Properties and Applications of Partially Deterministic Polytopes

TL;DR

This work develops a general mathematical framework for partially deterministic polytopes, extending the Bell-local and no-signalling sets to arbitrary numbers of parties, inputs, and deterministically constrained inputs. It introduces the construction of partially deterministic polytopes $\mathbf{PD}(S,M')$ via convex hulls of partially predictable no-signalling behaviours, and provides a complete vertex description in terms of deterministic subbehaviours and NS-extremes. A central contribution is the generalization of Fine's theorem to a partial-determinism setting, together with the notions of partial uncorrelatedness and partial factorizability, which yield equivalent characterizations and a robust geometric view via restriction maps and composable sets. The framework is applied to device-independent inseparability witnesses, broadcast-locality, and Local Friendliness sequential Wigner’s friend scenarios, revealing how PD polytopes serve as unifying tools across diverse contexts and enabling new witnesses and equivalence-class classifications. The results offer a versatile, structure-based approach to boundary problems in quantum correlations, with potential extensions to broader composable-causality settings and multipartite nonclassicality notions.

Abstract

The assumption of a deterministic local hidden variable model constrains the experimentally accessible statistics in a Bell experiment to be contained in the Bell-local polytope. But what if the outputs for only a subset of the measurements at each site are predetermined by the model? In this work, we thoroughly explore this concept of `partial determinism', allowing for arbitrary numbers of parties, inputs and outputs per site. The resulting objects form new classes of convex polytopes which recover the Bell and the no-signalling polytopes as special cases. Nontrivial equivalence classes of partially deterministic models arise, which we classify completely. In particular, the Bell polytope for any scenario can be expressed in multiple different ways in terms of local partially deterministic models. This allows us to generalise Fine's theorem, recovering the original formulation as a special case, but finding new constraints otherwise. We discuss scenarios with different physical motivations, which do not require the causal structure of the Bell scenario, and where classes of partially deterministic polytopes are relevant. Our example applications include device-independent quantum state inseparability witnesses, classes of broadcast-local polytopes, and Local Friendliness scenarios in quantum foundations. We also point out instances in previous literature where classes of related objects have been studied. In the case of correlations compatible with the Local Friendliness assumptions, we find a one-to-one correspondence between partially deterministic polytopes and sequential extended Wigner's friend scenarios so that every partially deterministic polytope has physical relevance. We discuss how the framework captures a broad class of non-classicality notions, and identify an even broader notion of `composable sets', of which partially deterministic polytopes are special cases.

Figures (10)

• Figure 1: An example of a correlation scenario with five parties. Each party is in possession of a black box, which takes an input $x_i$ and produces an output $a_{x_i}$. It is assumed that the parties perform their interventions at space-like separation, but whatever is inside their boxes may have originated from a common source.
• Figure 2: An illustration of the geometry of the set behaviour $\mathbf{E}(S)$, in a trivial scenario where the behaviour $\wp$ consists of a single probability distribution $\wp(a^k)$, $k\in \{1,2,3\}$. The vectors $\vec{r}$ representing the behaviour are necessarily contained in the cube $\mathscr{C}^3 = [0,1]^3$ (red dashed lines) since each probability satisfies $1\geq \wp(a^k)\geq 0$. The behaviour are further constrained by the normalization constraints, which bounds the appropriate vectors to be in the set $\{\mathrm{Norm}\} = \{\vec{r}\in \mathbb{R}^3 : \sum_{k}r_{a^k}=1\}$ which defines a plane in $\mathbb{R}^3$. The set $\mathbf{E}(S)$ is the intersection $\mathscr{C}^3\cap \{\mathrm{Norm}\}$, a two-dimensional polytope (triangle) contained in the cube. The set $\mathbf{E}(S)$ is seen to be contained in the convex hull of the three predictable vertices $\wp(a^k)\in \{0,1\}$ which are a subset of the vertices of the cube $\mathscr{C}$. The facets of the set $\mathbf{E}(S)$ are the $1$-dimensional edges (light blue) corresponding to the positivity constraints $r_{a^k} \geq 0$. Since $\mathrm{dim}(\mathbf{E}(S)) = 2 <3$, the representation of the facet defining inequalities is not unique. For example constraining $\vec{r}$ to the hyperplane $r_{a^1}=0$ for the polytope $\mathbf{E}(S)$ is equivalent to $r_{a^2} +r_{a^3} = 1$, as may be seen by using the normalization constraints.
• Figure 3: A bipartite scenario $S = (I,M,O), I= \{1,2\}$, is restricted to the scenario $S_{|M'}$ as defined by the collection $M'$ with $M_1'=\emptyset$ and $\emptyset\neq M_2'\subset M_2$. The scenario $S_{|M'} = (I_{|M'}, M_{|M'}, O_{|M'})$ contains only one agent $i =2$, with some subset $M_2'$ of the inputs of agent $i=2$ in scenario $S$. The restriction map $R_{|M'}$ represented in the figure maps the bipartite behaviour $\wp^{S}$ to a well-defined behaviour $\wp^{S_{|M'}}$ essentially by marginalizing over agent $i=1$, and omitting distributions for which $x_1 \notin M_1'$.
• Figure 4: Illustration of the action of the behaviour product. The scenarios $S'=(I',M',O')$, which contains a single agent $i=3$, and $S'^{\perp}=(I'^{\perp}, M'^{\perp}, O'^{\perp})$, which contains three agents $i\in \{1,2,3\}$ form a disjoint bipartition of the tripartite scenario $S = (I,M,O)$. The behaviour product takes behaviour $\wp^{S'}, \wp^{S'^{\perp}}$ defined over the subscenarios $S', S'^{\perp}$ and produces a well-defined behaviour $\wp^S$ on scenario $S$. The expansion of the distributions $\wp^S(\vec{a}|\vec{x})$ is context-dependent, with each distribution given as the products of marginals $\wp^{S'}(\vec{a}_{F_{\vec{x}}}|\vec{x}_{F_{\vec{x}}})$ and $\wp^{S'^{\perp}}(\vec{a}_{I\setminus F_{\vec{x}}}|\vec{x}_{I\setminus F_{\vec{x}}})$, with $F_{\vec{x}} = \{ i\in I : x_i \in M_i'\}$. In the situation depicted above, the general formula gives $\wp^S(\vec{a}|\vec{x}) = \wp^{S'}(a_3|x_3)\wp^{S'^{\perp}}(a_1a_2|x_1x_2)$ if $x_3 \in M_3'$ and $\wp^{S}(\vec{a}|\vec{x}) = \wp^{S'^{\perp}}(\vec{a}|\vec{x})$ otherwise, owing to the fact that the scenario $S'$ contains only the agent $i=3$, with input set $M_3'$. Note that expressions such as $\wp^{S'^{\perp}}(\vec{a}|\vec{x})$ should more precisely be understood as referring to the 'matching' $\wp^{S'^{\perp}}(\vec{a}'^{\perp}=\vec{a}_{I'}|\vec{x}'^{\perp}=\vec{x}_{I'})$. The precise notation is generally dropped for convenience, when there is no risk of confusion.
• Figure 5: Illustration of the equivalence classes of all the partially deterministic polytopes of Example \ref{['EXAMPLE:PartialDeterminismInBipartitetHREE-INPUTscenario']}. The bipartite scenario $S=(I,M,O)$, $I=\{1,2\}$ with three inputs per site allows 17 distinct equivalence classes of partially deterministic polytopes to be defined. Six of those classes correspond to situations where only one input $x_i^j, i\in I, 1\leq j \leq |M_i|$ is deterministic, with each such class represented as a brown polyhedron with $\{x_i^j\}$ labelling the deterministic input. There are also 9 classes of partially deterministic polytopes for which pairs $x_i^j, x^{j'}_{i'}$, $i,i'\in I, i \neq i'$ of inputs are deterministic, represented in a similar way on a row above the class of Bell polytopes $\mathbf{B}(S)$. Each partially deterministic polytope with two deterministic inputs, is a strict subset of exactly two partially deterministic polytopes with one input. In the figure, if a path (sequence of dashed lines) exists from bottom to top (or top to bottom), then every object in that path can be ordered by the strict subset (or superset) relation. If two objects are not connected by such a path, then neither set is fully contained in the other. In particular, the quantum set $\mathbf{Q}(S)$ obeys $\mathbf{B}(S)\subsetneq \mathbf{Q}(S) \subsetneq \mathbf{NS}(S)$, but neither contains or is contained in any of the other partially deterministic polytopes. Every equivalence class has exactly one element, except for the class corresponding to the Bell-polytope which has 48 elements including the trivial fully deterministic case $M'=M$.

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