_{B,1},\\{E_{v,i}^{(1)}:v\in[n_{A}],i\in[k_{A}]\\},\\{F_{w,j}^{(1)}:w\in[n_{B}],j\in[k_{B}]\\}),$
$\displaystyle\mathscr{S}_{2}$
$\displaystyle=(\psi_{2}\in\mathcal{H}_{A,2}\otimes\mathcal{H}_{B,2},\\{E_{v,i}^{(2)}:v\in[n_{A}],i\in[k_{A}]\\},\\{F_{w,j}^{(2)}:w\in[n_{B}],j\in[k_{B}]\\}),$
$\displaystyle\mathscr{S}_{3}$
$\displaystyle=(\psi_{3}\in\mathcal{H}_{A,3}\otimes\mathcal{H}_{B,3},\\{E_{v,i}^{(3)}:v\in[n_{A}],i\in[k_{A}]\\},\\{F_{w,j}^{(3)}:w\in[n_{B}],j\in[k_{B}]\\}).$

Since $\mathscr{S}_{1}$ is a local $\epsilon_{1}$-dilation of
$\mathscr{S}_{2}$, there are isometries
$V_{A}\colon\mathcal{H}_{A,2}\to\mathcal{H}_{A,1}\otimes\mathcal{K}_{A,1}$ and
$V_{B}\colon\mathcal{H}_{B,2}\to\mathcal{H}_{B,1}\otimes\mathcal{K}_{B,1}$ for
some finite-dimensional Hilbert spaces $\mathcal{K}_{A,1},\mathcal{K}_{B,1}$,
and there exist a quantum state
$\psi_{\mathrm{junk},1}\in\mathcal{K}_{A,1}\otimes\mathcal{K}_{B,1}$ such that
for all $v,w,i,j$,

(4.3) $\displaystyle(V_{A}\otimes V_{B})(E_{v,i}^{(2)}\otimes
F_{w,j}^{(2)})\psi_{2}\approx_{\epsilon_{1}}(E_{v,i}^{(1)}\otimes
F_{w,j}^{(1)})\psi_{1}\otimes\psi_{\mathrm{junk},1}.$

Similarly, since $\mathscr{S}_{2}$ is a local $\epsilon_{2}$-dilation of
$\mathscr{S}_{3}$, there are isometries
$W_{A}\colon\mathcal{H}_{A,3}\to\mathcal{H}_{A,2}\otimes\mathcal{K}_{A,2}$ and
$W_{B}\colon\mathcal{H}_{B,3}\to\mathcal{H}_{B,2}\otimes\mathcal{K}_{B,2}$ for
some finite-dimensional Hilbert spaces $\mathcal{K}_{A,2},\mathcal{K}_{B,2}$,
and there exist a quantum state
$\psi_{\mathrm{junk},2}\in\mathcal{K}_{A,2}\otimes\mathcal{K}_{B,2}$ such that
for all $v,w,i,j$,

(4.4) $\displaystyle(W_{A}\otimes W_{B})(E_{v,i}^{(3)}\otimes
F_{w,j}^{(3)})\psi_{3}\approx_{\epsilon_{2}}(E_{v,i}^{(2)}\otimes
F_{w,j}^{(2)})\psi_{2}\otimes\psi_{\mathrm{junk},2}.$

Define $U_{A}=(V_{A}\otimes I_{\mathcal{K}_{A,2}})\circ W_{A}$, and
$U_{B}=(V_{B}\otimes I_{\mathcal{K}_{B,2}})\circ W_{B}$, and
$\psi_{\mathrm{junk}}=\psi_{\mathrm{junk},1}\otimes\psi_{\mathrm{junk},2}$.
Clearly $U_{A}$ and $U_{B}$ are isometries and that for all $v,w,i,j$

$\displaystyle(U_{A}\otimes U_{B})(E_{v,i}^{(3)}$ $\displaystyle\otimes
F_{w,j}^{(3)})\psi_{3}$ $\displaystyle=(V_{A}\otimes
I_{\mathcal{K}_{A,2}}\otimes V_{B}\otimes I_{\mathcal{K}_{B,2}})(W_{A}\otimes
W_{B})(E_{v,i}^{(3)}\otimes F_{w,j}^{(3)})\psi_{3}$
$\displaystyle\approx_{\epsilon_{2}}(V_{A}\otimes V_{B}\otimes
I_{\mathcal{K}_{A,2}}\otimes I_{\mathcal{K}_{B,2}})((E_{v,i}^{(2)}\otimes
F_{w,j}^{(2)})\psi_{2}\otimes\psi_{\mathrm{junk},2})$
$\displaystyle\approx_{\epsilon_{1}}(E_{v,i}^{(1)}\otimes
F_{w,j}^{(1)})\psi_{1}\otimes\psi_{\mathrm{junk},1}\otimes\psi_{\mathrm{junk},2},$

where we used (4.4) in the second approximation, and (4.3) in the third
approximation. ∎

While working with synchronous quantum correlations one usually takes the
quantum state to be full Schmidt rank. We show that this reduction step can
also be captured with local dilation. We first need a lemma which is folklore.

######  Lemma 4.8.

Let $X\in\mathbb{M}_{d_{A}},Y\in\mathbb{M}_{d_{B}}$ and
$\psi\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$ be such that $(X\otimes
I_{d_{B}})\psi=(I_{d_{A}}\otimes Y)\psi$. Then, $\mathrm{supp}_{A}(\psi)$ is
an invariant subspace of $X$, and $\mathrm{supp}_{B}(\psi)$ is an invariant
subspace of $Y$.

###### Proof.

If $\psi=\sum_{i=1}^{r}\alpha_{i}\xi_{i}\otimes\eta_{i}$ is a Schmidt
decomposition, then the matrix $D=\sum_{i=1}^{r}\alpha_{i}\xi_{i}\eta_{i}^{T}$
satisfies $\mathrm{vec}(D)=\psi$. Note that $\mathrm{supp}_{A}(\psi)$ is the
column space of $D$. To show that $\mathrm{supp}_{A}(\psi)$ is invariant under
$X$, it suffices to show that $X\xi_{i}\in\mathrm{supp}_{A}(\psi)$ for all
$i\in[r]$. Using Identity (2.3), we see that $(X\otimes
I_{d_{B}})\psi=(I_{d_{A}}\otimes Y)\psi$ is equivalent to $XD=DY^{T}$. Thus,
for $i\in[r]$,

$\displaystyle
X\xi_{i}=X\mathopen{}\mathclose{{}\left(\alpha_{i}^{-1}D\overline{\eta}_{i}}\right)=\alpha_{i}^{-1}(XD)(\overline{\eta}_{i})=\alpha_{i}^{-1}(DY^{T})(\overline{\eta}_{i})\in\mathrm{supp}_{A}(\psi),$

as required. The invariance of $\mathrm{supp}_{B}(\psi)$ under $Y$ follows
similarly. ∎

######  Lemma 4.9.

Let $p=(p(i,j|v,w))\in C_{q}^{s}(n,k)$ be a synchronous quantum correlation.
Let
$\mathscr{S}=(\psi\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}},\\{E_{v,i}\\},\\{F_{w,j}\\})$
be a strategy inducing $p$. Then, there exists a strategy
$\mathscr{S}^{\prime}=(\psi^{\prime}\in\mathbb{C}^{r}\otimes\mathbb{C}^{r},\\{E_{v,i}^{\prime}\\},\\{F_{w,j}^{\prime}\\})$
where $\psi^{\prime}$ is of full Schmidt rank and $\mathscr{S}^{\prime}$ is a
local dilation of $\mathscr{S}$.

###### Proof.

Consider a Schmidt decomposition
$\psi=\sum_{l=1}^{r}\alpha_{l}\xi_{l}\otimes\eta_{l}$, and let
$\iota_{A}\colon\mathbb{C}^{r}\to\mathbb{C}^{d_{A}}$ and
$\iota_{B}\colon\mathbb{C}^{r}\to\mathbb{C}^{d_{B}}$ be isometries given by
$\iota_{A}=\sum_{l=1}^{r}\xi_{l}e_{l}^{*}$ and
$\iota_{B}=\sum_{l=1}^{r}\eta_{l}e_{l}^{*}$. Then, the operators

(4.5) $\displaystyle E_{v,i}^{\prime}=\iota_{A}^{*}E_{v,i}\iota_{A},\qquad
F_{w,j}^{\prime}=\iota_{B}^{*}F_{w,j}\iota_{B},$

are themselves positive, and moreover, for each $v$ and $w$, the sets
$\\{E_{v,i}^{\prime}:i\in[k]\\}$ and $\\{F_{w,j}^{\prime}:j\in[k]\\}$ form
POVMs. With $\psi^{\prime}=(\iota_{A}^{*}\otimes\iota_{B}^{*})\psi$, these
above operators constitute the strategy $\mathscr{S}^{\prime}$, and clearly
$\psi^{\prime}$ is of full Schmidt rank.

To see that $\mathscr{S}^{\prime}$ is a local dilation of $\mathscr{S}$, set
$\mathcal{K}_{A}=\mathbb{C}^{d_{A}}$ and $\mathcal{K}_{B}=\mathbb{C}^{d_{B}}$
and define isometries
$V_{A}\colon\mathbb{C}^{d_{A}}\to\mathbb{C}^{r}\otimes\mathbb{C}^{d_{A}}$ and
$V_{B}\colon\mathbb{C}^{d_{B}}\to\mathbb{C}^{r}\otimes\mathbb{C}^{d_{B}}$, by

$\displaystyle V_{A}(\xi)$
$\displaystyle=\iota_{A}^{*}(\xi)\otimes\xi_{1}+e_{1}\otimes(I_{d_{A}}-\iota_{A}\iota_{A}^{*})(\xi),$
$\displaystyle V_{B}(\eta)$
$\displaystyle=\iota_{B}^{*}(\eta)\otimes\eta_{1}+e_{1}\otimes(I_{d_{B}}-\iota_{B}\iota_{B}^{*})(\eta),$

where $\xi\in\mathbb{C}^{d_{A}}$ and $\eta\in\mathbb{C}^{d_{B}}$. By Corollary
3.6(a), we have $(E_{v,i}\otimes I_{d_{B}})\psi=(I_{d_{A}}\otimes
F_{v,i})\psi$ for all $v,i$. Then using Lemma 4.8, $\mathrm{supp}_{A}(\psi)$
is invariant under each $E_{v,i}$, and $\mathrm{supp}_{B}(\psi)$ is invariant
under each $F_{w,j}$, and since $\iota_{A}\iota_{A}^{*}$ is the projection
onto $\mathrm{supp}_{A}(\psi)$ and $\iota_{B}\iota_{B}^{*}$ is the projection
onto $\mathrm{supp}_{B}(\psi)$, we have

$\displaystyle(V_{A}\otimes V_{B})(E_{v,i}\otimes
F_{w,j})\psi=(\iota_{A}\otimes\iota_{B})^{*}((E_{v,i}\otimes
F_{w,j})\psi)\otimes(e_{1}\otimes e_{1}).$

But then, using the invariance property,

$\displaystyle(\iota_{A}\otimes\iota_{B})^{*}((E_{v,i}\otimes F_{w,j})\psi)$
$\displaystyle=(\iota_{A}^{*}\otimes\iota_{B}^{*})(\iota_{A}\iota_{A}^{*}\otimes\iota_{B}\iota_{B}^{*})(E_{v,i}\otimes
F_{w,j})\psi$
$\displaystyle=(\iota_{A}^{*}\otimes\iota_{B}^{*})(\iota_{A}\iota_{A}^{*}\otimes\iota_{B}\iota_{B}^{*})(E_{v,i}\otimes
F_{w,j})(\iota_{A}\iota_{A}^{*}\otimes\iota_{B}\iota_{B}^{*})\psi$
$\displaystyle=((E_{v,i}^{\prime}\otimes F_{w,j}^{\prime})\psi^{\prime}),$

as desired. ∎

##  5\. Correlations from projections adding up to scalar times identity

We are interested in self-testing projections which sum up to (some specific)
scalar times the identity. Such kind of projections have been studied in
detail in [KRS02], and we collect some of the material from there as needed.

For $n\in\mathbb{N}$, let $\Sigma_{n}$ be the set of all scalars $x$ such that
there exist $n$ projections $R_{1},\dots,R_{n}\in\mathfrak{B}(\mathcal{H})$,
for some Hilbert space $\mathcal{H}$ (possibly infinite dimensional), such
that $R_{1}+\dots+R_{n}=xI_{\mathcal{H}}$. For $n\leq 3$, it can be shown that

$\displaystyle\Sigma_{1}=\\{0,1\\},\qquad\Sigma_{2}=\\{0,1,2\\},\qquad\Sigma_{3}=\mathopen{}\mathclose{{}\left\\{0,1,\frac{3}{2},2,3}\right\\}.$

For $n\geq 4$, the set $\Sigma_{n}$ is described in [KRS02]. In particular,
they show that the set $\Sigma_{4}$ is countably infinite; whereas for all
$n\geq 5$, the set $\Sigma_{n}$ is uncountable with a nondegenerate interval
subset. For our purpose, we do not require the full description of the set
$\Sigma_{n}$. The following theorem is a distillation of Proposition 2,
Theorem 3, and Theorem 4 from the aforementioned work.

For a definition of a universal C∗-algebra we refer the reader to Appendix A.

######  Theorem 5.1.

For $n=3$, set $\Lambda_{3}=\\{\frac{3}{2}\\}$; and for $n\geq 4$, set
$\Lambda_{n}=\\{x_{l}\\}_{l=0}^{\infty}$ with the sequence defined recursively
as follows: $x_{0}=0$, and for all $l\geq 1$,

(5.1) $\displaystyle x_{l}=1+\frac{1}{n-1-x_{l-1}}.$

Then $\Lambda_{n}\subseteq\Sigma_{n}$ for all $n\geq 3$. Moreover, if
$\mathscr{P}_{n,x}$ is the universal $C^{*}$-algebra with generators
$r_{1},\dots,r_{n}$ and relations given as follows (here $1$ is the unit of
the algebra)

(5.2)
$\displaystyle\mathscr{P}_{n,x}\coloneqq\mathrm{C}^{*}\mathopen{}\mathclose{{}\left\langle
r_{1},\dots,r_{n}\,\big{|}\,r_{v}=r_{v}^{*}=r_{v}^{2},\;\forall\;v\in[n],\sum_{v=1}^{n}r_{v}=x1}\right\rangle,$

then for $x\in\Lambda_{n}$ ($n\geq 3$), there is a unique irreducible
representation (up to unitary equivalence) of the $C^{*}$-algebra
$\mathscr{P}_{n,x}$. Furthermore, this representation is of dimension $d$,
where $x=\frac{b}{d}$ is in its lowest terms.

The C∗-algebra $\mathscr{P}_{n,x}$ has the universal property that whenever
there exist projections $R_{1},\dots,R_{n}\in\mathfrak{B}(\mathcal{H})$ with
$\sum_{v=1}^{n}R_{v}=xI_{\mathcal{H}}$, then there exists a representation
$\pi\colon\mathscr{P}_{n,x}\to\mathfrak{B}(\mathcal{H})$ with
$\pi(r_{v})=R_{v}$ for all $v\in[n]$.

If $x\in\\{x_{l}\\}_{l=0}^{\infty}$ with $x=\frac{b}{d}$ in its lowest terms,
the uniqueness property of $\mathscr{P}_{n,x}$ implies that if
$\widetilde{P}_{1},\dots,\widetilde{P}_{n}$ are projections in
$\mathbb{M}_{d}$ satisfying
$\widetilde{P}_{1}+\dots+\widetilde{P}_{n}=xI_{d}$, and
$R_{1},\dots,R_{n}\in\mathfrak{B}(\mathcal{H})$ are projections such that
$R_{1}+\dots+R_{n}=xI_{\mathcal{H}}$, then there exists a Hilbert space
$\mathcal{K}$ and a unitary
$U\colon\mathcal{H}\rightarrow\mathbb{C}^{d}\otimes\mathcal{K}$ such that
$UR_{v}U^{*}=\widetilde{P}_{v}\otimes I_{\mathcal{K}}$ for all $v\in[n]$.

We are now ready to define the synchronous quantum correlations that we will
be self-testing.

######  Definition 5.2.

Fix $n\geq 3$ and $x\in\Lambda_{n}$ with $x=\frac{b}{d}$ in its lowest terms.
Let $\widetilde{P}_{1},\dots,\widetilde{P}_{n}$ be projections in
$\mathbb{M}_{d}$ (by Theorem 5.1) such that
$\widetilde{P}_{1}+\dots+\widetilde{P}_{n}=xI_{d}$. Consider the synchronous
correlation $\widetilde{p}_{n,x}=(\widetilde{p}_{n,x}(i,j|v,w))\in
C^{s}_{q}(n,2)$ induced by the strategy

(5.3)
$\displaystyle\widetilde{\mathscr{S}}\coloneqq\mathopen{}\mathclose{{}\left(\varphi_{d}\in\mathbb{C}^{d}\otimes\mathbb{C}^{d},\mathopen{}\mathclose{{}\left\\{\widetilde{P}_{v},I_{d}-\widetilde{P}_{v}}\right\\}_{v=1}^{n},\mathopen{}\mathclose{{}\left\\{\widetilde{P}_{v}^{T},I_{d}-\widetilde{P}_{v}^{T}}\right\\}_{v=1}^{n}}\right),$

where $\varphi_{d}$ is the maximally entangled state. We shall refer to this
strategy as the _canonical strategy_ for $\widetilde{p}_{n,x}$.

We shall use the uniqueness result from Theorem 5.1 to show that the strategy
$\widetilde{\mathscr{S}}$ given in Definition 5.2 is essentially the unique
strategy which induces the correlation $\widetilde{p}_{n,x}$—more precisely,
$\widetilde{p}_{n,x}$ self-tests the strategy $\widetilde{\mathscr{S}}$. This
result, together with robustness, will be the focus of next section. First we
show how the uniqueness result provides an explicit description of the
C∗-algebra $\mathscr{P}_{n,x}$.

######  Lemma 5.3.

Let $n\geq 3$ and $x\in\Lambda_{n}$ with $x=\frac{b}{d}$ in lowest terms. Then
$\mathscr{P}_{n,x}\simeq\mathbb{M}_{d}$. In particular, if
$R_{1},\dots,R_{n}\in\mathbb{M}_{d}$ are projections such that
$\sum_{v=1}^{n}R_{v}=xI_{d}$, then $\\{R_{v}\\}_{v=1}^{n}$ generate
$\mathbb{M}_{d}$ as a $C^{*}$-algebra.

###### Proof.

By Theorem 5.1, there is an irreducible representation
$\pi\colon\mathscr{P}_{n,x}\to\mathfrak{B}(\mathbb{C}^{d})$. As $\pi$ is
irreducible, $\pi$ must be surjective (using that $\mathbb{C}^{d}$ is finite-
dimensional) [EGH+11, Theorem 3.2.2]. We will show that $\pi$ is injective and
hence is an isomorphism. If this is not the case, then $\ker(\pi)$ is a non-
zero C∗-algebra and hence admits an irreducible representation
$\phi_{0}\colon\ker(\pi)\to\mathfrak{B}(\mathcal{H})$ on a (possibly infinite-
dimensional) Hilbert space $\mathcal{H}$ [Mur90, Theorem 5.1.12]. Then
$\phi_{0}$ extends to an irreducible representation
$\phi\colon\mathscr{P}_{n,x}\rightarrow\mathfrak{B}(\mathcal{H})$ [Mur90,
Theorem 5.1.13] which is necessarily not unitarily equivalent to $\pi$ since
$\ker(\pi)\neq\ker(\phi)$. This contradicts the uniqueness of the irreducible
representation of $\mathscr{P}_{n,x}$ and hence shows that $\pi$ is injective.
∎

We now collect some further consequences of the uniqueness property in Theorem
5.1. The following lemma tells us that the projections $\\{R_{v}\\}_{v=1}^{n}$
obtained through a representation of $\mathscr{P}_{n,x}$ have the same trace
and are “symmetrically distributed”.

######  Lemma 5.4.

Let $n\geq 3$ and $x\in\Lambda_{n}$ with $x=\frac{b}{d}$ in its lowest terms.
Let $R_{1},\dots,R_{n}\in\mathbb{M}_{d}$ be projections such that
$\sum_{v=1}^{n}R_{v}=xI_{d}$. Then,

(5.4)
$\displaystyle\mathrm{tr}_{d}(R_{v}R_{w})=\begin{cases}\frac{x}{n}&\text{ if
}v=w,\\\ \frac{x(x-1)}{n(n-1)}&\text{ if }v\neq w.\end{cases}$

More generally, if $R_{1},\dots,R_{n}\in\mathbb{M}_{k}$ (for some
$k\in\mathbb{N}$) are projections such that $\sum_{v=1}^{n}R_{v}=xI_{k}$, then
$\operatorname{Tr}(R_{v})$ is constant for all $v\in[n]$, and
$\operatorname{Tr}(R_{v}R_{w})$ is constant for all $v\neq w$.

###### Proof.

Since $\sum_{v=1}^{n}R_{v}=xI_{d}$, by the universal property of
$\mathscr{P}_{n,x}$, there exists a representation
$\pi\colon\mathscr{P}_{n,x}\to\mathbb{M}_{d}$ such that $\pi(r_{v})=R_{v}$ for
all $v\in[n]$. By Theorem 5.1 this representation is unique (up to unitary
equivalence) and irreducible.

Now, choose a pair $(v,w)$ with $v\neq w$ and define a new representation
$\sigma_{v,w}\colon\mathscr{P}_{n,x}\to\mathbb{M}_{d}$ by

$\displaystyle\sigma_{v,w}(r_{k})$ $\displaystyle=\begin{cases}R_{k}&\text{ if
}k\neq v\text{ and }k\neq w,\\\ R_{w}&\text{ if }k=v,\\\ R_{v}&\text{ if
}k=w.\\\ \end{cases}$

By the uniqueness of the irreducible representation $\pi$ (Theorem 5.1), there
exists a unitary $U\in\mathbb{M}_{d}$ such that
$\sigma_{v,w}(r_{k})=U\pi(r_{k})U^{*}$ for all $k$. But then,

$\displaystyle\operatorname{Tr}(R_{v})=\operatorname{Tr}(\sigma_{v,w}(r_{w}))=\operatorname{Tr}(U\pi(r_{w})U^{*})=\operatorname{Tr}(\pi(r_{w}))=\operatorname{Tr}(R_{w}).$

Additionally, since $R_{1}+\dots+R_{n}=xI_{d}$, it follows that
$\mathrm{tr}_{d}(R_{v})=\frac{x}{n}$ for all $v\in[n]$.

Similarly, one can show that $\operatorname{Tr}(R_{v}R_{w})$ is constant for
all $v\neq w$. Moreover, squaring $\sum_{v=1}^{n}R_{v}=xI_{d}$, and taking
normalized trace yields that
$\mathrm{tr}_{d}(R_{v}R_{w})=\frac{x(x-1)}{n(n-1)}$, for all $v\neq w$.

For the final statement, let
$\pi\colon\mathscr{P}_{n,x}\rightarrow\mathbb{M}_{k}$ be the representation
given by $\pi(r_{v})=R_{v}$ for all $v\in[n]$. If
$\pi_{0}\colon\mathscr{P}_{n,x}\rightarrow\mathbb{M}_{d}$ is the unique
irreducible representation, then $d$ divides $k$ and $\pi$ is unitarily
equivalent to $\pi_{0}^{\oplus(k/d)}$. So,
$\mathrm{Tr}(R_{v}R_{w})=k\mathrm{tr}_{d}(\pi_{0}(r_{v}r_{w}))$ for all
$v,w\in[n]$. The result follows from the first part of the lemma. ∎

######  Remark 5.5 (The correlation table for $\widetilde{p}_{n,x}$).

Let $n\geq 3$ and $x\in\Lambda_{n}$ with $x=\frac{b}{d}$ in lowest terms. Let
$\widetilde{p}_{n,x}$ be the quantum correlation as in Definition 5.2. Then,

(5.5)
$\displaystyle\widetilde{p}_{n,x}(1,1|v,w)=\mathopen{}\mathclose{{}\left\langle{\mathopen{}\mathclose{{}\left(\widetilde{P}_{v}\otimes\widetilde{P}_{w}^{T}}\right)\varphi_{d}},{\varphi_{d}}}\right\rangle=\mathrm{tr}_{d}(\widetilde{P}_{v}\widetilde{P}_{w})=\begin{cases}\frac{x}{n}&\text{
if }v=w,\\\ \frac{x(x-1)}{n(n-1)}&\text{ if }v\neq w,\end{cases}$

where the second equality follows from Identity (2.5) and the third equality
follows from Lemma 5.4. Note that with these values of
$\widetilde{p}_{n,x}(1,1|v,w)$, all other values of
$\widetilde{p}_{n,x}(i,j|v,w)$ can be deduced using the synchronous condition.
Indeed, for all $v\neq w$,

$\displaystyle\widetilde{p}_{n,x}(1,2|v,w)=\widetilde{p}_{n,x}(2,1|v,w)$
$\displaystyle=\frac{x}{n}-\widetilde{p}_{n,x}(1,1|v,w)\text{ and}$
$\displaystyle\widetilde{p}_{n,x}(2,2|v,w)$
$\displaystyle=1-\frac{2x}{n}+\widetilde{p}_{n,x}(1,1|v,w),$

and for $v=w$,

$\displaystyle\widetilde{p}_{n,x}(1,2|v,v)$
$\displaystyle=\widetilde{p}_{n,x}(2,1|v,v)=0\text{ and}$
$\displaystyle\widetilde{p}_{n,x}(2,2|v,v)$
$\displaystyle=1-\widetilde{p}_{n,x}(1,1|v,v).$

For $n\geq 3$ and $x=\frac{n}{n-1}\in\Lambda_{n}$, Lemma 5.4 implies that
$\operatorname{Tr}(R_{v})=1$ for projections
$R_{1},\dots,R_{n}\in\mathbb{M}_{n-1}$ satisfying
$\sum_{v=1}^{n}R_{v}=\frac{n}{n-1}I_{n-1}$. Thus, each $R_{v}$ is the
projection onto the span of some unit vector $\xi_{v}\in\mathbb{C}^{n-1}$. The
following corollary is then straightforward to verify.

######  Corollary 5.6.

Let $n\geq 3$. Projections $\\{R_{v}\\}_{v=1}^{n}\subseteq\mathbb{M}_{n-1}$
which sum up to $\frac{n}{n-1}I_{n-1}$ give rise to $n$ unit vectors
$\\{\xi_{v}\\}_{v=1}^{n}\subseteq\mathbb{C}^{n-1}$ such that
$|\mathopen{}\mathclose{{}\left\langle{\xi_{v}},{\xi_{w}}}\right\rangle|=\frac{1}{n-1}$
for all $v\neq w$, and conversely.

Together with unitary equivalence, the property that
$|\mathopen{}\mathclose{{}\left\langle{\xi_{v}},{\xi_{w}}}\right\rangle|=\frac{1}{n-1}$
for all $v\neq w$ implies that the vectors can be taken as the vertices of a
regular $(n-1)$-simplex in $\mathbb{R}^{n-1}$ centered at the origin. For
instance, in dimension $2$, (i.e., $n=3$ in Corollary 5.6), the vertices of a
regular $2$-simplex (a triangle) centered at origin are

$\displaystyle\xi_{1}=\begin{bmatrix}1\\\
0\end{bmatrix},\qquad\xi_{2}=\frac{1}{2}\begin{bmatrix}1\\\
-\sqrt{3}\end{bmatrix},\qquad\xi_{3}=\frac{1}{2}\begin{bmatrix}-1\\\
-\sqrt{3}\end{bmatrix}.$

Then the projections $R_{1},R_{2},R_{3}$ are onto the span of
$\xi_{1},\xi_{2},\xi_{3}$, respectively, and are given by

(5.6) $\displaystyle R_{1}=\begin{bmatrix}1&0\\\ 0&0\end{bmatrix},\quad
R_{2}=\frac{1}{4}\begin{bmatrix}1&-\sqrt{3}\\\ -\sqrt{3}&3\end{bmatrix},\quad
R_{3}=\frac{1}{4}\begin{bmatrix}1&\sqrt{3}\\\ \sqrt{3}&3\end{bmatrix}.$

For other $x\in\Lambda_{n}$ ($n\geq 4$), the projections adding up to $xI$ can
be constructed using a recursive method described in [KRS02], but it is not
clear what natural geometric picture we may associate to such projections. For
the sake of completeness, we describe this construction for $n=4$ in Appendix
B.

Finally, we require a result on the maximum eigenvalue of a certain matrix
obtained from projections adding up to a scalar times identity.

######  Lemma 5.7.

Let $n\geq 3$ and $x\in\Lambda_{n}$ with $x=\frac{b}{d}$ in lowest terms. Let
$R_{1},\dots,R_{n}\in\mathbb{M}_{d}$ be projections such that
$\sum_{v=1}^{n}R_{v}=xI_{d}$. If $N=\sum_{v=1}^{n}R_{v}\otimes R_{v}^{T}$,
then $x$ is the largest eigenvalue of $N$, and further, the eigenspace
corresponding to $x$ is the one-dimensional space spanned by the maximally
entangled state $\varphi_{d}$.

###### Proof.

With $D=\frac{1}{\sqrt{d}}I_{d}$, we have $\mathrm{vec}(D)=\varphi_{d}$. Using
Identity (2.3),

$\displaystyle
N\varphi_{d}=N\mathrm{vec}(D)=\sum_{v=1}^{n}\mathrm{vec}(R_{v}DR_{v})=x\mathrm{vec}(D)=x\varphi_{d}.$

Thus $\varphi_{d}$ is an eigenvector of $N$ with eigenvalue $x$.

Since $N\geq 0$, the largest eigenvalue of $N$ is $\|N\|$. Since

$\displaystyle N=\sum_{v=1}^{n}R_{v}\otimes
R_{v}^{T}\leq\sum_{v=1}^{n}R_{v}\otimes I_{d}=xI_{d}\otimes I_{d}$

it follows that $\|N\|\leq x$. Then
$\|N\|=\sup_{\|\xi\|=1}|\mathopen{}\mathclose{{}\left\langle{N\xi},{\xi}}\right\rangle|$
coupled with
$\mathopen{}\mathclose{{}\left\langle{N\varphi_{d}},{\varphi_{d}}}\right\rangle=x$,
yields $\|N\|=x$.

We now show that the eigenspace corresponding to the eigenvalue $x$ is one-
dimensional. Let $\varphi\in\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ be a unit
vector such that $N\varphi=x\varphi$. Let $B\in\mathbb{M}_{d}$ be such that
$\mathrm{vec}(B)=\varphi$. Then $N\varphi=x\varphi$ is equivalent to
$N\mathrm{vec}(B)=x\mathrm{vec}(B)$, which in turn, using Identity (2.3), is
equivalent to

(5.7) $\displaystyle\sum_{v=1}^{n}R_{v}BR_{v}=xB.$

Now consider the unital quantum channel
$\Phi\colon\mathbb{M}_{d}\to\mathbb{M}_{d}$ defined by

(5.8) $\displaystyle X\mapsto\frac{1}{x}\sum_{v=1}^{n}R_{v}XR_{v},\qquad
X\in\mathbb{M}_{d}.$

Then Equation (5.7) implies that $B$ is a fixed point of the quantum channel.
By [Wat18, Theorem 4.25], it follows that $R_{v}B=BR_{v}$ for all $v\in[n]$.
By Lemma 5.3, the projections $\\{R_{v}\\}_{v=1}^{n}$ generate the whole of
$\mathbb{M}_{d}$. Hence, $B$ commutes with every element of $\mathbb{M}_{d}$,
and we must have $B=\lambda I_{d}$, for some $\lambda\in\mathbb{C}$. In other
words, $\varphi=\mathrm{vec}(B)=(\lambda\sqrt{d})\varphi_{d}$, as required. ∎

##  6\. Robust self-testing of projections adding up to scalar times identity

We now set out to prove that the correlation $\widetilde{p}_{n,x}$ of
Definition 5.2 robustly self-tests the canonical strategy
$\widetilde{\mathscr{S}}$ therein.

We first show in Theorem 6.1 that if a strategy $\mathscr{S}$ induces the
correlation $\widetilde{p}_{n,x}$ then $\widetilde{\mathscr{S}}$ must be a
local dilation of $\mathscr{S}$. For this, we prove that the measurement
operators of $\mathscr{S}$ form a representation of the algebra
$\mathscr{P}_{n,x}$. Then invoking the irreducibility assumption we get
isometries relating the strategy $\mathscr{S}$ in hand with the canonical one.

To handle the robust case, we first formally define a suitable notion of an
“approximate” representation. Then we show that if we have a strategy which
induces a correlation within $\epsilon$-distance of $\widetilde{p}_{n,x}$,
then we get an “approximate” representation of the relation
$r_{1}+\dots+r_{n}=x1$. Finally, to relate the approximate strategy with the
canonical one, we establish an analogue of Gowers–Hatami Theorem for the
C∗-algebra $\mathscr{P}_{n,x}$.

######  Theorem 6.1.

Let $n\geq 3$ and $x\in\Lambda_{n}$. The synchronous correlation
$\widetilde{p}_{n,x}\in C_{q}^{s}(n,2)$ induced by the strategy
$\widetilde{\mathscr{S}}$ in Definition 5.2 self-tests
$\widetilde{\mathscr{S}}$.

###### Proof.

Let
$\mathscr{S}=(\psi\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}},\\{P_{v},I_{d_{A}}-P_{v}\\}_{v=1}^{n},\\{Q_{w},I_{d_{B}}-Q_{w}\\}_{w=1}^{n})$
be a strategy which induces correlation $\widetilde{p}_{n,x}$. Using the
transitivity of local dilations (Lemma 4.7) and Lemma 4.9 we may assume that
$\psi$ is of full Schmidt rank $r$, i.e., $d_{A}=d_{B}=r$. The “moreover” part
of Corollary 3.6 then implies that $P_{v}$ and $Q_{v}$ are projections for all
$v\in[n]$.

Set $\rho=\psi\psi^{*}$ and let $\rho_{A}$ and $\rho_{B}$ be its reduced
density matrices. Clearly,
$\mathopen{}\mathclose{{}\left\langle{I_{r}},{I_{r}}}\right\rangle_{\rho_{A}}=1$,
and if $P\coloneqq\sum_{v=1}^{n}P_{v}$, then using the property $(P_{v}\otimes
I_{r})\psi=(I_{r}\otimes Q_{v})\psi$ of Corollary 3.6, we compute

$\displaystyle\mathopen{}\mathclose{{}\left\langle{P},{I_{r}}}\right\rangle_{\rho_{A}}$
$\displaystyle=\sum_{v=1}^{n}\mathopen{}\mathclose{{}\left\langle{(P_{v}\otimes
I_{r})\psi},{\psi}}\right\rangle=\sum_{v=1}^{n}\mathopen{}\mathclose{{}\left\langle{(P_{v}\otimes
Q_{v})\psi},{\psi}}\right\rangle=\sum_{v=1}^{n}\widetilde{p}_{n,x}(1,1|v,v)=x,$
and
$\displaystyle\mathopen{}\mathclose{{}\left\langle{P},{P}}\right\rangle_{\rho_{A}}$
$\displaystyle=\sum_{v,w=1}^{n}\mathopen{}\mathclose{{}\left\langle{(P_{v}P_{w}\otimes
I_{r})\psi},{\psi}}\right\rangle=\sum_{v,w=1}^{n}\mathopen{}\mathclose{{}\left\langle{(P_{v}\otimes
I_{r})\psi},{(P_{w}\otimes I_{r})\psi}}\right\rangle$
$\displaystyle=\sum_{v,w=1}^{n}\mathopen{}\mathclose{{}\left\langle{(P_{v}\otimes
I_{r})\psi},{(I_{r}\otimes
Q_{w})\psi}}\right\rangle=\sum_{v,w=1}^{n}\mathopen{}\mathclose{{}\left\langle{(P_{v}\otimes
Q_{w})\psi},{\psi}}\right\rangle$
$\displaystyle=\sum_{v,w=1}^{n}\widetilde{p}_{n,x}(1,1|v,w)=n\frac{x}{n}+n(n-1)\frac{x(x-1)}{n(n-1)}=x^{2}.$

Therefore,
$\mathopen{}\mathclose{{}\left\langle{P},{I_{r}}}\right\rangle_{\rho_{A}}=\|P\|_{\rho_{A}}\|I_{r}\|_{\rho_{A}}=x$.
That is, equality holds in a Cauchy–Schwarz inequality, and since $\psi$ is of
full Schmidt rank, this equality implies $P=\sum_{v=1}^{n}P_{v}=xI_{r}$.
Similarly, we can show that $\sum_{w=1}^{n}Q_{w}=xI_{r}$.

Invoking the uniqueness of the irreducible representation of
$\mathscr{P}_{n,x}$, it follows that there exist unitaries
$U_{A}\colon\mathbb{C}^{r}\to\mathbb{C}^{d}\otimes\mathcal{K}_{A}$ and
$U_{B}:\mathbb{C}^{r}\to\mathbb{C}^{d}\otimes\mathcal{K}_{B}$ for some Hilbert
spaces $\mathcal{K}_{A},\mathcal{K}_{B}$ such that
$U_{A}P_{v}U_{A}^{*}=\widetilde{P}_{v}\otimes I_{\mathcal{K}_{A}}$ and
$U_{B}Q_{v}U_{B}^{*}=\widetilde{P}_{v}^{T}\otimes I_{\mathcal{K}_{B}}$ for all
$v\in[n]$. Set $U=U_{A}\otimes U_{B}$. Then, we have

$\displaystyle x$
$\displaystyle=\sum_{v=1}^{n}\mathopen{}\mathclose{{}\left\langle{(P_{v}\otimes
Q_{v})\psi},{\psi}}\right\rangle$
$\displaystyle=\sum_{v=1}^{n}\mathopen{}\mathclose{{}\left\langle{(\widetilde{P}_{v}\otimes\widetilde{P}_{v}^{T}\otimes
I_{\mathcal{K}_{A}}\otimes I_{\mathcal{K}_{B}})U\psi},{U\psi}}\right\rangle$
$\displaystyle=\mathopen{}\mathclose{{}\left\langle{(N\otimes
I_{\mathcal{K}_{A}}\otimes I_{\mathcal{K}_{B}})U\psi},{U\psi}}\right\rangle,$

where $N=\sum_{v=1}^{n}\widetilde{P}_{v}\otimes\widetilde{P}_{v}^{T}$. Then
using Lemma 5.7, $x$ is also the largest eigenvalue of
$N\otimes(I_{\mathcal{K}_{A}}\otimes I_{\mathcal{K}_{B}})$ with eigenspace
$\\{\varphi_{d}\otimes\psi^{\prime}:\psi^{\prime}\in\mathcal{K}_{A}\otimes\mathcal{K}_{B}\\}$.
The above computation implies, by means of Cauchy–Schwarz inequality, that
$U\psi$ is an eigenvector of $N\otimes(I_{\mathcal{K}_{A}}\otimes
I_{\mathcal{K}_{B}})$ corresponding to the eigenvalue $x$, and therefore
$U\psi=\varphi_{d}\otimes\psi_{\mathrm{junk}}$ for some unit vector
$\psi_{\mathrm{junk}}\in\mathcal{K}_{A}\otimes\mathcal{K}_{B}$. Finally, for
all $v,w$,

$\displaystyle U(P_{v}\otimes Q_{w})\psi$ $\displaystyle=U(P_{v}\otimes
Q_{w})U^{*}U\psi$
$\displaystyle=\big{(}(\widetilde{P}_{v}\otimes\widetilde{P}_{w}^{T})\otimes(I_{\mathcal{K}_{A}}\otimes
I_{\mathcal{K}_{B}})\big{)}(\varphi_{d}\otimes\psi_{\mathrm{junk}})$
$\displaystyle=\big{(}(\widetilde{P}_{v}\otimes\widetilde{P}_{w}^{T})\varphi_{d}\big{)}\otimes\psi_{\mathrm{junk}},$

as required. ∎

With an eye towards the robust self-testing result, we first define what we
mean by an “approximate” representation for a relation. This is an analogue of
an “approximate” group representation.

######  Definition 6.2.

A _polynomial relation_ in $n$ variables is a relation of the form

(6.1) $\displaystyle
f\mathopen{}\mathclose{{}\left(x_{1},\dots,x_{n},x_{1}^{*},\dots,x_{n}^{*}}\right)=0$

where $f$ is a polynomial in $2n$ non-commuting variables
$x_{1},\dots,x_{n},x_{1}^{*},\dots,x_{n}^{*}$. Given $\delta\geq 0$, a density
matrix $\sigma\in\mathbb{M}_{r}$, and polynomial relations
$(f_{1},\ldots,f_{k})$ as in Expression (6.1), an $n$-tuple of matrices
$\mathopen{}\mathclose{{}\left(X_{1},\dots,X_{n}}\right)$ in $\mathbb{M}_{r}$
is called a $(\delta,\sigma)$-_representation_ of relations
$(f_{1},\ldots,f_{k})$ if

$\displaystyle\max_{i\in[k]}\mathopen{}\mathclose{{}\left\|f_{i}\mathopen{}\mathclose{{}\left(X_{1},\dots,X_{n},X_{1}^{*},\dots,X_{n}^{*}}\right)}\right\|_{\sigma}\leq\delta.$

######  Lemma 6.3.

Let $n\geq 3$ and $x\in\Lambda_{n}$. Let $\widetilde{p}_{n,x}\in
C_{q}^{s}(n,2)$ be as in Definition 5.2. Let $p\in C_{q}(n,2)$ and suppose it
is induced by a strategy

$\displaystyle\mathopen{}\mathclose{{}\left(\psi\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}},\mathopen{}\mathclose{{}\left\\{E_{v},I_{d_{A}}-E_{v}}\right\\}_{v=1}^{n},\mathopen{}\mathclose{{}\left\\{F_{w},I_{d_{B}}-F_{w}}\right\\}_{w=1}^{n}}\right).$

Let $\rho_{A}$ and $\rho_{B}$ be the reduced density matrices of
$\rho=\psi\psi^{*}$. If $\|p-\widetilde{p}_{n,x}\|_{1}\leq\delta$ for some
$\delta\geq 0$, then the tuple $(E_{1},\dots,E_{n})$ is a
$(C\delta^{\frac{1}{4}},\rho_{A})$-representation and $(F_{1},\dots,F_{n})$ is
a $(C\delta^{\frac{1}{4}},\rho_{B})$-representation of the relations given in
(5.2), where

(6.2) $\displaystyle C\coloneqq\sqrt{n^{2}+(1+2x)\sqrt{\delta}}.$

###### Proof.

Using triangle inequality in comparing marginals,

$\displaystyle\bigg{|}\sum_{v=1}^{n}\big{(}p_{A}(1|v)$
$\displaystyle-(\widetilde{p}_{n,x})_{A}(1|v)\big{)}\bigg{|}$
$\displaystyle=\mathopen{}\mathclose{{}\left|\sum_{v=1}^{n}\big{(}p(1,1|v,v)+p(1,2|v,v)-\widetilde{p}_{n,x}(1,1|v,v)-\widetilde{p}_{n,x}(1,2|v,v)\big{)}}\right|$
$\displaystyle\leq\sum_{v=1}^{n}|p(1,1|v,v)-\widetilde{p}_{n,x}(1,1|v,v)|+|p(1,2|v,v)-\widetilde{p}_{n,x}(1,2|v,v)|$
$\displaystyle\leq\|p-\widetilde{p}_{n,x}\|_{1}\leq\delta.$

Since $\sum_{v=1}^{n}(\widetilde{p}_{n,x})_{A}(1|v)=x$, we get

(6.3) $\displaystyle\sum_{v=1}^{n}p_{A}(1|v)\geq x-\delta.$

Similarly, we establish

(6.4)
$\displaystyle\mathopen{}\mathclose{{}\left|x^{2}-\sum_{v,w=1}^{n}p(1,1|v,w)}\right|\leq\delta.$

On the other hand, using triangle inequality and Lemma 3.5(a),

$\displaystyle\sum_{v,w=1}^{n}\mathopen{}\mathclose{{}\left\langle{(E_{v}E_{w}\otimes
I_{d_{B}})\psi},{\psi}}\right\rangle$
$\displaystyle=\sum_{v,w=1}^{n}\mathopen{}\mathclose{{}\left\langle{(E_{v}\otimes
I_{d_{B}})\psi},{(E_{w}\otimes I_{d_{B}})\psi}}\right\rangle$
$\displaystyle\approx_{n^{2}\sqrt{\delta}}\sum_{v,w=1}^{n}\mathopen{}\mathclose{{}\left\langle{(E_{v}\otimes
I_{d_{B}})\psi},{(I_{d_{A}}\otimes F_{w})\psi}}\right\rangle$
$\displaystyle=\sum_{v,w=1}^{n}\mathopen{}\mathclose{{}\left\langle{(E_{v}\otimes
F_{w})\psi},{\psi}}\right\rangle=\sum_{v,w=1}^{n}p(1,1|v,w).$

Therefore,

(6.5)
$\displaystyle\mathopen{}\mathclose{{}\left|\sum_{v,w=1}^{n}\mathopen{}\mathclose{{}\left\langle{(E_{v}E_{w}\otimes
I_{d_{B}})\psi},{\psi}}\right\rangle-\sum_{v,w=1}^{n}p(1,1|v,w)}\right|\leq
n^{2}\sqrt{\delta}.$

Using this inequality together with Inequality (6.4) and an application of
triangle inequality, we get

(6.6)
$\displaystyle\mathopen{}\mathclose{{}\left|x^{2}-\sum_{v,w=1}^{n}\mathopen{}\mathclose{{}\left\langle{(E_{v}E_{w}\otimes
I_{d_{B}})\psi},{\psi}}\right\rangle}\right|\leq\delta+n^{2}\sqrt{\delta}.$

Finally, if $E\coloneqq E_{1}+\dots+E_{n}$, then using Inequalities (6.3) and
(6.6),

$\displaystyle\|E-xI_{d_{A}}\|_{\rho_{A}}^{2}$
$\displaystyle=\sum_{v,w=1}^{n}\mathopen{}\mathclose{{}\left\langle{(E_{v}E_{w}\otimes
I_{d_{B}})\psi},{\psi}}\right\rangle-2x\sum_{v=1}^{n}p_{A}(1|v)+x^{2}$
$\displaystyle\leq
x^{2}+\delta+n^{2}\sqrt{\delta}-2x(x-\delta)+x^{2}=(1+2x)\delta+n^{2}\sqrt{\delta}.$

Similarly, one proves the statement about $F=F_{1}+\dots+F_{n}$. ∎

We now supply with an analogue of Gowers–Hatami Theorem as promised. Our
presentation of Gowers–Hatami Theorem mimics the one presented in [Vid18,
Theorem 12] (which is a slightly general version of [Gow17, Theorem 15.2] and
was originally published in [GK17]). We remark that the relation between
$\epsilon$ and $\delta$ in Theorem 6.5 is non-constructive. We first recall a
standard fact from C∗-algebra theory.

######  Lemma 6.4.

If $\tau$ is a tracial state on a unital $C^{*}$-algebra $\mathfrak{A}$, then
$\mathfrak{N}_{\tau}=\\{a\in\mathfrak{A}:\tau(a^{*}a)=0\\}$ is a closed two-
sided ideal of $\mathfrak{A}$.

###### Proof.

Applying the Cauchy–Schwarz inequality to the sesquilinear form
$(a,b)\mapsto\tau(b^{*}a)$ on $\mathfrak{A}$ shows $\mathfrak{N}_{\tau}$ is a
subspace of $\mathfrak{A}$, which is closed as $\tau$ is continuous. For
$z\in\mathfrak{A}$ and $a\in\mathfrak{N}_{\tau}$, using the inequality
$z^{*}z\leq\|z^{*}z\|1_{\mathfrak{A}}$, we have

$\displaystyle\tau((za)^{*}(za))=\tau(a^{*}z^{*}za)\leq\|z^{*}z\|\tau(a^{*}a)=0,$

and hence $za\in\mathfrak{N}_{\tau}$. This shows $\mathfrak{N}_{\tau}$ is a
left-ideal. As $\tau(a^{*}a)=\tau(aa^{*})$ for all $a\in\mathfrak{A}$,
$\mathfrak{N}_{\tau}$ is self-adjoint and hence is also a right ideal. ∎

######  Theorem 6.5 (Analogue of Gowers–Hatami).

Let $n\geq 3$ and $x\in\Lambda_{n}$ with $x=\frac{b}{d}$ in lowest terms. Let
$\\{\widetilde{P}_{v}\\}_{v=1}^{n}\subseteq\mathbb{M}_{d}$ be an irreducible
representation of $\mathscr{P}_{n,x}$. Then, given $\epsilon>0$, there exist
$\delta>0$ and $m\in\mathbb{N}$ such that the following property is satisfied:

(6.7) for all $r\in\mathbb{N}$, density matrix $\rho\in\mathbb{M}_{r}$, and
positive contractions $E_{1},\dots,E_{n}\in\mathbb{M}_{r}$ satisfying (a)
$\mathopen{}\mathclose{{}\left\|E_{v}^{2}-E_{v}}\right\|_{\rho}\leq\delta$ for
all $v\in n$, (b)
$\mathopen{}\mathclose{{}\left\|xI_{r}-\sum_{v=1}^{n}E_{v}}\right\|_{\rho}\leq\delta$,
and (c)
$\mathopen{}\mathclose{{}\left|\operatorname{Tr}_{\rho}(W_{1}W_{2}-W_{2}W_{1})}\right|\leq\delta$
for monomials $W_{1}$ and $W_{2}$ of degree at most $m$ in the noncommuting
variables $E_{1},\dots,E_{n}$, there are $s\in\mathbb{N}$ and an isometry
$V\colon\mathbb{C}^{r}\to\mathbb{C}^{d}\otimes\mathbb{C}^{s}$ such that for
all $v\in[n]$, we have $\big{\|}E_{v}-V^{*}(\widetilde{P}_{v}\otimes
I_{s})V\big{\|}_{\rho}\leq\epsilon.$

###### Proof.

Suppose the result is false and fix a counterexample $n\geq 3$,
$x\in\Lambda_{n}$, and $\epsilon>0$. For each natural number $k\geq 1$, fix
$\delta_{k}>0$ and $m_{k}\in\mathbb{N}$ such that
$\lim_{k\to\infty}\delta_{k}=0$ and $\lim_{k\to\infty}m_{k}=\infty$. For each
$k\geq 1$, fix $r_{k}\in\mathbb{N}$, a density matrix
$\rho_{k}\in\mathbb{M}_{r_{k}}$, and positive contractions
$E_{1,k},\dots,E_{n,k}\in\mathbb{M}_{r_{k}}$, such that

  1. (ak)

$\big{\|}E_{i,k}^{2}-E_{i,k}\big{\|}_{\rho_{k}}\leq\delta_{k}$ for all
$i\in[n]$,

  2. (bk)

$\mathopen{}\mathclose{{}\left\|xI_{r_{k}}-\sum_{i=1}^{n}E_{i,k}}\right\|_{\rho_{k}}\leq\delta_{k}$,
and

  3. (ck)

$\mathopen{}\mathclose{{}\left|\operatorname{Tr}_{\rho_{k}}(W_{1}W_{2}-W_{2}W_{1})}\right|\leq\delta_{k}$
for monomials $W_{1},W_{2}$ of degree at most $m_{k}$ in the variables
$E_{1,k},\dots,E_{n,k}$,

but such that for all natural numbers $s_{k}\geq 1$ and isometries
$V_{k}\colon\mathbb{C}^{r_{k}}\to\mathbb{C}^{d}\otimes\mathbb{C}^{s_{k}}$,
there is an $i\in[n]$ with

(6.8) $\displaystyle\big{\|}E_{i,k}-V_{k}^{*}(\widetilde{P}_{i}\otimes
I_{s_{k}})V_{k}\big{\|}_{\rho_{k}}>\epsilon.$

Let $\mathfrak{A}$ be the universal unital C∗-algebra generated by $n$
contractions $A_{1},\dots,A_{n}$. For each $k\in\mathbb{N}$, by the universal
property of $\mathfrak{A}$ there exists a representation
$\pi_{k}\colon\mathfrak{A}\to\mathbb{M}_{r_{k}}$ such that
$\pi_{k}(A_{i})=E_{i,k}$ for all $i\in[n]$. Note that
$(\operatorname{Tr}_{\rho_{k}}\circ\pi_{k})(AB-BA)\to 0$ for all
$A,B\in\mathfrak{A}$. If $\tau$ is a weak∗-limit point of the set of states
$\\{\operatorname{Tr}_{\rho_{k}}\circ\pi_{k}\\}_{k=1}^{\infty}$ on
$\mathfrak{A}$, then $\tau$ is a tracial state on $\mathfrak{A}$. Passing to a
subsequence, we may assume
$(\operatorname{Tr}_{\rho_{k}}\circ\pi_{k})(A)\to\tau(A)$ for all
$A\in\mathfrak{A}$.

Let $q\colon\mathfrak{A}\to\mathfrak{A}/\mathfrak{N}_{\tau}$ be the quotient
map where $\mathfrak{N}_{\tau}=\\{a\in\mathfrak{A}:\tau(a^{*}a)=0\\}$. For
$1\leq i\leq n$, we have

$\displaystyle\tau\mathopen{}\mathclose{{}\left((A_{i}^{2}-A_{i})^{*}(A_{i}^{2}-A_{i})}\right)$
$\displaystyle=\lim_{k\to\infty}\mathopen{}\mathclose{{}\left(\operatorname{Tr}_{\rho_{k}}\circ\pi_{k}}\right)\mathopen{}\mathclose{{}\left((A_{i}^{2}-A_{i})^{*}(A_{i}^{2}-A_{i})}\right)$
$\displaystyle=\lim_{k\to\infty}\operatorname{Tr}_{\rho_{k}}\mathopen{}\mathclose{{}\left((E_{i,k}^{2}-E_{i,k})^{*}(E_{i,k}^{2}-E_{i,k})}\right)$
$\displaystyle=\lim_{k\to\infty}\|E_{i,k}^{2}-E_{i,k}\|_{\rho_{k}}^{2}=0.$

So $A_{i}^{2}-A_{i}\in\mathfrak{N}_{\tau}$, and hence $q(A_{i})$ is a
projection for all $1\leq i\leq n$. Similarly,

$\displaystyle\tau\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(xI_{\mathfrak{A}}-\sum_{i=1}^{n}A_{i}}\right)^{*}\mathopen{}\mathclose{{}\left(xI_{\mathfrak{A}}-\sum_{i=1}^{n}A_{i}}\right)}\right)=0,$

and hence $\sum_{i=1}^{n}q(A_{i})=xI_{\mathfrak{A}/\mathfrak{N}_{\tau}}$.
Therefore, there is a representation
$\theta\colon\mathscr{P}_{n,x}\to\mathfrak{A}/\mathfrak{N}_{\tau}$ such that
$\theta(r_{i})=q(A_{i})$ for $1\leq i\leq n$ where $r_{1},\dots,r_{n}$ are the
generators of $\mathscr{P}_{n,x}$.

Since $\theta\colon\mathscr{P}_{n,x}\to\mathfrak{A}/\mathfrak{N}_{\tau}$ is a
$*$-homomorphism, there is a unital, completely positive map
$\widetilde{\theta}\colon\mathscr{P}_{n,x}\to\mathfrak{A}$ such that
$q\widetilde{\theta}=\theta$.333This is a special case of the Choi–Effros
lifting theorem [CE76]. The special case needed here is [Arv77, Lemma 3.3]
using that $\mathscr{P}_{n,x}$ is a full matrix algebra by Theorem 5.3. Define
$\widetilde{\pi}_{k}\colon\mathscr{P}_{n,x}\to\mathbb{M}_{r_{k}}$ by
$\widetilde{\pi}_{k}=\pi_{k}\circ\widetilde{\theta}$. Then
$\widetilde{\pi}_{k}$ is unital and completely positive. For each $1\leq i\leq
n$, we have $q(\widetilde{\theta}(r_{i}))=\theta(r_{i})=q(A_{i})$, and hence
$\widetilde{\theta}(r_{i})-A_{i}\in\mathfrak{N}_{\tau}$. In particular,

$\displaystyle\lim_{k\to\infty}\mathopen{}\mathclose{{}\left\|\widetilde{\pi}_{k}(r_{i})-E_{i,k}}\right\|_{\rho_{k}}^{2}=\tau\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\widetilde{\theta}(r_{i})-A_{i}}\right)^{*}\mathopen{}\mathclose{{}\left(\widetilde{\theta}(r_{i})-A_{i}}\right)}\right)=0.$

For each $k\geq 1$, as $\widetilde{\pi}_{k}$ is a unital, completely positive
map between finite-dimensional C∗-algebras, Stinespring’s Theorem [Pau02,
Theorem 4.1] produces a finite-dimensional Hilbert space $\mathcal{H}_{k}$, a
$*$-homomorphism
$\sigma_{k}\colon\mathscr{P}_{n,x}\to\mathfrak{B}(\mathcal{H}_{k})$, and an
isometry $W_{k}\colon\mathbb{C}^{r_{k}}\to\mathcal{H}_{k}$ such that
$W_{k}^{*}\sigma_{k}(a)W_{k}=\widetilde{\pi}_{k}(a)$ for all
$a\in\mathscr{P}_{n,x}$. In particular, for each $k\geq 1$ and for all
$i\in[n]$, we have

(6.9)
$\displaystyle\lim_{k\to\infty}\mathopen{}\mathclose{{}\left\|W_{k}^{*}\sigma_{k}(r_{i})W_{k}-E_{i,k}}\right\|_{\rho_{k}}=0.$

Since $\mathscr{P}_{n,x}$ has a unique irreducible representation
$\mathscr{P}_{n,x}\to\mathbb{M}_{d}$ given by $r_{i}\mapsto\widetilde{P}_{i}$,
for each $k\geq 1$, there are a natural number $s_{k}\geq 1$ and unitary
$U_{k}\colon\mathcal{H}_{k}\to\mathbb{C}^{d}\otimes\mathbb{C}^{s_{k}}$ such
that $U_{k}^{*}(\widetilde{P}_{i}\otimes I_{s_{k}})U_{k}=\sigma_{k}(r_{i})$
for all $i\in[n]$. Define
$V_{k}=U_{k}W_{k}\colon\mathbb{C}^{r_{k}}\to\mathbb{C}^{d}\otimes\mathbb{C}^{s_{k}}$.
Then $V_{k}$ is an isometry, and for all $i\in[n]$, we can rewrite Equation
(6.9) as

$\displaystyle\lim_{k\to\infty}\big{\|}V_{k}^{*}(\widetilde{P}_{i}\otimes
I_{s_{k}})V_{k}-E_{i,k}\big{\|}_{\rho_{k}}=0.$

In particular, for some large $k\geq 1$, we have
$\big{\|}V_{k}^{*}(\widetilde{P}_{i}\otimes
I_{s_{k}})V_{k}-E_{i,k}\big{\|}_{\rho_{k}}\leq\epsilon$ for all $i\in[n]$,
which is a contradiction to Inequality (6.8). ∎

######  Remark 6.6.

The proof of Theorem 6.5 is by contradiction and therefore the relation
between $\epsilon$ and $\delta$ is not known. To find the explicit dependence
of $\delta$ on $\epsilon$ is left as an open problem.

With the analogue of the Gowers–Hatami Theorem in hand, we now proceed to
prove the claimed robustness result. We first need three intermediate lemmas.

######  Lemma 6.7.

Let $A\in\mathbb{M}_{d}$ be a positive matrix having at least two distinct
eigenvalues and let $\lambda_{1}>\lambda_{2}>\dots>\lambda_{l}\geq 0$ be the
distinct eigenvalues of $A$. Let $\xi\in\mathbb{C}^{d}$ be a unit vector such
that
$\lambda_{1}-\epsilon\leq\mathopen{}\mathclose{{}\left\langle{A\xi},{\xi}}\right\rangle$
for some $\epsilon\geq 0$. If $Q_{1}$ is the projection onto the eigenspace
corresponding to $\lambda_{1}$, then

$\displaystyle\mathopen{}\mathclose{{}\left\|Q_{1}\xi}\right\|^{2}\geq
1-\frac{\epsilon}{\lambda_{1}-\lambda_{2}}.$

###### Proof.

By the Spectral Theorem, $A=\sum_{i=1}^{l}\lambda_{i}Q_{i}$ where $Q_{i}$ be
the projection onto the eigenspace of $A$ corresponding to the eigenvalue
$\lambda_{i}$. Then,
$\mathopen{}\mathclose{{}\left\langle{A\xi},{\xi}}\right\rangle=\sum_{i=1}^{l}\lambda_{i}\mathopen{}\mathclose{{}\left\|Q_{i}\xi}\right\|^{2}$,
and hence

$\displaystyle\epsilon$
$\displaystyle\geq\lambda_{1}-\sum_{i=1}^{l}\lambda_{i}\mathopen{}\mathclose{{}\left\|Q_{i}\xi}\right\|^{2}=\lambda_{1}\sum_{i=1}^{l}\mathopen{}\mathclose{{}\left\|Q_{i}\xi}\right\|^{2}-\sum_{i=1}^{l}\lambda_{i}\mathopen{}\mathclose{{}\left\|Q_{i}\xi}\right\|^{2}=\sum_{i=2}^{l}(\lambda_{1}-\lambda_{i})\mathopen{}\mathclose{{}\left\|Q_{i}\xi}\right\|^{2}$
$\displaystyle\geq(\lambda_{1}-\lambda_{2})\sum_{i=2}^{l}\mathopen{}\mathclose{{}\left\|Q_{i}\xi}\right\|^{2}=(\lambda_{1}-\lambda_{2})\mathopen{}\mathclose{{}\left(1-\mathopen{}\mathclose{{}\left\|Q_{1}\xi}\right\|^{2}}\right).$

Rearranging this inequality gives our desired inequality. ∎

######  Lemma 6.8.

Let $\psi\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$ be a unit vector and
let $X_{1},X_{2}\in\mathbb{M}_{d_{A}}$ and $Y_{1},Y_{2}\in\mathbb{M}_{d_{B}}$.
Let $\rho_{A},\rho_{B}$ be the reduced density matrices of the density matrix
$\rho=\psi\psi^{*}$. Then

(6.10) $\displaystyle|\mathopen{}\mathclose{{}\left\langle{(X_{1}\otimes
Y_{1}-X_{2}\otimes Y_{2})\psi},{\psi}}\right\rangle|$
$\displaystyle\leq\|X_{1}-X_{2}\|_{\rho_{A}}\|Y_{2}^{*}\|_{\rho_{B}}+\|Y_{1}-Y_{2}\|_{\rho_{B}}\|X_{1}^{*}\|_{\rho_{A}},$
and (6.11) $\displaystyle\|(X_{1}\otimes Y_{1}-X_{2}\otimes Y_{2})\psi\|$
$\displaystyle\leq\|X_{1}-X_{2}\|_{\rho_{A}}\|Y_{2}\|+\|Y_{1}-Y_{2}\|_{\rho_{B}}\|X_{1}\|.$

###### Proof.

Using the triangle inequality and Cauchy–Schwarz inequality,

$\displaystyle\mathopen{}\mathclose{{}\left|\mathopen{}\mathclose{{}\left\langle{(X_{1}\otimes
Y_{1}-X_{2}\otimes Y_{2})\psi},{\psi}}\right\rangle}\right|$
$\displaystyle\leq\mathopen{}\mathclose{{}\left|\mathopen{}\mathclose{{}\left\langle{(I_{d_{A}}\otimes(Y_{1}-Y_{2}))\psi},{(X_{1}^{*}\otimes
I_{d_{B}})\psi}}\right\rangle}\right|$
$\displaystyle\qquad+\mathopen{}\mathclose{{}\left|\mathopen{}\mathclose{{}\left\langle{((X_{1}-X_{2})\otimes
I_{d_{B}})\psi},{(I_{d_{A}}\otimes Y_{2}^{*})\psi}}\right\rangle}\right|$
$\displaystyle\leq\|(I_{d_{A}}\otimes(Y_{1}-Y_{2}))\psi\|\|(X_{1}^{*}\otimes
I_{d_{B}})\psi\|$ $\displaystyle\qquad+\|((X_{1}-X_{2})\otimes
I_{d_{B}})\psi\|\|(I_{d_{A}}\otimes Y_{2}^{*})\psi\|$
$\displaystyle=\|Y_{1}-Y_{2}\|_{\rho_{B}}\|X_{1}^{*}\|_{\rho_{A}}+\|X_{1}-X_{2}\|_{\rho_{A}}\|Y_{2}^{*}\|_{\rho_{B}},$

where we used identities (2.1) and (2.2) in the last equation.

The second claimed inequality is derived as follows:

$\displaystyle\|(X_{1}\otimes Y_{1}-X_{2}\otimes Y_{2})\psi\|$
$\displaystyle\leq\|(X_{1}\otimes(Y_{1}-Y_{2}))\psi\|+\|((X_{1}-X_{2})\otimes
Y_{2})\psi\|$ $\displaystyle\leq\|X_{1}\otimes
I_{d_{B}}\|\|(I_{d_{A}}\otimes(Y_{1}-Y_{2}))\psi\|$
$\displaystyle\qquad+\|I_{d_{A}}\otimes Y_{2}\|\|((X_{1}-X_{2})\otimes
I_{d_{B}})\psi\|$
$\displaystyle=\|X_{1}\|\|Y_{1}-Y_{2}\|_{\rho_{B}}+\|Y_{2}\|\|X_{1}-X_{2}\|_{\rho_{A}},$

where the first inequality follows from the triangle inequality, and the
second follows from the definition of the operator norm. ∎

######  Lemma 6.9.

Let $\xi,\eta\in\mathbb{C}^{d}$ and let $P\in\mathbb{M}_{d}$ be a projection.
Suppose
$\mathopen{}\mathclose{{}\left|\mathopen{}\mathclose{{}\left\|\xi}\right\|^{2}-\mathopen{}\mathclose{{}\left\|\eta}\right\|^{2}}\right|\leq\epsilon_{1}$
and $\mathopen{}\mathclose{{}\left\|\xi-P\eta}\right\|\leq\epsilon_{2}$ for
some $\epsilon_{1},\epsilon_{2}\geq 0$. Then

(6.12)
$\displaystyle\mathopen{}\mathclose{{}\left\|\xi-\eta}\right\|\leq\epsilon_{2}+\sqrt{\epsilon_{1}+\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left\|\xi}\right\|+\mathopen{}\mathclose{{}\left\|\eta}\right\|}\right)\epsilon_{2}}.$

###### Proof.

Let $P^{\perp}=I_{d}-P$. Then
$\mathopen{}\mathclose{{}\left\|\eta}\right\|^{2}=\mathopen{}\mathclose{{}\left\|P\eta}\right\|^{2}+\mathopen{}\mathclose{{}\left\|P^{\perp}\eta}\right\|^{2}$,
and thus

$\displaystyle\mathopen{}\mathclose{{}\left\|P^{\perp}\eta}\right\|^{2}$
$\displaystyle=\mathopen{}\mathclose{{}\left\|\eta}\right\|^{2}-\mathopen{}\mathclose{{}\left\|P\eta}\right\|^{2}\leq\epsilon_{1}+\mathopen{}\mathclose{{}\left\|\xi}\right\|^{2}-\mathopen{}\mathclose{{}\left\|P\eta}\right\|^{2}=\epsilon_{1}+(\mathopen{}\mathclose{{}\left\|\xi}\right\|+\mathopen{}\mathclose{{}\left\|P\eta}\right\|)(\mathopen{}\mathclose{{}\left\|\xi}\right\|-\mathopen{}\mathclose{{}\left\|P\eta}\right\|)$
$\displaystyle\leq\epsilon_{1}+(\mathopen{}\mathclose{{}\left\|\xi}\right\|+\mathopen{}\mathclose{{}\left\|P\eta}\right\|)\mathopen{}\mathclose{{}\left\|\xi-P\eta}\right\|\leq\epsilon_{1}+(\mathopen{}\mathclose{{}\left\|\xi}\right\|+\mathopen{}\mathclose{{}\left\|\eta}\right\|)\epsilon_{2}.$

Thus,
$\mathopen{}\mathclose{{}\left\|P^{\perp}\eta}\right\|\leq\sqrt{\epsilon_{1}+(\mathopen{}\mathclose{{}\left\|\xi}\right\|+\mathopen{}\mathclose{{}\left\|P\eta}\right\|)\epsilon_{2}},$
using which in
$\mathopen{}\mathclose{{}\left\|\xi-\eta}\right\|\leq\mathopen{}\mathclose{{}\left\|\xi-P\eta}\right\|+\mathopen{}\mathclose{{}\left\|P^{\perp}\eta}\right\|$
proves the lemma. ∎

Finally, we show that the correlation $\widetilde{p}_{n,x}\in C_{q}^{s}(n,2)$
as in Definition 5.2 robustly self-tests the strategy
$\widetilde{\mathscr{S}}$ mentioned therein.

######  Theorem 6.10 (Main result).

Let $n\geq 3$ and $x\in\Lambda_{n}$. The synchronous correlation
$\widetilde{p}_{n,x}\in C_{q}^{s}(n,2)$ induced by the strategy
$\widetilde{\mathscr{S}}$ in Definition 5.2 self-tests
$\widetilde{\mathscr{S}}$ robustly.

###### Proof.

Theorem 6.1 states that $\widetilde{p}_{n,x}$ self-tests
$\widetilde{\mathscr{S}}$. We must show that for all $\epsilon>0$, there is a
$\delta>0$ such that if $p\in C_{q}(n,2)$ is a correlation induced by a
strategy $\mathscr{S}$ and $\|p-\widetilde{p}_{n,x}\|_{1}\leq\delta$, then
$\widetilde{\mathscr{S}}$ is a local $\epsilon$-dilation of $\mathscr{S}$. Fix
$\epsilon>0$.

With the notation of Definition 5.2, let

$\displaystyle\widetilde{N}=\sum_{v=1}^{n}\widetilde{P}_{v}\otimes\widetilde{P}_{v}^{T}\in\mathbb{M}_{d}\otimes\mathbb{M}_{d}$

and note that $\widetilde{N}$ is positive. By Lemma 5.7, $x$ is the largest
eigenvalue of $\widetilde{N}$, and the eigenspace corresponding to $x$ is the
one-dimensional space spanned by the maximally entangled state $\varphi_{d}$.
In particular, $\widetilde{N}$ has more than one eigenvalue—let
$\lambda_{2}\geq 0$ be the second largest eigenvalue of $\widetilde{N}$.

Let $\epsilon^{\prime}>0$ be such that
$\epsilon^{\prime}<(x-\lambda_{2})/(2n+1)$ and

(6.13) $\displaystyle
2\epsilon^{\prime}+\beta+\sqrt{5\epsilon^{\prime}+4\sqrt{\epsilon^{\prime}}+2\beta}\leq\epsilon\qquad\text{where}\qquad\beta\coloneqq\sqrt{\frac{2(2n+1)\epsilon^{\prime}}{x-\lambda_{2}}}.$

Apply Theorem 6.5 to produce $m\in\mathbb{N}$ and $\delta^{\prime}>0$ such
that Condition (6.7) holds with $\delta^{\prime}$ and $\epsilon^{\prime}$ in
place of $\delta$ and $\epsilon$. Then define

(6.14)
$\displaystyle\delta\coloneqq\min\mathopen{}\mathclose{{}\left\\{\epsilon^{\prime},\mathopen{}\mathclose{{}\left(\frac{\delta^{\prime}}{n+1}}\right)^{4},\mathopen{}\mathclose{{}\left(\frac{\delta^{\prime}}{2m}}\right)^{2}}\right\\}>0.$

Let $p\in C_{q}(n,2)$ be a correlation with
$\|p-\widetilde{p}_{n,x}\|_{1}\leq\delta$ and let

(6.15)
$\displaystyle\mathscr{S}=\big{(}\psi\in\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}},\\{E_{v},I_{d_{A}}-E_{v}\\}_{v=1}^{n},\\{F_{w},I_{d_{B}}-F_{w}\\}_{w=1}^{n}\big{)}$

be a strategy inducing $p$. We will show $\widetilde{\mathscr{S}}$ is a local
$\epsilon$-dilation of $\mathscr{S}$.

Let $\rho=\psi\psi^{*}\in\mathbb{M}_{d_{A}}\otimes\mathbb{M}_{d_{B}}$ and let
$\rho_{A}\in\mathbb{M}_{d_{A}}$ and $\rho_{B}\in\mathbb{M}_{d_{B}}$ be the
reduced density matrices of $\rho$. By Lemma 6.3, $(E_{1},\ldots,E_{n})$ is a
$(C\delta^{\frac{1}{4}},\rho_{A})$-representation of the relations given in
(5.2), where

$C\coloneqq\sqrt{n^{2}+(1+2x)\sqrt{\delta}}.$

As $x\leq n$, we have
$\delta\leq\epsilon^{\prime}<(x-\lambda_{2})/(2n+1)<\frac{1}{2}$. Now, since
$x\leq n$ and $\delta\leq 1$, we have

$C\leq\sqrt{n^{2}+2n+1}=n+1.$

Therefore, $C\delta^{\frac{1}{4}}\leq\delta^{\prime}$ by the choice of
$\delta$. Hence $(E_{1},\ldots,E_{n})$ is a
$(\delta^{\prime},\rho_{A})$-representation of the relations given in (5.2).
Therefore, (a) and (b) of Condition 6.7 hold with
$(\delta^{\prime},\rho_{A},d_{A})$ in place of $(\delta,\rho,r)$. Since
$2m\sqrt{\delta}\leq\delta^{\prime}$, Lemma 3.7 implies that part (c) of
Condition 6.7 also holds.

The conclusion of Condition 6.7 provides $r_{A}\in\mathbb{N}$ and an isometry
$V_{A}\colon\mathbb{C}^{d_{A}}\rightarrow\mathbb{C}^{d}\otimes\mathbb{C}^{r_{A}}$
with

(6.16) $\|E_{v}-V_{A}^{*}(\widetilde{P}_{v}\otimes
I_{r_{A}})V_{A}\|_{\rho_{A}}\leq\epsilon^{\prime}\qquad\text{for all
$v\in[n]$}.$

Similarly, there are $r_{B}\in\mathbb{N}$ and an isometry
$V_{B}\colon\mathbb{C}^{d_{B}}\rightarrow\mathbb{C}^{d}\otimes\mathbb{C}^{r_{B}}$
such that

(6.17) $\|F_{v}-V_{B}^{*}(\widetilde{P}_{v}^{T}\otimes
I_{r_{B}})V_{B}\|_{\rho_{B}}\leq\epsilon^{\prime}\qquad\text{for all
$v\in[n]$}.$

We work towards constructing a quantum state
$\psi_{\mathrm{junk}}\in\mathbb{C}^{r_{A}}\otimes\mathbb{C}^{r_{B}}$ such that

(6.18) $(V_{A}\otimes
V_{B})\psi\approx_{\epsilon}\varphi_{d}\otimes\psi_{\mathrm{junk}}.$

Note that Inequalities (6.16) and (6.17) together with Lemma 6.8 imply

$\displaystyle\mathopen{}\mathclose{{}\left|\mathopen{}\mathclose{{}\left\langle{\big{(}E_{v}\otimes
F_{w}-(V_{A}\otimes
V_{B})^{*}(\widetilde{P}_{v}\otimes\widetilde{P}_{w}^{T}\otimes
I_{r_{A}}\otimes I_{r_{B}})(V_{A}\otimes
V_{B})\big{)}\psi},{\psi}}\right\rangle}\right|$ (6.19)
$\displaystyle\leq\|E_{v}-V_{A}^{*}(\widetilde{P}_{v}\otimes
I_{r_{A}})V_{A}\|_{\rho_{A}}+\|F_{w}-V_{B}^{*}(\widetilde{P}_{w}^{T}\otimes
I_{r_{B}})V_{B}\|_{\rho_{B}}\leq 2\epsilon^{\prime}.$

Let $N=\sum_{v=1}^{n}E_{v}\otimes
F_{v}\in\mathbb{M}_{d_{A}}\otimes\mathbb{M}_{d_{B}}$. Taking $v=w$ in
Inequality (6.19) and summing over $v\in[n]$ yields

(6.20)
$\mathopen{}\mathclose{{}\left|\mathopen{}\mathclose{{}\left\langle{N\psi},{\psi}}\right\rangle-\mathopen{}\mathclose{{}\left\langle{(\widetilde{N}\otimes
I_{r_{A}}\otimes I_{r_{B}})(V_{A}\otimes V_{B})\big{)}\psi},{(V_{A}\otimes
V_{B})\psi}}\right\rangle}\right|\leq 2n\epsilon^{\prime}.$

Note that
$\mathopen{}\mathclose{{}\left\langle{N\psi},{\psi}}\right\rangle=\sum_{v=1}^{n}p(1,1|v,v)$.
Since $\|p-\widetilde{p}_{n,x}\|_{1}\leq\delta\leq\epsilon^{\prime}$ and
$\sum_{v=1}^{n}\widetilde{p}_{n,x}(1,1|v,v)=x$, Inequality 6.20 implies

(6.21) $\mathopen{}\mathclose{{}\left\langle{(\widetilde{N}\otimes
I_{r_{A}}\otimes I_{r_{B}})(V_{A}\otimes V_{B})\big{)}\psi},{(V_{A}\otimes
V_{B})\psi}}\right\rangle\geq x-(2n+1)\epsilon^{\prime}.$

Recall that by Lemma 5.7, $x$ is the largest eigenvalue of $\widetilde{N}$,
and hence also of $\widetilde{N}\otimes I_{r_{A}}\otimes I_{r_{B}}$. Let $Q$
be the projection on to the eigenspace of $\widetilde{N}\otimes
I_{r_{A}}\otimes I_{r_{B}}$ corresponding to $x$. Equation (6.21) and Lemma
6.7 yield

(6.22) $\alpha\coloneqq\|Q(V_{A}\otimes
V_{B})\psi\|\geq\mathopen{}\mathclose{{}\left(1-\frac{(2n+1)\epsilon^{\prime}}{x-\lambda_{2}}}\right)^{1/2}>0,$

where the last inequality holds by the choice of $\epsilon^{\prime}$. Then
$\alpha^{-1}Q(V_{A}\otimes V_{B})\psi$ is a unit vector in the eigenspace of
$\widetilde{N}\otimes I_{r_{A}}\otimes I_{r_{B}}$ corresponding to $x$. As the
eigenspace of $\widetilde{N}$ is spanned by $\varphi_{d}$, there is a unique
quantum state
$\psi_{\mathrm{junk}}\in\mathbb{C}^{r_{A}}\otimes\mathbb{C}^{r_{B}}$ such that
$Q(V_{A}\otimes V_{B})\psi=\alpha\varphi_{d}\otimes\psi_{\mathrm{junk}}$.

To simplify notation, let $\psi^{\prime}=(V_{A}\otimes V_{B})\psi$, and note
that

$\displaystyle\big{\|}\psi^{\prime}-\|Q\psi^{\prime}\|^{-1}Q\psi^{\prime}\big{\|}^{2}$
$\displaystyle=\|(1-Q)\psi^{\prime}\|^{2}+\big{|}1-\|Q\psi^{\prime}\|^{-1}\big{|}^{2}\|Q\psi^{\prime}\|^{2}$
$\displaystyle=1-\|Q\psi^{\prime}\|^{2}+(\|Q\psi^{\prime}\|^{-1}-1)^{2}\|Q\psi^{\prime}\|^{2}$
$\displaystyle=1-\|Q\psi^{\prime}\|^{2}+1-2\|Q\psi^{\prime}\|+\|Q\psi^{\prime}\|^{2}$
$\displaystyle=2(1-\|Q\psi^{\prime}\|).$

Therefore,

$\|(V_{A}\otimes
V_{B})\psi-\varphi_{d}\otimes\psi_{\mathrm{junk}}\|\leq\sqrt{2(1-\alpha)}\leq\sqrt{2(1-\alpha^{2})}.$

Combining this inequality with Inequality (6.22),

(6.23) $\|(V_{A}\otimes
V_{B})\psi-\varphi_{d}\otimes\psi_{\mathrm{junk}}\|\leq\sqrt{\frac{2(2n+1)\epsilon^{\prime}}{x-\lambda_{2}}}=\beta.$

In particular, Estimate (6.18) holds.

We now work to show

(6.24) $\displaystyle(V_{A}\otimes V_{B})(E_{v}\otimes
F_{w})\psi\approx_{\epsilon}\big{(}(\widetilde{P}_{v}\otimes\widetilde{P}_{w}^{T})\varphi_{d}\big{)}\otimes\psi_{\mathrm{junk}}$

for all $v,w\in[n]$. Fix $v,w\in[n]$ and, to declutter the notation, define

$V\coloneqq V_{A}\otimes V_{B}\qquad\text{and}\qquad
P_{v,w}\coloneqq\widetilde{P}_{v}\otimes\widetilde{P}_{w}^{T}\otimes
I_{r_{A}}\otimes I_{r_{B}}.$

By Inequalities (6.11), (6.16), and (6.17), we have

$\|V^{*}P_{v,w}V\psi-(E_{v}\otimes
F_{w})\psi\|\leq\|V_{A}^{*}(\widetilde{P}_{v}\otimes
I_{r_{A}})V_{A}-E_{v}\|_{\rho_{A}}+\|V_{B}^{*}(\widetilde{P}_{w}^{T}\otimes
I_{r_{B}})V_{B}-F_{w}\|_{\rho_{B}}\leq 2\epsilon^{\prime}.$

Since $V$ is an isometry, this implies

$\|VV^{*}P_{v,w}V\psi-V(E_{v}\otimes F_{w})\psi\|\leq 2\epsilon^{\prime}.$

Combining this with Inequality (6.23) and the triangle inequality, we have

(6.25) $\|V(E_{v}\otimes F_{w})\psi-
VV^{*}P_{v,w}(\varphi_{d}\otimes\psi_{\mathrm{junk}})\|\leq
2\epsilon^{\prime}+\beta.$

Note that

(6.26)
$\|P_{v,w}(\varphi_{d}\otimes\psi_{\mathrm{junk}})\|^{2}=\mathopen{}\mathclose{{}\left\langle{\mathopen{}\mathclose{{}\left(\widetilde{P}_{v}\otimes\widetilde{P}_{w}^{T}}\right)\varphi_{d}},{\varphi_{d}}}\right\rangle=\widetilde{p}_{n,x}(1,1|v,v).$

Further, since $p(1,1|v,w)=\mathopen{}\mathclose{{}\left\langle{(E_{v}\otimes
F_{w})\psi},{\psi}}\right\rangle$, Inequality (6.10) and parts (d) and (e) of
Lemma 3.5 imply

(6.27) $\mathopen{}\mathclose{{}\left|\|(E_{v}\otimes
F_{w})\psi\|^{2}-p(1,1|v,w)}\right|\leq\|E_{v}-E_{v}^{2}\|_{\rho_{A}}+\|F_{w}-F_{w}^{2}\|_{\rho_{B}}\leq
4\sqrt{\delta}\leq 4\sqrt{\epsilon^{\prime}}.$

Since $V$ is an isometry and
$\|p-\widetilde{p}_{n,x}\|_{1}\leq\delta\leq\epsilon^{\prime}$, Equation
(6.26) and Inequality (6.27) imply

(6.28) $\mathopen{}\mathclose{{}\left|\|V(E_{v}\otimes
F_{w})\psi\|^{2}-\|P_{v,w}(\varphi_{d}\otimes\psi_{\mathrm{junk}})\|^{2}}\right|\leq\epsilon^{\prime}+4\sqrt{\epsilon^{\prime}}.$

As $VV^{*}$ is a projection, Inequalities (6.25) and (6.28) and Lemma 6.9
imply

$\|V(E_{v}\otimes F_{w})\psi-
P_{v,w}(\varphi_{d}\otimes\psi_{\mathrm{junk}})\|\leq
2\epsilon^{\prime}+\beta+\sqrt{5\epsilon^{\prime}+4\sqrt{\epsilon^{\prime}}+2\beta}\leq\epsilon,$

and this proves Estimate (6.24). Similarly, for each $v,w\in[n]$, using the
estimates

$\displaystyle\big{\|}(I_{d_{A}}-E_{v})-V_{A}^{*}\big{(}(I_{d_{A}}-\widetilde{P}_{v})\otimes
I_{r_{A}}\big{)}V_{A}\big{\|}_{\rho_{A}}$
$\displaystyle\leq\epsilon^{\prime}\quad\text{and}$
$\displaystyle\big{\|}(I_{d_{B}}-F_{w})-V_{B}^{*}\big{(}(I_{d_{B}}-\widetilde{P}_{w}^{T})\otimes
I_{r_{B}}\big{)}V_{B}\big{\|}_{\rho_{B}}$
$\displaystyle\leq\epsilon^{\prime},$

which follow from Inequalities (6.16) and (6.17), one can show

$\displaystyle(V_{A}\otimes
V_{B})\big{(}E_{v}\otimes(I_{d_{B}}-F_{w})\big{)}\psi$
$\displaystyle\approx_{\epsilon}\Big{(}\big{(}\widetilde{P}_{v}\otimes(I_{d}-\widetilde{P}_{w}^{T})\big{)}\varphi_{d}\Big{)}\otimes\psi_{\mathrm{junk}},$
$\displaystyle(V_{A}\otimes V_{B})\big{(}(I_{d_{A}}-E_{v})\otimes
F_{w}\big{)}\psi$
$\displaystyle\approx_{\epsilon}\Big{(}\big{(}(I_{d}-\widetilde{P}_{v})\otimes\widetilde{P}_{w}^{T}\big{)}\varphi_{d}\Big{)}\otimes\psi_{\mathrm{junk}},\text{
and}$ $\displaystyle(V_{A}\otimes
V_{B})\big{(}(I_{d_{A}}-E_{v})\otimes(I_{d_{B}}-F_{w})\big{)}\psi$
$\displaystyle\approx_{\epsilon}\Big{(}\big{(}(I_{d}-\widetilde{P}_{v})\otimes(I_{d}-\widetilde{P}_{w}^{T})\big{)}\varphi_{d}\big{)}\otimes\psi_{\mathrm{junk}}.$

These last three estimates together with Estimates (6.18) and (6.24) show that
$\widetilde{\mathscr{S}}$ is a local $\epsilon$-dilation of $\mathscr{S}$. ∎

##  7\. Implications

We discuss some corollaries which result from the specific case of four
projections adding up to scalar times the identity.

######  Corollary 7.1.

Let $n=4$. For any $k\in\mathbb{N}$, there exists four rank $k$ projections
$\widetilde{P}_{k,1},\widetilde{P}_{k,2},\widetilde{P}_{k,3},\widetilde{P}_{k,4}$
in $\mathbb{M}_{2k+1}$ such that
$\widetilde{P}_{k,1}+\widetilde{P}_{k,2}+\widetilde{P}_{k,3}+\widetilde{P}_{k,4}=\frac{4k}{2k+1}I_{2k+1}$.
Each of the following quantum strategies

$\displaystyle\widetilde{\mathscr{S}}_{k}=(\varphi_{2k+1},\\{\widetilde{P}_{k,v},I_{2k+1}-\widetilde{P}_{k,v}\\}_{v=1}^{4},\\{\widetilde{P}_{k,w}^{T},I_{2k+1}-\widetilde{P}_{k,w}^{T}\\}_{w=1}^{4}),$

can be robustly self-tested from the correlations $\widetilde{p}_{4,k}$ that
each strategy induces.

###### Proof.

By the recurrence relation given in Theorem 5.1 it is readily computed that
$\Lambda_{4}=\mathopen{}\mathclose{{}\left\\{\frac{4k}{2k+1}}\right\\}_{k=0}^{\infty}$.
Observe that each fraction $\frac{4k}{2k+1}$ is already in lowest terms. Hence
by Theorem 5.1 there exist four projections
$\widetilde{P}_{k,1},\widetilde{P}_{k,2},\widetilde{P}_{k,3},\widetilde{P}_{k,4}$
in $\mathbb{M}_{2k+1}$ such that
$\widetilde{P}_{k,1}+\widetilde{P}_{k,2}+\widetilde{P}_{k,3}+\widetilde{P}_{k,4}=\frac{4k}{2k+1}I_{2k+1}$.
By Lemma 5.4, we observe that each of these projections is of rank $k$. Rest
of the corollary then follows from Theorem 6.10. ∎

The following two observations immediately follow from Corollary 7.1.

######  Corollary 7.2.

The maximally entangled state $\varphi_{d}$ in each odd dimension $d\geq 3$
can be robustly self-tested by quantum correlations with four inputs and two
outputs.

######  Corollary 7.3.

Given any natural number $k$ there exist four projections of rank $k$ which
can be robustly self-tested by quantum correlations with four inputs and two
outputs.

In general, a quantum state is represented by a density matrix
$\rho\in\mathbb{M}_{d}$. This reduces to the usual vector state formulation if
$\rho$ is rank one. Thus, in general, a description of a quantum strategy is
given by

$\displaystyle\mathscr{S}=\mathopen{}\mathclose{{}\left(\rho\in\mathbb{M}_{d_{A}}\otimes\mathbb{M}_{d_{B}},\\{E_{v,i}:v\in[n_{A}],i\in[k_{A}]\\},\\{F_{w,j}:w\in[k_{B}],j\in[k_{B}]\\}}\right),$

and its induced quantum correlation is given by
$p(i,j|v,w)=\operatorname{Tr}((E_{v,i}\otimes F_{w,j})\rho)$. One can show
that this quantum correlation $p\in C_{q}(n_{A},n_{B},k_{A},k_{B})$ induced by
$\mathscr{S}$ can also be obtained by a quantum strategy
$\mathscr{S}^{\prime}$ where the quantum state is now given by a unit vector
by using the notion of _purification_ [NC00, Section 2.5].

One can formulate a definition of self-testing which takes this more general
view of quantum states in to account. Indeed, this is how self-testing is
defined in [GKW+18, Definition C.1]. It is then proved there [GKW+18,
Proposition C.1] that if a quantum correlation $\widetilde{p}\in
C_{q}(n_{A},n_{B},k_{A},k_{B})$ self-tests a quantum strategy, then
$\widetilde{p}$ must be an extreme point of $C_{q}(n_{A},n_{B},k_{A},k_{B})$.

While in this article we don’t work444Also, we do not know whether Definition
4.5 is equivalent to the definition given in [GKW+18]. with the definition in
[GKW+18], one can show that all of our results can be suitably generalized.
More precisely, let $n\geq 3$ and $x\in\Lambda_{n}$ with $x=\frac{b}{d}$ in
lowest terms. Let $\widetilde{p}_{n,x}\in C_{q}^{s}(n,2)$ and
$\widetilde{\mathscr{S}}$ be as in Definition 5.2. One can show that
$\widetilde{p}_{n,x}$ self-tests the strategy $\widetilde{\mathscr{S}}$ in the
following sense as well. Consider a quantum strategy
$(\rho\in\mathbb{M}_{d_{A}}\otimes\mathbb{M}_{d_{B}},\\{E_{v},I_{d_{A}}-E_{v}\\}_{v=1}^{n},\\{F_{w},I_{d_{B}}-F_{w}\\}_{w=1}^{n})$
which also induces $\widetilde{p}_{n,x}$ and where $\rho$ is a density matrix.
Then, there exist isometries
$V_{A}\colon\mathbb{C}^{d_{A}}\to\mathbb{C}^{d}\otimes\mathcal{K}_{A}$ and
$V_{B}\colon\mathbb{C}^{d_{B}}\to\mathbb{C}^{d}\otimes\mathcal{K}_{B}$ for
some finite-dimensional Hilbert spaces $\mathcal{K}_{A},\mathcal{K}_{B}$, and
a density matrix
$\rho_{\mathrm{junk}}\in\mathfrak{B}(\mathcal{K}_{A}\otimes\mathcal{K}_{B})$
such that for all $v,w$, we have

(7.1) $\displaystyle(V_{A}\otimes V_{B})((E_{v}\otimes
F_{w})\rho)(V_{A}\otimes
V_{B})^{*}=((\widetilde{P}_{v}\otimes\widetilde{P}_{w}^{T})\widetilde{\rho})\otimes\rho_{\mathrm{junk}},$

where $\widetilde{\rho}=\varphi_{d}\varphi_{d}^{*}$. With this result, it
follows that for all $n\geq 3$ and $x\in\Lambda_{n}$, the quantum correlation
$\widetilde{p}_{n,x}$ is an extreme point of $C_{q}(n,2)$.

#### Acknowledgements.

L. M. and J. P. are supported by the Villum Fonden via the QMATH Centre of
Excellence (Grant No. 10059). C. S. is partially supported by NSF Grant
DMS-2000129. We thank the QIP 2021 referees for their feedback and Honghao Fu
for clarifying his results in [Fu19].

##  Appendix A A short introduction to C*-algebras

In this appendix we give some overview of C∗-algebras that we have used in
this article. Some general references on this subject are [Mur90, Pau02].

A _C ∗-algebra_ $\mathfrak{A}$ is a subset of the set of bounded linear
operators $\mathfrak{B}(\mathcal{H})$ for some Hilbert space $\mathcal{H}$
satisfying the following properties:

  1. (1)

$\mathfrak{A}$ is a unital (complex) algebra, that is, for
$S,T\in\mathfrak{A}$ and $\lambda\in\mathbb{C}$, we have $S+\lambda
T\in\mathfrak{A}$ and $\lambda S$, and there is an element
$1_{\mathfrak{A}}\in\mathfrak{A}$ which serves as the unit element for
$\mathfrak{A}$,

  2. (2)

$\mathfrak{A}$ is $*$-closed, that is, if $S\in\mathfrak{A}$, then its adjoint
$S^{*}\in\mathfrak{A}$ as well, and,

  3. (3)

$\mathfrak{A}$ is closed in the operator norm.

Sometimes, such a C∗-algebra is also called a _concrete C ∗-algebra_ or a _C
∗-algebra of operators_ to distinguish it from an abstract C∗-algebra which we
describe next.

Let $\mathfrak{A}$ be a unital Banach algebra, that is, $\mathfrak{A}$ is an
algebra equipped with a norm $\mathopen{}\mathclose{{}\left\|.}\right\|$
satisfying
$\mathopen{}\mathclose{{}\left\|ab}\right\|\leq\mathopen{}\mathclose{{}\left\|a}\right\|\mathopen{}\mathclose{{}\left\|b}\right\|$
for all $a,b\in\mathfrak{A}$, and $\mathfrak{A}$ is complete (every Cauchy
sequence converges). Assume that $\mathfrak{A}$ has an involution operation
$*:\mathfrak{A}\to\mathfrak{A}$ which satisfies
$1_{\mathfrak{A}}^{*}=1_{\mathfrak{A}}$ and

$\displaystyle(a+\lambda
b)^{*}=a^{*}+\overline{\lambda}b^{*},\qquad(ab)^{*}=b^{*}a^{*},\qquad\qquad(a^{*})^{*}=a,$

for all $a,b\in\mathfrak{A}$ and $\lambda\in\mathbb{C}$. We say that
$\mathfrak{A}$ is an _abstract C ∗-algebra_ if
$\mathopen{}\mathclose{{}\left\|a^{*}a}\right\|=\mathopen{}\mathclose{{}\left\|a}\right\|^{2}$
for all $a\in\mathfrak{A}$.

A linear map $\pi:\mathfrak{A}\to\mathfrak{B}$ between two abstract
C∗-algebras is called a _$*$ -homomorphism_ if it is unital
$\pi(1_{\mathfrak{A}})=1_{\mathfrak{B}}$, and $\pi(ab)=\pi(a)\pi(b)$ and
$\pi(a^{*})=\pi(a)^{*}$ for all $a,b\in\mathfrak{A}$.

Observe that $\mathfrak{B}(\mathcal{H})$ with the operator norm is an abstract
C∗-algebra and hence every concrete C∗-algebra is an abstract C∗-algebra.
Conversely, every abstract C∗-algebra can be identified with a concrete
C∗-algebra on some Hilbert space:

######  Theorem A.1 (Gelfand-Naimark-Segal).

Let $\mathfrak{A}$ be an abstract C∗-algebra. Then, there is some Hilbert
space $\mathcal{H}$ and an isometric $*$-homomorphism
$\pi:\mathfrak{A}\to\mathfrak{B}(\mathcal{H})$.

Thus, one may identify an abstract C∗-algebra $\mathfrak{A}$ with the concrete
C∗-algebra $\mathfrak{B}:=\pi(\mathfrak{A})$. A $*$-homomorphism from a
C∗-algebra $\mathfrak{A}$ into some $\mathfrak{B}(\mathcal{H})$ is called a
_representation_ of $\mathfrak{A}$.

A _state_ on a C∗-algebra $\mathfrak{A}$ is a linear functional
$\phi:\mathfrak{A}\to\mathbb{C}$ such that $\phi(a^{*}a)\geq 0$ for all
$a\in\mathfrak{A}$ and $\phi(1_{\mathfrak{A}})=1$. A state $\phi$ is called
_tracial_ if $\phi(ab)=\phi(ba)$ for all $a,b\in\mathfrak{A}$. We let
$S(\mathfrak{A})$ denote the set of all states on $\mathfrak{A}$. We say that
a net of states $(\phi_{\lambda})_{\lambda\in\Lambda}$ converges to a state
$\phi$ if $\lim_{\lambda}|\phi_{\lambda}(a)-\phi(a)|=0$ for all
$a\in\mathfrak{A}$, that is, if $\phi_{\lambda}$ converges pointwise to
$\phi$. This gives rise to a topology on $S(\mathfrak{A})$ called the _weak
∗-topology_, with respect to which $S(\mathfrak{A})$ is compact.

Let $\pi_{1}:\mathfrak{A}\to\mathfrak{B}(\mathcal{H}_{1})$ and
$\pi_{2}:\mathfrak{A}\to\mathfrak{B}(\mathcal{H}_{2})$ be two representations
of a C∗-algebra $\mathfrak{A}$. We say $\pi_{1}$ and $\pi_{2}$ are _unitarily
equivalent_ if there is some unitary $U:\mathcal{H}_{1}\to\mathcal{H}_{2}$
such that $\pi_{2}(a)=U\pi_{1}(a)U^{*}$ for all $a\in\mathfrak{A}$. A
representation $\pi:\mathfrak{A}\to\mathfrak{B}(\mathcal{H})$ is said to be
_irreducible_ if there is no invariant closed subspace of $\pi(\mathfrak{A})$
apart from $\\{0\\}$ and $\mathcal{H}$.

Give a non-empty set $A\subseteq\mathfrak{B}(\mathcal{H})$, we define the _C
∗-algebra generated by $A$_, denoted $C^{*}(A)$, to be the smallest
$C^{*}$-algebra containing $A$.

Let $G=\\{a_{1},\dots,a_{n}\\}$ be a non-empty set of _generators_ , and let
$R=\\{p_{1},\dots,p_{m}\\}$ be a finite set of _relations_ where $p_{i}$ is a
polynomial in $2n$ noncommuting variables
$a_{1},\dots,a_{n},a_{1}^{*},\dots,a_{n}^{*}$. Let $\mathfrak{A}$ be the free
$*$-algebra generated by $G$. Then an $n$-tuple $(T_{1},\dots,T_{n})$ in some
$\mathfrak{B}(\mathcal{H})$ satsfiying the relations gives rise to a
$*$-representation of $\mathfrak{A}$. For $a\in\mathfrak{A}$, define
$\mathopen{}\mathclose{{}\left\|a}\right\|=\sup\\{\mathopen{}\mathclose{{}\left\|\pi(a)}\right\|:\pi\text{
is a *-representation of }(G,R)\\}$. If $\|a_{i}\|<\infty$ for all $i\in[n]$,
then $\|\cdot\|$ is a C∗-seminorm on $\mathfrak{A}$, that is,
$\mathopen{}\mathclose{{}\left\|a^{*}a}\right\|=\mathopen{}\mathclose{{}\left\|a}\right\|^{2}$
for all $a\in\mathfrak{A}$. Letting
$N=\\{a\in\mathfrak{A}:\mathopen{}\mathclose{{}\left\|a}\right\|=0\\}$, the
completion of $\mathfrak{A}/N$ with respect to the induced norm is called the
_universal C ∗-algebra_ on $(G,R)$, denoted by $C^{*}(G,R)$. The universal
property says that whenever there is an $n$-tuple $(A_{1},\dots,A_{n})$ of
operators in $\mathfrak{B}(\mathcal{H})$ for some Hilbert space $\mathcal{H}$
satisfying the relations $R$, there is a representation
$\pi:C^{*}(G,R)\to\mathfrak{B}(\mathcal{H})$ with $\pi(a_{i})=A_{i}$ for all
$i\in[n]$. For more information on universal C∗-algebras we refer the reader
to [Bla06, Section II.8.3].

##  Appendix B A recipe to generate four projections adding up to scalars
times identity

For each $k\in\mathbb{N}$, we present a recipe to generate four projections
$\widetilde{P}_{k,1},\dots,\widetilde{P}_{k,4}$ such that
$\widetilde{P}_{k,1}+\dots+\widetilde{P}_{k,4}=\frac{4k}{2k+1}$ (see Corollary
7.1). This is a slightly simplified presentation from [KRS02]. We follow the
following iterative procedure.

We begin with four unit vectors
$\xi_{1},\xi_{2},\xi_{3},\xi_{4}\in\mathbb{R}^{3}$ which form the vertices of
a regular tetrahedron. These vectors satisfy
$\mathopen{}\mathclose{{}\left\langle{\xi_{i}},{\xi_{j}}}\right\rangle=\frac{-1}{3}$
for all $i\neq j$, and are given by

$\displaystyle\xi_{1}=\begin{bmatrix}1\\\ 0\\\
0\end{bmatrix},\quad\xi_{2}=\begin{bmatrix}\frac{-1}{3}\\\
\frac{2\sqrt{2}}{3}\\\
0\end{bmatrix},\quad\xi_{3}=\begin{bmatrix}\frac{-1}{3}\\\
\frac{-\sqrt{2}}{3}\\\
\sqrt{\frac{2}{3}}\end{bmatrix},\quad\xi_{4}=\begin{bmatrix}\frac{-1}{3}\\\
\frac{-\sqrt{2}}{3}\\\ -\sqrt{\frac{2}{3}}\end{bmatrix}.$

Set $\widetilde{P}_{1,v}=\xi_{v}\xi_{v}^{*}$ to be the rank-1 projection on
the subspace spanned by $\xi_{v}$, for $v\in[4]$. Then

$\displaystyle\widetilde{P}_{1,1}=\begin{bmatrix}1&0&0\\\ 0&0&0\\\
0&0&0\end{bmatrix},\quad\widetilde{P}_{1,2}=\begin{bmatrix}\frac{1}{9}&\frac{-2\sqrt{2}}{9}&0\\\
\frac{-2\sqrt{2}}{9}&\frac{8}{9}&0\\\
0&0&0\end{bmatrix},\quad\widetilde{P}_{1,3}=\begin{bmatrix}\frac{1}{9}&\frac{\sqrt{2}}{9}&\frac{-\sqrt{2}}{3\sqrt{3}}\\\
\frac{\sqrt{2}}{9}&\frac{2}{9}&\frac{-2}{3\sqrt{3}}\\\
\frac{-\sqrt{2}}{3\sqrt{3}}&\frac{-2}{3\sqrt{3}}&\frac{2}{3}\end{bmatrix},\quad\widetilde{P}_{1,4}=\begin{bmatrix}\frac{1}{9}&\frac{\sqrt{2}}{9}&\frac{\sqrt{2}}{3\sqrt{3}}\\\
\frac{\sqrt{2}}{9}&\frac{2}{9}&\frac{2}{3\sqrt{3}}\\\
\frac{\sqrt{2}}{3\sqrt{3}}&\frac{2}{3\sqrt{3}}&\frac{2}{3}\end{bmatrix}.$

One readily checks that
$\widetilde{P}_{1,1}+\dots+\widetilde{P}_{1,4}=\frac{4}{3}I_{3}$.

More generally, for $k\geq 1$, we show how to construct projections
$\widetilde{P}_{k+1,1},\dots,\widetilde{P}_{k+1,4}$ which sum up to
$\frac{4(k+1)}{2(k+1)+1}$ from the set of projections
$\widetilde{P}_{k,1},\dots,\widetilde{P}_{k,4}$. We begin with projections
$\widetilde{P}_{k,1},\dots,\widetilde{P}_{k,4}$ satisfying
$\widetilde{P}_{k,1}+\dots+\widetilde{P}_{k,1}=\frac{4k}{2k+1}I_{2k+1}$. Set
$\widetilde{Q}_{k,v}=I_{2k+1}-\widetilde{P}_{k,v}$. Observe that
$\mathrm{rank}(\widetilde{Q}_{k,v})=k+1$ since
$\mathrm{rank}(\widetilde{P}_{k,v})=k$ (Corollary 5.4). Thus if
$\widetilde{Q}_{k,v}=U_{k,v}DV_{k,v}^{*}$ is the singular decomposition with
$D=I_{k+1}\oplus 0_{k}$, we pick the first $k+1$ columns from each of
$U_{k,v}$ and join them laterally to form a larger matrix $\Gamma_{1}$ of size
$(2k+1)\times 4(k+1)$:

$\displaystyle\Gamma_{1}=\begin{bmatrix}\text{first }k+1\text{ columns of
}U_{k,1}&\dots&\text{first }k+1\text{ columns of }U_{k,4}\end{bmatrix}$

The null space of $\Gamma_{1}$ is then spanned by an orthonormal basis
$\\{\eta_{1},\dots,\eta_{2k+3}\\}$, and therefore

$\displaystyle\Gamma_{2}:=\begin{bmatrix}\eta_{1}^{T}\\\ \vdots\\\
\eta_{2k+3}^{T}\end{bmatrix}$

is a $(2k+3)\times 4(k+1)$ matrix. Let $R_{1}$ be the first $k+1$ columns of
$\Gamma_{2}$, $R_{2}$ be the second $k+1$ columns of $\Gamma_{2}$, and so on.
Finally, set

$\displaystyle\widetilde{P}_{k+1,v}=\frac{4(k+1)}{2(k+1)+1}R_{v}R_{v}^{*}$

for all $v\in[4]$, to get the desired set of projections.

We illustrate the procedure given above by constructing projections
$\widetilde{P}_{2,1},\dots,\widetilde{P}_{2,4}$ in $\mathbb{M}_{5}$ such that
$\widetilde{P}_{2,1}+\dots+\widetilde{P}_{2,4}=\frac{8}{5}I_{5}$.

Set $\widetilde{Q}_{1,v}=I_{3}-\widetilde{P}_{1,v}$ for each $v\in[4]$. Notice
that each $\widetilde{Q}_{1,v}$ is a rank-2 projection and is given by

$\displaystyle\widetilde{Q}_{1,1}=\begin{bmatrix}0&0&0\\\ 0&1&0\\\
0&0&1\end{bmatrix},\quad\widetilde{Q}_{1,2}=\begin{bmatrix}\frac{8}{9}&\frac{2\sqrt{2}}{9}&0\\\
\frac{2\sqrt{2}}{9}&\frac{1}{9}&0\\\
0&0&1\end{bmatrix},\quad\widetilde{Q}_{1,3}=\begin{bmatrix}\frac{8}{9}&\frac{-\sqrt{2}}{9}&\frac{\sqrt{2}}{3\sqrt{3}}\\\
\frac{-\sqrt{2}}{9}&\frac{7}{9}&\frac{2}{3\sqrt{3}}\\\
\frac{\sqrt{2}}{3\sqrt{3}}&\frac{2}{3\sqrt{3}}&\frac{1}{3}\end{bmatrix},\quad\widetilde{Q}_{1,4}=\begin{bmatrix}\frac{8}{9}&\frac{-\sqrt{2}}{9}&\frac{-\sqrt{2}}{3\sqrt{3}}\\\
\frac{-\sqrt{2}}{9}&\frac{7}{9}&\frac{-2}{3\sqrt{3}}\\\
\frac{-\sqrt{2}}{3\sqrt{3}}&\frac{-2}{3\sqrt{3}}&\frac{1}{3}\end{bmatrix}.$

For each $v\in[4]$, let $\widetilde{Q}_{1,v}=U_{1,v}DV^{*}_{1,v}$ be the
singular value decomposition of $\widetilde{Q}_{1,v}$, where
$D=\mathrm{diag}(1,1,0)$ and

$\displaystyle U_{1,1}$ $\displaystyle=\begin{bmatrix}0&0&1\\\ 0&1&0\\\
1&0&0\end{bmatrix},\qquad
U_{1,2}=\begin{bmatrix}0&\frac{2\sqrt{2}}{3}&\frac{-1}{3}\\\
0&\frac{1}{3}&\frac{2\sqrt{2}}{3}\\\ 1&0&0\end{bmatrix},$ $\displaystyle
U_{1,3}$
$\displaystyle=\begin{bmatrix}\sqrt{\frac{6}{7}}&\frac{-\sqrt{2}}{3\sqrt{7}}&\frac{-1}{3}\\\
0&\frac{\sqrt{7}}{3}&\frac{-\sqrt{2}}{3}\\\
\frac{1}{\sqrt{7}}&\frac{2}{\sqrt{21}}&\sqrt{\frac{2}{3}}\end{bmatrix},\qquad
U_{1,4}=\begin{bmatrix}-\sqrt{\frac{6}{7}}&\frac{-\sqrt{2}}{3\sqrt{7}}&\frac{1}{3}\\\
0&\frac{\sqrt{7}}{3}&\frac{\sqrt{2}}{3}\\\
\frac{1}{\sqrt{7}}&\frac{-2}{\sqrt{21}}&\sqrt{\frac{2}{3}}\end{bmatrix}.$

We pick the first two columns of $U_{1,v}$ (which span the column space of
$\widetilde{Q}_{1,v}$) and join them laterally to form a larger matrix:

$\displaystyle\Gamma_{1}=\begin{bmatrix}0&0&0&\frac{2\sqrt{2}}{3}&\sqrt{\frac{6}{7}}&\frac{-\sqrt{2}}{3\sqrt{7}}&-\sqrt{\frac{6}{7}}&\frac{-\sqrt{2}}{3\sqrt{7}}\\\
0&1&0&\frac{1}{3}&0&\frac{\sqrt{7}}{3}&0&\frac{\sqrt{7}}{3}\\\
1&0&1&0&\frac{1}{\sqrt{7}}&\frac{2}{\sqrt{21}}&\frac{1}{\sqrt{7}}&-\frac{2}{\sqrt{21}}\\\
\end{bmatrix}.$

Let $\\{\eta_{1},\dots,\eta_{5}\\}$ be an orthonormal basis of the null space
of $\Gamma_{1}$. Form the matrix

$\displaystyle\Gamma_{2}=\begin{bmatrix}\eta_{1}^{T}\\\ \vdots\\\
\eta_{5}^{T}\end{bmatrix}=\begin{bmatrix}2\sqrt{\frac{2}{89}}&-5\sqrt{\frac{3}{178}}&0&\sqrt{\frac{3}{178}}&0&0&0&\sqrt{\frac{42}{89}}\\\
-3\sqrt{\frac{10}{979}}&-7\sqrt{\frac{3}{9790}}&0&37\sqrt{\frac{3}{9790}}&0&0&\sqrt{\frac{178}{385}}&-8\sqrt{\frac{6}{34265}}\\\
-2\sqrt{\frac{30}{913}}&-\frac{\sqrt{\frac{83}{110}}}{2}&0&-\frac{17}{2\sqrt{9130}}&0&\frac{\sqrt{\frac{385}{166}}}{2}&-2\sqrt{\frac{42}{4565}}&-\frac{1}{2}\mathopen{}\mathclose{{}\left(17\sqrt{\frac{7}{9130}}}\right)\\\
-\frac{6}{\sqrt{415}}&0&0&-\sqrt{\frac{15}{83}}&\frac{\sqrt{\frac{83}{35}}}{2}&-\frac{1}{2}\mathopen{}\mathclose{{}\left(3\sqrt{\frac{3}{2905}}}\right)&\frac{33}{2\sqrt{2905}}&\frac{13\sqrt{\frac{3}{2905}}}{2}\\\
-\frac{3}{2\sqrt{10}}&0&\frac{\sqrt{\frac{5}{2}}}{2}&0&-\frac{3}{2\sqrt{70}}&-\sqrt{\frac{3}{70}}&-\frac{3}{2\sqrt{70}}&\sqrt{\frac{3}{70}}\end{bmatrix}.$

Set $R_{1}$ to be the first two columns of $\Gamma_{2}$, $R_{2}$ to be the
third and fourth columns of $\Gamma_{2}$, and so on. Finally, set
$\widetilde{P}_{2,v}=\frac{8}{5}R_{v}R_{v}^{*}$ for all $v\in[4]$, which are
given by

$\displaystyle\widetilde{P}_{2,1}$
$\displaystyle=\begin{bmatrix}\frac{364}{445}&-\frac{12}{89\sqrt{55}}&102\sqrt{\frac{3}{406285}}&\frac{-96}{5}\sqrt{\frac{2}{36935}}&-\frac{24}{5\sqrt{445}}\\\
-\frac{12}{89\sqrt{55}}&\frac{4188}{24475}&\frac{3562}{275}\sqrt{\frac{3}{7387}}&\frac{144}{5}\sqrt{\frac{2}{81257}}&\frac{36}{5\sqrt{979}}\\\
102\sqrt{\frac{3}{406285}}&\frac{3562}{275}\sqrt{\frac{3}{7387}}&\frac{11689}{22825}&\frac{96}{415}\sqrt{\frac{6}{11}}&\frac{24}{5}\sqrt{\frac{3}{913}}\\\
\frac{-96}{5}\sqrt{\frac{2}{36935}}&\frac{144}{5}\sqrt{\frac{2}{81257}}&\frac{96}{415}\sqrt{\frac{6}{11}}&\frac{288}{2075}&\frac{36}{25}\sqrt{\frac{2}{83}}\\\
-\frac{24}{5\sqrt{445}}&\frac{36}{5\sqrt{979}}&\frac{24}{5}\sqrt{\frac{3}{913}}&\frac{36}{25}\sqrt{\frac{2}{83}}&\frac{9}{25}\end{bmatrix},$
$\displaystyle\widetilde{P}_{2,2}$
$\displaystyle=\begin{bmatrix}\frac{12}{445}&\frac{444}{445\sqrt{55}}&\frac{-34}{5}\sqrt{\frac{3}{406285}}&-12\sqrt{\frac{2}{36935}}&0\\\
\frac{444}{445\sqrt{55}}&\frac{16428}{24475}&\frac{-1258}{275}\sqrt{\frac{3}{7387}}&\frac{-444}{5}\sqrt{\frac{2}{81257}}&0\\\
\frac{-34}{5}\sqrt{\frac{3}{406285}}&\frac{-1258}{275}\sqrt{\frac{3}{7387}}&\frac{289}{22825}&\frac{34}{415}\sqrt{\frac{6}{11}}&0\\\
-12\sqrt{\frac{2}{36935}}&-\frac{444}{5}\sqrt{\frac{2}{81257}}&\frac{34}{415}\sqrt{\frac{6}{11}}&\frac{24}{83}&0\\\
0&0&0&0&1\end{bmatrix},$ $\displaystyle\widetilde{P}_{2,3}$
$\displaystyle=\begin{bmatrix}0&0&0&0&0\\\ 0&0&0&0&0\\\
0&0&\frac{77}{83}&\frac{-3\sqrt{66}}{415}&\frac{-2}{5}\sqrt{\frac{33}{83}}\\\
0&0&\frac{-3\sqrt{66}}{415}&\frac{1976}{2075}&\frac{-33}{25}\sqrt{\frac{2}{83}}\\\
0&0&\frac{-2}{5}\sqrt{\frac{33}{83}}&\frac{-33}{25}\sqrt{\frac{2}{83}}&\frac{3}{25}\end{bmatrix},$
$\displaystyle\widetilde{P}_{2,4}$
$\displaystyle=\begin{bmatrix}\frac{336}{445}&-\frac{384}{445\sqrt{55}}&\frac{-476}{5}\sqrt{\frac{3}{406285}}&\frac{156}{5}\sqrt{\frac{2}{36935}}&\frac{24}{5\sqrt{445}}\\\
-\frac{384}{445\sqrt{55}}&\frac{18544}{24475}&\frac{-2304}{275}\sqrt{\frac{3}{7387}}&60\sqrt{\frac{2}{81257}}&-\frac{36}{5\sqrt{979}}\\\
\frac{-476}{5}\sqrt{\frac{3}{406285}}&\frac{-2304}{275}\sqrt{\frac{3}{7387}}&\frac{3367}{22825}&\frac{-97}{415}\sqrt{\frac{6}{11}}&\frac{-2}{5}\sqrt{\frac{3}{913}}\\\
\frac{156}{5}\sqrt{\frac{2}{36935}}&60\sqrt{\frac{2}{81257}}&\frac{-97}{415}\sqrt{\frac{6}{11}}&\frac{456}{2075}&\frac{-3}{25}\sqrt{\frac{2}{83}}\\\
\frac{24}{5\sqrt{445}}&-\frac{36}{5\sqrt{979}}&\frac{-2}{5}\sqrt{\frac{3}{913}}&\frac{-3}{25}\sqrt{\frac{2}{83}}&\frac{3}{25}\end{bmatrix}.$

Then, $\widetilde{P}_{2,1},\dots,\widetilde{P}_{2,4}$ are projections such
that $\widetilde{P}_{2,1}+\dots+\widetilde{P}_{2,4}=\frac{8}{5}I_{5}$.

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