Continuous-Variable Nonlocality and Contextuality

Abstract

Contextuality is a non-classical behaviour that can be exhibited by quantum systems. It is increasingly studied for its relationship to quantum-over-classical advantages in informatic tasks. To date, it has largely been studied in discrete-variable scenarios, where observables take values in discrete and usually finite sets. Practically, on the other hand, continuous-variable scenarios offer some of the most promising candidates for implementing quantum computations and informatic protocols. Here we set out a framework for treating contextuality in continuous-variable scenarios. It is shown that the Fine–Abramsky–Brandenburger theorem extends to this setting, an important consequence of which is that Bell nonlocality can be viewed as a special case of contextuality, as in the discrete case. The contextual fraction, a quantifiable measure of contextuality that bears a precise relationship to Bell inequality violations and quantum advantages, is also defined in this setting. It is shown to be a non-increasing monotone with respect to classical operations that include binning to discretise data. Finally, we consider how the contextual fraction can be formulated as an infinite linear program. Through Lasserre relaxations, we are able to express this infinite linear program as a hierarchy of semi-definite programs that allow to calculate the contextual fraction with increasing accuracy.

Appendices

Appendices

Linear Programs in Standard Form

This appendix may be of particular interest to readers familiar with global optimisation. We express the problems (P-CF) and (D-CF) in the standard form of infinite linear programming [17, IV–(7.1)]. We recall them below:

$$\begin{aligned} \hbox {(P-CF)}\left\{ \begin{aligned}&\quad \text {Find } \mu \in \mathbb {M}_{\pm }({\varvec{O}}_X) \\&\quad \text {maximising } \mu (O_X) \\&\quad \text {subject to:} \\&\quad \begin{aligned}&\qquad \forall C \in \mathcal {M},\; \mu |_C \;\le \; e_C \\&\qquad \mu \;\ge \; 0 \text { .}\end{aligned} \end{aligned} \right.&\end{aligned}$$

$$\begin{aligned} \hbox {(D-CF)}\left\{ \begin{aligned}&\quad \text {Find } ({f_C})_{C \in \mathcal {M}} \in \prod _{C \in \mathcal {M}} C(O_C) \\&\quad \text {minimising } \int _{O_C}f_C\;\,\mathrm {d}\,e_C \\&\quad \text {subject to:} \\&\quad \begin{aligned}&\qquad \sum _{C \in \mathcal {M}} f_C \circ \rho ^X_C \;\ge \; \mathbf {1}_{O_X} \\&\qquad \forall C \in \mathcal {M},\; f_C \;\ge \; \mathbf {0}_{O_C} \text { .}\end{aligned} \end{aligned} \right.&\end{aligned}$$

Problems (P-CF) and (D-CF) are indeed infinite linear programs as both the objective and the constraints are linear with respect to the unknown measure $\mu \in \mathbb {M}_{\pm }({{\varvec{O}}}_X)$. To write (P-CF) in the standard form [17], we introduce the following spaces:

• $\displaystyle E_1 :=\mathbb {M}_{\pm }({\varvec{O}}_X)$.

• $\displaystyle F_1 :=C(O_X)$, the dual space of $E_1$.

• $\displaystyle E_2 :=\prod _{C \in \mathcal {M}} \mathbb {M}_{\pm }({\varvec{O}}_C)$.

• $\displaystyle F_2 :=\prod _{C \in \mathcal {M}} C(O_C)$, the dual space of $E_2$.

The dualities $\langle - , - \rangle _1 : E_1 \times F_1 \longrightarrow \mathbb {R}$ and $\langle - , - \rangle _2 : E_2 \times F_2 \longrightarrow \mathbb {R}$ are defined as follows:

$$\begin{aligned}&\forall \mu \in E_1, \; f \in F_1,&\langle \mu ,f \rangle _1 :=\int _{O_X}f\;\,\mathrm {d}\,\mu \\&\forall (\nu _C) \in E_2, \; (f_C) \in F_2,&\langle (\nu _C),(f_C) \rangle _2 :=\sum _{C \in \mathcal {M}} \int _{O_C}f_C\;\,\mathrm {d}\,\nu _C\, , \end{aligned}$$

where, for simplicity, we have omitted $C \in \mathcal {M}$ as a subscript for the families of functions. We fix $K_1$ to be the convex cone of positive measures in $E_1 = \mathbb {M}_{\pm }({\varvec{O}}_C)$ and $K_2$ to be the convex cone of families of positive measures in $E_2 = \prod _{C \in \mathcal {M}} \mathbb {M}_{\pm }({\varvec{O}}_C)$. Then $K_1^*$ is the convex cone of positive function in $F_1 = C(O_X)$ and $K_2^*$ is the convex cone of families of positive functions in $F_2 = \prod _{C \in \mathcal {M}} C(O_C)$.

Let $A :E_1 \longrightarrow E_2$ be the following linear transformation: for $\mu \in E_1$

$$\begin{aligned} A(\mu ) := (\mu |_C)_{C \in \mathcal {M}} \in E_2 \text{. } \end{aligned}$$

We also define the linear transformation $A^* :F_2 \longrightarrow F_1$ as follows: for $(f_C) \in F_2$,

$$\begin{aligned} A^*((f_C)) := \sum _{C \in \mathcal {M}} f_C \circ \rho ^X_C \in F_1 \text{. } \end{aligned}$$

We can verify that $A^*$ is the dual transformation of A: given $\mu \in E_1$ and $(f_C) \in F_2$, we have

$$\begin{aligned} \langle A(\mu ), (f_C) \rangle _2&= \langle (\mu |_C), (f_C) \rangle _2 \end{aligned}$$

(47)

$$\begin{aligned}&= \sum _{C \in \mathcal {M}} \int _{O_C}f_C\;\,\mathrm {d}\,\mu |_C \end{aligned}$$

(48)

$$\begin{aligned}&= \int _{O_X}\sum _{C \in \mathcal {M}} f_C \circ \rho ^X_C \;\,\mathrm {d}\,\mu \end{aligned}$$

(49)

$$\begin{aligned}&= \langle \mu , \sum _{C \in \mathcal {M}} f_C \circ \rho ^X_C \rangle _1 \end{aligned}$$

(50)

$$\begin{aligned}&= \langle \mu , A^*((f_C)) \rangle _1 \text { .}\end{aligned}$$

(51)

Now fixing the vector function in the objective to be $c :=- {\varvec{1}}_{O_X} \in F_1$ and the vector in the constraints to be $b := (-e_C)_{C \in \mathcal {M}} \in E_2$, the program (P-CF) (resp. (D-CF)) can be expressed as in the standard form given in [17]:

$$\begin{aligned} \hbox {LP}\left\{ \begin{aligned}&\quad \text {Find } e_1 \in E_1 \\&\quad \text {minimising } \langle e_1,c \rangle _1 \\&\quad \text {subject to:} \\&\quad \begin{aligned}&A(e_1) \ge _{K_2} b \\&e_1 \ge _{K_1} 0 \text { .}\end{aligned} \end{aligned} \right.&\end{aligned}$$

$$\begin{aligned} \hbox {(D-LP)}\left\{ \begin{aligned}&\quad \text {Find } f_2 \in F_2 \\&\quad \text {maximising } \langle b,f_2 \rangle _2 \\&\quad \text {subject to:} \\&\quad \begin{aligned}&A^*(f_2) \le _{K_1^*} c \\&f_2 \ge _{K_2} 0 \text { .}\end{aligned} \end{aligned} \right.&\end{aligned}$$

Note that the minus sign in the vectors c and b was added because we chose the primal program in the standard form to be a minimisation problem while the primal program (P-CF) at hand is a maximisation problem.

Proof of Proposition 2: Zero Duality Gap

In this appendix we give a full proof of Proposition 2; i.e. that strong duality holds between problems (P-CF) and (D-CF).

Proof

To show strong duality, we rely on [17, Theorem 7.2]. Because $\mu _0= {\varvec{0}}_{{\varvec{O}}_X}$—the measure that assigns 0 to every measurable set of ${\varvec{O}}_X$— is a feasible solution for (P-CF) and the noncontextual fraction lies between 0 and 1, (P-CF) is consistent with finite value. Thus it suffices to show that the following cone

$$\begin{aligned} \mathcal {K}= \{ \left( A(\mu ), \langle \mu ,c \rangle _1 \right) \, \mid \, \mu \in K_1 \} = \{ \left( \, (\mu |_C)_C, \mu (O_X)\, \right) \, \mid \, \mu \in K_{1} \} \end{aligned}$$

(52)

is weakly closed in $E_2 \oplus \mathbb {R}$ (i.e. closed in the weak topology of $K_1$) where we recall that $K_1$ is the convex cone of positive measures in $E_1 = \mathbb {M}_{\pm }({\varvec{O}}_X)$.

We first notice that the linear transformation A is a bounded linear operator and thus continuous. Boundedness comes from the fact that for all $\mu \in K_1$,

$$\begin{aligned} \left\Vert A(\mu )\right\Vert _{E_2}&= \left\Vert (\mu |_C)_C\right\Vert _{E_2} \end{aligned}$$

(53)

$$\begin{aligned}&= \sum _{C \in \mathcal {M}} \left\Vert \mu |_C\right\Vert _{\mathbb {M}_{\pm }({\varvec{O}}_C)} \end{aligned}$$

(54)

$$\begin{aligned}&\le \sum _{C \in \mathcal {M}} \left\Vert \mu \right\Vert _{E_1} \end{aligned}$$

(55)

$$\begin{aligned}&= \vert \mathcal {M}\vert \left\Vert \mu \right\Vert _{E_1} \text { ,}\end{aligned}$$

(56)

where we take the strong topology—i.e. the norm induced by the total variation distance—on finite-signed measure spaces. It is defined as:

$$\begin{aligned} \left\Vert \mu \right\Vert _{\mathbb {M}_{\pm }({\varvec{U}})} = \vert \mu \vert (U)\text { .}\end{aligned}$$

We also equip the finite product space $E_2 = \prod _{C \in \mathcal {M}} \mathbb {M}_{\pm }({\varvec{O}}_C)$ with the norm obtained by summing¹⁸ the individual total variation norms. The inequality in Eq. (55) is due to the fact that $\mu \in K_1$ so this is a positive measure and thus $\left\| \mu |_C\right\| _{\mathbb {M}_{\pm }({\varvec{O}}_C)} \le \left\| \mu \right\| _{E_1}$. This, of course, extends to the weak topology.

Secondly, we consider a sequence $(\mu ^k)_{k \in \mathbb {N}}$ in $K_1$ and we want to show that the accumulation point $((\varTheta _C)_C,\lambda ) = \lim _{k \rightarrow \infty } \left( A(\mu ^k), \langle \mu ^k,c \rangle _1 \right)$ belongs to $\mathcal {K}$, where $\varTheta = (\varTheta _C)_C \in E_2$ and $\lambda \in \mathbb {R}$. If we consider the product of indicator functions $(\mathbf {1}_{O_C})_C \in F_2$ then $\langle A(\mu ^k), (\mathbf {1}_{O_C}) \rangle _2 = \sum _{C \in \mathcal {M}} \mu |_C^k(O_C) \longrightarrow _{k} \sum _{C \in \mathcal {M}} \varTheta _C(O_C) < \infty$ as $\varTheta _C$ is a finite measure for all maximal contexts $C \in \mathcal {M}$. Then, because $\mathcal {M}$ is a covering family of X, we have that, for all $k \in \mathbb {N} \ \mu ^k(O_X) \le \sum _{C \in \mathcal {M}} \mu ^k|_C(O_C) < \infty$. Since $(\mu ^k) \in K_1^{\mathbb {N}^{}}$ is a sequence of positive measures, this implies that $(\mu ^k)$ is bounded. Next, by weak-$*$ compactness of the unit ball (Alaoglu’s theorem [58]), there exists a subsequence $(\mu ^{k_i})_{i}$ that converges weakly to an element $\omega \in K_1$. By continuity of A, we obtain that the accumulation point is such that $((\varTheta _C)_C,\lambda ) = \left( A(\omega ), \langle \omega ,c \rangle _1 \right) \in \mathcal {K}$. $\quad \square$

The Lasserre–Parrilo Hierarchy

Below we introduce the Lasserre–Parrilo hierarchy for relaxing infinite-dimensional linear programs known as Generalised Moment Problems [53, 54, 69]. We start by giving insightful results: Sect. C.1 provides results concerning the representation of positive polynomials while Sect. C.2 provides results to understand when a sequence can be represented by a Borel measure.

Notation, terminology

We work in $\mathbb {R}^d$ for $d \in \mathbb {N}^*$. We fix ${\varvec{K}}$ to be a generic Borel measurable subspace of ${\varvec{\mathbb {R}}}^d$. Below we fix some multi-index notations. Let $\mathbb {R}[1]{[}{\varvec{x}}[1]{]}$ denote the ring of real polynomials in the variables ${\varvec{x}} \in \mathbb {R}^d$, and let $\mathbb {R}[1]{[}{\varvec{x}} [1]{]}_{k}\subset \mathbb {R}[1]{[}{\varvec{x}}[1]{]}$ contain those polynomials of total degree at most k. The latter forms a vector space of dimension $s(k) :=\left( {\begin{array}{c}d+k\\ k\end{array}}\right)$, with a canonical basis consisting of monomials ${\varvec{x}}^{{\varvec{\alpha }}} :=x_1^{\alpha _1}\cdots x_d^{\alpha _d}$ indexed by the set $\mathbb {N}^d_k :=\left\{ {\varvec{\alpha }}\in \mathbb {N}^d \mid |{\varvec{\alpha }}|\le k\right\}$ where $|{\varvec{\alpha }}|:=\sum _{i=1}^d\alpha _i$. For $k \in \mathbb {N}$, ${\varvec{x}} \in \mathbb {R}^d$, we define ${\varvec{\mathrm {v}}}_k({\varvec{x}}) :=({\varvec{x}}^{{\varvec{\alpha }}})_{\vert {\varvec{\alpha }}\vert \le d} = (1,x_1,\dots ,x_n,x_1^2,x_1x_2,\dots ,x_n^k)^T$ the vector of monomials of total degree less or equal than k.

Any $p \in \mathbb {R}[1]{[}{\varvec{x}} [1]{]}_{k}$ is associated with a vector of coefficients ${\varvec{p}} :=(p_{{\varvec{\alpha }}}) \in \mathbb {R}^{s(k)}$ via expansion in the canonical basis as $p({\varvec{x}}) = \sum _{{\varvec{\alpha }}\in \mathbb {N}^d_k} p_{{\varvec{\alpha }}}{\varvec{x}}^{{\varvec{\alpha }}}$.

Moment problem in probability

Given a finite set of indices $\varGamma$, a set of reals $\mathopen { \{ }\gamma _j : j \in \varGamma \mathclose { \} }$ and functions $h_j :K \longrightarrow \mathbb {R}$, $j \in \varGamma$, that are integrable with respect to every measure $\mu \in \mathbb {M}_{\pm }({\varvec{K}})$, the corresponding Global Moment Problem (GMP) can be expressed as:

$$\begin{aligned} \hbox {(GMP)}\left\{ \begin{aligned}&\quad \text {Find } \mu \in \mathbb {M}_{\pm }({\varvec{K}}) \\&\quad \text {maximising } \mu ({\varvec{K}}) \\&\quad \text {subject to:} \\&\qquad \qquad \begin{aligned}&\forall j \in \varGamma ,\quad \int _{K}h_j\;\,\mathrm {d}\,\mu \le \gamma _j \text { .}\end{aligned} \end{aligned} \right.&\end{aligned}$$

It dual program can be expressed as:

$$\begin{aligned} \hbox {(D-GMP)}\left\{ \begin{aligned}&\quad \text {Find } {\varvec{\lambda }}\in \mathbb {R}^\varGamma \\&\quad \text {minimising } \sum _{j \in \varGamma } \gamma _j \lambda _j \\&\quad \text {subject to:} \\&\quad \begin{aligned}&\qquad \forall {\varvec{x}} \in K,\quad \sum _{j \in \varGamma } \lambda _j h_j({\varvec{x}}) - {\varvec{x}} \ge 0 \\&\qquad \forall j \in \varGamma , \quad \lambda _j \ge 0 \text { .}\end{aligned} \end{aligned} \right.&\end{aligned}$$

1.1 Positive polynomials and sum-of-squares

Here we present the link between positive polynomials and sum-of-squares representation so that we can derive a converging hierarchy of restriction problems for program (D-GMP).

Definition 14

A polynomial $p \in \mathbb {R}[1]{[}{\varvec{x}}[1]{]}$ is a sum-of-squares (SOS) polynomial if there exists a finite family of polynomials $({q_i})_{i \in I}$ such that $p = \sum _{i\in I} q_i^2$.

SOS polynomials are widely used in convex optimisation. We will denote by $\varSigma ^2 \mathbb {R}[1]{[}{\varvec{x}} [1]{]}\subset \mathbb {R}[1]{[}{\varvec{x}}[1]{]}$ the set of (multivariate) SOS polynomials, and $\varSigma ^2 \mathbb {R}[1]{[}{\varvec{x}} [1]{]}_{k}\subset \varSigma ^2 \mathbb {R}[1]{[}{\varvec{x}} [1]{]}$ the set of SOS polynomials of degree at most 2k. The following proposition hints towards the reason why it is desirable to be able to look for a sum-of-squares decomposition: it can be cast as a semidefinite optimisation problem.

Proposition 3

(Prop. 2.1, [54]). A polynomial $p \in \mathbb {R}[1]{[}{\varvec{x}}[1]{]}_{2k}$ has a sum-of-squares decomposition if and only if there exists a real symmetric positive semidefinite matrix $Q \in \textsf {Sym}_{s(k)}$ such that $\forall {\varvec{x}} \in \mathbb {R}^d$, $p({\varvec{x}}) = {\varvec{\mathrm {v}}}_k({\varvec{x}})^T Q {\varvec{\mathrm {v}}}_k({\varvec{x}})$.

Then we will be looking at conditions under which a nonnegative polynomial can be expressed as a sum-of-squares polynomial. This is in essence the question raised by Hilbert in his 17$^\text {th}$ conjecture [44].

Definition 15

For a family $q = (q_j)_{j\in \mathopen { \{ }1, \ldots , m\mathclose { \} }}$ of polynomials, the set:

$$\begin{aligned} Q(q) :=\left\{ \sum _{j=0}^m \sigma _j q_j \mid (\sigma _j)_{j\in \mathopen { \{ }0,\ldots , m\mathclose { \} }} \subset \varSigma ^2 \mathbb {R}[1]{[}{\varvec{x}} [1]{]}\right\} \end{aligned}$$

(57)

is a convex cone in $\mathbb {R}[1]{[}{\varvec{x}}[1]{]}$ called the quadratic module generated by the family q with, for convenience, $q_0 = 1$ added. For $k \in \mathbb {N}$, we define $Q_k(q)$ to be the quadratic module Q(q) where we further impose that $(\sigma _j)_{j\in \mathopen { \{ }0,\ldots , m\mathclose { \} }} \subset \varSigma ^2 \mathbb {R}[1]{[}{\varvec{x}} [1]{]}_{k}$ i.e. we limit the degree of SOS polynomials.

In the family of polynomials $(g_j)_{j\in \mathopen { \{ }1, \ldots , m\mathclose { \} }}$ we add $g_0 = 1$ for convenience.

Assumptions 1

Let $K \subset \mathbb {R}^d$. We make the following three assumptions on K.

• (i) Suppose K is a basic semi-algebraic set i.e. there exists a family of polynomials $g = (g_j)_{j\in \mathopen { \{ }1, \ldots , m\mathclose { \} }} \in \mathbb {R}[1]{[}{\varvec{x}}[1]{]}^m$ of degrees deg($g_j$) respectively such that:

  $$\begin{aligned} K :=\left\{ {\varvec{x}} \in \mathbb {R}^d \mid \forall j = 1,\dots ,m, \; g_j({\varvec{x}}) \ge 0\right\} . \end{aligned}$$

  (58)

• (ii) Further suppose that K is compact.

• (iii) Finally suppose that there exists $u \in Q(q)$ such that the level set $\left\{ {\varvec{x}} \in \mathbb {R}^d \mid u({\varvec{x}}) \ge 0\right\}$ is compact.

The following theorem is the key result that we will exploit for deriving the hierarchy of SDP restrictions for the dual program (GMP).

Theorem 6

(Putinar’s Positivellensatz [73]). Let $K \subset \mathbb {R}^d$ satisfy Assumptions 1. If $p \in \mathbb {R}[1]{[}{\varvec{x}}[1]{]}$ is strictly positive on K then $p \in Q(g)$, that is

$$\begin{aligned} p = \sum _{j=0}^m \sigma _j g_j \end{aligned}$$

(59)

for some sum-of-squares polynomials $\sigma _j \in \varSigma ^2 \mathbb {R}[1]{[}{\varvec{x}} [1]{]}$ for $j=0,1,\dots ,m$.

A proof can also be found in [56].

Using the results above and Assumption 1, one can derive a hierarchy of SDPs [54] which provide a converging sequence of optimal values towards the value of program (D-GMP):

1.2 Moment sequences and moment matrices

In this subsection, we want to understand why the program (P-CF) can be relaxed so that a converging hierarchy of SDPs can be derived. The program (P-CF) is essentially a maximisation problem on finite-signed Borel measures with additional constraints such as the fact that these are proper measures (i.e. they are nonnegative). We will represent a measure by its moment sequence and find conditions for which this moment sequence has a (unique) representing Borel measure.

Definition 16

Given a sequence ${\varvec{y}} = (y_{{\varvec{\alpha }}})_{{\varvec{\alpha }}\in \mathbb {N}^d}\in \mathbb {R}^{\mathbb {N}^d}$, we define the linear functional $L_{{\varvec{y}}} :\mathbb {R}[1]{[}{\varvec{x}}[1]{]} \longrightarrow \mathbb {R}$ by

$$\begin{aligned} L_{{\varvec{y}}}(p) :=\sum _{{\varvec{\alpha }}\in \mathbb {N}^d}p_{{\varvec{\alpha }}} y_{{\varvec{\alpha }}}. \end{aligned}$$

(60)

Definition 17

Given a measure $\mu \in \mathbb {M}({\varvec{K}})$, its moment sequence ${\varvec{y}} = (y_{{\varvec{\alpha }}})_{{\varvec{\alpha }}\in \mathbb {N}^d} \in \mathbb {R}^{\mathbb {N}^d}$ is given by

$$\begin{aligned} y_{{\varvec{\alpha }}} :=\int _{{\varvec{K}}}{\varvec{x}}^{{\varvec{\alpha }}}\;\,\mathrm {d}\,\mu ({\varvec{x}}) \text { .} \end{aligned}$$

(61)

We say that ${\varvec{y}}$ has a unique representing measure $\mu$ when there exists a unique $\mu$ such that Eq. (61) holds. If $\mu$ is unique then we say it is determinate (i.e. determined by its moments).

The linear functional $L_{{\varvec{y}}}$ then gives integration of polynomials with respect to $\mu$ i.e. for any $p \in \mathbb {R}[1]{[}{\varvec{x}}[1]{]}$:

$$\begin{aligned} \begin{aligned} L_{{\varvec{y}}}(p)&= \sum _{{\varvec{\alpha }}\in \mathbb {N}^d}p_{{\varvec{\alpha }}} y_{{\varvec{\alpha }}} = \sum _{{\varvec{\alpha }}\in \mathbb {N}^d}p_{{\varvec{\alpha }}} \int _{{\varvec{K}}}{\varvec{x}}^{{\varvec{\alpha }}}\;\,\mathrm {d}\,\mu ({\varvec{x}}) = \int _{{\varvec{K}}}\sum _{{\varvec{\alpha }}\in \mathbb {N}^d}p_{{\varvec{\alpha }}} {\varvec{x}}^{{\varvec{\alpha }}}\;\,\mathrm {d}\,\mu ({\varvec{x}}) \\&= \int _{{\varvec{K}}}p({\varvec{x}})\;\,\mathrm {d}\,\mu ({\varvec{x}}) \\&= \int _{{\varvec{K}}}p\;\,\mathrm {d}\,\mu , \end{aligned} \end{aligned}$$

(62)

where we reversed summation and integration because the sum is finite since p is a polynomial.

The following theorem is often used in optimisation theory over measures as it provides a necessary and sufficient condition for a sequence to have a representing measure.

Theorem 7

(Riesz-Haviland [41]). Let ${\varvec{y}} = (y_{{\varvec{\alpha }}})_{{\varvec{\alpha }}\in \mathbb {N}^d} \in \mathbb {R}^{\mathbb {N}^d}$ and suppose that $K \subseteq \mathbb {R}^d$ is closed. Then ${\varvec{y}}$ has a representation (nonnegative) measure i.e. there exists $\mu$ a measure on K such that:

$$\begin{aligned} \forall {\varvec{\alpha }}\in \mathbb {N}^d, \; \int _{K}{\varvec{x}}^{{\varvec{\alpha }}}\;\,\mathrm {d}\,\mu = y_{{\varvec{\alpha }}} \end{aligned}$$

if and only if $L_{{\varvec{y}}}(p) \ge 0$ for all polynomials $p \in \mathbb {R}[1]{[}{\varvec{x}}[1]{]}$ nonnegative on K.

We recall that for $k \in \mathbb {N}$, $s(k) = \left( {\begin{array}{c}d+k\\ k\end{array}}\right)$.

Definition 18

For each $k \in \mathbb {N}$, the moment matrix of order k, $M_k({\varvec{y}}) \in \textsf {Sym}_{s(k)}$, of a truncated sequence $(y_{{\varvec{\alpha }}})_{{\varvec{\alpha }}\in \mathbb {N}^d_{2k}}$ is the $s(k) \times s(k)$ symmetric matrix with rows and columns indexed by $\mathbb {N}^d_k$ (i.e. by the canonical basis for $\mathbb {R}[1]{[}{\varvec{x}} [1]{]}_{k}$) defined as follows: for any ${\varvec{\alpha }}, {\varvec{\beta }}\in \mathbb {N}^d_k$,

$$\begin{aligned} \left( M_k({\varvec{y}})\right) _{{\varvec{\alpha }},{\varvec{\beta }}} :=L_{{\varvec{y}}}({\varvec{x}}^{{\varvec{\alpha }}+ {\varvec{\beta }}}) = y_{{\varvec{\alpha }}+ {\varvec{\beta }}} \text { .}\end{aligned}$$

(63)

Definition 19

Given a polynomial $p \in \mathbb {R}[1]{[}{\varvec{x}}[1]{]}$, the localising matrix $M_k(p {\varvec{y}}) \in \textsf {Mat}_{s(k)}(\mathbb {R})$ of a moment sequence $(y_{{\varvec{\alpha }}})_{{\varvec{\alpha }}\in \mathbb {N}^d} \in \mathbb {R}^{\mathbb {N}^d}$ is defined by: for all ${\varvec{\alpha }}, {\varvec{\beta }}\in \mathbb {N}^d_{k}$,

$$\begin{aligned} \left( M_k(p {\varvec{y}})\right) _{{\varvec{\alpha }}, {\varvec{\beta }}} :=L_{{\varvec{y}}}(p(x) x^{{\varvec{\alpha }}+ {\varvec{\beta }}}) = \sum _{\gamma \in \mathbb {N}^d} p_{\gamma } y_{{\varvec{\alpha }}+ {\varvec{\beta }}+ \gamma } \text { .}\end{aligned}$$

(64)

The localising matrix reduces to the moment matrix for $p=1$. For well-defined moment sequences, i.e. sequences that have a representing finite Borel measure, moment matrices and localising matrices are positive semidefinite, which provides insight on the reason why problem (P-CF) can be relaxed to a problem with positive semidefiniteness constraints.

Proposition 4

Let ${\varvec{y}} = (y_{{\varvec{\alpha }}})_{{\varvec{\alpha }}\in \mathbb {N}^d} \in \mathbb {R}^{\mathbb {N}^d}$ be a sequence of moments for some finite Borel measure $\mu$ on ${\varvec{K}}$. Then for all $k\in \mathbb {N}$, $M_k({\varvec{y}}) \succeq 0$. If $\mu$ has support contained in the set $\left\{ {\varvec{x}} \in K \mid g({\varvec{x}}) \ge 0\right\}$ for some polynomial $g \in \mathbb {R}[1]{[}{\varvec{x}}[1]{]}$ then, for all $k \in \mathbb {N}$, $M_k(g {\varvec{y}}) \succeq 0$.

Proof

Let ${\varvec{y}} = (y_{{\varvec{\alpha }}})$ be the moment sequence of a given Borel measure $\mu$ on ${\varvec{K}}$. Fix $k \in \mathbb {N}$. For any vector ${\varvec{v}} \in \mathbb {R}^{s(k)}$ (noting that ${\varvec{v}}$ is canonically associated with a polynomial $v \in \mathbb {R}[1]{[}{\varvec{x}} [1]{]}_{k}$ in the basis $({\varvec{x}}^{{\varvec{\alpha }}})$):

$$\begin{aligned} {\varvec{v}}^T M_k({\varvec{y}}) {\varvec{v}}&= \sum _{{\varvec{\alpha }},{\varvec{\beta }}\in \mathbb {N}^d_k} v_{{\varvec{\alpha }}} y_{{\varvec{\alpha }}+ {\varvec{\beta }}} v_{{\varvec{\beta }}} \end{aligned}$$

(65)

$$\begin{aligned}&= \sum _{{\varvec{\alpha }}, {\varvec{\beta }}\in \mathbb {N}^d_k} v_{{\varvec{\alpha }}} v_{{\varvec{\beta }}} \int _{K}{\varvec{x}}^{{\varvec{\alpha }}+ {\varvec{\beta }}}\;\,\mathrm {d}\,\mu \end{aligned}$$

(66)

$$\begin{aligned}&= \int _{K}\left( \sum _{{\varvec{\alpha }}\in \mathbb {N}^d_k} v_{{\varvec{\alpha }}} {\varvec{x}}^{{\varvec{\alpha }}} \right) ^2\;\,\mathrm {d}\,\mu \end{aligned}$$

(67)

$$\begin{aligned}&= \int _{K}v^2({\varvec{x}})\;\,\mathrm {d}\,\mu \ge 0 \text { .} \end{aligned}$$

(68)

Thus $M_{k}({\varvec{y}}) \succeq 0$.

Similarly we can prove that the localising matrix $M_k(g {\varvec{y}})$ is positive semidefinite when g is a nonnegative polynomial on the support of $\mu$. Indeed for all ${\varvec{v}} \in \mathbb {R}^{s(k)}$:

$$\begin{aligned} {\varvec{v}}^T M_k(g {\varvec{y}}) {\varvec{v}} = \int _{K}v^2({\varvec{x}}) g({\varvec{x}})\;\,\mathrm {d}\,\mu \ge 0 \text { ,}\end{aligned}$$

(69)

which concludes the proof. $\quad \square$

The following theorem, which is the dual facet of Theorem 6, is the key result for deriving the hierarchy of SDP relaxations for the primal problem (P-CF). It provides a necessary and sufficient condition for a sequence to have a representing measure.

Theorem 8

(Theorem 3.8 [54]). Let ${\varvec{y}} = (y_{{\varvec{\alpha }}})_{{\varvec{\alpha }}\in \mathbb {N}^d} \in \mathbb {R}^{\mathbb {N}^d}$ be a given infinite sequence in $\mathbb {R}$. Let $K \subset \mathbb {R}^d$ satisfy Assumptions 1. Then ${\varvec{y}}$ has a finite Borel representing measure with support contained in K if and only if:

$$\begin{aligned}&M_k({\varvec{y}}) \succeq 0, \; \forall k \in \mathbb {N}, \end{aligned}$$

(70)

$$\begin{aligned}&M_k(g_j {\varvec{y}}) \succeq 0, \; \forall j=1,\dots ,m , \; \forall k \in \mathbb {N}. \end{aligned}$$

(71)

Using the results above and Assumption 1, one can derive a hierarchy of SDPs [54] which provide a converging sequence of optimal values towards the value of program (GMP):

We refer readers to [54] for the proof of convergence of the hierarchies given by programs ($\hbox {GMP}^k$) and (D-$\hbox {GMP}^k$).

Duality between programs (SDP-$\hbox {CF}^k$) and (DSDP-$\hbox {CF}^k$)

As mentioned above, we chose to derive programs (SDP-$\hbox {CF}^k$) and (DSDP-$\hbox {CF}^{k}$) using dual arguments. These programs should therefore be dual to one another, which will immediately provide weak duality. We prove this for completeness.

Proposition 5

The program (DSDP-$\hbox {CF}^{k}$) is the dual formulation of the program (SDP-$\hbox {CF}^k$).

Proof

We start by rewriting $M_k({\varvec{y}})$ as $\sum _{{\varvec{\alpha }}\in \mathbb {N}^d_k} {\varvec{y}}_{{\varvec{\alpha }}} A_{{\varvec{\alpha }}}$ and $M_{k-1}(g_j {\varvec{y}})$ as $\sum _{{\varvec{\alpha }}\in \mathbb {N}^d_k} y_{{\varvec{\alpha }}} B^j_{{\varvec{\alpha }}}$ for $1 \le j \le m$ and for appropriate real symmetric matrices $A_{{\varvec{\alpha }}}$ and $(B^j_{{\varvec{\alpha }}})_j$. For instance, in the basis $({\varvec{x}}^{{\varvec{\alpha }}})$:

$$\begin{aligned} ( A_{{\varvec{\alpha }}} )_{{\varvec{s}}, {\varvec{t}}} = \bigg ( {\left\{ \begin{array}{ll} 1 \text { if } {\varvec{s}}+{\varvec{t}} = {\varvec{\alpha }}\\ 0 \text { otherwise} \end{array}\right. } \bigg )_{{\varvec{s}}, {\varvec{t}}} \text { .}\end{aligned}$$

From $A_{{\varvec{\alpha }}}$, we also extract $A_{{\varvec{\alpha }}}^C$ for $C \in \mathcal {M}$ in order to rewrite $M_k({\varvec{y}}|_C)$ as $\sum _{\alpha \in \mathbb {N}^d_k} {\varvec{y}}_{{\varvec{\alpha }}} A_{{\varvec{\alpha }}}^C$. This amounts to identifying which matrices $(A_{{\varvec{\alpha }}})$ contribute to a given context $C \in \mathcal {M}$. Then (SDP-$\hbox {CF}^k$) can be rewritten as:

Via the Lagrangian method, this is equivalent to:

$$\begin{aligned} \sup _{{\varvec{y}} \in \mathbb {R}^{s(k)}} \; \inf _{ \begin{array}{c} (X^C), (Y_j), Z \\ \text { SDP matrices} \end{array}} \; \mathcal {L}({\varvec{y}},(X_C),Y,(Z_j)), \end{aligned}$$

(72)

with

$$\begin{aligned} \begin{aligned} \mathcal {L}({\varvec{y}},(X^C),(Y_j),Z)&= y_{{\varvec{0}}} \\&\quad + \sum _{C \in \mathcal {M}} \text {Tr}(M_k({\varvec{y}}^{e,C}) X^C) - \sum _{C \in \mathcal {M}} \sum _{{\varvec{\alpha }}\in \mathbb {N}^d_k} y_{{\varvec{\alpha }}} \text {Tr}(A_{{\varvec{\alpha }}}^C X^C) \\&\quad + \sum _{j=1}^m \sum _{{\varvec{\alpha }}\in \mathbb {N}^d_k} y_{{\varvec{\alpha }}} \text {Tr}(B_{{\varvec{\alpha }}}^j Y_j) \\&\quad + \sum _{{\varvec{\alpha }}\in \mathbb {N}^d_k} y_{{\varvec{\alpha }}} \text {Tr}(A_{{\varvec{\alpha }}} Z) \text { .}\end{aligned} \end{aligned}$$

(73)

The dual program corresponds to

$$\begin{aligned} \inf _{ \begin{array}{c} (X^C), (Y_j), Z \\ \text { SDP matrices} \end{array}} \; \sup _{{\varvec{y}} \in \mathbb {R}^{s(k)}} \; \mathcal {L}({\varvec{y}},(X^C),(Y_j),Z) \text { .}\end{aligned}$$

(74)

We rewrite the Lagrangian as:

$$\begin{aligned} \mathcal {L}({\varvec{y}},(X_C),Y,(Z_j))= & {} \sum _{C \in \mathcal {M}} \text {Tr}(M_k({\varvec{y}}^{e,C}) X^C) \nonumber \\&+ \sum _{{\varvec{\alpha }}\in \mathbb {N}^d_k} y_{{\varvec{\alpha }}} \left( \delta _{{\varvec{\alpha }}, {\varvec{0}}} - \sum _{C \in \mathcal {M}} \text {Tr}(A_{{\varvec{\alpha }}}^C X^C) + \sum _{j=1}^m \text {Tr}(B_{{\varvec{\alpha }}}^j Y_j) + \text {Tr}(A_{{\varvec{\alpha }}} Z) \right) \text { .}\quad \quad \end{aligned}$$

(75)

Then the dual program of (SDP-$\hbox {CF}^k$) reads:

$$\begin{aligned} \left\{ \begin{aligned}&\quad \text {Find } ({X^C})_{C \in \mathcal {M}}, ({Y_j})_{j=1,\dots ,m} \text { and } Z \text { SDP matrices } \\&\quad \text {maximising } \sum _{C \in \mathcal {M}} \text {Tr}(M_k({\varvec{y}}^{e,C}) X^C) \\&\quad \text {subject to:} \\&\quad \begin{aligned}&\forall {\varvec{\alpha }}\in \mathbb {N}^d_k, \; \sum _{C \in \mathcal {M}} \text {Tr}(A_{{\varvec{\alpha }}}^C X^C) - \sum _{j=1}^m \text {Tr}(B_{{\varvec{\alpha }}}^j Y_j) - \text {Tr}(A_{{\varvec{\alpha }}} Z) = \delta _{{\varvec{\alpha }}, {\varvec{0}}} \text { .}\end{aligned} \end{aligned} \right.&\end{aligned}$$

We finally show that the above program exactly corresponds to (DSDP-$\hbox {CF}^{k}$). We start with the objective. For all $C \in \mathcal {M}$, with $X^C$ a positive semidefinite matrix:

$$\begin{aligned} \text {Tr}(M_k({\varvec{y}}^{e,C}) X^C)&= \sum _{{\varvec{\alpha }}} \sum _{{\varvec{\beta }}} (M_k({\varvec{y}}^{e,C}))_{{\varvec{\alpha }}{\varvec{\beta }}} (X^C)_{{\varvec{\beta }}{\varvec{\alpha }}} \end{aligned}$$

(76)

$$\begin{aligned}&= \sum _{{\varvec{\alpha }}} \sum _{{\varvec{\beta }}} {\varvec{y}}^{e,C}_{{\varvec{\alpha }}+ {\varvec{\beta }}} (X^C)_{{\varvec{\alpha }}{\varvec{\beta }}} \end{aligned}$$

(77)

$$\begin{aligned}&= \sum _{{\varvec{\alpha }}} \sum _{{\varvec{\beta }}} \int _{\mathcal {O}_C}{\varvec{x}}^{{\varvec{\alpha }}+ {\varvec{\beta }}}\;\,\mathrm {d}\,e_C (X^C)_{{\varvec{\alpha }}{\varvec{\beta }}} \end{aligned}$$

(78)

$$\begin{aligned}&= \int _{\mathcal {O}_C} {\varvec{\mathrm {v}}}_k({\varvec{x}})^T X^C {\varvec{\mathrm {v}}}_k({\varvec{x}})\;\,\mathrm {d}\,e_C \end{aligned}$$

(79)

$$\begin{aligned}&= \int _{\mathcal {O}_C} p_C \;\,\mathrm {d}\,e_C, \end{aligned}$$

(80)

with for all $C \in \mathcal {M}$, $p_C \in \varSigma ^2 \mathbb {R}[1]{[}{\varvec{x}} [1]{]}_{k}$ a sum-of-squares polynomial via Proposition 3 and where we used ${\varvec{\mathrm {v}}}_k({\varvec{x}})$ the vectors of monomials of maximal total degree k.

Now, to retrieve the constraint, we multiply each side by ${\varvec{x}}^{{\varvec{\alpha }}}$ and we sum for all ${\varvec{\alpha }}$:

$$\begin{aligned} \sum _{C \in \mathcal {M}} \text {Tr}(\sum _{{\varvec{\alpha }}} {\varvec{x}}^{{\varvec{\alpha }}} A^C_{{\varvec{\alpha }}} X^C) - {\varvec{1}} = \sum _{j=1}^m \text {Tr}(\sum _{{\varvec{\alpha }}} {\varvec{x}}^{{\varvec{\alpha }}} B_{{\varvec{\alpha }}}^j Y_j) + \text {Tr}(\sum _{{\varvec{\alpha }}} {\varvec{x}}^{{\varvec{\alpha }}} A_{{\varvec{\alpha }}} Z) \end{aligned}$$

(81)

Recalling the definition of moment and localising matrices:

$$\begin{aligned}&\sum _{{\varvec{\alpha }}} A^C_{{\varvec{\alpha }}} {\varvec{x}}^{{\varvec{\alpha }}} = {\varvec{\mathrm {v}}}_k({\varvec{x}}) {\varvec{\mathrm {v}}}_k({\varvec{x}})^T \end{aligned}$$

(82)

$$\begin{aligned}&\sum _{{\varvec{\alpha }}} B_{{\varvec{\alpha }}}^j \mathrm {v}_k({\varvec{x}}) \mathrm {v}_k({\varvec{x}})^T = g_j({\varvec{x}}) {\varvec{\mathrm {v}}}_{k-1}({\varvec{x}}) {\varvec{\mathrm {v}}}_{k-1}({\varvec{x}})^T, \; \forall j=1,\dots ,m \end{aligned}$$

(83)

Thus, by Proposition 3, for appropriate sum-of-squares polynomials $(\sigma _j)_{j=0,1,\dots ,m} \subset \mathbb {R}[1]{[}{\varvec{x}}[1]{]}_{k-1}$:

$$\begin{aligned}&\text {Tr}(\sum _{{\varvec{\alpha }}} {\varvec{x}}^{{\varvec{\alpha }}} A^C_{{\varvec{\alpha }}} X^C) = p_C \circ \rho _C^X ({\varvec{x}}) \end{aligned}$$

(84)

$$\begin{aligned}&\text {Tr}(\sum _{{\varvec{\alpha }}} {\varvec{x}}^{{\varvec{\alpha }}} B_{{\varvec{\alpha }}}^j Y_j) = g_j({\varvec{x}}) \sigma _j({\varvec{x}}) \end{aligned}$$

(85)

$$\begin{aligned}&\text {Tr}(\sum _{{\varvec{\alpha }}} {\varvec{x}}^{{\varvec{\alpha }}} A_{{\varvec{\alpha }}} Z) = \sigma _0({\varvec{x}}) \end{aligned}$$

(86)

This is exactly the constraint in (DSDP-$\hbox {CF}^{k}$) with, for convenience, $g_0 = 1$. $\quad \square$