arXiv subjects

Mirko Navara

Publications and source records attributed to Mirko Navara.

9 recordsLinked to original sources

arXiv2026

Pseudocontexts forced by finite context hypergraphs

A context is a complete set of mutually exclusive outcomes whose probabilities sum to one. We define a pseudocontext as two disjoint groups, with no mutually exclusive pair within either group, whose total probabilities must nevertheless be equal solely because of how the contexts overlap. We give an exact finite test for this property and show that the equality is independent of the chosen coordinates and probability model. Applying the test in three dimensions, we obtain a 15-outcome real example with groups of three and a 20-outcome complex example with groups of two. Under this definition, the sharp minimum number of outcomes in each target group is two over the complex field and three over the real field.

quant-ph↗

arXiv2025

Construction of Kochen-Specker Sets from Mutually Unbiased Bases

We give a constructive analysis of Kochen-Specker (KS) contextuality at the level of complete orthogonal bases (contexts). In dimension three, three noncomputational mutually unbiased bases generate a union of 165 rays and 130 contexts. Exact provenance accounting gives a symmetric decomposition: the three 69-ray lineages share $21$ rays and $10$ contexts, while each contributes 48 exclusive rays and $40$ lineage-specific contexts. Each 69-ray, 50-context lineage is a KS configuration, and this framework clarifies the relation among the Yu-Oh, Harding-Salinas Schmeis, and Cabello constructions. In dimensions four and five, we give parametrized orthogonality gadgets whose connector contexts force a central ray to be maximally unbiased relative to the computational basis. For the concrete four-dimensional example, the 20-ray gadget and the Cabello 18-ray set are informationally equivalent subsets of the 24-ray Peres-Mermin eigensystem, although their context hypergraphs have different two-valued-state behavior. The results show why both the rays and the complete incidence structure of their contexts are needed in assessments of KS contextuality.

quant-ph↗

arXiv2024

Exploring Quantum Contextuality with the Quantum Moebius-Escher-Penrose hypergraph

This paper presents the quantum Moebius-Escher-Penrose hypergraph, drawing inspiration from paradoxical constructs such as the Moebius strip and Penrose's `impossible objects'. The hypergraph is constructed using faithful orthogonal representations in Hilbert space, thereby embedding the graph within a quantum framework. Additionally, a quasi-classical realization is achieved through two-valued states and partition logic, leading to an embedding within a Boolean algebra. This dual representation delineates the distinctions between classical and quantum embeddings, with a particular focus on contextuality, highlighted by violations of exclusivity and completeness, quantified through classical and quantum probabilities. The study also examines violations of Boole's conditions of possible experience using correlation polytopes, underscoring the inherent contextuality of the hypergraph. These results offer deeper insights into quantum contextuality and its intricate relationship with classical logic structures.

quant-ph↗

arXiv2024

Maximum Likelihood Estimators of Quantum Probabilities

Classical probability theory is based on assumptions which are often violated in practice. Therefore quantum probability is a proposed alternative not only in quantum physics, but also in other sciences. However, so far it mostly criticizes the classical approach, but does not suggest a working alternative. Maximum likelihood estimators were given very low attention in this context. We show that they can be correctly defined and their computation in closed form is feasible at least in some cases.

quant-ph↗

arXiv2023

Form of Contextuality Predicting Probabilistic Equivalence between Two Sets of Three Mutually Noncommuting Observables

We introduce a contextual quantum system comprising mutually complementary observables organized into two or more collections of pseudocontexts with the same probability sums of outcomes. These pseudocontexts constitute non-orthogonal bases within the Hilbert space, featuring a state-independent sum of probabilities. In other words, regardless of the initial state preparation, the total probability remains constant but may be distinct from unity. The measurement contextuality in this setup arises from the quantum realizations of the hypergraph, which adhere to a specific bound on the linear combination of probabilities. In contrast, classical realizations can surpass this bound. The violation of quantum bounds stems from the inability of classical ontological models, specifically the set-theoretic representation of the hypergraph corresponding to the quantum observables' collections, to adhere to and explain the observed statistics.

quant-ph↗

arXiv2021

Quantum logics close to Boolean algebras

We consider orthomodular posets endowed with a symmetric difference. We call them ODPs. Expressed in the quantum logic language, we consider quantum logics with an XOR-type connective. We study three classes of "almost Boolean" ODPs, two of them defined by requiring rather specific behaviour of infima and the third by a Boolean-like behaviour of Frink ideals. We establish a (rather surprising) inclusion between the three classes, shadding thus light on their intrinsic properties. (More details can be found in the Introduction that follows.) Let us only note that the orthomodular posets pursued here, though close to Boolean algebras (i.e., close to standard quantum logics), still have a potential for an arbitrarily high degree of non-compatibility and hence they may enrich the studies of mathematical foundations of quantum mechanics.

quant-ph↗

arXiv2017

Boolean subalgebras of orthoalgebras

We develop a direct method to recover an orthoalgebra from its poset of Boolean subalgebras. For this a new notion of direction is introduced. Directions are also used to characterize in purely order-theoretic terms those posets that are isomorphic to the poset of Boolean subalgebras of an orthoalgebra. These posets are characterized by simple conditions defining orthodomains and the additional requirement of having enough directions. Excepting pathologies involving maximal Boolean subalgebras of four elements, it is shown that there is an equivalence between the category of orthoalgebras and the category of orthodomains with enough directions with morphisms suitably defined. Furthermore, we develop a representation of orthodomains with enough directions, and hence of orthoalgebras, as certain hypergraphs. This hypergraph approach extends the technique of Greechie diagrams and resembles projective geometry. Using such hypergraphs, every orthomodular poset can be represented by a set of points and lines where each line contains exactly three points.

math.QA↗

arXiv2010

Subalgebras of orthomodular lattices

Sachs showed that a Boolean algebra is determined by its lattice of subalgebras. We establish the corresponding result for orthomodular lattices. We show that an orthomodular lattice L is determined by its lattice of subalgebras Sub(L), as well as by its poset of Boolean subalgebras BSub(L). The domain BSub(L) has recently found use in an approach to the foundations of quantum mechanics initiated by Butterfield and Isham, at least in the case where L is the orthomodular lattice of projections of a Hilbert space, or von Neumann algebra. The results here may add some additional perspective to this line of work.

math-ph↗

arXiv2004

Extending states on finite concrete logics

In this note we collect several observations on state extensions. They may be instrumental to anyone who pursues the theory of quantum logics. In particular, we find out when extensions (resp. signed extensions) exist in the "concrete" concrete logic of all even-element subsets of an even-element set. We also mildly add to the study of difference-closed logics by finding an extension theorem for subadditive states.

math-ph↗