# Hierarchies for Semidefinite Optimization in $\mathcal{C}^{\star}$-Algebras

Gereon Koßmann<EMAIL_ADDRESS>Institute for Quantum
Information, RWTH Aachen University, Aachen, Germany  René Schwonnek
<EMAIL_ADDRESS>Naturwissenschaftlich–Technische Fakultät,
Universität Siegen, Germany  Institut für Theoretische Physik, Leibniz
Universität Hannover, Germany  Jonathan Steinberg<EMAIL_ADDRESS>siegen.de Naturwissenschaftlich–Technische Fakultät, Universität Siegen,
Germany

###### Abstract

Semidefinite Optimization has become a standard technique in the landscape of
Mathematical Programming that has many applications in finite dimensional
Quantum Information Theory. This paper presents a way for finite-dimensional
relaxations of general cone programs on $\mathcal{C}^{\star}$-algebras which
have structurally similar properties to ordinary cone programs, only putting
the notion of positivity at the core of optimization. We show that well-known
hierarchies for generalized problems like NPA[1] but also Lasserre’s
hierarchy[2] and to some extend symmetry reductions of generic SDPs by de-
Klerk et al. [3] can be considered from a general point of view of
$\mathcal{C}^{\star}$-algebras in combination to optimization problems.

###### Abstract

The class of Semidefinite Programs on matrices has a natural extension to
conic optimization problems on $\mathcal{C}^{\star}$-algebras. Especially in
the field of Quantum Information Theory many relevant optimization problems
can be cast in this way. Finding exact solutions to these problems is however
a notoriously hard task.

In this paper we investigate a construction for finite-dimensional relaxations
of general cone programs on $\mathcal{C}^{\star}$-algebras. This construction
gives outer bounds and can be grasped in terms of positive maps and basic
linear algebra. We show that well-known hierarchies like NPA[1] but also
Lasserre’s hierarchy[2] and to some extend symmetry reductions of generic SDPs
by de-Klerk et al. [3] can be formulated from this perspective.

##  I Introduction

In recent years, it has become increasingly apparent that many optimization
problems can be represented as cone programs. In particular the community
figured out that there is an incredible strong interplay between cone programs
and convex optimization problems whereby the convex set of feasible points is
not directly accessible[2, 4, 5]. In quantum information the most prominent
representative of this type of problem is the optimization task over separable
states. Optimizing over separable states is a convex optimization problem but
the description of the set of separable states is computationally hard[5].
However, the solution is a hierarchy of efficient SDPs and the complexity of
the problem is translated to a convergent hierarchy of convex optimization
problems. Even from another point of view, one can ask whether two parties in
a quantum experiment can achieve certain probability distributions. This task
was solved by the famous NPA hierarchy[1, 6]. What all these hierarchies have
in common is that they map the structure of states to a finite space and have
a kind of mapping of positive elements to positive elements, because in
quantum information theory, a generic cone structure of positivity exists due
to the state space of quantum systems. However, due to the existence of mixed
states, more general forms of positivity have to be considered.

In this work, we focus on this idea of pursuing positivity in the most general
structure of $\mathcal{C}^{\star}$-algebras and assign a
$\mathcal{C}^{\star}$-algebra to the description of a quantum experiment. In
the simplest and yet also most general form, this is done by rules such as how
two or more parties are spatially related[7]. This top down approach to the
description of an experiment in nature inevitably leads to questions about the
state space and properties of observables with respect to states. For example,
if one wants to find the optimal state for a given observable over all states,
this is directly a generalized form of a SDP: Optimize over all positive
functionals so that they are normalized. Usually this problem is hard to
tackle because the $\mathcal{C}^{\star}$-algebra to be considered is infinite
dimensional.

In this paper, we present a generalized formalism that is both mathematically
sound with respect to the rules of the system (i.e., the spatial partitioning
of the parties) and capable of solving general optimization problems.

The success and popularity can be mainly ascribed to two different points.
First, a huge variety of problems can be formulated (at least approximately)
in the framework of semidefinite programming. Second, there exists algorithms,
like interior-point methods, which allow for an efficient computation of the
solution. More formally, a SDP describes the task of optimizing a given linear
functional over the cone of positive (semidefinite) matrices of a fixed
dimension $n$, under linear constraints.

In addition to our motivation in quantum information, we note in passing that
many well-known problems can be formulated as $\mathcal{C}^{\star}$-SDP. In
particular, many typical benchmark examples for quantum computing such as
Boolean optimization can be understood in a general way as
$\mathcal{C}^{\star}$-SDP.

In this work, we investigate a generalization of semidefinite programming from
the algebra of matrices to an arbitrary $\mathcal{C}^{\star}$-algebra, which
is obtained by replacing the cone of positive semidefinite matrices with the
cone of positive semidefinite functionals.

Further, we show that the finite dimensional approximations can be rephrased
as a conventional matrix SDP, turning them into a efficient solvable problem.
Subsequent, we discuss its relations to well known hieraries as those of
Lasserre [2] and Navascues-Pironio-Acin (NPA) [1].

##  II Preliminaries and introductory example

We start our considerations with a simple example coming from an operational
point of view. Consider the following generic problem in quantum theory:
$H\in\mathcal{B}(\mathcal{H}_{1})$ is a (selfadjoint) bounded operator on a
Hilbert space $\mathcal{H}_{1}$ and $\mathcal{B}_{1}(\mathcal{H}_{1})^{+}$ is
the set of all positive trace-class operators. We aim to minimize the
expectation value of

$\displaystyle\min$ $\displaystyle\operatorname{tr}[\rho H]$ (1) s.th.
$\displaystyle\operatorname{tr}[\rho]=1$ (2) $\displaystyle\rho\in$
$\displaystyle\mathcal{B}_{1}(\mathcal{H}_{1})^{+}.$ (3)

This problem is inherently hard, because $H$ is in general an operator on a
infinite dimensional Hilbert space and apparently there is a priori no
efficient way to handle the set of states.

If we now assume to have access to a quantum channel111i.e. a completely
positive and trace-preserving linear map.

$\displaystyle
T:\mathcal{B}_{1}(\mathcal{H}_{1})\to\mathcal{B}_{1}(\mathcal{H}_{2})$

and we know an operator $M\in\mathcal{B}(\mathcal{H}_{2})$ such that
$T^{\star}(M)=H$, then it follows that the optimization problem (1) is
equivalent to222Recall that if the channel $T$ is trace-preserving the dual
mapping is unital.

$\displaystyle\min$ $\displaystyle\operatorname{tr}[\sigma M]$ s.th.
$\displaystyle\operatorname{tr}[\sigma]=1$ $\displaystyle\sigma\in T($
$\displaystyle\mathcal{B}_{1}(\mathcal{H}_{1})^{+})\subset\mathcal{B}_{1}(\mathcal{H}_{2})^{+}.$

Now it can happen that the Hilbert space $\mathcal{H}_{2}$ is even finite
dimensional. We have shifted the entire difficulty of the problem to the
description of $T(\mathcal{B}_{1}(\mathcal{H}_{1})^{+})$, i.e. the image of
the quantum channel $T$. Nevertheless, this set contains global information of
the cone $\mathcal{B}_{1}(\mathcal{H}_{1})^{+}$ and is therefore in general
hard to describe. However it could be contained in a finite dimensional space.
We can make an illustrative example: consider an entanglement breaking
channel, which can be given by[8, 9]

$\displaystyle\rho\mapsto\sum_{k}\sigma_{k}\operatorname{tr}[F_{k}\rho]$

for density operators
$(\sigma_{k})_{k}\in\mathcal{B}_{1}(\mathcal{H}_{2})^{+}$ and a POVM
$\\{F_{k}\\}_{k}\subset\mathcal{B}(\mathcal{H}_{1})$. It is clear that the
dual mapping is given by

$\displaystyle M\mapsto\sum_{k}\operatorname{tr}[\sigma_{k}M]F_{k}.$

Therefore, if we assume that $\\{F_{k}\\}_{k}$ corresponds to the spectral
decomposition of $H$ and $M$ is chosen in a way that
$\operatorname{tr}[\sigma_{k}M]$ is the $k$th eigenvalue, then we can conclude
that this is in general an infinite series and therefore the cone of
interesting density operators

$\displaystyle\text{cone}(\sigma_{k}\ |\ k\in\mathbb{N})$

is hard to describe although it is contained in a possible finite dimensional
space. This example provides us with some nice theoretical insights. First of
all it tells us how to convert at least theoretically infinite dimensional
optimization problems to finite dimensional spaces. Moreover if we can
approximate $T(\mathcal{B}_{1}(\mathcal{H}_{1})^{+})$ appropriately with
linear constraints, then we can relax the problem with a standard semidefinite
program. As it turns out, in the example above we have chosen the state space
as the set of trace-class operators, which would correspond to the normal
states. In order to avoid this discussion about duality, we abstract to
general $\mathcal{C}^{\star}$-algebras and consider the dual as the generic
state space. In addition, as we will conclude, our considerations are not
restricted to the special class of quantum channels. Positivity of linear maps
will be the central property of this paper in comparison to completely
positivity and trace-preserving maps.

The structure of the document is as follows: in section III we introduce into
our notion of $\mathcal{C}^{\star}$-SDP. In section IV we show how to relax
general $\mathcal{C}^{\star}$-SDP and in section V we show a convergence
result for particular assumptions. In the last section section VI we give a
small overview about the connection to symmetry reductions in this context.

##  III Semidefinite optimization on $\mathcal{C}^{\star}$-algebras

We generalize now our toy example from section II to a general language for so
called _$\mathcal{C}^{\star}$ -SDP_ and define what we mean by optimality in
this context.

###  III.1 Conventional SDPs

Formally, an instance of a conventional SDP is given by a set of $m+1$
selfadjoint matrices $\\{F_{0},F_{1},\dots,F_{m}\\}$, all coming from a common
space $\mathbb{C}^{n\times n}$, and a set of real numbers
$\\{f_{1},\dots,f_{m}\\}$ that together define the optimization problem

$\displaystyle\min\quad$ $\displaystyle\text{tr}\left(\rho F_{0}\right)$ (4)
s.th. $\displaystyle\text{tr}\left(\rho F_{i}\right)\leq f_{i}$
$\displaystyle\forall i\in\\{1,\dots,m\\}$ $\displaystyle\rho\geq 0$
$\displaystyle\qquad,$

which is conventionally referred to as the primal form of a semidefinite
program. The optimization above runs over the cone of all positive selfadjoint
matrices, i.e. those matrices $\rho$ for which equivalently[10]

  * (i)

all eigenvalues are positive and $\rho$ is self-adjoint

  * (ii)

all expectations $\text{tr}\left(\rho XX^{*}\right)$ are positive

  * (iii)

a decomposition $\rho=XX^{*}$ exists.

It should be noted that the cone of positive semidefinite (PSD) matrices plays
two roles in the case of conventional SDPs. On one hand, PSD matrices are the
positive elements of $\mathbb{C}^{n\times n}$ from a purely algebraic
perspective. This notion of positivity only depends on the form of the element
itself. This corresponds to the definition (iii) with $\rho=XX^{*}$ or to
definition (i) by using the spectrum of $\rho$. On the other hand, PSD
matrices are also used to label the set of positive linear functionals, see
definition (ii). Indeed, by Riesz representation it can be shown that any
positive linear functional $\omega:\mathbb{C}^{n\times
n}\rightarrow\mathbb{C}$ is of the from $\omega(X)=\text{tr}\left(\rho
X\right)$ for a positive $\rho$. Consequently, each of those properties
(i)-(iii) could in principle give rise to a variety of different
generalizations. Initially the class of optimization problems that is
considered in this work could be understood as a generalization of property
(ii): we generalize the matrices $\\{F_{0},F_{1},\dots,F_{m}\\}$ by self-
adjoint elements of a $C^{*}$-algebra $\mathcal{A}$ and consider optimizations
over the cone of functionals that are positive semidefinite on all elements of
the form $XX^{*}$.

###  III.2 Generalized SDPs

With respect to the usual matrix product the vector space $\mathbb{C}^{n\times
n}$ naturally admits the structure of an algebra, on which it is somewhat
natural and well motivated by many applications to consider the hermitian
conjugation

$\displaystyle X^{*}:=X^{\dagger}=\overline{X}^{T}$ (5)

as an involution and

$\displaystyle\|X\|:=\sup\\{\,\text{tr}\left(\rho X\right)\,|\,\rho\geq
0\text{ and }\text{tr}\left(\rho\right)=1\\}$ (6)

as a norm. In the following we will write
$M_{n}(\mathbb{C}):=(\mathbb{C}^{n\times n},\cdot,\phantom{x}^{*},\|\cdot\|)$
in order to denote $\mathbb{C}^{n\times n}$ equipped with this extra
structure. Further this norm shares the property of being submultiplicative,
that is for any $X,Y\in M_{n}(\mathbb{C})$ one has $\|XY\|\leq\|X\|\,\|Y\|$
and turns $M_{n}(\mathbb{C})$ into a complete space.

The abstraction of this structure, i.e. $M_{n}(\mathbb{C})$ is nowadays known
as a $\mathcal{C}^{\star}$-algebra $\mathcal{A}$: a Banach, i.e. norm
complete, algebra equipped with a $*$-involution that fulfills the so called
$\mathcal{C}^{\star}$-property $\|XX^{*}\|=\|X\|^{2}$[10]. The deceptively
simple axioms turn out to be extremely powerful and forcing a rigid structure
on a $\mathcal{C}^{\star}$-algebra. In particular, it is possible to retrieve
the initial example of $M_{n}(\mathbb{C})$ from the abstract definition in
case of finite dimensionality: any finite dimensional
$\mathcal{C}^{\star}$-algebra is isomorphic to a direct sum of matrix algebras
(Artin-Wedderburn Theorem).

However, in a general algebra $\mathcal{A}$, the three definitions (i)-(iii)
are not longer equivalent and we have to make a decision for defining what a
generalized SDP is. Given a general algebra $\mathcal{A}$ it makes sense to
call an element $F\in\mathcal{A}$ selfadjoint, if $F=F^{*}$ holds with respect
to the $*$-involution. Furthermore, we get a generalization of positive
semidefiniteness by generalizing property (iii), i.e. we say that an element
$P\in\mathcal{A}$ is positive semidefinite, written $P\geq 0$, if it admits a
decomposition $P=XX^{*}$. Alternatively one could think about generalizing
property (i) by demanding $P$ to have a non-negative spectrum. However, it
turns out that this definition would coincide with $P=XX^{*}$, which is in
fact a distinct property of $\mathcal{C}^{\star}$-algebras333Indeed, for the
more general concept of a $*$-algebra, where one drops the requirement of a
complete norm, there exist cases where $\rho=XX^{*}$ can have a negative
eigenvalues.. The set of all those positive elements forms a closed convex
cone in $\mathcal{A}$, which will be denoted by
$\mathcal{P}_{+}(\mathcal{A})$. The dual of this cone will be denoted by
$\mathcal{P}_{+}(\mathcal{A})^{\star}$ and can be defined algebraically[10] by

$\displaystyle\mathcal{P}_{+}(\mathcal{A})^{\star}:=\\{\rho:\mathcal{A}\mapsto\mathbb{C}\
|\ \text{ linear},\quad\rho(XX^{*})\geq 0\quad\forall X\in\mathcal{A}\\}.$ (7)

Now we are able to state the generalized version of a conventional SDP with
operations on an abstract $\mathcal{C}^{\star}$-algebra. Motivated by
applications in which $\rho$ takes the role of a quantum state, we take the
path of replacing $\mathcal{P}_{+}(\mathbb{C}^{n\times n})^{\star}$ by
$\mathcal{P}_{+}(\mathcal{\mathcal{A}})^{\star}$ and therefore consider
optimization problems of the following type:

######  Definition 1.

For $m\in\mathbb{N}$, let $\\{F_{0},F_{1},\dots F_{m}\\}$ be a list of
selfadjoint elements from a $\mathcal{C}^{\star}$-algebra $\mathcal{A}$ and
let $\\{f_{1},\dots,f_{m}\\}$ be a list of real numbers. The conic
optimization problem

$\displaystyle\inf\quad$ $\displaystyle\omega(F_{0})$ (8) s.th.
$\displaystyle\omega(F_{i})\leq f_{i}$ $\displaystyle\forall
i\in\\{1,\dots,m\\}$
$\displaystyle\omega\in\mathcal{P}_{+}(\mathcal{\mathcal{A}})^{\star}$ (9)

is called the primal form of a semidefinite program on $\mathcal{A}$, short
$\mathcal{C}^{\star}$-SDP.

Same as in the case of ordinary SDP, we can define feasible points.

######  Definition 2.

A $\mathcal{C}^{\star}$-SDP is called feasible, if there is an element
$\omega\in\mathcal{P}_{+}(\mathcal{\mathcal{A}})^{\star}$ with

$\displaystyle\omega(F_{i})\leq f_{i}$ $\displaystyle\forall
i\in\\{1,\dots,m\\}.$ (10)

Accordingly an optimal solution is an element $\tilde{\omega}$, such that for
all feasible solutions
$\omega\in\mathcal{P}_{+}(\mathcal{\mathcal{A}})^{\star}$

$\omega(F_{0})\geq\tilde{\omega}(F_{0})$ (11)

holds true.

Because well-known finite dimensional SDPs are included in this
generalization, we must be appropriately careful when adding a value to the
program. This is not important for the first statements, since we first
specify bounds for the value. It only gets interesting when we define and use
the dual program. Therefore, the value (possibly infinity) of a
$\mathcal{C}^{\star}$-SDP is given by

$\tilde{c}:=\inf\\{\omega(F_{0})\ |\ \omega(F_{i})\geq f_{i}\quad\forall
i\in\\{1,\dots,m\\}\\},\quad\omega\in\mathcal{P}_{+}(\mathcal{\mathcal{A}})^{\star}.$
(12)

Our original interest on this class of problems stems from the huge variety of
well-known problem types that can obtained as particular instances of (8). In
the next section we present examples of how those problems can be rephrased.

However, for understanding the theoretical insights the interested reader can
skip the next section.

###  III.3 Examples of $\mathcal{C}^{\star}$-SDPs

In the following we will rephrase two important optimization problems, namely
the problem of polynomial optimization and the problem of optimizing over the
set of quantum correlations as a $\mathcal{C}^{\star}$-SDP.

######  Example 3 (The Generalized Moment Problem).

We consider a the generalized moment problem (GMP), which can be solved
Lasserre’s famous polynomial optimization method [2]. Without sake of
completeness we introduce concepts from Lasserre’s hierarchy which are
important for us later in this work. Consider a polynomial $f$ in $d$
variables and a compact set $K\subset\mathbb{R}^{d}$ and the optimization task

$\inf\\{f(x)\ |\ x\in K\\}.$ (13)

One can show[2] that (13) is equivalent to the following optimization task:

$\displaystyle\inf\quad$ $\displaystyle\mu(f)$ s.th. $\displaystyle\mu(1)=1$
$\displaystyle\mu\in\mathcal{M}(K)_{+}$

whereby $\mathcal{M}(K)_{+}$ is the set of positive Borel measures; i.e. the
topological dual space of $C(K)$. In general a GMP is given by

$\displaystyle\rho_{\text{mom}}=\sup_{\mu\in\mathcal{M}(K)_{+}}$
$\displaystyle\int_{K}fd\mu$ (14) s.th. $\displaystyle\int_{K}h_{j}d\mu\leq
h_{j}\quad\forall 1\leq j\leq m$

for multivariate polynomials $f,h_{1},\ldots,h_{m}$ and real numbers
$\gamma_{1},\ldots,\gamma_{m}$.

one possible point of view would be the following. Consider the set of finite
positive Borel measures and in particular the _moment sequence_
$y=(y_{\alpha})_{\alpha\in\mathbb{N}^{n}}$ for each of them given by

$\displaystyle y_{\alpha}=\int_{K}x^{\alpha}d\mu.$

If we introduce the functional

$\displaystyle L_{y}:\mathbb{R}[x]$ $\displaystyle\to\mathbb{R}$
$\displaystyle f$ $\displaystyle\mapsto
L_{y}(f):=\sum_{\alpha\in\mathbb{N}^{n}}f_{\alpha}y_{\alpha}$

for $f=\sum_{\alpha}f_{\alpha}x^{\alpha}$, so we can reformulate the
optimization problem (14) to

$\displaystyle\sup_{y\ \text{moment sequence}}$ $\displaystyle L_{y}(f)$ s.th.
$\displaystyle L_{y}(h_{j})\leq\gamma_{j}\quad\forall 1\leq j\leq m.$

The question whether $y$ is moment sequence can be answered with linear
constraints of the following type. Define for a multivariate polynomial in $n$
variables _moment matrices_ to be

$\displaystyle
M_{r}(uy)_{\alpha,\beta}:=(L_{y}(ux^{\alpha}x^{\beta}))_{\alpha,\beta}$

whereby we cut the degree up to $r$. Then Theorem $3.8$[11] states that $y$
has a finite Borel representing measure with support contained in a semi-
algebraic set $K=\\{x\in\mathbb{R}^{n}\ |\ g_{j}(x)\geq 0\quad\forall 1\leq
j\leq m$444There is in addition a compactness criteria for technical reasons.
if and only if

$\displaystyle L_{y}(f^{2}g_{J})\geq 0\quad\forall
J\subset\\{1,\ldots,m\\}\quad\forall f\in\mathbb{R}[x]$

or equivalently

$\displaystyle M_{r}(g_{J}y)\geq 0\quad\forall
J\subset\\{1,\ldots,m\\}\quad\forall r\in\mathbb{N}.$

One can think about these constraints as semidefinite constraints for $y$
being a moment sequence. In particular we can introduce the dual program

$\displaystyle\rho_{\text{pop}}=\inf_{\lambda}\sum_{j=1}^{m}$
$\displaystyle\gamma_{j}\lambda_{j}$ (15) $\displaystyle\text{s.th.}\
\sum_{j=1}^{m}$ $\displaystyle\lambda_{j}h_{j}(x)-f(x)\geq 0\quad\forall x\in
K$ (16) $\displaystyle\lambda_{j}\geq 0$ $\displaystyle\quad\forall 1\leq
j\leq m.$ (17)

I.e. the problem is equivalent in checking whether a polynomial is positive
over $K$. For basic semi-algebraic sets there is Putinar’s
Positivstellensatz[12] which tells us how positive polynomials look like on
these type of sets and therefore one can basically relax the dual problem (15)
to a finite level version and solve the corresponding semidefinite program.

The interesting observation of this story is that from a global point of view
(14) is something what we will later call $\mathcal{C}^{\star}$-SDP, because
it is an optimization problem over the dual space of the
$\mathcal{C}^{\star}$-algebra $C(K)$ for the compact set $K$. Apparently this
does not solve the description of the dual space of $C(K)$, i.e. due to Riesz
famous representation theorem $\mathcal{M}(K)$, but in general it leads the
eyes to the question whether one can decide positivity in a
$\mathcal{C}^{\star}$-algebra. And this is theoretically easily done: in a
$\mathcal{C}^{\star}$-algebra $\mathcal{A}$ an element $x\in\mathcal{A}$ is
positive if and only if there is a $\star$-root, i.e. there is
$y\in\mathcal{A}$ such that $x=y^{\star}y$. That is, there is somehow the
difference of being positive in terms of polynomials, which is apparently on
the one hand a hard question and solved by Putinar’s Positivstellensatz and on
the other hand it seems to be quite easy because we have roots at our
disposal.

######  Example 4 (NPA).

Essentially the NPA hierarchy[1] examines the question whether there is a
quantum system for a given joint probability distribution. To be concrete,
consider a bipartite experiment and a bipartite probability distribution. The
question is whether the bipartite probability distribution can be generated by
local measurements of two parties. In a mathematical sense this ansatz works
with free $\star$-algebras. It chooses operators and works then with
representations on Hilbert spaces of this free $\star$-algebras.

Comparing to this ansatz we assume in the paper presented here always the
existence of a $\mathcal{C}^{\star}$-algebra at the beginning. This clarifies
mathematically a lot, because we have a direct notion of positivity which is a
priori not clear in NPA (one has to construct positivity from
representations). Moreover, we have a simple notion of ’quadratic module’, it
will be a set of squares. However, the relaxed semidefinite programs in the
end can become the same in particular cases.

In conclusion all the selected examples among many others presented above have
the common structure of a $\mathcal{C}^{\star}$-algebra and an optimization
over a feasible region in the dual cone. Therefore it is worth to construct a
theory how to solve in general such problems assuming only the global
structure.

##  IV Finite dimensional relaxations of generalized SDPs

###  IV.1 The joint numerical range

In contrast to conventional SDPs, which are considered to be efficiently
solvable, finding the solution to a generalized SDP may turn out to be a
challenging task.

In the following we will introduce methods for approximating a generalized
$\mathcal{C}^{\star}$-SDP by a finite dimensional one. The leading intuition
for the existence of such a finite approximation to a
$\mathcal{C}^{\star}$-SDP (8) stems from the observation that, even though we
may consider a wild and potentially infinite dimensional algebra
$\mathcal{A}$, all the relevant information for an optimization like (8) is in
principle encoded in the finite (indeed not more that $N$ dimensional)
subspace $\mathcal{F}\subset\mathcal{A}$ that is spanned by
$\vec{F}:=\\{F_{0},\dots,F_{N}\\}$ and in an appropriate subset of its dual
$\mathcal{F}^{\star}$. It is easy to see that we can, at least formally,
replace the optimization in (8) by only considering functionals from the
convex cone

$\displaystyle\mathcal{F}_{+}^{\star}:=\mathcal{P}_{+}(\mathcal{\mathcal{A}})^{\star}|_{\mathcal{F}}\subset\mathcal{F}^{\star}.$
(18)

Indeed, suppose we have a functional
$\omega\in\mathcal{P}_{+}(\mathcal{\mathcal{A}})^{\star}$ that is a solution
of (8). By definition, $\omega$ fulfills all constraints and thus also
$\omega|_{\mathcal{F}}$ does. Clearly, we also have
$\omega(F_{0})=\omega|_{\mathcal{F}}(F_{0})$, hence the optimal value is
independent of whether the restricted version $\omega|_{\mathcal{F}}$ or the
full supported functional $\omega$ is considered. There is also a more direct
way to see the finite dimensional object standing behind an generalized SDP.
Consider the set

$\displaystyle\operatorname{JNC}_{\vec{F}}:=\\{\vec{y}_{\omega}=(\omega(F_{0}),\dots,\omega(F_{N}))|\omega\in\mathcal{P}_{+}(\mathcal{\mathcal{A}})^{\star}\\}\subset\mathbb{R}^{N}$
(19)

to which we will refer as joint numerical cone (JNC). This name is chosen in
correspondence to the well-known concept of a (convex) joint numerical range,
which is defined similarly to (19).555 The only difference is that in the
definition of the (convex) joint numerical range $\omega$ runs over all
normalized states of a unital $\mathcal{C}^{\star}$-algebra. Moreover, the
joint numerical range can be seen as a cone base for the JNC.
$\operatorname{JNC}_{\vec{F}}$ can, indeed, be understood as embedding of
$\mathcal{F}^{\star}_{+}$ into $\mathbb{R}^{N}$, in the sense that for any
$f\in\mathcal{F}$ parameterized by coefficients $\vec{\lambda}$ with
$f=\sum\lambda_{\mu}F_{\mu}$, we have

$\displaystyle\omega(f)=\sum\lambda_{\mu}\omega(F_{\mu})=\vec{\lambda}.\vec{y}_{\omega}$
(20)

Accordingly we can state any general SDP as explicit conic program

$\displaystyle\inf\quad$ $\displaystyle\vec{y}.\mathbf{e}_{0}$ (21) s.th.
$\displaystyle\vec{y}\leq\vec{f}$
$\displaystyle\vec{y}\in\operatorname{JNC}_{\vec{F}}$ (22)

over the finite dimensional cone $\operatorname{JNC}_{\vec{F}}$.

Despite that this view is giving some nice perspective on the problem, it is
not yet doing any work. Here, the caveat is that the exact shape of
$\mathcal{F}_{+}^{\star}$ or $\operatorname{JNC}_{\vec{F}}$ is generally
inaccessible, since it still contains global information about the cone
$\mathcal{P}_{+}(\mathcal{\mathcal{A}})^{\star}$. It is worth to stress that
the shape of the cone $\mathcal{F}_{+}^{\star}$ has a priori no common
properties with the usual positive cone in a finite dimensional vector space.
It inherits a term of positivity from the global structure. Roughly our main
task is to track positivity through the whole approximation. The mechanism of
the following approximation technique can be therefore understood as a
construction that provides us with numerically accessible outer approximations
to $\mathcal{F}_{+}^{\star}$ or equivalently $\operatorname{JNC}_{\vec{F}}$.

###  IV.2 Construction of a Relaxation

In this section we discuss how to approximate the value of a
$\mathcal{C}^{\star}$-SDP with theoretical methods and a finite dimensional
relaxation. Recall from (7) that the cone
$\mathcal{P}_{+}(\mathcal{\mathcal{A}})^{\star}$ is the dual cone of
$\mathcal{P}_{+}(\mathcal{\mathcal{A}})$. It means that inner approximations
of the cone $\mathcal{P}_{+}(\mathcal{\mathcal{A}})$ lead to outer
approximations of the cone $\mathcal{P}_{+}(\mathcal{\mathcal{A}})^{\star}$.
But outer approximations of the dual cone are giving lower bounds to a
$\mathcal{C}^{\star}$-SDP.

In order to connect to the two very nice points of view in section II and
subsection IV.1 we have to attenuate the ideas from section II and abstract
the ideas from subsection IV.1. In section II we enforce the map $T$ to be a
quantum channel, i.e. completely positive and trace-preserving. As it turns
out firstly it is very difficult to find the right channel which dual maps a
known family of operator $\\{M_{0},M_{1},\dots,M_{m}\\}$ to operators
$\\{F_{0},F_{1},\ldots,F_{m}\\}$ for a $\mathcal{C}^{\star}$-SDP. Secondly it
is not clear how to appropriate describe the set of states after applying the
channel.

To circumvent the first problem we use the dual point of view and construct a
linear mapping $\phi$ which is not completely positive and trace-preserving
anymore from a finite dimensional space into the algebra $\mathcal{A}$ of
interest. Since positivity of a map implies positivity of the dual, we can
make similar arguments even without completely positivity.

In a first step we can choose random positive elements from the
$\mathcal{C}^{\star}$-algebra to approximate the positive cone
$\mathcal{P}_{+}(\mathcal{A})$. But then we need an assignment between these
abstract elements and a cone in $\mathbb{C}^{n\times n}$ for an appropriate
size $n$. The important point is that we need a way to translate positivity to
a finite calculable system. The following fundamental observation, which is in
fact a generalization of the idea of ’sum of squares’[2] in polynomial
optimization or in the same manner from the $\star$-isomorphism in symmetry
reductions[3] in generic SDPs gives a way for translating positivity to a
controllable cone.

######  Lemma 5 (Fundamental Lemma).

Let $\mathcal{A}$ be a $\mathcal{C}^{\star}$-algebra and consider a set

$\displaystyle\gamma=\\{\gamma_{1},\dots,\gamma_{n}\\}\subset\mathcal{A}.$
(23)

Then the mapping

$\displaystyle\phi:\mathbb{C}^{n\times n}$ $\displaystyle\to\mathcal{A}$ (24)
$\displaystyle M$
$\displaystyle\mapsto\sum_{i,j=1}^{n}m_{ij}\gamma_{i}\gamma_{j}^{\star}$ (25)

is linear and positive. In particular let
$x=yy^{\star}\in\mathcal{P}_{+}(\mathcal{\mathcal{A}})$ and
$y\in\operatorname{span}(\gamma)$. Then there exists $M\geq 0$ such that
$\phi(M)=x$.

###### Proof.

Linearity is clear. We have to show positivity.
Positive elements are hermitian in $\mathbb{C}^{n\times n}$ and therefore each
positive elements have a spectral decomposition. Therefore let
$M\in\mathbb{C}^{n\times n}$ be positive. Then

$M=\sum_{k}p_{k}P_{k}$

with $p_{k}\in\mathbb{R}_{>0}$ and $P_{k}=c^{(k)}(c^{(k)})^{\star}$ one
dimensional projections. Therefore

$\displaystyle\phi(M)$
$\displaystyle=\sum_{k}p_{k}\phi(c^{(k)}(c^{(k)})^{\star})$ (26)
$\displaystyle=\sum_{k}p_{k}\sum_{i,j}c^{k}_{i}\overline{c^{k}}_{j}\gamma_{i}\gamma_{j}^{\star}$
(27)
$\displaystyle=\sum_{k}p_{k}(\sum_{i}c^{k}_{i}\gamma_{i})(\sum_{j}c^{k}_{j}\gamma_{j})^{\star}.$
(28)

But then we see that each sumand over $k$ is a square in $\mathbb{A}$ and
therefore positive. Positive sums of positive elements are positive which
concludes the proof. ∎

The 5 shows us that for a given sequence of elements
$\gamma\subset\mathcal{A}$ there is a way of translating structure from a
$\mathcal{C}^{\star}$-algebra to a finite dimensional space. But in general to
use the preimage of $\phi$ for an approximation needs the structure of the
kernel of $\phi$ because the kernel contains the important inner structure of
the algebra $\mathcal{A}$. In other words the kernel decides whether two
parameterized elements $M,M^{\prime}\in C^{n\times n}$ are equal to each
other. To get a working approximation we have to have in mind that this is
precisely the point where we have to involve the knowledge about the algebra.
For example the relations of generators of the algebra.

We are now in the situation to present a working approximation. Recall that,
in fact, we want to approximate $\mathcal{F}_{+}^{\star}$ from subsection
IV.1. For this goal we optimally need to build a hierarchy666actually always
intersected with $\mathcal{F}$ of numerically accessible cones

$\Sigma_{2}^{(1)}\subset\ldots\subset\Sigma_{2}^{(n)}\subset\Sigma_{2}^{(n+1)}\subset\ldots\subset\mathcal{P}_{+}(\mathcal{\mathcal{A}}).$

These cones should be the images of positive cones in $\mathbb{C}^{n\times n}$
in 5 for different sizes of $n$. The sequence of dual cones, as defined in (7)
leads then to a sequence

$(\Sigma_{2}^{(1)})^{\star}\supset\ldots\supset(\Sigma_{2}^{(n)})^{\star}\supset(\Sigma_{2}^{(n+1)})^{\star}\supset\ldots\supset\mathcal{P}_{+}(\mathcal{\mathcal{A}})^{\star}.$
(29)

These are outer approximations of
$\mathcal{P}_{+}(\mathcal{\mathcal{A}})^{\star}$. To construct such sequences
of cones we start with a sequence of elements in the
$\mathcal{C}^{\star}$-algebra $\mathcal{A}$

$\displaystyle\gamma=\\{\gamma_{1},\dots,\gamma_{n}\\}.$ (30)

These elements $\gamma$ span a subspace $V_{n}\subset\mathcal{A}$, i.e.

$\displaystyle
V_{n}:=\operatorname{span}(\gamma)=\left\\{v\in\mathcal{A}\middle|v=\sum_{i=1}^{n}v_{i}\gamma_{i},\text{
for some }v_{1},\dots,v_{n}\in\mathbb{C}\right\\},$ (31)

from which we can construct a space777the notation
$\operatorname{conv}(\Omega)$ denotes the convex hull of a set $\omega$

$\displaystyle Q_{n}:=\operatorname{conv}\\{vw^{\star}\ |\ v,w\in
V\\}=\operatorname{span}\\{\gamma_{i}\gamma_{j}^{\star}\\}=\operatorname{Im}(\phi)$
(32)

that contains all sums of the products that could be build from elements from
$V_{n}$. To ensure that we get senseful statements about $\mathcal{F}$ we have
to consider sequences $\gamma$ that are constructed such that the subspace
$\mathcal{F}$ is contained in $Q_{n}$, because then the cone
$\mathcal{F}_{+}^{\star}=\mathcal{P}_{+}(\mathcal{\mathcal{A}})^{\star}\cap\mathcal{F}^{\star}$
is contained in (29). Recall that this is exactly the difficulty with our
channel in section II. Starting from the functionals, in general we can not
control which element is the right preimage under the dual mapping $T^{\star}$
there. In unital algebras this requirement could always be fulfilled by
appending $\\{F_{0},\dots,F_{N},\mathbb{I}\\}$ to an initial sequence
$\gamma_{0}$. An alternative, which also works on non-unital algebras, would
be to append the square roots $\sqrt{F_{i}}$.

Then we consider the cone

$\displaystyle\Sigma_{2}^{(n)}=\operatorname{conv}\\{vv^{\star}\ |\ v\in V\\}$
(33)

of ’sums of squares’ in $Q_{n}$. Indeed, as $\text{dim}(Q)\leq n^{2}$ and
$\Sigma_{2}\subset Q$, we also have $\text{dim}(\Sigma_{2})\leq n^{2}<\infty$.

Since in a $\mathcal{C}^{\star}$ algebra, all squares, as well as their convex
hull, are positive we obtain

$\displaystyle\Sigma_{2}^{(n)}\subseteq\mathcal{P}^{+}(\mathcal{A}),$ (34)

which implies (29) for the corresponding dual cones. This however implies that
the finite cone $\Sigma_{2}^{\star}|_{\mathcal{F}}$ (that no longer contains
inaccessible global information) is a super set

$\displaystyle(\Sigma_{2}^{(n)})^{\star}|_{\mathcal{F}}\supseteq\mathcal{F}_{+}^{\star}.$
(35)

Hence we can conclude

######  Proposition 1.

The value of a $\mathcal{C}^{\star}$-SDP (8), defined by the sets
$\\{F_{0},\dots,F_{m}\\}$ and $\\{f_{1},\dots,f_{m}\\}$, is bounded from below
by the optimization

$\displaystyle\inf\quad$ $\displaystyle\omega(F_{0})$ (36) s.th.
$\displaystyle\omega(F_{i})\leq f_{i}$ $\displaystyle\forall
i\in\\{1,\dots,m\\}$
$\displaystyle\omega\in(\Sigma_{2}^{(n)})^{\star}|_{\mathcal{F}}$
$\displaystyle\quad.$

We want to mention that there are somehow some open questions from a practical
point of view, which we tackle in the next section. In particular, we did not
talk about any properties of the sequence $\gamma$ and their properties of
approximating the positive cone of the $\mathcal{C}^{\star}$-algebra.

###  IV.3 The approximation as a conventional SDP

For this section we choose a fixed set $\gamma$ and corresponding spaces $V,Q$
and cone $\Sigma_{2}$.

The optimization (36) is in fact not only finite from a formal perspective
rather than also from actual practicality since it can be explicitly
formulated by a matrix valued SDP of $n\times n$ matrices, as we will show
next.

Recall the mapping $\phi$ from 5, defined by

$\displaystyle\phi:\mathbb{C}^{n\times n}\rightarrow
Q\quad\phi(|i\rangle\langle j|)=\gamma_{i}\gamma_{j}^{\star},$ (37)

i.e. the map that maps a matrix $M=\sum m_{ij}|i\rangle\langle j|$ to an
algebra element

$\displaystyle\phi(M)=\sum m_{ij}\gamma_{i}\gamma_{j}^{\star}.$ (38)

This map is surjective, since (38) merely describes the span of
$\gamma_{i}\gamma_{j}^{\star}$, i.e., $Q$. Let
$\operatorname{ker}(\phi)\subset\mathbb{C}^{n\times n}$ be the kernel of
$\phi$ and denote by $\pi$ the projection map $\mathbb{C}^{n\times
n}\rightarrow\mathbb{C}^{n\times n}/\operatorname{ker}(\phi)$, which assigns
to each element its coset. We write
$[x]:=\\{x+k|k\in\operatorname{ker}(\phi)\\}=\pi^{-1}(\phi(x))$ denote the
equivalence class induces by $\operatorname{ker}(\phi)$. As $\phi$ is
surjective, we obtain from isomorphism theorem, that the map $\hat{\phi}$ is
an isomorphism.

${\mathbb{C}^{n\times n}}$${Q}$${\hat{Q}}$$\pi$$\phi$$\exists!\ \hat{\phi}$
Figure 1: We define $\hat{Q}:=\mathbb{C}^{n\times
n}/\operatorname{ker}(\phi)$. The figure tells us that we get an isomorphism
between $Q$ and $\hat{Q}$ from the isomorphism theorem. In other words this
means that we constructed an image of $Q$ with the right relations between
elements in $Q$ in a accessible space $\mathbb{C}^{n\times n}$.

Now we can state the main proposition of this section.

######  Proposition 2 (Identification of $Q^{\star}$).

An element $g\in(\mathbb{C}^{n\times n})^{\star}$ can be identified up to
equivalence with a linear functional in $Q^{\star}$ if and only if $g$
annihilates $\operatorname{ker}(\phi)$ or in other words

$g(M)=0\quad\forall M\in\ker(\phi).$

###### Proof.

The map in Figure 1 shows us that $\hat{\phi}:\hat{Q}\rightarrow Q$ that acts
on $\hat{Q}$ via

$\displaystyle\hat{\phi}(x):=\phi([x])$ (39)

is an isomorphism. To describe $\hat{\Sigma}_{2}^{\star}$ we need to include
the isomorphism $\phi$ in an appropriate manner in our discussion. For this
purpose we consider the dual mapping

$\displaystyle\hat{\phi}^{\star}:Q^{\star}$ $\displaystyle\to\hat{Q}^{\star}$
(40) $\displaystyle q^{\prime}$ $\displaystyle\mapsto q^{\prime}\circ\phi.$
(41)

Basic linear algebra states that $\hat{\phi}^{\star}$ is an isomorphism.
Furthermore a well-known result from functional analysis states that

$\displaystyle\pi^{\star}:(\mathbb{C}^{n\times
n}/\operatorname{ker}(\phi))^{\star}$
$\displaystyle\to\operatorname{\ker}(\phi)^{\perp}$ (42)
$\displaystyle\hat{q}^{\prime}$ $\displaystyle\mapsto\hat{q}^{\prime}\circ\pi$
(43)

is an isomorphism as well. Therefore, we have the chain of isomorphisms

$\displaystyle
Q^{\star}\stackrel{{\scriptstyle\hat{\phi}^{\star}}}{{\longrightarrow}}\hat{Q}^{\star}\stackrel{{\scriptstyle\pi^{\star}}}{{\longrightarrow}}\operatorname{\ker}(\phi)^{\perp},$
(44)

whereby the annihilator is defined via

$\displaystyle\operatorname{ker}(\phi)^{\perp}:=\\{g\in(\mathbb{C}^{n\times
n})^{\star}\ |\ g(M)=0\quad\forall M\in\operatorname{ker}(\phi)\\}.$ (45)

∎

The proof of 2 shows that quotient spaces become subspaces in duality. This is
an important observation throughout the whole discussion.

This simple but very powerful 2 leads us to an abstract but nice description
of elements in $\hat{Q}^{\star}$ and therefore of the moment matrices as
generalizations of polynomial optimization in 3:

######  Corollary 6.

The relation between linear functionals from $Q^{\star}$ and $\hat{Q}^{\star}$
is given by

$\operatorname{tr}(\Gamma^{\sigma}M)=\sigma(\phi(M)).$

For $\Gamma^{\sigma}:=\phi^{\star}(\sigma)$. These matrices are also called
moment matrices in other hierarchies.

###### Proof.

This is clear from the definition of the dual mapping. ∎

Including subspaces in a SDP is therefore simple, because constraints of the
manner

$\operatorname{tr}(\rho K)=0$

correspond to functionals $\operatorname{tr}(\rho\cdot)$ which have the
subspaces spanned by $K$ in the kernel. Therefore, if we add all generators of
the kernel of $\phi$, we create the annihilator.

For our purpose there is only left an open discussion about the kernel of
$\phi$ and how to include such a kernel. But as we will see, it is in some
sense the task of a physicist to know his experiment in such a manner that he
can decide whether two experimental setups, i.e. mathematical models are equal
to each other. Our examples will demonstrate this impressively. In conclusion
of the discussion we can state the following main theorem of our paper:

######  Theorem 7 (Relaxation).

The relaxation (36) of a $\mathcal{C}^{\star}$-SDP that is build from a finite
sequence $\\{\gamma_{1},\dots,\gamma_{n}\\}$ can be formulated as an $n\times
n$ matrix valued SDP, and therefore efficiently computed. A concrete form of
this SDP is obtained from any set of matrices $M_{F_{i}}$ with

$\displaystyle\hat{\phi}([M_{F_{i}}])=F_{i}$ (46)

and any set of $m_{k}$ matrices $\\{K_{i}\\}$ with

$\displaystyle\operatorname{span}\\{K_{j}\\}=\operatorname{ker}(\phi)$ (47)

and given by

$\displaystyle\inf\quad$ $\displaystyle\operatorname{tr}(\rho M_{F_{0}})$ (48)
s.th. $\displaystyle\operatorname{tr}(\rho M_{F_{i}})\geq f_{i}$
$\displaystyle\forall i\in\\{1,\dots,m\\}$
$\displaystyle\operatorname{tr}(\rho K_{j})=0$ $\displaystyle\forall
K_{j}\in\\{1,\dots,m_{k}\\}$ $\displaystyle\rho\geq 0$

###  IV.4 CHSH worked out

We close this section with a familiar example.

######  Example 8 (The algebra generated by two projections).

One of the most famous examples of abstract language in quantum theory is the
algebra generated by two projections. Consider a set of generators

$\displaystyle\mathcal{G}:=\\{1,A_{0},A_{1},B_{0},B_{1}\\}$

which should fulfill the following relations

$\displaystyle\mathcal{R}:=\\{A_{i}^{2}=1,B_{i}^{2}=1,B_{i}^{\star}=B_{i},A_{i}^{\star}=A_{i},[A_{i},B_{j}]=0\
|\ 1\leq i,j\leq 2\\}.$

It is well known that the algebra spanned by the generators and relations
$C(\mathcal{G}\ |\ \mathcal{R})$ becomes a $\mathcal{C}^{\star}$-algebra[13].
Considering now a Bell inequality[14]

$\displaystyle F_{0}:=A_{0}B_{0}+A_{1}B_{1}+A_{0}B_{1}-A_{1}B_{0}$

leads directly to a $\mathcal{C}^{\star}$-SDP in the following sense:

$\displaystyle\inf\quad$ $\displaystyle\omega(F_{0})$ (49) s.th.
$\displaystyle\omega(1)=1$
$\displaystyle\omega\in\mathcal{P}_{+}(\mathcal{\mathcal{A}})^{\star}$
$\displaystyle\qquad.$ (50)

The interpretation of this problem is to optimize over all states for finding
the maximal violation of the Bell inequality above (CHSH type).
Solving this example with our method needs a sequence $\gamma$. One can choose
for example in a first step

$\displaystyle\gamma=(1,A_{0},A_{1},B_{0},B_{1},A_{0}B_{1},A_{0}A_{1},B_{0}B_{1}).$

Furthermore the relations $\mathcal{R}$ become directly elements in the kernel
of $\phi$ in the discussion above. Therefore working out the kernel is not
difficult in this example for short sequences $\gamma$. In conclusion one can
directly verify with the rules of Theorem 7 that the optimal value $\sqrt{2}2$
will be achieved in a nummerical example.

##  V Convergence of the hierarchy

In this section we aim to clearify under which circumstances we can prove
convergence of the hierarchy by iterating Theorem 7. The question now is how
the optimal value of the relaxation of the $\mathcal{C}^{\star}$-SDP relates
to its true value.

For this we need the notion of base property.

######  Definition 9 (Base Property).

Let $(\gamma^{(n)})_{n\geq 1}$ be a sequence in $\mathcal{A}$. Now define

$V_{n}:=\text{span}(\\{\gamma^{(1)},\ldots,\gamma^{(n)}\\}).$

We say that $(\gamma^{(n)})_{n\geq 1}$ has the base property, if for any
$\epsilon>0$ and any $a\in\mathcal{A}$ there exists $n_{0}\in\mathbb{N}$ such
that for all $n\geq n_{0}$ there is $b\in V_{n}$ with $||a-b||<\epsilon$.

We emphasize at this point that there is an obvious way to translate the base
property into separability of the $\mathcal{C}^{\star}$-algebra. Namely choose
a sequence with base property and consider the vector space $V_{n}$ spanned by
the subfield $\mathbb{Q}+i\mathbb{Q}$ of $\mathbb{C}$ (this is in fact the
field extension $\mathbb{Q}(i)$).

For example all $\mathcal{C}^{\star}$-algebras with finite set of generators
are automatically separable and fulfill therefore the base property.

######  Theorem 10.

We consider a $\mathcal{C}^{\star}$-SDP

$\displaystyle c^{\star}=$ $\displaystyle\inf\rho(F_{0})$
$\displaystyle\text{s.th.}\ \rho(F_{i})\leq f_{i}$

and assume that we have base property, strong duality at all levels, i.e. in
particular

$\displaystyle c^{\star}=$ $\displaystyle\sup_{\lambda}f^{T}\lambda$
$\displaystyle\text{s.th.}\
\sum_{i}\lambda_{i}F_{i}-F_{0}\in\mathcal{P}_{+}(\mathcal{\mathcal{A}}).$

Then we have $|c^{\star}-c^{n}|\xrightarrow{n\rightarrow\infty}0$ with value
$c^{n}$ for each level.

###### Proof.

Suppose that $x\in\mathcal{P}^{+}(\mathcal{A})$. By the
$\mathcal{C}^{\star}$-property from subsection III.1, there exists
$z\in\mathcal{A}$ such that $x=z^{*}z$. By the base property, we can conclude
that for sufficient large $n\in\mathbb{N}$ there exists $y\in V_{n}$ such that
$||y-z||<\epsilon$. In particular we have that there is $d\in\mathcal{A}$ with
$||d||<\epsilon$ such that $y+d=z$. Further $y^{\star}y\in\Sigma^{(n)}_{2}$.
Therefore we have

$\displaystyle||y^{\star}y-z^{\star}z||=||(z-d)^{\star}(z-d)-z^{\star}z||=||d^{\star}z-z^{\star}d+d^{\star}d||\leq||d^{\star}z||+||z^{\star}d||+||d^{\star}d||.$
(51)

By the submultiplicativity of the norm it follows that the left hand side can
be done sufficiently small by choosing $\varepsilon>0$ in dependence of $z$
small enough. This means that we have

$\displaystyle\Sigma_{2}^{(n)}\subset\Sigma_{2}^{(n+1)}\subset\ldots\subset\mathcal{P}_{+}(\mathcal{\mathcal{A}})\quad
n\to\infty$

as a limit of sets. Consider for

$\displaystyle L:\mathbb{R}^{m}$ $\displaystyle\to\mathcal{A}$
$\displaystyle\lambda$ $\displaystyle\mapsto\sum_{i}\lambda_{i}F_{i}-F_{0}$

the sets

$\displaystyle\Omega^{(n)}:=L^{-1}(\Sigma_{2}^{(n)}\cap\text{im}(L))$

and in particular

$\displaystyle\Omega:=L^{-1}(\mathcal{P}(\mathcal{A})\cap\text{im}(L)),$

the set of all $\lambda\in\mathbb{R}^{n}$ such that $L(\lambda)$ is positive
in $\mathcal{A}$. Then we have in particular

$\displaystyle\Sigma_{2}^{(n)}\cap\text{im}(L)\subset\ldots\subset\mathcal{P}(\mathcal{A})\cap\text{im}(L)\quad
n\to\infty.$

By strong duality on all levels we conclude

$\displaystyle c^{n}:=\sup_{\omega\in\Omega^{(n)}}f^{T}\omega$

and

$\displaystyle c^{\star}:=\sup_{\omega\in\Omega}f^{T}\omega.$ (52)

The sequence $(c^{n})$ is monotone and bounded and therefore convergent
towards $\sup_{n}c^{n}$. Assume for contradiction that there exists $\delta>0$
such that

$\displaystyle|\sup_{n}c^{n}-c^{\star}|\geq\delta.$

Then we have in particular that for all $n\in\mathbb{N}$

$\displaystyle|\sup_{\omega\in\Omega^{(n)}}f^{T}\omega-c^{\star}|\geq\delta$

and therefore

$\displaystyle|\sup_{\omega\in\bigcup_{n}\Omega^{(n)}}f^{T}\omega-c^{\star}|\geq\delta$

a contradiction to (52). ∎

This result has some metaphysical description. What we have done so far is
build up on the assumption of having a $\mathcal{C}^{\star}$-algebra at hand.
This means, that there is a notion of convergence inherently coming from the
property of being a complete normed space. If we compare this with e.g. the
NPA-hierarchy or even Lasserre’s great polynomial optimization work, there is
from a first view not clear what one means with ’convergence’ in a topological
sense.

##  VI Symmetry reductions of SDPs as a special case of
$\mathcal{C}^{\star}$-SDP

Let us repeat the framework for symmetry reduction in SDPs from a theoretical
perspective for finite groups. For this purpose consider a general SDP over a
finite dimensional Hilbert space $\mathcal{H}$ of the following form

$\displaystyle\max\ $ $\displaystyle\text{tr}[\rho F_{0}]$ (53) s.th.
$\displaystyle\text{tr}[\rho F_{j}]\leq f_{j}\quad 1\leq j\leq m$ (54)
$\displaystyle\rho\succeq 0.$ (55)

A (finite) symmetry of this SDP is now a finite group $G$ with a
representation888i.e. a group homomorphism

$\displaystyle\Phi:G\to\text{GL}(\mathcal{H})$

such that

$\displaystyle\Phi(g)F_{j}\Phi(g)^{\dagger}$ $\displaystyle=F_{j}\quad 1\leq
j\leq m$ $\displaystyle\Phi(g)F_{0}\Phi(g)^{\dagger}$ $\displaystyle=F_{0}.$

We assume further that the representation is unitary, which means that
$\Phi(g)^{-1}=\Phi(g)^{\dagger}$ for all $g\in G$. This assumption is for
finite groups without loss of generality. The cyclicity of the trace now
yields that a SDP which contains a finite symmetry has a solution which admits
this symmetry.

If we now assume access to a Hilbert-Schmidt orthonormal basis of the
invariant subspace
$\text{End}_{\mathbb{C}G}(\mathcal{H})\subset\text{End}_{\mathbb{C}}(\mathcal{H})$

$\displaystyle\gamma:=(B_{1},\ldots,B_{m})\subset\mathcal{B}(\mathcal{H})$
(56)

we can build up our theory in the same manner as e.g. in [3]. Define for

$\displaystyle B_{i}B_{j}=\sum_{k}\lambda_{ijk}B_{k}.$

the matrices $L_{k}\in\mathbb{C}^{m\times m}$ with the rule

$\displaystyle(L_{k})_{i,j}:=\lambda_{ijk}.$

Applying 5 yields then the mapping

$\displaystyle\phi:\mathbb{C}^{m\times m}$
$\displaystyle\to\text{End}_{\mathbb{C}G}(\mathcal{H})\subset\text{End}_{\mathbb{C}}(\mathcal{H})$
$\displaystyle L_{k}$
$\displaystyle\mapsto\sum_{i,j}\lambda_{ijk}B_{i}B_{j}^{\dagger}.$

Therefore it is easy to show that the optimal solution $\rho_{\text{opt}}$ is
contained in $\text{End}_{\mathbb{C}G}(\mathcal{H})$, which is the set of all
linear operators which commute with the operators $\Phi(g)$, i.e. the set of
intertwiner and fulfills Slater’s condition if the original SDP fulfilled
Slater’s condition. This observation states that the effective dimension of
the SDP is in fact the $\mathbb{C}$-vector space dimension

$\displaystyle
m:=\text{dim}_{\mathbb{C}}\text{End}_{\mathbb{C}G}(\mathcal{H}).$

Therefore, in this case our hierarchy basically has been finished in one step,
because the optimal solution is accessible in the dual space of
$\text{End}_{\mathbb{C}G}(\mathcal{H})$. Furthermore there is no kernel to
work out. The difficulty is really in defining the basis
$(B_{1},\ldots,B_{m})$. For this task, however, there are methods as presented
in de-Klerk et al.[3].

##  VII Conclusion

In this paper we have taken a new look at known problems of optimization in
quantum information theory. Our relaxation for $\mathcal{C}^{\star}$-SDPs is
based on the fundamental observation that we are in fact only interested in
the values of all positive functionals of a $\mathcal{C}^{\star}$-algebra on a
finite vector space. Thus, all functionals of interest are characterized by a
convex set in the dual space of $\mathbb{C}^{n\times n}$. This set can now be
relaxed. We have chosen the way via the sum of squares in the style of
Lasserre’s hierarchy. Remarkably it, however, follows directly from our
discussion how we can incorporate additional knowledge about, say, positive
elements: one could either add them to $\Sigma_{2}^{(n)}$ directly or write
them as SDP constraints. In contrast to Lasserre’s hierarchy, we can not
directly talk about local criteria about e.g. the dual space of a particular
$\mathcal{C}^{\star}$-algebra $C(K)$ for a compact set $K$. But our general
formulation gives hope that one may find in future local criteria for the
$\mathcal{C}^{\star}$-algebra. In general one can apparently include the ideas
of Lasserre, but then one has to choose the sequence $\gamma$, i.e. the basis
of the vector space $Q$ in this case, to be polynomials. Then there is no way
to present elements other than polynomials exactly. As this work pointed out,
the idea of moment matrices can be generalized and is extremely powerful. We
hope, that this general point of view will stimulate a new discussion about
possible relaxations for general $\mathcal{C}^{\star}$-SDP in particular in
quantum information theory. There is new work done by Klep et al.[15], which
is able to tackle nonlinear constraints in a $\mathcal{C}^{\star}$-SDP similar
notion. This has particular importance in causal structure problem as e.g. in
Ligthart et al. [16, 17]

##  VIII Acknowledgements

We thank Mario Berta, Tobias J. Osborne and Julius Zeiss for helpful
discussions. We thank David Gross and Laurens Ligthart for organizing the
workshop on SDP-hierarchies in May 2023 in Cologne. RS acknowledges financial
support by the BMBF project ATIQ.

## References

  * Navascues, Pironio, and Acin [2007] M. Navascues, S. Pironio, and A. Acin, “Bounding the Set of Quantum Correlations,” Phys. Rev. Lett. 98, 010401 (2007).
  * Lasserre [2001] J. B. Lasserre, “Global optimization with polynomials and the problem of moments,” SIAM Journal on Optimization 11, 796–817 (2001).
  * De Klerk, Pasechnik, and Schrijver [2007] E. De Klerk, D. V. Pasechnik, and A. Schrijver, “Reduction of symmetric semidefinite programs using the regular-representation,” Mathematical programming 109, 613–624 (2007).
  * M.Berta, Fawzi, and Scholz [2015] M.Berta, O. Fawzi, and V. B. Scholz, “Quantum bilinear optimization,” SIAM J. Optim. 26, 1529–1564 (2015).
  * Doherty, Parrilo, and Spedalieri [2004] A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, “Complete family of separability criteria,” Physical Review A 69 (2004), 10.1103/physreva.69.022308.
  * Navascues, Pironio, and Acin [2008] M. Navascues, S. Pironio, and A. Acin, “A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations,” New J. Phys. 10, 073013 (2008).
  * RAEBURN and SINCLAIR [1989a] I. RAEBURN and A. M. SINCLAIR, “The c*-algebra generated by two projections,” Mathematica Scandinavica 65, 278–290 (1989a).
  * Horodecki, Shor, and Ruskai [2003] M. Horodecki, P. W. Shor, and M. B. Ruskai, “Entanglement breaking channels,” Reviews in Mathematical Physics 15, 629–641 (2003).
  * Holevo, Shirokov, and Werner [2005] A. S. Holevo, M. E. Shirokov, and R. F. Werner, “Separability and entanglement-breaking in infinite dimensions,” (2005), arXiv:quant-ph/0504204 [quant-ph] .
  * Blackadar [2006] B. Blackadar,  _Operator Algebras_ (Springer, 2006).
  * Lasserre [2009] J. B. Lasserre, “The generalized moment problem,” in  _Moments, Positive Polynomials and Their Applications_ , Imperial College Press optimization series (IMPERIAL COLLEGE PRESS, 2009) pp. 3–13.
  * Putinar [1993] M. Putinar, “Positive polynomials on compact semi-algebraic sets,” Indiana University Mathematics Journal 42, 969–984 (1993).
  * RAEBURN and SINCLAIR [1989b] I. RAEBURN and A. M. SINCLAIR, “The c*-algebra generated by two projections,” Mathematica Scandinavica 65, 278–290 (1989b).
  * Bell [1964] J. S. Bell, “On the Einstein Podolsky Rosen paradox,” Physics 1, 195–200 (1964).
  * Klep  _et al._ [2023] I. Klep, V. Magron, J. Volčič, and J. Wang, “State polynomials: positivity, optimization and nonlinear bell inequalities,” (2023), arXiv:2301.12513 [math.FA] .
  * Ligthart and Gross [2023] L. T. Ligthart and D. Gross, “The inflation hierarchy and the polarization hierarchy are complete for the quantum bilocal scenario,” Journal of Mathematical Physics 64 (2023), 10.1063/5.0143792.
  * Ligthart, Gachechiladze, and Gross [2023] L. T. Ligthart, M. Gachechiladze, and D. Gross, “A convergent inflation hierarchy for quantum causal structures,” Communications in Mathematical Physics 401, 2673–2714 (2023).