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arXiv:2210.10186

Mermin polytopes in quantum computation and foundations

Cihan Okay, Ho Yiu Chung, Selman Ipek

Published Oct 18, 2022• 1 citation
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Abstract

Mermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes $\text{MP}_β$ obtained from the Mermin scenario, parametrized by a function $β$ on the set of contexts. Up to combinatorial isomorphism, there are two types of polytopes $\text{MP}_0$ and $\text{MP}_1$ depending on the parity of $β$. Our main result is the classification of the vertices of these two polytopes. In addition, we describe the graph associated with the polytopes. All the vertices of $\text{MP}_0$ turn out to be deterministic. This result provides a new topological proof of a celebrated result of Fine characterizing noncontextual distributions on the CHSH scenario. $\text{MP}_1$ can be seen as a nonlocal toy version of $Λ$-polytopes, a class of polytopes introduced for the simulation of universal quantum computation. In the $2$-qubit case, we provide a decomposition of the $Λ$-polytope using $\text{MP}_1$, whose vertices are classified, and the nonsignaling polytope of the $(2,3,2)$ Bell scenario, whose vertices are well-known.

DOI
10.48550/arxiv.2210.10186

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Mermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes MPβ\text{MP}_βMPβ​ obtained from the Mermin scenario, parametrized by a function βββ on the set of contexts. Up to combinatorial isomorphism, there are two types of polytopes MP0\text{MP}_0MP0​ and MP1\text{MP}_1MP1​ depending on the parity of βββ. Our main result is the classification of the vertices of these two polytopes. In addition, we describe the graph associated with the polytopes. All the vertices of MP0\text{MP}_0MP0​ turn out to be deterministic. This result provides a new topological proof of a celebrated result of Fine characterizing noncontextual distributions on the CHSH scenario. MP1\text{MP}_1MP1​ can be seen as a nonlocal toy version of ΛΛΛ-polytopes, a class of polytopes introduced for the simulation of universal quantum computation. In the 222-qubit case, we provide a decomposition of the ΛΛΛ-polytope using MP1\text{MP}_1MP1​, whose vertices are classified, and the nonsignaling polytope of the (2,3,2)(2,3,2)(2,3,2) Bell scenario, whose vertices are well-known.
Mermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes $\MP_\beta$ obtained from the Mermin scenario, parametrized by a function β\betaβ on the set of contexts. Up to combinatorial isomorphism, there are two types of polytopes $\MP_0$ and $\MP_1$ depending on the parity of β\betaβ. Our main result is the classification of the vertices of these two polytopes. In addition, we describe the graph associated with the polytopes. All the vertices of $\MP_0$ turn out to be deterministic. This result provides a new topological proof of a celebrated result of Fine characterizing noncontextual distributions on the CHSH scenario. $\MP_1$ can be seen as a nonlocal toy version of Λ\LambdaΛ-polytopes, a class of polytopes introduced for the simulation of universal quantum computation. In the 222-qubit case, we provide a decomposition of the Λ\LambdaΛ-polytope using $\MP_1$, whose vertices are classified, and the nonsignaling polytope of the (2,3,2)(2,3,2)(2,3,2) Bell scenario, whose vertices are well-known.
The degenerate vertices of the 222-qubit ΛΛΛ-polytope and their update rules
Recently, a class of objects, known as ΛΛΛ-polytopes, were introduced for classically simulating universal quantum computation with magic states. In ΛΛΛ-simulation, the probabilistic update of ΛΛΛ vertices under Pauli measurement yields dynamics consistent with quantum mechanics. Thus, an important open problem in the study of ΛΛΛ-polytopes is characterizing its vertices and determining their update rules. In this paper, we obtain and describe the update of all degenerate vertices of Λ2Λ_{2}Λ2​, the 222-qubit ΛΛΛ polytope. Our approach exploits the fact that Λ2Λ_{2}Λ2​ projects to a well-understood polytope MP\text{MP}MP consisting of distributions on the Mermin square scenario. More precisely, we study the ``classical" polytope MP‾\overline{\text{MP}}MP, which is MP\text{MP}MP intersected by the polytope defined by a set of Clauser-Horne-Shimony-Holt (CHSH) inequalities. Owing to a duality between CHSH inequalities and vertices of MP\text{MP}MP we utilize a streamlined version of the double-description method for vertex enumeration to obtain certain vertices of MP‾\overline{\text{MP}}MP.
Quantum correlations forms a subset of the set of nonsignaling boxes. This allows us to characterize quantum correlations as a convex combination of the extremal boxes of the nonsignaling polytope which are Popescu-Rohrlich boxes (maximally nonlocal boxes) and local deterministic boxes. There exists multiple decomposition of quantum correlations in the context of the nonsignaling polytope. I find that the existence of Popescu-Rohrlich box decomposition for local boxes associates two notions of discord which capture nonclassicality of quantum correlations originating from Bell nonlocality and EPR-steering. I introduce, Bell and Mermin discord, and demonstrate that any bipartite nonsignaling box admits a three-way decomposition. This decomposition allows us to isolate the origin of nonclassicality into three disjoint sources: a Popescu-Rohrlich box, a maximally local box that detects EPR-steering, and a classical correlation. Interestingly, I show that all non-null quantum discord states which are neither classical-quantum states nor quantum-classical states can give rise to nonclassical correlations which have non-null Bell and/or Mermin discord for suitable noncommuting measurements. I introduce two notions of genuine discord, which are the generalizations of Bell and Mermin discord to the multipartite boxes, to characterize the presence of genuine nonclassicality in multipartite quantum correlations.
We develop tools for characterizing vertices of fiber products of polytopes and apply them to simplicial distribution polytopes, a class of probability polytopes arising in quantum foundations and quantum information. In the theory of simplicial distributions, a pair of simplicial sets encoding measurement and outcome spaces determines a convex polytope of compatible probability assignments. Our first results give geometric criteria for detecting vertices of fiber products in terms of support data. These results are obtained in the more general framework of inverse limits of diagrams of polytopes in standard form, and they translate to corresponding criteria for simplicial distributions on arbitrary colimits of measurement spaces. We then focus on one-dimensional measurement spaces, where simplicial distributions recover and generalize local marginal polytopes in graphical models. In this setting, our sharpest results concern dipole graphs, for which we obtain a complete characterization of vertices and refine it to a graph-theoretic criterion. These characterizations are reminiscent of the classical support-graph criteria for transportation polytopes, but they arise in a richer class of polytopes in which vertex structure depends not only on support acyclicity but also on additional geometric compatibility data. Using the collapsing method from simplicial topology, we transfer the dipole characterization to rose graphs and obtain analogous results there. Finally, we apply collapsing to complete bipartite graphs, which encode physically relevant bipartite Bell scenarios, and more generally to arbitrary connected graphs, yielding lower bounds on the number of vertices.
Kochen--Specker contextuality is a fundamental feature of quantum mechanics and a crucial resource for quantum computational advantage and reduction of communication complexity. Its presence is witnessed in empirical data by the violation of noncontextuality inequalities. However, all known noncontextuality inequalities corresponding to facets of noncontextual polytopes are either Bell inequalities or refer to cyclic or state-independent contextuality scenarios. We introduce a general method for lifting noncontextuality inequalities, deriving facets of noncontextual polytopes for more complex scenarios from known facets of simpler subscenarios. Concretely, starting from an arbitrary scenario, the addition of a new measurement or a new outcome preserves the facet-defining nature of any noncontextuality inequality. This extends the results of Pironio [J. Math. Phys. 46, 062112 (2005)] from Bell nonlocality scenarios to contextuality scenarios, unifying liftings of Bell and noncontextuality inequalities. Our method produces facet-defining noncontextuality inequalities in all scenarios with contextual correlations, and we present examples of facet-defining noncontextuality inequalities for scenarios where no examples were known. Our results shed light on the structure of noncontextuality polytopes and the relationship between such polytopes across different scenarios.
We investigate the Λ\LambdaΛ-polytopes, a convex-linear structure recently defined and applied to the classical simulation of quantum computation with magic states by sampling. There is one such polytope, Λn\Lambda_nΛn​, for every number nnn of qubits. We establish two properties of the family {Λn,n∈N}\{\Lambda_n, n\in \mathbb{N}\}{Λn​,n∈N}, namely (i) Any extremal point (vertex) Aα∈ΛmA_\alpha \in \Lambda_mAα​∈Λm​ can be used to construct vertices in Λn\Lambda_nΛn​, for all n>mn>mn>m. (ii) For vertices obtained through this mapping, the classical simulation of quantum computation with magic states can be efficiently reduced to the classical simulation based on the preimage AαA_\alphaAα​. In addition, we describe a new class of vertices in Λ2\Lambda_2Λ2​ which is outside the known classification. While the hardness of classical simulation remains an open problem for most extremal points of Λn\Lambda_nΛn​, the above results extend efficient classical simulation of quantum computations beyond the presently known range.