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Quantumness via Discrete Structures

Ravi Kunjwal

TL;DR

The work develops a program to quantify quantum nonclassicality using discrete combinatorial structures—graphs and hypergraphs—across three fronts: contextuality (including KS/contextuality and generalized contextuality), causality (indefinite causal order via process matrices), and measurement incompatibility (joint measurability structures). It introduces hypergraph-based frameworks (CSW/AFLS) and new invariants (weighted max-predictability, Corr) to produce noise-robust witnesses and connect KS-contextuality to quantum communication tasks. It further elucidates the interplay between contextuality and entanglement, clarifies Gleason's and KS theorems in multiparty settings, and formalizes a hierarchy of correlation sets under definite and indefinite causal order, including antinomicity and LO-PF. The results collectively advance principled, operational criteria for quantumness and supply constructive methods to realize and distinguish quantum from almost-quantum correlations in single and multipartite scenarios, with implications for quantum computation and communication. The thesis hence provides a coherent, graph-theoretic lens on quantumness that spans foundational insights and practical information-processing tasks.

Abstract

Quantum theory departs from classical probabilistic theories in foundational ways. These departures--termed quantumness here--power quantum information and computation. This thesis charts the role of discrete structures in assessing quantumness, synthesizing elements of my postdoctoral research through this lens. After an introduction to the necessary background concepts, I present my work under three broad categories. First, I present work on contextuality that extensively relies on (undirected) graphs and hypergraphs as the discrete structures of interest; more specifically, it relies on invariants associated with them. This work includes Kochen-Specker (KS) contextuality and its operationalization to generalized contextuality, expressed via (hyper)graph-theoretic frameworks. I also present work on KS-contextuality in multiqubit systems and an application of generalized contextuality to a one-shot communication task, both of which rely on hypergraphs. Second, I present work on causality, where the discrete structures of interest are directed graphs. This includes work on indefinite causal order, specifically its connections to the gap between local operations and classical communication (LOCC) and separable operations (SEP), and a device-independent notion of nonclassicality--termed antinomicity--that generalizes Bell nonlocality without global causal assumptions. Finally, I present work on the incompatibility of quantum measurements, its connection to Bell nonlocality, and its role in discriminating between quantum and almost quantum correlations in the single-system setting. The discrete structures of interest here are hypergraphs that model joint measurability relations between quantum measurements. I conclude with a summary and an overview of work that is not covered in this thesis.

Quantumness via Discrete Structures

TL;DR

The work develops a program to quantify quantum nonclassicality using discrete combinatorial structures—graphs and hypergraphs—across three fronts: contextuality (including KS/contextuality and generalized contextuality), causality (indefinite causal order via process matrices), and measurement incompatibility (joint measurability structures). It introduces hypergraph-based frameworks (CSW/AFLS) and new invariants (weighted max-predictability, Corr) to produce noise-robust witnesses and connect KS-contextuality to quantum communication tasks. It further elucidates the interplay between contextuality and entanglement, clarifies Gleason's and KS theorems in multiparty settings, and formalizes a hierarchy of correlation sets under definite and indefinite causal order, including antinomicity and LO-PF. The results collectively advance principled, operational criteria for quantumness and supply constructive methods to realize and distinguish quantum from almost-quantum correlations in single and multipartite scenarios, with implications for quantum computation and communication. The thesis hence provides a coherent, graph-theoretic lens on quantumness that spans foundational insights and practical information-processing tasks.

Abstract

Quantum theory departs from classical probabilistic theories in foundational ways. These departures--termed quantumness here--power quantum information and computation. This thesis charts the role of discrete structures in assessing quantumness, synthesizing elements of my postdoctoral research through this lens. After an introduction to the necessary background concepts, I present my work under three broad categories. First, I present work on contextuality that extensively relies on (undirected) graphs and hypergraphs as the discrete structures of interest; more specifically, it relies on invariants associated with them. This work includes Kochen-Specker (KS) contextuality and its operationalization to generalized contextuality, expressed via (hyper)graph-theoretic frameworks. I also present work on KS-contextuality in multiqubit systems and an application of generalized contextuality to a one-shot communication task, both of which rely on hypergraphs. Second, I present work on causality, where the discrete structures of interest are directed graphs. This includes work on indefinite causal order, specifically its connections to the gap between local operations and classical communication (LOCC) and separable operations (SEP), and a device-independent notion of nonclassicality--termed antinomicity--that generalizes Bell nonlocality without global causal assumptions. Finally, I present work on the incompatibility of quantum measurements, its connection to Bell nonlocality, and its role in discriminating between quantum and almost quantum correlations in the single-system setting. The discrete structures of interest here are hypergraphs that model joint measurability relations between quantum measurements. I conclude with a summary and an overview of work that is not covered in this thesis.

Paper Structure

This paper contains 66 sections, 8 theorems, 73 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Background
  3. The mathematics of quantum theory
  4. What is an operational theory?
  5. What is operational quantum theory?
  6. Dirac notation
  7. Preparations, measurements, transformations
  8. The plan
  9. Graphs and hypergraphs for contextuality
  10. Introduction
  11. Generalized contextuality
  12. Kochen-Specker contextuality
  13. The CSW and AFLS frameworks
  14. A logical proof of the KS theorem
  15. A statistical proof of the KS theorem
  16. ...and 51 more sections

Key Result

Theorem 2.1

Given a separable Hilbert space $\mathcal{H}$ of dimension at least three, there does not exist a map such that for any complete set of rays $\{| \psi_1 \rangle,| \psi_2 \rangle,\ldots\}$, we have $c(| \psi_j \rangle)=1$ for exactly one value of $j\in\{1,2,\ldots\}$.Such a map $c$ is also called a KS-colouring, with the "colours" referring to the values $0$ and $1$.

Figures (11)

  • Figure 1: Schematic of a multi-instrument: the task of an operational theory is to describe its behaviour.
  • Figure 2: A prepare-and-measure experiment.
  • Figure 3: The $18$-ray contextuality scenario, $\Gamma_{18}$, used in Ref. CEGA96.
  • Figure 4: The KCBS contextuality scenario, $\Gamma_5$KCBS.
  • Figure 5: The KCBS construction KCBS.
  • ...and 6 more figures

Theorems & Definitions (16)

  • Theorem 2.1: Kochen--Specker KS67
  • Definition 2.1: Consistent Exclusivity.
  • Definition 2.2: Independence number.
  • Definition 2.3: Lovasz theta number.
  • Definition 2.4: Fractional packing number.
  • Theorem 3.1: Gleason Gleason57
  • Theorem : Kochen--Specker KS67
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • ...and 6 more