Optimization Over Trace Polynomials

Abstract

Motivated by recent progress in quantum information theory, this article aims at optimizing trace polynomials, i.e., polynomials in noncommuting variables and traces of their products. A novel Positivstellensatz certifying positivity of trace polynomials subject to trace constraints is presented, and a hierarchy of semidefinite relaxations converging monotonically to the optimum of a trace polynomial subject to tracial constraints is provided. This hierarchy can be seen as a tracial analog of the Pironio, Navascués and Acín scheme (Pironio et al. in New J. Phys. 10(7):073013, 2008) for optimization of noncommutative polynomials. The Gelfand–Naimark–Segal (GNS) construction is applied to extract optimizers of the trace optimization problem if flatness and extremality conditions are satisfied. These conditions are sufficient to obtain finite convergence of our hierarchy. The results obtained are applied to violations of polynomial Bell inequalities in quantum information theory. The main techniques used in this paper are inspired by real algebraic geometry, operator theory, and noncommutative algebra.

Alternative Proof of Theorem 4.4

Proof of (i)$\Rightarrow$(iii). Assume $a+\varepsilon \notin {{\mathcal {M}}^{\mathrm{cyc}}}$ for some $\varepsilon >0$. Let $U=\{p\in {{\,\mathrm{Sym}\,}}{\mathbb {T}}\mid {{\,\mathrm{tr}\,}}(p)=0 \}$. Then ${{\mathcal {M}}^{\mathrm{cyc}}}+U$ is a convex cone in ${{\,\mathrm{Sym}\,}}{\mathbb {T}}$. Since a is a pure trace polynomial, we have $a+\varepsilon \notin {{\mathcal {M}}^{\mathrm{cyc}}}+U$. Since ${{\mathcal {M}}^{\mathrm{cyc}}}$ is archimedean, for every $p\in {{\,\mathrm{Sym}\,}}{\mathbb {T}}$ there exists $\delta >0$ such that $1\pm \delta p\in {{\mathcal {M}}^{\mathrm{cyc}}}$, which in terms of [3, Definition III.1.6] means that 1 is an algebraic interior point of the cone ${{\mathcal {M}}^{\mathrm{cyc}}}+U$ in ${{\,\mathrm{Sym}\,}}{\mathbb {T}}$. By the Eidelheit–Kakutani separation theorem [3, Corollary III.1.7] there is a nonzero ${\mathbb {R}}$-linear functional $L_0:{{\,\mathrm{Sym}\,}}{\mathbb {T}}\rightarrow {\mathbb {R}}$ satisfying $L_0({{\mathcal {M}}^{\mathrm{cyc}}}+U)\subseteq {\mathbb {R}}_{\ge 0}$ and $L_0(a+\varepsilon )\le 0$. In particular, $L_0(U)=\{0\}$. Moreover, $L_0(1)>0$ because ${{\mathcal {M}}^{\mathrm{cyc}}}$ is archimedean, so after rescaling we can assume $L_0(1)=1$. Let $L:{\mathbb {T}}\rightarrow {\mathbb {R}}$ be the symmetric extension of $L_0$, i.e., $L(p)=\frac{1}{2} L_0(p+p^\star )$ for $p\in {\mathbb {T}}$. Note that

$$\begin{aligned} L(p)=L({{\,\mathrm{tr}\,}}(p)) \end{aligned}$$

(A.1)

for all $p\in {\mathbb {T}}$, and in particular $L(pq)=L(qp)$ for all $p,q\in {\mathbb {T}}$.

Now consider the set ${\mathcal {C}}$ of all symmetric linear functionals $L':{\mathbb {T}}\rightarrow {\mathbb {R}}$ satisfying $L'({{\mathcal {M}}^{\mathrm{cyc}}}+U)\subseteq {\mathbb {R}}_{\ge 0}$ and $L'(1)=1$. This set is nonempty because $L\in {\mathcal {C}}$. Endow ${\mathbb {T}}$ with the norm

$$\begin{aligned} \Vert p\Vert =\max \left\{ \Vert p({\underline{X}})\Vert \mid n\in {\mathbb {N}}, {\underline{X}}\in {\mathbb {S}}_n^k, \Vert X_j\Vert \le 1\right\} . \end{aligned}$$

This is indeed a norm because no nonzero trace polynomial vanishes on matrices of all finite sizes. By the Banach–Alaoglu theorem [3, Theorem III.2.9], the convex set ${\mathcal {C}}$ is weak*-compact. Thus by the Krein–Milman theorem [3, Theorem III.4.1] we may assume that our separating functional L is an extreme point of ${\mathcal {C}}$.

On ${\mathbb {T}}$ we define a semi-scalar product $\langle p, q\rangle =L(pq^\star )$. By the Cauchy–Schwarz inequality for semi-scalar products,

$$\begin{aligned} {\mathcal {N}}=\left\{ q \in {\mathbb {T}}\mid L(qq^\star )=0\right\} \end{aligned}$$

is a linear subspace of ${\mathbb {T}}$. Let $p,q\in {\mathbb {T}}$. Since ${{\mathcal {M}}^{\mathrm{cyc}}}$ is archimedean, there exists $\delta >0$ such that $1-\delta p p^\star \in {{\mathcal {M}}^{\mathrm{cyc}}}$ and therefore

$$\begin{aligned} 0\le L(q(1-\delta p p^\star )q^\star )=L(q q^\star )-\delta L(qpp^\star q^\star )\le L(q q^\star ). \end{aligned}$$

(A.2)

In particular, $q\in {\mathcal {N}}$ implies $qp\in {\mathcal {N}}$, so ${\mathcal {N}}$ is a left ideal. Furthermore, $L({\mathcal {N}})=\{0\}$: if $L(qq^\star )=0$, then for every $\delta >0$,

$$\begin{aligned} 0\le L((\delta \pm q)(\delta \pm q)^\star )=\delta (\delta \pm 2 L(q)) \end{aligned}$$

and hence $L(q)=0$. Let ${{\overline{p}}}=p+{\mathcal {N}}$ denote the residue class of $p\in {\mathbb {T}}$ in ${\mathbb {T}}/{\mathcal {N}}$. Because ${\mathcal {N}}$ is a left ideal, we can define linear maps

$$\begin{aligned} \chi _p :{\mathbb {T}}/{\mathcal {N}}\rightarrow {\mathbb {T}}/{\mathcal {N}},\qquad {{\overline{q}}}\mapsto {\overline{pq}} \end{aligned}$$

for $p\in {\mathbb {T}}$, which are bounded by (A.2).

Now

$$\begin{aligned} \langle {{\overline{p}}},{{\overline{q}}}\rangle =L(pq^\star ) \end{aligned}$$

(A.3)

is a scalar product on ${\mathbb {T}}/{\mathcal {N}}$, and we let H denote the completion of ${\mathbb {T}}/{\mathcal {N}}$ with respect to this scalar product. Each $\chi _p$ extends to a bounded operator ${{\hat{\chi }}}_p$ on H, and the map

$$\begin{aligned} \pi : {\mathbb {T}}\rightarrow {\mathcal {B}}(H), \qquad p\mapsto {{\hat{\chi }}}_p \end{aligned}$$

(A.4)

is clearly a $\star$-representation with $\ker \pi ={\mathcal {N}}$. Let ${\mathcal {F}}$ be the closure of $\pi ({\mathbb {T}})$ in ${\mathcal {B}}(H)$ with respect to the weak operator topology. The map

$$\begin{aligned} \tau :\pi ({\mathbb {T}})\rightarrow {\mathbb {R}},\qquad {{\hat{\chi }}}_p\mapsto L(p) \end{aligned}$$

is a faithful tracial state on $\pi ({\mathbb {T}})$ by $L({\mathcal {N}})=\{0\}$ and (A.1). Since

$$\begin{aligned} \tau ({{\hat{\chi }}}_p)=\langle {{\overline{p}}},{\overline{1}}\rangle , \end{aligned}$$

$\tau$ extends uniquely to a faithful normal tracial state on ${\mathcal {F}}$.

Next we claim that $\pi (\text{ T})={\mathbb {R}}$. Observe that ${\overline{1}}\in H$ is a cyclic vector for $\pi$ by construction and $L(p)=\langle \pi (p){\overline{1}},{\overline{1}}\rangle$. Suppose $\pi (\text{ T})\ne {\mathbb {R}}$. If ${\mathcal {E}}$ denotes the weak closure of $\pi (\text{ T})$ in ${\mathcal {F}}$, then ${\mathcal {E}}$ is a central von Neumann subalgebra of ${\mathcal {F}}$; since ${\mathcal {E}}\ne {\mathbb {R}}$ and all the elements of ${\mathcal {E}}$ are self-adjoint, there is a nontrivial projection $P\in {\mathcal {E}}$. Since ${\overline{1}}$ is cyclic for $\pi$, we have $P{\overline{1}}\ne 0$ and $(1-P){\overline{1}}\ne 0$. Hence we can define linear functionals $L_i$ on ${\mathbb {T}}$ by

$$\begin{aligned} L_1(p)=\frac{\langle \pi (p)P{\overline{1}},P{\overline{1}}\rangle }{\Vert P{\overline{1}}\Vert ^2} \qquad \text {and}\qquad L_2(p)=\frac{\langle \pi (p)(1-P){\overline{1}},(1-P){\overline{1}}\rangle }{\Vert (1-P){\overline{1}}\Vert ^2} \end{aligned}$$

for all $p\in {\mathbb {T}}$. One easily checks that L is a convex combination of $L_1$ and $L_2$, $L_i(1)=1$ and $L_i({{\mathcal {M}}^{\mathrm{cyc}}})={\mathbb {R}}_{\ge 0}$. Furthermore, since P is a weak limit of $\{\pi (s_n)\}_n$ for some $s_n\in \text{ T }$ and

$$\begin{aligned} \langle \pi (p-{{\,\mathrm{tr}\,}}(p))\pi (s_n){\overline{1}},{\overline{1}}\rangle =L(s_n(p-{{\,\mathrm{tr}\,}}(p)))=L(s_np-{{\,\mathrm{tr}\,}}(s_np)))=0, \end{aligned}$$

we also have

$$\begin{aligned} \langle \pi (p-{{\,\mathrm{tr}\,}}(p))P{\overline{1}},P{\overline{1}}\rangle = \langle \pi (p-{{\,\mathrm{tr}\,}}(p))P{\overline{1}},{\overline{1}}\rangle =0 \end{aligned}$$

so $L_i(U)=\{0\}$. Therefore $L_i\in {\mathcal {C}}$, so $L=L_1=L_2$ by the extreme property of L. Then for $\lambda =\Vert P{\overline{1}}\Vert ^2$,

$$\begin{aligned} \langle \pi (p){\overline{1}},\lambda {\overline{1}}\rangle =\lambda \langle \pi (p){\overline{1}},{\overline{1}}\rangle =\langle \pi (p)P{\overline{1}},P{\overline{1}}\rangle =\langle P\pi (p){\overline{1}},P{\overline{1}}\rangle =\langle \pi (p){\overline{1}},P{\overline{1}}\rangle \end{aligned}$$

for all $p\in {\mathbb {T}}$. Therefore $P{\overline{1}}=\lambda {\overline{1}}$ since ${\overline{1}}$ is a cyclic vector for $\pi$. So $\lambda \in \{0,1\}$ since P is a projection, a contradiction.

Let ${\underline{X}}:=({{\hat{\chi }}}_{x_1},\dots ,{{\hat{\chi }}}_{x_n})$. This is a tuple of self-adjoint operators in ${\mathcal {F}}$, and $\pi (\text{ T})={\mathbb {R}}$ implies $p( {\underline{X}})={{\hat{\chi }}}_p$ for all $p\in {\mathbb {T}}$. Therefore ${\underline{X}}\in {\mathcal {D}}_{{\mathcal {M}}^{\mathrm{cyc}}}^{{\mathcal {F}},\tau }$ by (A.3) and $L({{\mathcal {M}}^{\mathrm{cyc}}}) \subseteq {\mathbb {R}}_{\ge 0}$. Finally $a({\underline{X}})=\tau ({{\hat{\chi }}}_a)=L(a)<0$.