text:
\section{Introduction}

The study of multi-prover games has led to many exciting results in
classical complexity theory. A one-round multi-prover \emph{cooperative game of incomplete information} is a game played by a verifier against two provers, Alice and Bob. The strategy of the verifier is fixed. He randomly chooses two questions according to some fixed probability distribution and sends one question to each prover. Alice and Bob then each return an answer to the verifier. The verifier decides whether to accept these answers on the basis of some pre-defined rules of the game that specify whether the given answers are winning answers for the questions sent. To win the game, Alice and Bob may thereby agree on any strategy beforehand, but they may no longer communicate once the game has started. The maximum probability with which Alice and Bob can cause the verifier to accept is known as the \emph{value} of the game. A simple example is the well-known CHSH game~\cite{chsh:nonlocal,cleve:nonlocal}. In this case, the questions and answers are bits. The verifier chooses questions $s \in \{0,1\}$ and $t \in \{0,1\}$ uniformly at random and sends $s$ to Alice and $t$ to Bob. In order to win the game, Alice and Bob must reply with bits $a, b \in \{0,1\}$ such that $s\wedge t = a \oplus b$, i.e., the logical AND of $s$ and $t$ should be equal to the XOR of $a$ and $b$. It is straightforward to verify that the CHSH game has value $3/4$.

Interactive proof systems have received considerable attention since
their introduction by Babai~\cite{babai:ip} and Goldwasser, Micali
and Rackoff~\cite{goldwasser:ip} in 1985.  Of special interest to us
are proof systems with \emph{multiple}
provers~\cite{babai:ipNexp,benor:ip,cai:ip,
feige:twoProverOneRound,feige:mip,lapidot:ip} as introduced by
Ben-Or, Goldwasser, Kilian and Widgerson~\cite{benor:ip}, which can
be described in terms of multi-prover games between a verifier, and
two or more provers. Whereas the provers are computationally
unbounded, the verifier is limited to probabilistic polynomial time.
Both the provers and the verifier have access to a common input
string $x$. The goal of the provers is to convince the verifier that
$x$ belongs to a pre-specified language $L$.  The verifier's aim, on
the other hand, is to determine whether the provers' claim is indeed
valid.  In each round, the verifier sends a poly$(|x|)$ size query to
the provers, who return a polynomial size answer. At the end of the
protocol, the verifier either accepts, meaning that he concludes $x
\in L$, or rejects, based on the messages exchanged and his own
private randomness. A language $L$ has a multi-prover interactive
proof system if there is a protocol such that, if $x \in L$, there
exist answers the provers can give which will cause the verifier to
accept with high probability. However, if $x \not \in L$, then there
exists no strategy for the provers that will only cause the verifier
to accept, except with very low probability.
Here, $x$ and $L$ lead to particular game.
Let MIP\index{MIP} denote the class of languages having a
multi-prover interactive proof system. It has been shown that
classical two-prover interactive proof systems are just as powerful
as proof systems involving more than two provers.
Indeed, Babai, Fortnow and
Lund~\cite{babai:ipNexp}, and Feige and Lov\'asz~\cite{feige:mip}
have shown that a language is in NEXP if and only if it has a
\emph{two}-prover one-round proof
system, i.e., MIP $=$ NEXP.

\subsection{Multi-prover games with entanglement}

In this paper, we study multi-prover games in a quantum setting. In
particular, we allow Alice and Bob to share an entangled quantum
state as part of their strategy. After receiving their questions, the
provers may perform any local measurement on their part of the
entangled state, and decide on an answer based on the outcome of
their measurement. All communication between the verifier and the
provers remains classical. It turns out that sharing entanglement can
increase the probability that the provers can cause the verifier to
accept, an effect known as quantum \emph{nonlocality}. For example,
if the provers share a maximally entangled state of two qubits they
can win the CHSH game (cause the verifier to accept) with probability
$p^*_{\mbox{\tiny CHSH}}\approx 85\% > 3/4$.

We write $\mbox{MIP}^*$ for the set of languages that have
interactive proofs with entangled provers. Very little is known about
MIP$^*$. Most importantly, it was not known before our work whether there exists an algorithm of any complexity for deciding membership in
MIP$^*$, except for extremely restricted classes of games. In particular, if we restrict to games where Alice and Bob each answer a single bit $a,b \in \{0,1\}$, and the verifier only looks at the XOR of these two bits, then the (entangled) value of the game can be computed in time polynomial in the number of questions~\cite{tsirel:separated,cleve:nonlocal}.  Let
$\oplus\mbox{MIP}[2]$\index{MIP[2]} denote the restricted class where
the verifier's output is a function of the XOR of two binary answers.
Then $\oplus\mbox{MIP}^*[2] \subseteq \mbox{EXP}$~\cite{cleve:nonlocal}, while it is known that
$\oplus\mbox{MIP} = \mbox{NEXP}$, for certain completeness and
soundness parameters~\cite{Hastad:502098}, i.e., the resulting
proof system is significantly weakened if the provers are allowed to
share entanglement. In fact, such a proof system can even be
simulated using just a single quantum prover, i.e.,
$\oplus\mbox{MIP}^*[2] \subseteq \mbox{QIP}(2) \subseteq
\mbox{EXP}$~\cite{wehner05c,kitaev&watrous:qip}.

Unfortunately, very little is known for more general games where we allow shared entanglement between the provers, and where Alice and Bob give much longer answers, or when we have more than two provers. Unlike in the classical case, general multi-prover games are not equivalent to two-prover games when the provers share entanglement~\cite{julia:mip}. For two-prover unique games (where for each pair of questions there exists exactly one pair of winning answers), it is known that we can approximate the value of a two-prover game to within a certain accuracy in polynomial time~\cite{oded:unique}. Masanes~\cite{masanes:nonlocal} has shown how to compute the value of multi-prover games where the questions to, and the answers from each prover are bits. But even for very small games with a very limited number of questions, the entangled value is typically unknown~\cite{massar:tsirel}.

Assuming that the provers share quantum entanglement is a reasonable
model because it captures the properties of a multi-prover game that
a verifier can enforce physically: while the verifier can enforce the
condition that the provers cannot communicate by ensuring that they are
spacelike-separated, he has no way to ensure that provers in a
quantum universe do not share entanglement. Multi-prover games with
entangled provers are also known as \emph{non-local games} with
entanglement. The $\emph{entangled value}$ of a game is the maximum
probability with which Alice and Bob can win using entanglement.
Here, we are concerned with the following question: How can we
compute the entangled value of non-local games with multiple provers?
And, how can we decide membership of MIP$^*$?

\subsection{Results}

\paragraph{Quantum moment problem.}

To reach our goal, we first introduce the \emph{quantum moment
problem} that is a generalization of our problem. Informally, the
quantum moment problems asks whether given a conditional probability
distributions and some polynomial constraints on observables, we can
find a quantum state and quantum measurements satisfying the
constraints, that provide us with the required probabilities. We may
use the constraints to impose certain restrictions on the form of our
quantum measurements. For example, we may want to demand that two
measurement operators act on two or more distant subsystems
independently. Determining whether there is an entangled strategy for
a multi-prover game that achieves a certain winning probability is a
special case of the quantum moment problem.

Other special cases of the quantum moment problem include the
so-called classical marginal problem~\cite{pitowsky91:_correl}, which
asks whether given certain marginal distributions, we can find a
joint distribution that has the desired marginals. Our problem is
also closely related to the \emph{quantum} marginal problem in which
the aim is to find a density matrix for a multipartite quantum system
that is consistent with a specified set of reduced density matrices
for specific subsystems. This problem is QMA-complete and has
attracted a lot of interest recently~\cite{matthias:qma}. A
special case is $N$-representability, an important problem with a
long history in quantum chemistry~\cite{klyachko}.
One key difference between our quantum moment problem and the quantum
marginal problem is that in the latter case  the dimension of the
quantum state, and its various subsystems, is specified. In the
quantum moment problem the aim is to find a state satisfying the
given constraints in a quantum system of any, possibly infinite,
dimension. Finally, it may also be possible to treat games with
\emph{quantum} verifier within the framework of the quantum moment
problem.

\paragraph{Refuting unsatisfiable instances.}
We describe a general way of proving that an instance of a quantum
moment problem is unsatisfiable. The proof follows from a recent result of
Helton and McCullough~\cite{helton04:_posit_for_non_commut_polyn}, a
Positivstellensatz for polynomials in noncommuting variables. The
choice of polynomials will define a particular instance of the
quantum moment problem, where the variables correspond to measurement
operators. In Helton and McCullough's result the noncommuting
variables are required to satisfy certain polynomial equality and
inequality constraints but can be evaluated in any quantum system,
even one of infinite dimensions. Informally, the Positivstellensatz states that any
such polynomial that is positive can be written a sum of squares, a
form that makes it obvious that the polynomial is positive. By
positive we mean that, whenever the constraints are satisfied, the polynomial is positive semidefinite, i.e. it has a positive expectation value
for all quantum states in whatever quantum system.
Such a representation as a sum of squares acts as a
certificate for the unsatisfiability of an instance of a quantum
moment problem. Certificates of this kind have often been used in the
theoretical physics literature to place very general bounds on
quantum correlations (see for
example~\cite{glauber1963a}).
Helton and McCullough's result shows that
such certificates are all that is ever required to demonstrate that
an instance of the quantum moment problem is unsatisfiable.

\paragraph{Tensor products and commutation.}
In order to apply the Positivstellensatz to obtain bounds on the
entangled values of two-prover games, we need to incorporate a
constraint in the corresponding quantum moment problem that ensures
Alice and Bob's measurements act on different subsystems $\mathcal{H}_A$ and $\mathcal{H}_B$. When Alice
and Bob share a quantum system of some finite dimension, this means
that one demands that the Hilbert space $\mathcal{H}$ describing this system
decomposes as $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$. Alice and Bob's
measurements should be of the form $A_s \otimes B_t$ with $A_s \in \mathcal{H}_A$
and $B_t \in \mathcal{H}_B$ for all questions $s$ and $t$. Unfortunately, we
can only apply the Positivstellensatz when the constraints are
polynomials in $A_s$ and $B_t$. Thus we need an additional trick to impose an
explicit tensor product structure.
To get around this problem, we
demand that for all $s$ and $t$ we have $[A_s,B_t]=0$, i.e.,
that all measurement operators of Alice commute with all those of
Bob. If the Hilbert space is finite-dimensional, then imposing the
commutativity constraints is actually \emph{equivalent} to demanding
a tensor product structure. This result is well-known in the
mathematical physics literature~\cite{summers:qftIndep}. Here, we
 provide a simple version of this argument accessible
from a computer science perspective, which directly applies to our
analysis of multi-prover games.

From a physics point of view, however, the usual requirement on
observables that can be measured in space-like separated regions is
that they should \emph{commute}, not that they should have a tensor
product factorization. Indeed, this commutativity requirement on
local observables is regarded by many as an axiom that should be
satisfied by any reasonable quantum mechanical theory of
nature~\cite{haag1996a}. Unfortunately, when the algebra of
observables cannot be represented on a finite-dimensional Hilbert
space it is an open question whether this commutativity property
implies the existence of a tensor product factorization. Our results
will provide bounds on the values of multi-prover games that are
valid for all quantum systems whenever the observables of different
players commute. We will refer to
the maximum probability of winning a game $G$ with (possibly)
infinite-dimensional operators satisfying commutativity constraints
as the
\emph{field-theoretic value} $\ftv(G)$ of the game.
It is an open question whether this is the same as
the usual entangled value of the game. Since MIP$^*$ was defined with
a tensor product structure in mind, we here define the class MIP$^f$,
where the tensor products are replaced by the commutative requirement. The
class MIP$^f$ seems more appropriate to our main motivation of
studying the power of multi-prover games where the provers are only
limited by what they can achieve physically. Restricting Alice and
Bob to sharing finite-dimensional systems does not seem natural from
a physical perspective.

\paragraph{A hierarchy of semidefinite programs.}
We know that the Positivstellensatz leads to certificates that tell
us when a particular quantum moment problem is unsatisfiable. But how
can we find such certificates? If we place a bound on the size of the
certificate, then the problem of determining whether there exists a
certificate of that size can be formulated as a semidefinite program
(SDP)~\cite{L.Vandenberghe:SR:1996,Boyd:04}.  In particular,
searching for certificates of increasing size yields a hierarchy of
SDPs. The resulting hierarchy is very similar to the one presented in
a groundbreaking paper of Navascu\'es, Pironio and
Ac\'in~\cite{navascues:_bound}, which partly motivated this work.

In many applications, including multi-prover games, we are not only
interested in whether a specific instance of the moment problem is
satisfiable but in finding the best possible bound on some linear
combination of moments. Once again fixing the size of the
certificates of infeasibility straightforwardly leads to a hierarchy
of SDPs that provide progressively tighter bounds. For multi-prover
games $G$, it was previously not known whether the solutions to this
hierarchy of SDPs converged to the entangled value of the game, which
we denote by $\omega^*(G)$. Here we \emph{almost} show this. What we
actually show is that the hierarchy converges to the
field-theoretic value (see above) of a non-local game $G$.
In the language of the quantum moment problem, we wish to know if there exists an entangled strategy for $G$
such that the provers win with some fixed probability $p$.
However, the
Positivstellensatz only yields a certificate that there is no
entangled strategy that wins with probability $p$ if there is also no
such strategy even \emph{with infinite-dimensional
  measurement operators}. If the measurement operators are
infinite-dimensional, then the commutativity constraints do not
necessarily imply the existence of a tensor product
structure.
In other words, we show that our hierarchy converges to the field-theoretic value of the game.

\paragraph{MIP$^*$ is recursive}

Since our hierarchy converges, we can compute the value of an
entangled game and hence obtain an algorithm for deciding membership
of MIP$^*$ (under the assumption that the optimal value is achieved
with finite-dimensional operators) and of MIP$^f$. This implies that
these classes are recursive.

\paragraph{Examples: The $I_{3322}$ inequality and Yao's inequality.}
Finally, we demonstrate the power of our technique by providing an
extremely simple, algorithmically constructed, certificate bounding
the value of a two-party Bell inequality~\cite{Bell:64a} known as the $I_{3322}$ inequality~\cite{D.Collins:JPA:2004}, and a multi-player non-local game suggested by Yao and collaborators~\cite{yao:inequality}.

\subsection{Open Questions}
With respect to the above discussion, it would be interesting to
know whether there are games $G$ such that $\omega^*(G)$ is strictly
less than $\ftv(G)$. Can it really help the provers to have
infinite-dimensional systems when the number of questions and answers
in the game are finite? One way to establish that there is no
advantage to having infinite-dimensional systems would be to `round'
the SDP hierarchy directly to a quantum strategy with finite
entanglement, bypassing the (nonconstructive) Positivstellensatz
altogether. For XOR-games, the first level of our hierarchy is tight
and it is well-known how a solution of the SDP can be transformed
into a quantum strategy via so-called Tsirelson's vector
construction~\cite{tsirel:original, tsirel:separated,tsirel:hadron}.
However, there exist many non-local games, for which the first level
of the hierarchy does not provide us with the optimal value of the
game, but merely gives us an upper bound.  This fact alone shows that
for general games, we cannot find such a nice embedding of vectors
into observables as can be done for XOR-games. However, something
similar may still be possible for restricted classes of games, exhibiting
a likewise special structure.

We also do not establish anything about the rate of convergence of
the SDP hierarchy. In some numerical experiments with small games,
the low levels of the SDP hierarchy do yield optimal solutions.
Establishing this in general would provide an upper bound on
$\mbox{MIP}^*$. We have made partial progress on this question by
proving convergence for a particular hierarchy of SDPs.

\subsection{Related work}
In Ref.~\cite{navascues:_bound}, a paper which partly inspired this
work, Navascu\'es, Pironio, and Ac{\'\i}n (NPA), defined a closely
related semidefinite programming hierarchy.  Subsequently, and
independently of us, NPA have proved that their semidefinite
programming hierarchy converges to the field-theoretic value of the
game~\cite{navascues08:long}. Our paper and theirs are complementary:
While our work emphasizes the connection with Positivstellensatz of
Helton and McCullough, NPA prove convergence directly. Their proof
has a number of advantages: most notably, when their hierarchy
converges to the field-theoretic value of the game at a finite level,
NPA obtain a bound on the dimension of the state required to
reproduce the correlations.
NPA have also shown that their new technique for proving
convergence can be extended to general polynomial optimization
problems in noncommutative variables~\cite{navascues08:general}.

Finally, our techniques have recently extended by
Ito, Hirotada, and Matsumoto to the case of games with quantum messages between
verifier and provers~\cite{Ito08}.

\subsection{Outline}

In Section~\ref{prelim}, we provide an introduction to non-local
games including all necessary definitions. Section~\ref{qmpDef} then
defines the quantum moment problem, and Section~\ref{section:tools} introduces our main tools.
In particular, Section~\ref{section:commutation} provides an explanation of why
we obtain a tensor product structure from commutation relations, and in
Section~\ref{section:satz}, we show that if a quantum moment problem
is unsatisfiable, we can find certificates of this fact using the
Positivstellensatz. We then use these tools in our SDP hierarchy in
Section~\ref{findingBounds} and conclude in
Section~\ref{Sec:Examples} with some explicit examples.

\section{Preliminaries}\label{prelim}

\subsection{Notation}

We assume general familiarity with the quantum
model~\cite{nielsen&chuang:qc}. In the following, we use $A^\dagger$
to denote the conjugate transpose of a matrix $A$.
A matrix is \emph{Hermitian} if and only if $A^\dagger = A$. We write
$A \geq 0$ to indicate that a matrix $A$ is \emph{positive
semidefinite}, i.e., it is Hermitian and has no negative eigenvalues.
We also use $A =
0$ to express that $A$ is the all-zero matrix and $A \neq 0$ to
indicate that $A$ has at least one non-zero entry. The $(i,j)$--entry
of $A$ will be denoted by $[A]_{i,j}$. For two matrices $A$ and $B$
we write their commutator as $[A,B] = AB - BA$. We use $\mathcal{H}$ to
denote a Hilbert space and $\mathcal{H}_{k}$ the Hilbert space belonging to
subsystem $k$. $\mathbb{I}_{k}$ is the identity on system $k$, and
$\mathbb{B}(\mathcal{H})$ denotes the set of all bounded operators on the Hilbert
space $\mathcal{H}$. Unless stated otherwise, we take all systems to be
finite-dimensional. We will also employ the shorthand
$$
\mathbb{B}(\mathcal{H})^{\times n} := \underbrace{\mathbb{B}(\mathcal{H}) \times \ldots \times \mathbb{B}(\mathcal{H})}_{n}
$$
for the $n$-fold Cartesian product of $\mathbb{B}(\mathcal{H})$,
and let $[n] := \{1,\ldots,n\}$. Furthermore, we will use $|\mathcal{S}|$
and $|\mathcal{L}|$ to denote the number of elements of a set $\mathcal{S}$ and
list $\mathcal{L}$ respectively.

For the purpose of subsequent discussion, we now note that a
Hermitian polynomial $p(X)$ in noncommutative variables $X =
(X_1,\ldots,X_k)$ is a \emph{sum of squares} (SOS) if there exist
polynomials (matrices) $r_j$ of appropriate dimension such that
$$
p(X) = \sum_j r_j^\dagger r_j.
$$
It is important to note that if $p(X)$ is an SOS polynomial, it is
also a positive semidefinite matrix, i.e., $p(X) \geq 0$.

\subsection{Games}

As an example application of the quantum moment
problem, we will consider cooperative games among $n$ parties. For
simplicity, we first describe the setting for only two parties,
henceforth called Alice and Bob. A generalization is straightforward.
Let $S$, $T$, $A$ and $B$ be finite sets, and $\pi$ a probability
distribution on $S \times T$. Let $V$ be a predicate on $S \times T
\times A \times B$. Then $G = G(V,\pi)$ is the following two-person
cooperative game: A pair of questions $(s,t) \in S \times T$ is
chosen at random according to the probability distribution $\pi$.
Then $s$ is sent to Alice, and $t$ to Bob. Upon receiving $s$, Alice
has to reply with an answer $a \in A$. Likewise, Bob has to reply to
question $t$ with an answer $b \in B$. They win if $V(s,t,a,b) = 1$
and lose otherwise. Alice and Bob may agree on any kind of strategy
beforehand, but they are no longer allowed to communicate once they
have received questions $s$ and $t$. The \emph{value} $\omega(G)$ of
a game $G$ is the maximum probability that Alice and Bob win the
game. In what follows, we will write $V(a,b|s,t)$ instead of
$V(s,t,a,b)$ to emphasize the fact that $a$ and $b$ are answers given
questions $s$ and $t$.

Here, we are particularly interested in non-local games and where
Alice and Bob are allowed to share an arbitrary entangled state
$\ket{\psi}$ to help them win the game. Let $\mathcal{H}_A$ and $\mathcal{H}_B$
denote the Hilbert spaces of Alice and Bob respectively. The state
$\ket{\psi} \in \mathcal{H}_A \otimes \mathcal{H}_B$ is part of the quantum
strategy that Alice and Bob can agree on beforehand. This means that
for each game, Alice and Bob can choose a specific $\ket{\psi}$ to
maximize their chance of success. In addition, Alice and Bob can
agree on quantum measurements where we may without loss of generality
assume that these are projective
measurements~\cite{cleve:nonlocal}.\footnote{By Neumark's theorem,
any generalized measurements described by positive-operator-valued measure can be implemented as projective
measurements in some higher dimensional Hilbert space. See, for
example, pp. 285 of Ref.~\cite{A.Peres:Book:1995}.} For each $s \in
S$, Alice has a projective measurement described by $ \{A_s^a: a \in
A\} $ on $\mathcal{H}_A$. For each $t \in T$, Bob has a projective
measurement described by $ \{B_t^b: b \in B\} $ on $\mathcal{H}_B$. For
questions $(s,t) \in S \times T$, Alice performs the measurement
corresponding to $s$ on her part of $\ket{\psi}$ which gives her
outcome $a$. Likewise, Bob performs the measurement corresponding to
$t$ on his part of $\ket{\psi}$ with outcome $b$. Both send their
outcome, $a$ and $b$, back to the verifier. The probability that
Alice and Bob answer $(a,b) \in A \times B$ is then given by
$$
\bra{\psi}A_s^a \otimes B_t^b\ket{\psi}.
$$
We can now define:

\begin{definition}\label{definition:1}
The \emph{entangled value} of a two-prover game with classical
verifier $G = G(\pi, V)$ is given by:
\begin{align}\label{Eq:omega*}
    \omega^*(G) =& \lim_{d \to \infty}
    \max_{\stackrel{\ket \psi \in \mathbb{C}^d \otimes \mathbb{C}^d}{\|\,\ket\psi\,\| =1}}
\max_{\aoq, B_t^b}
    \sum_{a,b,s,t} \pi(s,t) V(a,b|s,t) \bra \psi \aoq \otimes B_t^b \ket \psi,
\end{align}
where $\aoq \in \mathbb{B}(\mathcal{H}_A)$ and $B_t^b \in \mathbb{B}(\mathcal{H}_B)$ for some
Hilbert space $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$,
satisfying $\aoq,B_t^b \geq 0$, $\sum_a \aoq = \mathbb{I}_A$, $\sum_b
B_t^b = \mathbb{I}_B$ for all $s \in S$ and $t \in T$.
\end{definition}

We also define a more general version of this statement, which will
provide an upper bound to the quantum value of the game:

\begin{definition}
\label{Dfn:ftv} The \emph{field-theoretic value} of a two-prover game
with classical verifier $G = G(\pi, V)$ is given by:
\begin{align}
    \ftv(G) =& \sup_{\aoq, B_t^b}
\Bigl\|
\sum_{a,b,s,t} \pi(s,t) V(a,b|s,t)
    \aoq B_t^b \Bigr\|,
\end{align}
where $\|O\|$ is the operator norm of $O$, $\aoq \in \mathbb{B}(\mathcal{H})$ and $B_t^b\in \mathbb{B}(\mathcal{H})$ for some Hilbert
space $\mathcal{H}$,
satisfying $\aoq, B_t^b \geq 0$,
$\sum_a \aoq = \sum_b B_t^b = \mathbb{I}$ for all $s,t$, and $[\aoq, B_t^b ] =
0$ for all $s\in S$, $t \in T$, $a \in A$, and $b \in B$.
\end{definition}

\begin{lemma}
  \label{lemma:1}
    Let $G = G(\pi, V)$ be a two-prover game with classical verifier.
    Then $\omega^*(G) \leq\ftv(G)$.
\end{lemma}
\begin{proof}
Let $\varepsilon > 0$. Choose $d$ sufficiently large so that there is a
normalized state $\ket \psi$ and operators $\aoq$, $B_t^b$ defining a
strategy with winning probability at least $\omega^*(G) - \varepsilon$. Let
$\hataoq = \aoq \otimes \mathbb{I}_B$ and $\hat B_t^b = \mathbb{I}_A \otimes B_t^b$.
Then $\hataoq$ and $\hat B_t^b$ are positive semidefinite operators on
$\mathbb{C}^{d^2}$ satisfying all the conditions in
Definition~\ref{Dfn:ftv}. Finally,
\begin{align*}
    \ftv(G) =& \sup_{\tilde{\aoq}, \tilde{B_t^b}} \Bigl\|\sum_{a,b,s,t} \pi(s,t) V(a,b|s,t)
    \tilde{\aoq} \tilde{B_t^b} \Bigr\|\\
    \geq&\, \Bigl\|\sum_{a,b,s,t} \pi(s,t) V(a,b|s,t)
    \hataoq \hat B_t^b \Bigr\|\\
    \geq&\, \bra \psi \Bigl( \sum_{a,b,s,t} \pi(s,t) V(a,b|s,t)
    \hataoq \hat B_t^b \Bigr) \ket \psi\\
    \geq&\, \omega^*(G) - \varepsilon.
\end{align*}
Since $\varepsilon$ was arbitrary, the result follows.
\end{proof}

In our examples, we will sometimes use the term \emph{Bell inequality}\cite{Bell:64a} to refer to a particular non-local game. This is an equivalent formulation, where we only consider terms of the form $\bra{\psi}\aoqB_t^b\ket{\psi}$. The value of the game can then be
obtained by averaging. In inequalities where Alice and Bob have, respectively, two measurement outcomes for each possible choice of measurement setting (i.e., $A = B = \{0,1\}$), their measurements can be described by observables of the form $A_s = A_s^0 - A_s^1$ and $B_t = B_t^0 - B_t^1$ respectively. In this case, we state inequalities in the form of the observables $A_s$ and $B_t$ where we will use the shorthand $\langle A_sB_s \rangle = \bra{\psi}A_s B_s\ket{\psi}$.

Note that it is straightforward to extend the above definitions to
the setting involving multiple players, but the resulting terms will
be much harder to read. When considering games among $N$ players
$P_1,\ldots,P_N$, let $S_1,\ldots,S_N$ and $A_1,\ldots,A_N$ be finite
sets corresponding to the possible questions and answers
respectively. Let $\pi$ be a probability distribution on $S_1 \times
\ldots \times S_N$, and let $V$ be a predicate on $A_1 \times \ldots
\times A_N \times S_1 \times \ldots \times S_N$. Then $G = G(V,\pi)$
is the following $N$-player cooperative game: A set of questions
$(s_1,\ldots,s_N) \in S_1 \times \ldots \times S_N$ is chosen at
random according to the probability distribution $\pi$. Player $P_j$
receives question $s_j$, and then responds with an answer $a_j \in A_j$. The players win if and only if $V(a_1,\ldots,a_N|s_1,\ldots,s_N) = 1$. Let $\ket{\psi}$ denote the players' choice of state, and let
$X_j := \{X_{s_j}^{a_j}\mid a_j \in A_j\}$ denote the positive-operator-valued measure(ment) (POVM) of player $P_j$ for question $s_j \in S_j$, i.e., $\sum_{a_j} X_{s_j}^{a_j}=\mathbb{I}_j$ and
$X_{s_j}^{a_j}\ge 0$ for all $a_j$. The value of the game can now be
written as
\begin{equation*}
    \omega^*(G) =\lim_{d\to\infty}
    \max_{\stackrel{\ket \psi \in \left(\mathbb{C}^d\right)^{\otimes N}}{\|\,\ket\psi\,\| =1}}
    \max_{X_1,\ldots,X_{N}} \!\!\sum_{s_1,\ldots,s_N}\!\!\!\pi(s_1,\ldots,s_N)\!\!
    \sum_{a_1,\ldots,a_N}\!\!\!\!V(a_1,\ldots,a_N|s_1,\ldots,s_N)
    \bra{\psi} X_{s_1}^{a_1} \otimes \ldots \otimes X_{s_N}^{a_N} \ket{\psi},
\end{equation*}
where the maximization is taken over all legitimate POVMs $X_j$ for
all $j \in [N]$. Similarly, we can now write the field-theoretic
value of the game as
\begin{equation*}
    \ftv(G) = \sup_{X_1,\ldots,X_N} \Bigl\|
    \sum_{s_1,\ldots,s_N}\!\!\! \pi(s_1,\ldots,s_N) \sum_{a_1,\ldots,a_N}\!\!\!
    V(a_1,\ldots,a_N|s_1,\ldots,s_N)\,X_{s_1}^{a_1}\ldots X_{s_N}^{a_N}
    \bigr\|,
\end{equation*}
where we now have $\sum_{a_j} X_{s_j}^{a_j}=\mathbb{I}$ for all $a_j,s_j,j$
and $[X_{s_j}^{a_j},X_{s_{j'}}^{a_{j'}}]=0$ for all $j\neq j'$.

\subsection{Proof systems}

Interactive proof systems can be phrased as such games. For
completeness, we here provide a definition of MIP. We refer to the
introduction and the previous section for an explanation of the notions of a tensor
product form vs. commutation relations.

\begin{definition}
For $0 \leq s < c \leq 1$, let $\oplus\mbox{MIP}^*_{c,s}[k]$ denote
the class of all languages $L$ recognized by a classical $k$-prover
interactive proof system with entanglement such that:
\begin{itemize}
\item The interaction between the verifier and the provers is
    limited to one round and classical communication. The
    verifier chooses $k$ questions from a finite set of possible
    questions, according to a fixed probability distribution
    known to the provers, and sends one question to each prover. Afterwards,
the provers may perform any measurement that
    has tensor product form on a shared state $\ket{\psi}$ that
    they have chosen ahead of time. Each prover returns an answer
    to the verifier, whose decision function is known to the
    provers.
\item If $x \in L$ then there exists a strategy for the provers
    for which the probability that the verifier accepts is at
    least $c$ (the \emph{completeness} parameter).
\item If $x \notin L$ then, whatever strategy the $k$ provers
    follow, the probability that the verifier accepts is at most
    $s$ (the \emph{soundness} parameter).
\end{itemize}
\end{definition}

\begin{definition}
For $0 \leq s < c \leq 1$, let $\oplus\mbox{MIP}^f_{c,s}[k]$ denote
the class corresponding to a modified version of the previous
definition: here we merely ask that the measurements operators
between the different players commute.
\end{definition}

\section{The quantum moment problem}\label{qmpDef}

\subsection{General form}
Let us now state the quantum moment problem in its most general form,
before explaining its connection to non-local games. Intuitively, the
quantum moment problem states that given a certain probability
distribution, is it possible to find quantum measurements and a state
that provide us with such a distribution?
\begin{definition}[\textbf{Quantum moment problem}]
Given a list of numbers $\mathcal{M} = (m_i \mid m_i \in [0,1])$, a set of
polynomial equations $\mathcal{R} = \{r = 0\mid r: \mathbb{B}(\mathcal{H})^{\times
|\mathcal{M}|} \rightarrow \mathbb{B}(\mathcal{H})\}$, and polynomial inequalities
$\mathcal{S} = \{s \geq 0 \mid s: \mathbb{B}(\mathcal{H})^{\times |\mathcal{M}|} \rightarrow
\mathbb{B}(\mathcal{H})\}$, does there exist said Hilbert space $\mathcal{H}$, operators
$M_i \in \mathbb{B}(\mathcal{H})$ and a state $\rho \in \mathbb{B}(\mathcal{H})$ such that
\begin{enumerate}
\item For all $m_i \in \mathcal{M}$, $\Tr(M_i \rho) = m_i$.
\item For all $r \in \mathcal{R}$, $r(M_1,\ldots,M_{|\mathcal{M}|}) = 0$.
\item For all $s \in \mathcal{S}$, $s(M_1,\ldots,M_{|\mathcal{M}|}) \geq 0$.
\end{enumerate}
\end{definition}

\subsection{Non-local games}\label{qmp:nonlocal}
In this paper, we are particularly interested in a special case of
the quantum moment problem, where we consider measurements on many
space-like separated systems as in the setting of non-local games.
For simplicity, we will explain the connection to non-local games for
only two such systems, Alice $\mathcal{H}_A$ and Bob $\mathcal{H}_B$, where it is
straightforward to extend our arguments to more than two. On each
system $\mathcal{H}_A$ and $\mathcal{H}_B$, we want to perform a finite set of
possible measurements $S$ and $T$ each of which has the same finite
set of outcomes $A$ and $B$ respectively. Let $m^A(a|s)$ and
$m^B(b|t)$ denote the probability that on systems $\mathcal{H}_A$ and
$\mathcal{H}_B$ we obtain outcomes $a \in A$ and $b \in B$ given measurement
settings $s \in S$ and $t \in T$ respectively. Furthermore, let
$m^{AB}(a|s,b|t)$ denote the joint probability of obtaining outcomes
$a$ and $b$ given settings $s$ and $t$ when performing measurements
on systems $\mathcal{H}_A$ and $\mathcal{H}_B$.

Informally, our question is now: Given probabilities
$m^{AB}(a|s,b|t)$, does there exist a shared state $\rho$ such that
we can find measurements on the individual systems $\mathcal{H}_A$ and $\mathcal{H}_B$
that lead to such probabilities? Let's first consider what polynomial
equations and inequalities we need to express our problem in the
above form. First of all, how can we express the fact that we want
our measurement operators to act on the individual systems $\mathcal{H}_A$ and
$\mathcal{H}_B$ alone? I.e, how can we ensure that the measurement operators
have tensor product form? We will show in Lemma~\ref{tensorProduct}
that we are guaranteed to observe such a tensor product form if and
only if for all $s \in S$, $a \in A$, $t \in T$ and $b \in B$ we have
$[A_s^a,B_t^b] = 0$, where we used $A_s^a$ and $B_t^b$ to denote the
measurement operators of Alice and Bob corresponding to measurement
settings $s$ and $t$ and outcomes $a$ and $b$ respectively. Hence, we
need to impose the polynomial equality constraints of the form
$[A_s^a,B_t^b] = 0$.

Furthermore, we want that for both systems $\mathcal{A}$ and $\mathcal{B}$, we obtain
a valid measurement for each measurement setting $s \in S$ and $t \in
T$. I.e., we impose further polynomial equality constraints for all
$s \in S$ and $t \in T$ of the form
$$
\sum_{a \in A} \aoq - \mathbb{I} = 0
\mbox{ and }
\sum_{b \in B} B_t^b - \mathbb{I} = 0,
$$
and finally the following polynomial inequality constraints
for all $a \in A$ and $b \in B$
$$
\aoq \geq 0 \mbox{ and }
B_t^b \geq 0.
$$
Recall, that we may restrict ourselves to considering projective measurements.
We may thus add the equality constraints
$$
(\aoq)^2 = \aoq \mbox{ and } (B_t^b)^2 = B_t^b,
$$
which automatically imply that $\aoq, B_t^b \geq 0$. For simplicity, we will later
use this constraint instead of the previous one.

In this paper, we are mainly concerned with the (weighted) average of
the probabilities of generating certain outcomes in a non-local game.
In other words, we wanted to know if there exist operators of the
above form such that
$$
\nu =  \Bigl\|\sum_{a,b,s,t} \pi(s,t) V(a,b|s,t) \aoq B_t^b\Bigr\|
$$
for some success probability $\nu$. Semidefinite programming will
allow us to turn the question of existence into an optimization
problem.

\section{Tools}\label{section:tools}

For our analysis we first need to introduce two key tools. The first
one allows us to deal with the fact that we want measurements to have
tensor product form. Our second tool is an extension of the
non-commutative Positivstellensatz of Helton and McCullough to the
field of complex numbers, from which we will derive a converging
hierarchy of semidefinite programs.

\subsection{Tensor product structure from commutation relations}\label{section:commutation}

We now first show that imposing commutativity constraints does indeed
give us the tensor product structure required for our analysis of
non-local games. It is well-known that the following statement holds
within the framework of quantum
mechanics~\cite{summers:qftIndep}.\footnote{an algebra of type-I}

In Appendix~\ref{Sec:Tensor}, we provide a simple version of this
argument accessible from a computer science perspective, which
directly applies to the task at hand.
\begin{lemma}\label{tensorProduct}
Let $\mathcal{H}$ be a finite-dimensional Hilbert space, and let $\{X_{s_j}^{a_j} \in
\mathbb{B}(\mathcal{H})\mid \mbox{ for all } j \in [N] \mbox{ and for all } s_j \in S_j, a_j \in A_j\}$.
Then the following two statements are equivalent:
\begin{enumerate}
\item For all $j,j' \in [N]$, $j\neq j'$, and all $s_j \in S_j$, $s_{j'} \in S_{j'}$, $a_j \in A_j$ and $a_{j'} \in {A_{j'}}$ it holds that
$[X_{s_j}^{a_j},X_{s_{j'}}^{a_{j'}}] = 0$.
\item There exist Hilbert spaces $\mathcal{H}_1, \ldots,\mathcal{H}_N$ such that $\mathcal{H}
    = \mathcal{H}_1 \otimes \ldots \otimes \mathcal{H}_N$ and for all $j \in [N]$, all $s_j \in S_j$, $a_j \in A_j$ we
    have $X_{s_j}^{a_j} \in \mathbb{B}(\mathcal{H}_j)$.
\end{enumerate}
\end{lemma}

\subsection{Positivstellensatz}\label{section:satz}

Our second tool, the Positivstellensatz (in combination with
semidefinite programming) will allow us to find certificates for the
fact a quantum moment problem is infeasible. For simplicity, we here
describe the Positivstellensatz from the perspective of non-local
games. An extension to the general quantum moment problem is possible
and will be provided in a longer version of this paper.
Our results
follow almost directly from Helton and McCullough's work and our
proof closely follows that in
Ref.~\cite{helton04:_posit_for_non_commut_polyn}.  We have chosen to
provide a complete proof of the Positivstellensatz for three reasons:
(i) the proof is more straightforward in our concrete setting, (ii)
Helton and McCullough's theorem is formulated for symmetric operators
over the field $\mathbb{R}$, and we need to work with Hermitian operators
over the field $\mathbb{C}$, and (iii) so we can highlight the
nonconstructive steps in the proof. We first define:

\begin{definition}[Convex Cone $\cC_\cP$]
  \label{definition:3}
    Let $\mathcal{P}$ be a collection of Hermitian  polynomials in
    (noncommutative) variables $\{X_{s_j}^{a_j}\}$. The \emph{convex cone}
    $\cC_\cP$ generated by $\mathcal{P}$ consists of polynomials of the form
  \begin{align}\label{eq:weightedSOS}
       q = \sum_{i=1}^M r_i^\dagger r_i +  \sum_{i=1}^N
       \sum_{j=1}^L s_{ij}^\dagger\,p_i\,s_{ij},
  \end{align}
where $p_i \in \mathcal{P}$, $M$, $N$ and $L$ are finite, and $r_i$, $s_{ij}$
are arbitrary polynomials.
\end{definition}
\noindent In the following, we will call Eq.~\eqref{eq:weightedSOS} a
\emph{weighted sum of squares} (WSOS) representation of $q$.

The purpose of the set $\mathcal{P}$ is to keep track of the constraints on
the measurement operators. Note that when considering the measurement
operators for non-local games, it is sufficient for us to restrict
ourselves to considering (measurement) operators that are positive
semidefinite. In particular, this means that all operators are
Hermitian. The Positivstellensatz as such does not require us to deal
only with Hermitian variables in the polynomials, but allows us to
use any noncommuting matrix variables. In the following, we will
always take our measurement operators to be of the form $X_{s}^{a} =
(\hat{X}_{s}^{a})^\dagger \hat{X}_{s}^{a}$. Clearly, $X_{s}^a$ is
itself a Hermitian polynomial in the variable $\hat{X}_s^a$. For
clarity of notation, we will omit this explicit expansion in the
future. Note that we will thus not impose the constraint that our
operators are Hermitian, and this
implicit expansion does not increase the size of our SDP.

We can write our constraints in terms of the following sets of
Hermitian polynomials. In the following, we will use the short hand
notation $O_{-j} := X_{s_1}^{a_1}\ldots X_{s_{j-1}}^{a_{j-1}}
X_{s_{j+1}}^{a_{j+1}} \ldots X_{s_N}^{a_N}$ where we leave indices
$s_j$ and $a_j$ implicit, to refer to a product of measurement
operators where we exclude player $j$. First, we want measurements on
different subsystems to commute. In the multi-party case, this gives
us the set of polynomials
$$
\mathcal{Q}_1 = \{i[X_{s_j}^{a_j},O_{-j}] \mid \mbox{ for all } s_j \in S_j,
a_j \in A_j \mbox{ and all } O_{-j}\}.
$$
Second, we want our operators to form valid measurements.
$$
\mathcal{Q}_2 = \bigcup_{j,s_j} \{\mathbb{I} - \sum_{a_j} X_{s_j}^{a_j}\}.
$$
Finally, by Neumark's theorem~\cite{A.Peres:Book:1995}, we may take
our measurement operators to be projectors, this gives
$$
\mathcal{Q}_3 = \bigcup_{j,s_j,a_j} \{(X_{s_j}^{a_j})^2 - X_{s_j}^{a_j}\}.
$$
It's not hard to see that these constraints actually give us orthogonality
of the projectors.
For clarity, however, we may also include the following sets of polynomials
$$
\mathcal{Q}_4 = \{i[X_{s_j}^{a_j}, X_{s_j}^{a_j'}] \mid
\text{for all $s_j\in S_j$ and all $a_j\neq a_j'$}\},
$$
$$
\mathcal{Q}_5 = \{X_{s_j}^{a_j}X_{s_j}^{a_j'}+X_{s_j}^{a_j'}X_{s_j}^{a_j}\mid
\text{for all $s_j\in S_j$ and all $a_j\neq a_j'$}\},
$$
which explicitly demand that projectors corresponding to the same
$s_j$ are orthogonal.

Let $\mathcal{Q} = \mathcal{Q}_1 \cup \mathcal{Q}_2 \cup \mathcal{Q}_3 \cup \mathcal{Q}_4 \cup \mathcal{Q}_5$ and let
$\mathcal{P} = \mathcal{Q} \cup (-\mathcal{Q})$. Note that all polynomials in $\mathcal{P}$ are
Hermitian. It is clear that if the measurement operator satisfy the
constraints, then the term
$$
\sum_{i,j} s_{ij}^\dagger\, p_j\,s_{ij}
$$
vanishes for arbitrary $p_j\in\mathcal{P}$ and arbitrary polynomial $s_{ij}$.
We are now ready to state the Positivstellensatz:

\begin{theorem}[Positivstellensatz]\label{theorem:satz}
    Let $G=G(\pi, V)$ be an $N$-prover game and let $\cC_\cP$ be the cone
    generated by the set $\mathcal{P}$ defined above. Set
  \begin{align}\label{Dfn:qnu}
    q_\nu = \nu \mathbb{I} -\!\!\sum_{s_1,\ldots,s_N}\!\!\! \pi(s_1,\ldots,s_N)
    \sum_{a_1,\ldots,a_N}\!\!\! V(a_1,\ldots,a_N|s_1,\ldots,s_N)
    X_{s_1}^{a_1}\ldots X_{s_N}^{a_N}.
  \end{align}
If $q_\nu > 0$, then $q_\nu \in \cC_\cP$, i.e.,
\begin{align}\label{Eq:qnu:SOS}
    \nu \mathbb{I} -\!\!\sum_{s_1,\ldots,s_N}\!\!\! \pi(s_1,\ldots,s_N)\!\!
    \sum_{a_1,\ldots,a_N}\!\!\!V(a_1,\ldots,a_N|s_1,\ldots,s_N)
    X_{s_1}^{a_1}\ldots X_{s_N}^{a_N}
    =\sum_i r_i^\dagger r_i +  \sum_{i,j} s_{ij}^\dagger\,p_i\,s_{ij},
\end{align}
for some $p_i\in\mathcal{P}$, and some polynomials $r_i$, $s_{ij}$.
\end{theorem}

\section{Finding upper bounds}\label{findingBounds}

We now show how we can approximate the optimal field-theoretic value
of a non-local game using semidefinite programming. We thereby
construct a converging hierarchy of SDPs, where each level in this
hierarchy gives us a better upper bound on the actual value of the
game. To this end we will use the Positivstellensatz of
Theorem~\ref{theorem:satz} in combination with the beautiful approach
of Parrilo~\cite{parrilo:thesis,parrilo:paper}. For simplicity, we
first describe everything for the two party setting; a generalization
is straightforward.

Recall from Definition~\ref{Dfn:ftv} that if for some real number
$\nu$ we have
\begin{equation}\label{Eq:qnu>0}
    q_\nu = \nu \mathbb{I} - \sum_{a,b,s,t} \pi(s,t) V(a,b|s,t)
    \aoq B_t^b \geq 0,
\end{equation}
and the operators $\{\aoq\}$ and $\{B_t^b\}$ form a valid measurement,
then $\nu \geq \omega^f(G)$ gives us an upper bound for the optimum
value of the game. When trying to find the optimal value of the game,
our task is thus to find the smallest $\nu$ for which $q_\nu \geq 0$
for any choice of measurement operators. Clearly, if we could express $q_\nu$ as an SOS for any choice of measurement operators $\{\aoq\}$ and $\{B_t^b\}$ then $q_\nu \geq 0$ and we would also have $\nu \geq \omega^f(G)$. Luckily, the Positivstellensatz of Theorem~\ref{theorem:satz} gives us almost the converse: \emph{if} $q_\nu > 0$, then $q_\nu$ can be written as a {\em weighted} sum of squares (WSOS). Recall from the previous section, that the purpose of the additional term in the weighted sums of squares representation is to deal with the constraint that we would like the operators $\{\aoq\}$ and $\{B_t^b\}$ to form a valid quantum measurement. Note that $q_\nu$ reduces to an SOS if we could express $q_\nu$ as a WSOS, i.e.,
\begin{equation}
    q_\nu = \nu \mathbb{I} - \sum_{a,b,s,t} \pi(s,t) V(a,b|s,t)=\sum_{i=1}^M r_i^\dagger r_i +
    \sum_{i=1}^N\sum_{j=1}^L s_{ij}^\dagger\, p_i\, s_{ij},
\end{equation}
for some polynomials $r_i$ and $s_{ij}$ in the variables $\{\aoq\}$
and $\{B_t^b\}$ in such a way that whenever the variables satisfy the constraints the second term in the above expansion vanishes.

It is not difficult to see that if $q_\nu > 0$, then
there exists \emph{no} strategy that achieves a winning
probability of $\nu$ or higher.
Applied to
our problem, the Positivstellensatz thus tells us that if there
exists \emph{no} strategy that achieves winning probability $\nu$,
then $q_\nu$ \emph{can} be written as a weighted sum of squares.
Intuitively, the WSOS representation of $q_\nu$ thus bears witness to
the fact that the set of measurement operators and states giving a
success probability higher than $\nu$ is empty. The advantage of this
procedure is that semidefinite programming can be used to test
whether polynomials (such as $q_\nu$) admit a representation as WSOS.
In Section~\ref{Sec:Examples}, we will look at some specific examples
of this approach (see also~\cite{parrilo:paper,sostools:manual} for
the analogous treatment for commutative variables).

When trying to find the optimal value of the game, our task is thus
to find the smallest $\nu$ for which $q_\nu$ admits a WSOS
representation. Hence, we want to
\begin{sdp}{minimize}{$\nu$}
&$q_\nu \in \cC_\cP.$
\end{sdp}
Recall that if $q_\nu \in \cC_\cP$, then $q_\nu$ is of the
form
\begin{equation}\label{Eq:qSOS}
    q_\nu =  \sum_{i=1}^M r_i^\dagger r_i +
    \sum_{i=1}^N\sum_{j=1}^L s_{ij}^\dagger\, p_i\, s_{ij},
\end{equation}
for some polynomials $r_i$ and $s_{ij}$ in the variables $\{\aoq\}$
and $\{B_t^b\}$. A point that is worth noting now is that in the above
optimization,
Eq.~\eqref{Eq:qSOS} is an identity true for all $\{\aoq\}$,
$\{B_t^b\}$, rather than an equation that is only true when
$\{\aoq\}$, $\{B_t^b\}$ correspond to projective measurements. In
this, the additional term is rather similar to the Lagrange
multipliers in more conventional constrained optimizations.

\subsection{SDP hierarchy}
The main difficulty now is that we do not know how large the WSOS
representation of $q_\nu$ has to be. That is, we do not know ahead of
time how large we need to choose the degree of the polynomials in the
representation. The techniques discussed above are therefore not
constructive and do not lead to a direct computation of
$\omega^f(G)$. However it is straightforward to find semidefinite
relaxations that provide upper bounds on $\omega^f(G)$. In this we
simply apply the methods of Parrilo~\cite{parrilo:thesis,
parrilo:paper} for the case of polynomials of commutative variables.
The main requirement is to fix an integer $n$ and look for a sum of
squares decomposition for $q_\nu$ that has a total degree of at most
$2n$. Letting $\nu = \omega^f(G) + \varepsilon$, this means that $\varepsilon$ may
not be made arbitrarily small but will always result in an upper
bound for $\omega^f(G)$. This upper bound can be computed as an SDP
using methods analogous to~\cite{parrilo:paper}. Consider the problem given above for $q_\nu$ as in Eq.~\eqref{Eq:qSOS}. Notice that all of the constraint polynomials $p_i$ defined in Section \ref{section:satz} have total degree less than or equal to 2 so we require that each $r_i$ is of total degree $n$ and each $s_i$ is of total degree at most $n-1$. The lowest level of the hierarchy has $n=1$ and corresponds to applying the method of Lagrange multipliers to finding the quantum value of the game. In the following, we use the term \emph{level $n$} to refer to a level of the hierarchy where the total degree of $q_\nu$ is $2n$ when concerned with a bipartite game. For a game $G$, denote the solution to the SDP at level $n$ as $\omega^{\text{sdp}}_{n}(G)$. It should be clear that if $q_\nu$ has a WSOS decomposition of degree $2n$, it must also have a WSOS decomposition with higher degree. As such, the optimum derived from the hierarchy of SDPs must obey the
following inequalities:
\begin{equation}\label{Eq:Omega:Convergence}
    \omega^\text{sdp}_1(G)\geq \omega^\text{sdp}_2(G) \geq \cdots \geq
    \omega^{\text{sdp}}_n(G).
\end{equation}

\begin{theorem}
  \label{theorem:4}
  The solutions to the SDP hierarchy converge to $\omega^{f}(G)$, i.e.,
  $\lim_{n\to\infty}\omega^{\text{sdp}}_n = \omega^f(G)$.
\end{theorem}

\begin{proof}
That $\omega^\text{sdp}_n (G) \geq \omega^f(G)$ follows from our
discussion above. To prove convergence, we use the Positivstellensatz
given by Theorem~\ref{theorem:satz}. Fix $\varepsilon>0$ and let $\nu =
\omega^f(G) + \varepsilon$ with $q_\nu$ defined as in Eq.~\eqref{Dfn:qnu}.
By Theorem~\ref{theorem:satz}, $q_\nu$ has a representation as a
WSOS,
\begin{equation*}
    q_\nu =  \sum_{i=1}^M r_i^\dagger r_i +
    \sum_{i=1}^N\sum_{j=1}^L s_{ij}^\dagger\, p_i\, s_{ij},
\end{equation*}
Let $2D$ be the maximum degree of any of the polynomials $r_i^\dagger
r_i$ and $s_{ij}^\dagger\, p_i\, s_{ij}$ that occurs in the above
expression. Then, if we consider a level $D$ SDP relaxation, we must
necessarily arrive at an optimum such that $\omega_{D}^\text{sdp}(G)
\le \omega^f(G)+\varepsilon$. Likewise, by choosing $\varepsilon$ arbitrarily close
to zero, there is a corresponding SDP with total degree $2n$ whose
optimum $\omega_{n}^\text{sdp}(G)$ is arbitrarily close to
$\omega^f(G)$. In particular, from Eq.~\eqref{Eq:Omega:Convergence},
we can see that as $n\to\infty$, the optimum of the SDP hierarchy
must converge to $\omega^f(G)$.
\end{proof}

In Section~\ref{Sec:I3322}, we provide a simple example of how the
degree of the polynomials can be increased when going from level 1 to
level 2. There are many connections of this semidefinite programming
hierarchy to other methods that can be used to bound the quantum
values of games. In particular, it can be shown that the dual
semidefinite programs to this hierarchy are equivalent to the moment
matrix methods of NPA~\cite{navascues:_bound}, thus showing that the
hierarchy of semidefinite programs discussed in that work converges
to the entangled value of the game.
Our example of the CHSH inequality below demonstrates this connection
explicitly. In relation to this, it is worth noting that the duality
between the two approaches (sum of squares and moment matrix) arises
also in the case of commutative variables where the moment matrix
methods of Laserre~\cite{lasserre01} are dual to the semidefinite
programs discussed by Parrilo~\cite{parrilo:paper}.

\subsection{Examples}\label{Sec:Examples}

\subsubsection{CHSH inequality}\label{example:CHSH}

We will now look at the simplest non-local game that is derived from
the CHSH inequality~\cite{chsh:nonlocal}. In particular, we will
illustrate how the tools that have we developed allow us to
prove that~\cite{tsirel:original}
$$
\mathcal{S}_{\rm CHSH} = \ext{A_1 B_1} + \ext{A_1 B_2} + \ext{A_2 B_1} - \ext{A_2 B_2} \leq 2 \sqrt{2},
$$
where $A_1,A_2$ and $B_1,B_2$ are observables with eigenvalues $\pm
1$ corresponding to Alice and Bob's measurement settings
respectively. First of all, note that since we are only interested in
the expectation values of the form $\ext{A_1 B_2}$ we may simplify
our problem: instead of dealing with the probabilities of individual
measurement outcomes, we are only interested in whether said expectation values can be obtained. Here, our constraints become much simpler and we only demand that $A_j^2 = \mathbb{I}$, $B_j^2 = \mathbb{I}$ and $[A_j,B_k] = 0$ for all $j,k \in \{1,2\}$. The Bell operator for the CHSH inequality is given by~\cite{S.L.Braunstein:PRL:1992}
$$
\mathcal{B}_{\rm CHSH} = A_1 B_1 + A_1 B_2 + A_2 B_1 - A_2 B_2.
$$
Hence, to find the optimum value our goal is to
\begin{sdp}{minimize}{$\nu$}
&$q_\nu = \nu\mathbb{I} - \mathcal{B}_{\rm CHSH} \in \cC_\cP$.
\end{sdp}
The constraint in the above optimization thereby amounts to
determining whether $q_\nu$ as written above can be cast in the form
of a WSOS, which reduces to an SOS for measurement operators
satisfying the constraints. The numerical package
SOSTOOLS~\cite{sostools} gives a frontend to other SDP solvers and
explains how to apply these techniques in the case of commutative
variables. Similar ideas can be applied here. However, for our simple
example, it is not hard to see how this problem can be recast in a
language that may be more familiar.

Since $\mathcal{B}_{\rm CHSH}$ is a noncommutative
polynomial of degree 2, the lowest level relaxation consists of
looking for a WSOS decomposition for $q_\nu$ that is of degree 2. To
this end, we shall consider a vector of monomial of degree 1, namely,
$z = (A_1,A_2,B_1,B_2)^\dagger$. Our goal is to find a
$4\times 4$ matrix $\Gamma$ such that $q_\nu = z^\dagger \Gamma z$
whenever the constraints are satisfied. I.e. we have
$[A_j,B_k] = 0$ for all $j,k \in \{1,2\}$, and
polynomials
\begin{equation}\label{Eq:p_j:CHSH}
    p^{(A)}_j:=\mathbb{I}- (A_j)^2,\quad
    p^{(B)}_j:=\mathbb{I}- (B_j)^2,\quad j=1,2,
\end{equation}
and their negations vanish.
Evidently, since we want $q_\nu$ to be a
Hermitian polynomial, and we want our commutation constraints to hold, we may without loss of generality take $\Gamma$
to be real and symmetric.
Note that this already takes care of the commutation constraints.
Moreover, since all remaining constraints are
quadratic, when looking for a WSOS decomposition for $q_\nu$, it
suffices to consider $s_{ij}$ in Eq.~\eqref{Eq:qSOS} as multiples of
$\mathbb{I}$. Let $\gamma_{ij} = [\Gamma]_{i,j}$, then
a small calculation shows that this amounts to requiring
\begin{eqnarray}
\nu &=& \gamma_{11} + \gamma_{22} + \gamma_{33} + \gamma_{44}\nonumber\\
0 &=& \gamma_{12} = \gamma_{21} = \gamma_{34} = \gamma_{43}\nonumber\\
-1 &=& 2\gamma_{13} = 2\gamma_{14} =2\gamma_{23}\nonumber\\
1 &=&  2\gamma_{24},
\label{Eq:constraints:gamma}
\end{eqnarray}
so that
\begin{equation}\label{Eq:qnu:CHSH:explicit}
    q_\nu=\nu\,\mathbb{I}-\mathcal{B}_{\rm CHSH}=z^\dagger\Gamma z+\sum_{j=1}^2
    \gamma_{jj}\, p_j^{(A)} +\sum_{j=3}^4 \gamma_{jj}\,
    p_{j-2}^{(B)}.
\end{equation}
Using the constraints given in Eq.~\eqref{Eq:constraints:gamma}, we
see that $\Gamma$ should be of the form
\begin{equation}\label{Eq:Gamma:CHSH}
    \Gamma = \frac{1}{2}\left(\begin{array}{cccc}
    2\gamma_{11} &0 &-1 &-1\\
    0 & 2\gamma_{22} & -1 & 1\\
    -1 & -1 & 2\gamma_{33} & 0\\
    -1 & 1 & 0 & 2\gamma_{44}
    \end{array}\right).
\end{equation}
Effectively, $\Gamma$ is the matrix obtained by expressing
$\nu\,\mathbb{I}-\mathcal{B}_{\rm CHSH}-\sum_{j=1}^2\gamma_{jj}\, p_j^{(A)}
-\sum_{j=3}^4 \gamma_{jj}\, p_{j-2}^{(B)}$ in the form of
$z^\dagger\Gamma z$. Now, if we can find a $\Gamma \geq 0$ that is of
this form, then whenever the polynomials given in
Eq.~\eqref{Eq:p_j:CHSH} vanish, $q_\nu = z^\dagger \Gamma z$ is an
SOS. To see this, note that in this case, we may write $\Gamma =
U^\dagger D U$, where $U$ is unitary and $D = \text{diag}(d_i)$ only
consists of nonnegative diagonal entries. Then we can write $q_\nu$
as $\sum_i d_i (Uz)_i^\dagger (Uz)_i$ which is clearly an SOS.
Conversely, note that if $q_\nu$ is an SOS, we can find such a matrix
$\Gamma$. Hence, we can rephrase our optimization problem as the SDP
\begin{sdp}{minimize}{$\Tr(\Gamma)$}
&$\Gamma \geq 0$.
\end{sdp}
This is, in fact, exactly the dual of the SDP corresponding to the first level of the SDP hierarchy given by NPA~\cite{navascues:_bound}, and the dual of the SDP for the special case of XOR games~\cite{wehner05d}. Solving this SDP, one
obtains
\begin{equation}\label{CHSHgamma}
      \Gamma = \frac{1}{2}\begin{pmatrix}
     \sqrt{2} & 0 & -1 & -1 \\
     0 & \sqrt{2} & -1 & 1 \\
     -1 & -1 & \sqrt{2} & 0 \\
     -1 & 1 & 0 & \sqrt{2}
    \end{pmatrix},
\end{equation}
which gives $2\sqrt{2}$ as an optimum. From here and
Eq.~\eqref{Eq:qnu:CHSH:explicit}, it is possible to write down a WSOS
decomposition for $\nu=2\sqrt{2}$ as
\begin{align}\label{eqNext:2}
    q_{2\sqrt{2}} = 2\sqrt{2}\,\mathbb{I} - \mathcal{B}_{\rm CHSH} =
    \frac1{2\sqrt{2}} (h_1^\dagger h_1 + h_2^\dagger h_2)
    +\frac{1}{\sqrt{2}}\sum_{j=1}^2 p_j^{(A)}
    +\frac{1}{\sqrt{2}}\sum_{j=3}^4  p_{j-2}^{(B)},
\end{align}
with $h_1 = A_1 + A_2 - \sqrt{2}\,B_1$ and $h_2 = A_1 - A_2
-{\sqrt{2}}\,B_2$. This immediately implies that whenever the
constraints are satisfied, $q_{2\sqrt{2}} \geq 0$ and hence
$\mathcal{B}_{\rm CHSH} \leq 2\sqrt{2}\,\mathbb{I}$. It is well known that for
the CHSH inequality, this bound can be
achieved~\cite{tsirel:original}.

\subsubsection{The $I_{3322}$ inequality}\label{Sec:I3322}

We now consider another example of a two-player game, where the first level of the
hierarchy does not give a tight bound.
The $I_{3322}$ inequality~\cite{D.Collins:JPA:2004} is a Bell
inequality phrased in terms of probabilities (not expectation values)
whereby Alice and Bob can each perform
one of three possible two-outcome measurements. Without loss of generality, the Bell
operator in this case can be written as:
\begin{align*}
    \mathcal{B}_{3322}=A^a_1(B^b_1+B^b_2+B^b_3)+A^a_2(B^b_1+B^b_2-B^b_3)
    +A^a_3(B^b_1-B^b_2)-A^a_1-2B^b_1-B^b_2,
\end{align*}
where $A^a_i$ and $B^b_j$ ($i,j=1,2,3$) are projectors corresponding
to, respectively, outcome $a$ of Alice's $i$-th measurement and
outcome $b$ of Bob's $j$-th measurement for some fixed $a$ and $b$.
To the best our knowledge, the maximum entangled value for
$\mathcal{B}_{3322}$, i.e., $\omega^*(I_{3322})$, is not known. The best
known lower bound on $\omega^*(I_{3322})$ is
0.25~\cite{D.Collins:JPA:2004}; some upper bounds
(0.375~\cite{Y.C.Liang:PRA:2007}, 0.3660~\cite{D.Avis:JPA:2006}) are
also known in the literature. Here, we will make use of the tools
that we have developed  to obtain a hierarchy of upper bounds on this
maximum. In analogous with the CHSH scenario, this corresponds to
solving the following SDP for some fixed degree of $q_\nu$:
\begin{align}
    &\text{minimize\ \ \ } \nu,\nonumber\\
    &\text{subject to\ \ } q_\nu=\nu\,\mathbb{I}-\mathcal{B}_{3322}\in\cC_\cP.
    \label{Eq:SDP:abstract}
\end{align}

In particular, since $\mathcal{B}_{3322}$ is a noncommutative polynomial of
degree 2, the lowest level SDP relaxation would correspond to
choosing a vector of monomials with degree at most one, i.e.,
\begin{equation}\label{Eq:z:1st:3322}
    z^\dagger=(\mathbb{I}, A^a_1, A^a_2, A^a_3, B^b_1, B^b_2, B^b_3).
\end{equation}
We can now proceed analogously to the CHSH case, where we will look
for a particular matrix $\Gamma$ restricted by our constraints,
namely, $(A^a_j)^2 = A^a_j$ and $(B^b_j)^2 = B^b_j$ for all $j \in
\{1,2,3\}$, where again for the purpose of implementation, we will
implicitly enforce the commutativity conditions $[A_i,B_j]=0$ for all
$i,j\in\{1,2,3\}$. Solving the corresponding SDP (Appendix~\ref{Sec:SDP:Lowest}), one obtains
$\omega_1^\text{sdp}(I_{3322})=3/8$, and the matrix
\begin{gather*}
    \Gamma=\frac{1}{2}\left(\begin{array}{rrrrrrr}
    \frac{3}{4} & 0 & -1 & -\frac{1}{2} & 1 &
    0 & -\frac{1}{2}\\
    0 & 2 & 0 & 0 & -1 & -1 & -1\\
    -1& 0 & 2 & 0 & -1& -1& 1\\
    -\frac{1}{2} & 0 & 0 & 1& -1& 1& 0\\
    1& -1& -1& -1& 2 & 0 & 0\\
    0 & -1& -1& 1& 0 & 2 & 0\\
    -\frac{1}{2} & -1& 1& 0 & 0 & 0 & 1\\
    \end{array}\right),
\end{gather*}
which provides a WSOS decomposition for $\nu=3/8$, i.e.,
\begin{equation}\label{Eq:3322:SOS}
    q_{3/8} = \frac{3}{8}\,\mathbb{I}-\mathcal{B}_{3322}=z^\dagger\Gamma z +\sum_i s_i^\dagger\,p_i s_i,
\end{equation}
where
\begin{gather}\label{Eq:p_i:3322}
    p_i=\left\{ \begin{array}{c@{\quad:\quad}l}
        A^a_i-\left(A^a_i\right)^2 & i=1,2,3,\\
        B^b_{i-3}-\left(B^b_{i-3}\right)^2 & i=4,5,6.\\
        \end{array} \right. ,\quad
    s_i=\left\{ \begin{array}{c@{\quad:\quad}l}
        1  & i=1,2,4,5,\\
        \frac{1}{\sqrt{2}} & i=3,6.\\
        \end{array} \right. .
\end{gather}

Given that $\omega_1^\text{sdp}(I_{3322})$ is far from the best known lower
bound on $\omega^*(I_{3322})$, it seems natural to also look at
higher level relaxations for $I_{3322}$. For the next level, we will
look for $q_\nu$ that is of degree at most 4. Note that we have many
options to extend the hierarchy. The easiest way to proceed is to
extend the vector $z$ by some monomials of degree two in the
measurement operators. For this we do not even have to consider using
all possible degree 2 monomials, but could consider only a subset of
them such as that given by the following vector
\begin{equation}\label{Eq:z:1st+AB}
    z^\dagger=(\mathbb{I}, A^a_1, A^a_2, A^a_3, B^b_1, B^b_2, B^b_3, A^a_1
    B^b_1, A^a_1B^b_2,A^a_1B^b_3,A^a_2B^b_1, A^a_2B^b_2,A^a_2B^b_3,
    A^a_3B^b_1, A^a_3B^b_2, A^a_3B^b_3).
\end{equation}
In particular, solving the corresponding SDP
(Appendix~\ref{Sec:SDP:Higher}) with $z$ given by
Eq.~\eqref{Eq:z:1st+AB} gives an optimum that is approximately
$0.251~470~90$ which is significantly less than $3/8=0.375$.
Clearly, we could increase the size of $z$ further by including all
relevant monomials of degree 2 or less.
\begin{equation*}
    z^\dagger=(\mathbb{I},A^a_1,A^a_2,\ldots,B^b_2,B^b_3,
    A^a_1A^a_2,A^a_1A^a_3, \ldots, A^a_3A^a_2, B^b_1B^b_2, B^b_1B^b_3,\ldots, B^b_3B^b_2,
    A^a_1B^b_1,A^a_1B^b_2,\ldots,A^a_3B^b_3).
\end{equation*}
Proceeding as before, we end up with the optimum of the second order
relaxation $\omega_2^\text{sdp}(I_{3322})\approx 0.250~939~72$. In the next
level, we would then include all monomials of degree 3 and less, and
this gives $\omega_3^\text{sdp}(I_{3322})\approx 0.250~875~56$.

\subsubsection{Yao's inequality}\label{yao}

Finally, we examine a well-known tripartite Bell
inequality~\cite{yao:inequality} among three provers: Alice, Bob and
Charlie. Each prover performs three possible measurements, each of
which has two possible outcomes. Similarly to the CHSH inequality
above, we may thus express each measurement as an observable with
eigenvalues $\pm 1$. For simplicity, let $A_1,A_2,A_3$, $B_1,B_2,B_3$ and
$C_1,C_2,C_3$ correspond to the observables of Alice, Bob and Charlie
respectively. Yao's inequality states that for any shared state
$\rho$ we have
\begin{equation}\label{Eq:YaoIneq}
    \mathcal{S}_{\rm Yao} = \ext{A_1 B_2 C_3} + \ext{A_2 B_3 C_1} +
    \ext{A_3 B_1 C_2} - \ext{A_1 B_3 C_2} - \ext{A_2 B_1 C_3} - \ext{A_3
    B_2 C_1} \leq 3 \sqrt{3}.
\end{equation}

We now provide a very simple proof of this inequality based on our
framework. First of all, note that since we are only interested in
the expectation values of the form $\ext{A_1 B_2 C_3}$ we may again
restrict ourselves to dealing only with expectation values in analogy
with the CHSH example presented above. Our constraints are also
analogous to the CHSH case. Among them, we have the following
constraint polynomials
\begin{equation}\label{Eq:Constraint:Yao}
    p^{(A)}_j:=\mathbb{I}- (A_j)^2,\quad
    p^{(B)}_j:=\mathbb{I}- (B_j)^2,\quad
    p^{(C)}_j:=\mathbb{I}- (C_j)^2,
\end{equation}
for $j=1,2,3$. Next, note that the Bell operator for Yao's inequality
can be written as [c.f. Eq.~\eqref{Eq:YaoIneq}]
\begin{equation}\label{Eq:BellOperator:Yao}
    \mathcal{B}_{\rm Yao} = A_1 B_2 C_3 + A_2 B_3 C_1 + A_3 B_1 C_2 - A_1 B_3 C_2 -
    A_2 B_1 C_3 - A_3 B_2 C_1,
\end{equation}
which is a noncommutative polynomial of degree 3.

For our task at hand, we will consider the following SDP relaxation
\begin{align*}
    &\text{minimize\ \ \ } \nu,\\
    &\text{subject to\ \ } q_\nu=\nu\,\mathbb{I}-\mathcal{B}_{\rm Yao}\in\cC_\cP,
\end{align*}
with $q_\nu$ being a polynomial of degree at most 6. As usual, we
will implicitly enforce the commutativity constraints, i.e.,
$[A_i,B_j]=0$, $[A_i,C_k]=0$, and $[B_j,C_k]=0$ for all
$i,j,k\in\{1,2,3\}$. With this assumption, it turns out that it
suffices to consider the following 25-element vector
\begin{equation}\label{Eq:z:NewBasis}
    z=
    \left(\begin{array}{c}
     \mathbb{I}     \\
     A_1B_2C_3 \\
     A_2B_3C_1 \\
     A_3B_1C_2 \\
     A_1B_3C_2 \\
     A_2B_1C_3 \\
     A_3B_2C_1 \\
     \end{array}\right)\oplus
     \left(\begin{array}{c}
     A_1B_1C_2 \\
     A_1B_2C_1 \\
     A_2B_1C_1 \\
     A_1B_2C_2 \\
     A_2B_1C_2 \\
     A_2B_2C_1 \\
     \end{array}\right)\oplus
     \left(\begin{array}{c}
     A_1B_1C_3 \\
     A_1B_3C_1 \\
     A_3B_1C_1 \\
     A_1B_3C_3 \\
     A_3B_1C_3 \\
     A_3B_3C_1 \\
     \end{array}\right)\oplus
     \left(\begin{array}{c}
     A_2B_2C_3 \\
     A_2B_3C_2 \\
     A_3B_2C_2 \\
     A_2B_3C_3 \\
     A_3B_2C_3 \\
     A_3B_3C_2 \\
     \end{array}\right).
\end{equation}
In this case, since the constraint polynomials given in
Eq.~\eqref{Eq:Constraint:Yao} are quadratic, when looking for a WSOS
decomposition for $q_\nu$, we will need to consider $s_{ij}$ in
Eq.~\eqref{Eq:qSOS} as an arbitrary polynomial of $A_i$, $B_j$ and
$C_k$ with degree at most 2. Proceeding in a way analogous to the 2nd
level relaxation for $I_{3322}$ inequality, one obtains the
$25\times25$ positive semidefinite matrix
$    \Gamma=\Gamma_{7\times7}\bigoplus_{i=1}^6\Gamma_{3\times3}$,
where
\begin{gather}\label{Eq:Gamma:Blks}
    \Gamma_{7\times7}:=\frac{1}{2}\left(\begin{array}{ccccccc}
    3\sqrt{3} & -1 & -1 & -1 & 1 &
    1 & 1\\
    -1 & \frac{1}{\sqrt{3}} & 0 & 0 & -\frac{1}{3\sqrt{3}} & -\frac{1}{3\sqrt{3}} & -\frac{1}{3\sqrt{3}}\\
    -1 & 0 & \frac{1}{\sqrt{3}} & 0 & -\frac{1}{3\sqrt{3}} & -\frac{1}{3\sqrt{3}} & -\frac{1}{3\sqrt{3}}\\
    -1 & 0 & 0 & \frac{1}{\sqrt{3}} & -\frac{1}{3\sqrt{3}} & -\frac{1}{3\sqrt{3}} & -\frac{1}{3\sqrt{3}}\\
    1 & -\frac{1}{3\sqrt{3}} & -\frac{1}{3\sqrt{3}} & -\frac{1}{3\sqrt{3}} & \frac{1}{\sqrt{3}} & 0 & 0\\
    1 & -\frac{1}{3\sqrt{3}} & -\frac{1}{3\sqrt{3}} & -\frac{1}{3\sqrt{3}} & 0 & \frac{1}{\sqrt{3}} & 0\\
    1 & -\frac{1}{3\sqrt{3}} & -\frac{1}{3\sqrt{3}} & -\frac{1}{3\sqrt{3}} & 0 & 0 & \frac{1}{\sqrt{3}}\\
    \end{array}\right),\quad
    \Gamma_{3\times3}:=\frac{1}{12\sqrt{3}}\left(\begin{array}{ccc}
    1 & 1 & 1\\
    1 & 1 & 1\\
    1 & 1 & 1
    \end{array}\right).
\end{gather}
From some simple calculations, it then follows that
whenever the constraints $A_i^2=B_j^2=C_k^2=\mathbb{I}$ are satisfied, we
have
\begin{equation}\label{Eq:Yao:SOSwoConstraints}
    3\sqrt{3}\,\mathbb{I} - \mathcal{B}_{\rm Yao}=z^\dagger\Gamma z=\frac{1}{6\sqrt{3}} \left(h_0^\dagger h_0
    +\sum_{j=1}^2 h_{+,j}^\dagger h_{+,j}
    +\sum_{j=1}^2 h_{-,j}^\dagger h_{-,j}
    +\frac{1}{2}\sum_{j,k=1,2,3} h_{j,k}^\dagger h_{j,k}\right),
\end{equation}
where
\begin{align*}
    h_0&=3\sqrt{3}\,\mathbb{I} - \mathcal{B}_{\rm Yao},\\
    h_{+,j}&=A_1B_2C_3 +\exp{i(2\pi j/3)}A_2B_3C_1 +\exp{i(4\pi j/3)}A_3B_1C_2,\\
    h_{-,j}&=A_1B_3C_2 +\exp{i(2\pi j/3)}A_2B_1C_3 +\exp{i(4\pi j/3)}A_3B_2C_1,\\
    h_{j,k}&=A_jB_jC_k + A_jB_kC_j + A_kB_jC_j.
\end{align*}
This makes it explicit that whenever the constraints are satisfied,
$3\sqrt{3}\,\mathbb{I} -\mathcal{B}_{3322}\ge0$ and therefore
$\mathcal{S}_{3322}\le3\sqrt{3}$. As a last remark, we note that the
constraint term $\sum_{i,j} s_{ij}^\dagger\, p_i\,s_{ij}$ could have
been included explicitly in the WSOS decomposition for
$q_{3\sqrt{3}}=3\sqrt{3}\,\mathbb{I}-\mathcal{B}_{\rm Yao}$ and we refer the reader to
Appendix~\ref{App:Yao} for details.

\section{Acknowledgements}
We thank Stefano Pironio and Tsuyoshi Ito for interesting discussions and for
sharing drafts of their papers~\cite{navascues08:long,Ito08} with us.
SW is supported by the National Science Foundation under contract
number PHY-0456720.

redpajama_set_name: RedPajamaArXiv


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How to Use this Edition
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Key Documents of German-Jewish History
A Digital Source Edition
Epochs and Events
A Hamburg Friendship. Erwin Panofsky's obituary on Aby Warburg
Emily J. Levine
Source Description
This source is an obituary first published by the art historian Erwin Panofsky in the Hamburger Fremdenblatt on 28.10.1929 following his elder colleague Aby Warburg's death on 26.10.1929. However, it is more than the record of an art historian's life. Born in Hannover in 1892, and educated in Berlin, Freiburg, and Munich, Panofsky was hired by the University of Hamburg in 1920, and he quickly climbed the ranks to become an Ordinarius professor in 1928 and a leading figure of the still nascent field of art history. Panofsky and Warburg were engaged, like many art historians of their day, in identifying whether art was an expression of its time period, a national characteristic, or artistic genius. Panofsky's obituary reveals the intellectual partnership between the two and the wider urban community of scholars that congregated around the Kulturwissenschaftliche Bibliothek Warburg (KBW) in Hamburg during the 1920s. Though the Hamburger Fremdenblatt was a daily newspaper with a wide audience, the intimate and detailed nature of the obituary was—like much of the activity of the KBW—presented with an insider audience in the mind. As a celebration of this partnership, Panofsky's obituary of Warburg offers a window into this distinct Hamburgian sub-culture of Weimar-era intellectual life in which German-Jewish scholars were able to play such an active role.
Aby Warburg, scholar with a true intellectual calling
Warburg's criterion of "passionate excitement of the soul"
Panofsky's relation to Aby Warburg
Aby Warburg's life work
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German historian of art and civilization, Aby (Abraham Moritz) Warburg was born the first of seven children to a banking family in Hamburg in 1866. When Aby was 13 and his younger brother Max only 12, Aby sold his share in his family's prominent bank to his brother, in exchange for Max's agreement to buy him books for the rest of his life. With roots in the 16th century, M.M. Warburg & Company became an international commercial business in the last decade of the 19th century under the leadership of Aby's brother Max. Aby Warburg, in turn, devoted himself wholeheartedly to his studies, producing scholarly writings that focused on the intellectual and social context of Renaissance art. His works such as Bildniskunst und Florentinisches Bürgertum (1902), Die Grablegung Rogers in den Uffizien (1903), and Francesco Sassettis letztwillige Verfügung (1907) were concerned with the relationship between classical antiquity and the Christian religion in the Renaissance. Later writings as Italienische Kunst und internationale Astrologie im Palazzo Schifanoja zu Ferrara (1912), Heidnisch-antike Weissagung in Wort und Bild zu Luthers Zeiten (1920) and Orientalisierende Astrologie (1926) revealed the importance of classical astrology in Renaissance art. In these works and in numerous essays, Warburg developed an interdisciplinary approach to art history, which investigated the psychological and cultural role of symbolism in general.
However, as Panofsky's text makes clear, Warburg was hardly a traditional scholar, either in profession or temperament. He founded a privately-funded library that was critical to a vibrant partnership between cultural philanthropy and scholarship, a partnership that placed Hamburg at the center of the cultural map of Europe in the 1920s. In his text, Panofsky adopts a famous aphorism from Leonardo de Vinci to describe Warburg, "He who is tied to a star will not turn back.", suggesting that his elder colleague was endowed with a special destiny and the burden that accompanied it. As Panofsky admiringly recounts, Warburg had a true intellectual calling to investigate the rich humanist tradition and its development over time. Panofsky opts for an unlikely choice of words since Warburg was never officially called (berufen) to a university as a professor but served the scholarly world, the new discipline of art history, and the newly founded university in an extra-university capacity his entire life. A half generation older than Panofsky, Warburg did not benefit from the new Weimar-era universities like that in Hamburg which was more open to hiring Jewish scholars. Instead, he relied for support on his childhood deal with his brother, which Max later recalled, "was certainly the most careless [business decision] of my life." Namely, because Aby's library, then a boyhood obsession, became the Kulturwissenschaftliche Bibliothek Warburg (KBW), which officially opened in Hamburg in 1926, and was funded on a yearly basis by the family bank. A collection of 60,000 volumes at its height, the KBW was quietly transferred to London in 1933 after Hitler came to power. In Panofsky's telling, the library was an expression of what was possible for German and German-Jewish scholars through new spaces of scholarship and sources of income in interwar Hamburg.
At the young age of twenty-five, Warburg published his dissertation, Bildniskunst und Florentinisches Bürgertum (1902), which utilized a wide array of texts to analyze early Renaissance art. Here Warburg first introduced his characteristic category, "passionate excitement of the soul," that paved the way for his scholarly path, which united research and formal analysis, iconographic meaning and exegesis. In his meticulous attention to details, Warburg read the properties of all human knowledge as the history of human emotions. Panofsky describes how Warburg traced the history of expressive formal tensions over time from the ancient through the modern periods where subversive tendencies were submerged, and "had to be forgotten, reclaimed, and overcome time and again" Amending Warburg's well-known observation that "Athens wants again and again to be reconquered afresh from Alexandria", Panofsky betrays the crucial distinction between the two: Warburg remained drawn to what Burckhardt called the "shadow" of human society while Panofsky focused on higher culture—a distinction that separated them as scholars and sent them on divergent intellectual paths and professional careers.
That Panofsky was asked to write the obituary is itself noteworthy. Other obituaries were written by closer friends, including Fritz Saxl and Gertrud Bing. For while Warburg was continuously overrun by his irrational demons—what scholars have identified as manic depresion—Panofsky was known to smooth over these tensions and elevate them in the presentation of artistic genius. To a certain degree, Panofsky was viewed as a "popularizer" of Warburg's ideas, which were more complex, nuanced, but for that, also less accessible than those of his younger colleague. Given this contrast, it is interesting that Warburg's lifelong struggle with mental illness receives a sensitive but honest treatment from a friend, a struggle that is glossed over in Ernst Gombrich's seminal biography of Warburg. Panofsky's account of Warburg's life and work rightly recognizes what is unique about his contribution: its interdisciplinarity. Enabled by the private institute that was unhindered by what Warburg called the "Grenzwächterei," Warburg began with a comparison of art works and poetry and extended out from "das Zentrum eines Kreises mit unendlich vielen Radien" until it included also occult, philosophy, language, mathematics, and science. Warburg's investigation led him to place the study of art in a wider context, and to widen what was considered legitimate art.
At the center of this productive interdisciplinary partnership was Warburg's library, which Warburg reminds us was "not just the instrument but also the vivid representation of his intellectual endeavour." Indeed Warburg scarcely published more than 200 pages. His library was arguably his greatest life's work, a representation of his vision, its collection, series, and the scholarly circle which he cultivated. Moreover, the library drew on its urban surrounding, as Panofsky observed, "without the same agony" that accompanied Warburg's written work. Warburg was known to have said that he was a "Hamburger at heart, Jew by blood, [and] Florentine in spirit." One sees this urban affinity reflected in Panofsky's delightful nod to the Hamburg port in his description of Warburg's colleagues as the "crew for his Columbus ship." Like its urban home, the library, too, was welcoming to outsiders and always looked to the Atlantic world rather than to German models for its organization as a privately-funded extra-university institute. In turn, what Panofsky called the "Hamburg School" responded with scholarship that highlighted the role that material conditions played for ideas and art, a contribution that would outlast its time and place even if many of these German Jewish scholars sadly would not.
Gertrud Bing, Aby M. Warburg (1958), in: Rivistia storica italiana 71 (1960).
Ernst Gombrich / Fritz Saxl, Aby Warburg: Intellectual Biography, Chicago 1986.
Emily J. Levine, Dreamland of Humanists: Warburg, Cassirer, Panofsky, and the Hamburg School, Chicago et al. 2013.
Karen Michels / Martin Warnke (eds.), Erwin Panofsky: Deutschsprachige Aufsätze. 2 vol.: Studien aus dem Warburg-Haus, Berlin 1999.
Aby Warburg, Die Erneuerung der heidnischen Antike: Kulturwissenschaftliche Beiträge zur Geschichte der europäischen Renaissance. 1.1 und 1.2, ed. by Horst Bredekamp / Michael Diers, in: Horst Bredekamp et al. (eds.), Gesammelte Schriften: Studienausgabe, Berlin, 1998.
This text is licensed under a Creative Commons Attribution - Non commercial - No Derivatives 4.0 International License. As long as the work is unedited and you give appropriate credit according to the Recommended Citation, you may reuse and redistribute the material in any medium or format for non-commercial purposes.
Emily J. Levine, Dr. phil., is associate professor at the University of North Carolina at Greensboro. In her research she deals with history of thought and science in 19th and 20th century-Europe and North America. Her first monography is titled "Dreamland of Humanists: Warburg, Cassirer, Panofsky and the Hamburg School".
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Emily J. Levine, A Hamburg Friendship. Erwin Panofsky's obituary on Aby Warburg, in: Key Documents of German-Jewish History, February 02, 2017. <https://dx.doi.org/10.23691/jgo:article-25.en.v1> [January 22, 2020].
Obituary for Aby Warburg in the Hamburger Fremdenblatt, October 28, 1929
Memory and Remembrance

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Regardless of whether or not LTC insurance is the right answer, every family should have a very frank discussion concerning potential illness or incapacity of family members. Although it is a difficult subject to broach, everyone should have some plan in place that is well understood in advance by all concerned. When adversity strikes - many times without warning - those decisions will have been already made with no surprises.
Eleanore K. Szymanski is a fee-only Certified Financial Planner Practitioner and principal of The Financial Planning Answerplace, LLC and EKS Associates of Princeton. You may send questions to her at Elly@TheFPAnswerplace.com or info@eksassoc.com.


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Christianity: General >
Advent [Lat.,=coming], season of the Christian ecclesiastical year preceding Christmas, lasting in the West from the Sunday nearest Nov. 30 (St. Andrew's Day) until Christmas Eve. In the Roman Catholic Church it is traditionally considered a season of penitence and fasting, to prepare for the holy day, and its liturgical color is purple. However, the Roman observance has always contained an element of joyful anticipation of Christmas, a feeling that prevails during this season in Western churches today. Originally Advent was seen as a time of preparation for the feast of Christ's nativity. But during the Middle Ages this meaning was extended to include preparation for Christ's second coming, as well as Christ's present coming through grace.
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Pine Knoll Shores History
Welcome to the history blog for the Town of Pine Knoll Shores, NC. Browse our site and discover the people, places and events that create the rich heritage of this unique coastal community. Come back often. New posts are added monthly.
Beach Town in a Forest
Beach Town in a Forest, Pine Knoll Shores located in Carteret County on North Carolina's Crysal Coast. Photo compliments of Bill Flexman and Dave Prutzman
Dairy Farming on Bogue Banks
Recently, in the process of clearing a lot on Holly Rd. of undergrowth, a Pine Knoll Shores property owner uncovered a remnant of Alice Hoffman's time on Bogue Banks. To understand the significance of this discovery, we must piece together some background information about Alice Hoffman.
Having lived in Paris since the 1880s, Alice, by 1914, found life in her beloved France becoming less appealing each day. Hostilities in Europe were closing in, the eastern portion of France was overrun and German forces were within 50 miles of Paris. The combatants were employing poison gas. The sound of artillery explosions were felt on the streets of Paris, rationing was in place and travel restrictions were enforced. In July 1914, she closed her home at 29 Avenue du Bois de Boulogne and left for the duration. When the U.S. became involved in WW I, Alice gave the use of this home to her niece Eleanor Butler Alexander Roosevelt since Eleanor's husband, Ted, the son of President Teddy Roosevelt, was serving with U.S. forces in France.
By 1915, Alice discovered Bogue Banks: "I feel that Bogue Banks is as safe a place to be as any to be found. It is too remote to attract gas bombs...." (Alice Hoffman's unpublished Autobiography) After acquiring 2,900 acres on Bogue Banks and expanding the small cabin originally built by John Royall, she embarked on making the land productive.
Before the end of World War I, she began a concerted effort to establish a dairy herd on Bogue Banks.
She understood the impact of the war in Europe, particularly the devastation to cattle and dairy herds, so she decided to raise Holstein-Friesian dairy cattle on Bogue Banks and send them to France after the war to help re-establish the French and Belgium dairy industry. The cattle were to be a gift to her second home.
It turns out that Bogue Banks is not particularly suited for raising cattle. She purchased the highest quality breeding stock she could find; hired experienced managers to help run the operation; spent extensively on feed, supplies, equipment, buildings, medical testing and research. She dealt with disputes about the pedigree of some of her breeding stock. Barns and corrals were built, and a large portion of the land was fenced. Acreage was cleared attempting to grow pasture.
Her livestock operations as well as most such operations in North Carolina and the rest of the US were periodically bedeviled with outbreaks of Bovine Tuberculosis. This scourge was a major state and federal government concern. For a period of time, dairy cattle in North Carolina were under quarantine and could not be removed from the state.
To combat an infection from tick-borne Taxes Fever, she had a dipping vat built. The animals were led through the trough while it was filled with an insecticide solution.
It is this dipping trough that was uncovered recently on Holly Rd. The concrete structure is 30 feet long. The inside dimensions are 36" wide at the top, narrowing to 30" at the bottom. At its low point, it is over five feet deep.
From her unpublished autobiography and material contained in the Alice Green Hoffman collection in the Joyner Library at East Carolina University and other historic documents, it is most probable the dipping vat was constructed in the late teens or early 1920s. This structure is in remarkably good condition given its age and considering that at the time all construction materials had to be brought to the island by boat. A bridge to Bogue Banks was not opened until 1928.
In spite of the many difficulties that presented themselves, she persevered, and in the early 1920s, sent a small group of breeding stock to France. Perhaps the offspring of her animals live on today and find their way back to us in the form of fine French cheeses—"Pine Knoll Shores Camembert".
This 1939 aerial shows the extent of the cleared area. The blue and red lines show the location of today's roads.
1964 Aerial by Piedmount Aviation showing first portion of Oakleaf Dr ending on the west at the remnants of Alice Hoffman's house. The acreage that had been cleared in the 1920s is filling in with new growth. The dipping vat is located at the southern edge of the cleared area west of Holly Rd.
1954 survey by Rivers & Associates identifies the Roosevelt property on Bogue Banks and on the north side o f the Sound. The mainland property was the location of Alice Hoffman's commercial dairy operation.
After her struggles raising cattle on Bogue Banks, in 1923, she began acquiring land on the north shore of Bogue Sound near the town of Mansfield. Eventually owning over 600 acres, land she hoped would be more ameanable to raising and maintaining a herd of dairy cows. On this property, Alice Hoffman established a commercial dairy operation, which included a healthy herd and a bottling plant. The dairy products were delivered to local stores and communities. As with many of her business ventures this too was a financial failure and, by the mid-30s was abandoned.
Post author: Walt Zaenker
To contact the author or the History Committee
Early Mayors
Jim Redfield
Ken Haller
Wayne Cleveland
Recent Exhibit
Coming Announcement
Museum Touch Screen
A Chronolog
Abonita
Atlantic Hotel
Banks, Outer Banks
Bogue Banks Library
Bogue Banks State Park, 1923
Boundaries, part 3 of 3
Deeds, part 1 of 3
Early Sittum History
Sittum Storms 1995-1999
John Royall
Hoop Pole Watershed
Landmarks, part 2 of 3
S.S. Pevencey
Theron R. Butler
Was Verrazzano Here
Alice Green Hoffman
91 Years, a Timeline
Alice's Financial Misadventures I
Alice's Financial Misadventures II
Alice's Financial Misadventures III
Alice in her Own Words
Alice's Last Will and Sad Testament
Alice's Teahouse
Alice the Gambler
Evolution of Hoffman Homestead
Flying to San Juan
Gabrielle Brard - Gabby
Georgina Pope Yeatman
Influence of Friends
Iver-Johnson Revolver
Kitchen, Food, and Drink
Life at Shore House
Locating Alice's House
Miss Porters School for Girls - under construction
Our First Conservationist
Pistols and Parrots
French Connection part I
From Paris to Bogue Banks, part II
United States Post Office
Why did Alice come to Bogue Banks
Roosevelts
Alice's Tie to the Roosevelts
Roosevelt Family Tree
Roosevelts Who Developed PKS
What Shall We Call This Place?
Bogue Banks
Beach Protection
Bogue, The Name
C.C.C.
Churches of Salter Path
Iron Steamer
Hurricane History
Links & Bridges
Radio Island
Traveling to Bogue Banks
Willis Seafood
World War II on Bogue Banks
Town of Pine Knoll Shores
A Chronology
A.C. Hall: Master Planner
A House Where Alice Lived
Brock Basin
Canal Building
Early Pine Knoll Shores
Early Town Clerks
Emergency Rescue Squad #65
Fire Fighting& Rescue
Main Street, PKS
Why Its All About Trees
Beachwalk
Beacon's Reach
Bogue Shore Club
McGinnis Point
Pine Knoll Townes
Pine Knoll Village
PKA & PIKSCO
Reefstone
Whaler Inn
Roosevelt Natural Area
Garner Park
Storms That Took Our Sittum
Time Period Listing
S.S. Pevensey
Late 1800s - Early 1900s
The Atlantic Hotel
The French Connection Part I
1915 - 1953, The Alice Hoffman Years
Alice's Last and Sad Testament
1940 - 1980s The Roosevelt Years
A.C.Hall: Master Planner
Beacons Reach
1970 Continued Growth
Fire Fighting & Rescue
"Beach Town in a Forest"