All analytic and optimizer properties of statistical strength

Let $S$ be any Bell experiment: it has a positive number $N$ of parties, each party $i$ has a positive number $m_i$ of settings, and setting $a$ at party $i$ has a positive number $r_{i,a}$ of outcomes. Write $\Omega=\prod_i\{0,\ldots,m_i-1\}$ for the setting tuples, $X_a=\prod_i\{0,\ldots,r_{i,a_i}-1\}$ for the outcome tuples at setting $a$, and $\mathcal V=\prod_i\prod_{b=0}^{m_i-1}\{0,\ldots,r_{i,b}-1\}$ for the assignments of an outcome to every setting of every party. For a finite set $E$, write $\Delta(E)$ for the nonnegative real functions on $E$ summing to $1$. Let $Q_a(x)$ be a real table satisfying the following assumptions: each $Q_a$ belongs to $\Delta(X_a)$; whenever two setting tuples agree on a subset of parties, their marginal probabilities for every outcome tuple on that subset agree; and there exist positive integers $d_i$, a positive semidefinite matrix $\rho$ of trace $1$ indexed by $\prod_i\{0,\ldots,d_i-1\}$, and positive semidefinite matrices $M_{i,b,u}$ indexed by $\{0,\ldots,d_i-1\}$, with $\sum_u M_{i,b,u}=I$, such that $Q_a(x)=\operatorname{Re}\operatorname{tr}((\bigotimes_i M_{i,a_i,x_i})\rho)$. For $p:\mathcal V\to\mathbb R$, define $P_p(a,x)=\sum_{v:\,v_i(a_i)=x_i\ \forall i}p(v)$. For real functions $q,r$ on the same finite set, define

where $[u]_+=\max\{u,0\}$, regarded as an extended nonnegative real; division by zero is $0$, and the total real logarithm used here has $\log_2 0=0$ and $\log_2(-u)=\log_2 u$ for $u>0$. Put $D_s(p)=D((s(a)Q_a(x))_{a,x}\Vert(s(a)P_p(a,x))_{a,x})$ and $F(s)=\inf_{p\in\Delta(\mathcal V)}D_s(p)$. Then all the following hold. (i) For every $s\in\Delta(\Omega)$ and $p\in\Delta(\mathcal V)$,

using extended nonnegative arithmetic, including $0\cdot\infty=0$. (ii) For every $s\in\Delta(\Omega)$, all $p,r\in\Delta(\mathcal V)$, and all real $\alpha,\beta\ge0$ with $\alpha+\beta=1$,

the function $p\mapsto D_s(p)$ is lower semicontinuous on $\Delta(\mathcal V)$; and at every $p\in\Delta(\mathcal V)$ with $p(v)>0$ for every $v$, its value is finite, it is continuous within that simplex at $p$, and $r\mapsto\operatorname{toReal}(D_s(r))$ is differentiable over $\mathbb R$ within that simplex at $p$, where $\operatorname{toReal}$ takes finite values to their real values and takes $\infty$ to $0$. (iii) For every $p\in\Delta(\mathcal V)$ such that $Q_a(x)>0$ implies $P_p(a,x)>0$ for every $a,x$, there exists a real linear functional $L:\mathbb R^\Omega\to\mathbb R$ satisfying $D_s(p)=[L(s)]_+$ for every $s\in\Delta(\Omega)$. (iv) For every $s\in\Delta(\Omega)$, some $p\in\Delta(\mathcal V)$ satisfies $D_s(p)=F(s)$. (v) Every $s\in\Delta(\Omega)$ satisfies $0\le F(s)<\infty$; there exists a single finite extended nonnegative number $B$ with $F(s)\le B$ for all such $s$; and $s\mapsto\operatorname{toReal}(F(s))$ is concave and continuous on $\Delta(\Omega)$. (vi) If $Q=P_p$ for some $p\in\Delta(\mathcal V)$, then $F(s)=0$ for every $s\in\Delta(\Omega)$; if no such $p$ exists, then $F(s)>0$ for every $s\in\Delta(\Omega)$ whose coordinates are all strictly positive. (vii) If there are exactly two parties and each has exactly two settings, then every $s\in\Delta(\Omega)$ having at least one zero coordinate satisfies $F(s)=0$. (viii) For every $s\in\Delta(\Omega)$ with all coordinates strictly positive, define

If there exist $a,x$ with $0<Q_a(x)<1$, then, for every $p\in\Delta(\mathcal V)$, the equality $D_s(p)=F(s)$ holds if and only if $Q_a(x)>0$ implies $P_p(a,x)>0$ for all $a,x$, $H_{s,p}(v)=1$ whenever $p(v)>0$, and $H_{s,p}(v)\le1$ whenever $p(v)=0$. Independently of that additional assumption $0<Q_a(x)<1$, any two $p,r\in\Delta(\mathcal V)$ attaining $F(s)$ both satisfy this support domination condition, and every $a,x$ with $Q_a(x)>0$ satisfies $P_p(a,x)=P_r(a,x)>0$. Setting and outcome sets of size $1$, and zero coordinates in the probability distributions, are included wherever the stated hypotheses permit them.