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Computational Companion to Information-Theoretic and Operational Measures of Quantum Contextuality

Ali Can Günhan, Z. Gedik

Published Jun 1, 2026• 0 citations
Publisher

Abstract

This volume is a computational and mathematical companion to the article "Information-Theoretic and Operational Measures of Quantum Contextuality". Rather than introducing new conceptual results, it provides complete derivations, independent cross-validations, and reproducible numerical implementations of every key result in the main article. Each quantity is established by two independent routes ---a direct matrix method using spin-1 operators, eigenvalue decompositions, and trace operations, and a geometric method based on the Majorana–Stellar representation on the Bloch–Poincaré sphere--- ensuring robustness against basis choices, degeneracies, and numerical artefacts. The treatment covers the KCBS geometry and observables, the mutual information energy, the operational contextuality measure D(G, ρ̂), the Majorana–Stellar formulation, and the associated uncertainty-optimization results, together with the accompanying Python reference implementation. It is intended for researchers and graduate students in quantum foundations, quantum information theory, and mathematical physics, assuming a working familiarity with finite-dimensional quantum mechanics. Published open access by Mersin University Press under a Creative Commons Attribution 4.0 International (CC BY 4.0) license.

DOI
10.5281/zenodo.20492278

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This volume is a computational and mathematical companion to the article "Information-Theoretic and Operational Measures of Quantum Contextuality". Rather than introducing new conceptual results, it provides complete derivations, independent cross-validations, and reproducible numerical implementations of every key result in the main article. Each quantity is established by two independent routes ---a direct matrix method using spin-1 operators, eigenvalue decompositions, and trace operations, and a geometric method based on the Majorana–Stellar representation on the Bloch–Poincaré sphere--- ensuring robustness against basis choices, degeneracies, and numerical artefacts. The treatment covers the KCBS geometry and observables, the mutual information energy, the operational contextuality measure D(G, ρ̂), the Majorana–Stellar formulation, and the associated uncertainty-optimization results, together with the accompanying Python reference implementation. It is intended for researchers and graduate students in quantum foundations, quantum information theory, and mathematical physics, assuming a working familiarity with finite-dimensional quantum mechanics. Published open access by Mersin University Press under a Creative Commons Attribution 4.0 International (CC BY 4.0) license.
We propose an information -- theoretic framework for quantifying Kochen-Specker contextuality. Two complementary measures are introduced: the mutual information energy, a state-independent quantity inspired by Onicescu's information energy that captures the geometric overlap between joint eigenspaces within a context; and an operational measure based on commutator expectation values that reflects contextual behavior at the level of measurement outcomes. We establish a hierarchy of bounds connecting these measures to the Robertson uncertainty relation, including spectral, purity-corrected, and operator norm estimates. The framework is applied to the Klyachko-Can-Binicioğlu-Shumovsky (KCBS) scenario for spin-1 systems, where all quantities admit closed-form expressions. The Majorana-stellar representation furnishes a common geometric platform on which both the operational measure and the uncertainty products can be analyzed. For spin-1, this representation yields a three-dimensional Euclidean-like visualization of the Hilbert space in which, states lying on a plane exhibit maximum uncertainty for the observable along the perpendicular direction; simultaneous optimization across all KCBS contexts singles out a unique state on the symmetry axis. Notably, states achieving the optimal sum of uncertainty products exhibit vanishing operational contextuality, while states with substantial operational contextuality satisfy a nontrivial Robertson bound -- the two extremes are achieved by distinct quantum states.
This largely self-contained book on the theory of quantum information focuses on precise mathematical formulations and proofs of fundamental facts that form the foundation of the subject. It is intended for graduate students and researchers in mathematics, computer science, and theoretical physics seeking to develop a thorough understanding of key results, proof techniques, and methodologies that are relevant to a wide range of research topics within the theory of quantum information and computation. The book is accessible to readers with an understanding of basic mathematics, including linear algebra, mathematical analysis, and probability theory. An introductory chapter summarizes these necessary mathematical prerequisites, and starting from this foundation, the book includes clear and complete proofs of all results it presents. Each subsequent chapter includes challenging exercises intended to help readers to develop their own skills for discovering proofs concerning the theory of quantum information.
This book introduces quantum computing and quantum communications at the undergraduate level for students in the physical sciences, engineering, and mathematics, assuming no prior knowledge of quantum mechanics. It is a self-contained guide assuming only that the reader is familiar with physics at the college level, calculus (up to and including ordinary differential equations), and some matrix algebra and complex numbers. The book brings the reader up to speed with fundamental concepts in quantum information processing and builds a working knowledge of basic quantum algorithms, quantum error correction, and quantum communication protocols. It covers various models of quantum computation and some of the most common physical realizations of qubits. There is a good number of practice problems and exercises that rely on computer programming with the Qiskit language. The book can be used to prepare students of physics, mathematics, electrical and computer engineering, computer science, optics and photonics, and mathematics for internships and research experiences in quantum information processing, both during and after their undergraduate studies. It also helps those who plan to apply to graduate school and do research in this area. Key Features: Self-contained coverage of quantum computing and quantum communications, from the motivations to the fundamentals and applications, including key concepts and contemporary topics. Contains practice problems and exercises, including some that require programming in Qiskit (the python-based, high-level language for programming quantum computers, adopted by nearly all quantum hardware providers and completely open source). Very light background knowledge assumed, making this book accessible to a variety of majors in the natural sciences, engineering, and mathematics.
This concise textbook introduces an innovative computational approach to quantum mechanics. Over the course of this engaging and informal book, students are encouraged to take an active role in learning key concepts by working through practical exercises. The book equips readers with some basic methodology and a toolbox of scientific computing methods, so they can use code to simulate and directly visualize how quantum particles behave. The important foundational elements of the wave function and the Schrödinger equation are first introduced, then the text gradually builds up to advanced topics including relativistic, open, and non-Hermitian quantum physics. This book assumes familiarity with basic mathematics and numerical methods, and can be used to support a two-semester advanced undergraduate course. Source code and solutions for every book exercise involving numerical implementation are provided in Python and MATLAB®, along with supplementary data. Additional problems are provided online for instructor use with locked solutions.
This concise textbook introduces an innovative computational approach to quantum mechanics. Over the course of this engaging and informal book, students are encouraged to take an active role in learning key concepts by working through practical exercises. The book equips readers with some basic methodology and a toolbox of scientific computing methods, so they can use code to simulate and directly visualize how quantum particles behave. The important foundational elements of the wave function and the Schrödinger equation are first introduced, then the text gradually builds up to advanced topics including relativistic, open, and non-Hermitian quantum physics. This book assumes familiarity with basic mathematics and numerical methods, and can be used to support a two-semester advanced undergraduate course. Source code and solutions for every book exercise involving numerical implementation are provided in Python and MATLAB®, along with supplementary data. Additional problems are provided online for instructor use with locked solutions.
This concise textbook introduces an innovative computational approach to quantum mechanics. Over the course of this engaging and informal book, students are encouraged to take an active role in learning key concepts by working through practical exercises. The book equips readers with some basic methodology and a toolbox of scientific computing methods, so they can use code to simulate and directly visualize how quantum particles behave. The important foundational elements of the wave function and the Schrödinger equation are first introduced, then the text gradually builds up to advanced topics including relativistic, open, and non-Hermitian quantum physics. This book assumes familiarity with basic mathematics and numerical methods, and can be used to support a two-semester advanced undergraduate course. Source code and solutions for every book exercise involving numerical implementation are provided in Python and MATLAB®, along with supplementary data. Additional problems are provided online for instructor use with locked solutions.