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Quantum Graphs: Different Perspectives

Dejan Dejan Stančević

Published Jul 4, 2024• 0 citations
OpenAlex

Abstract

In this thesis, we investigate two different notions of quantum graphs. The first approach is through quantum adjacency matrices, while, the second approach is through bimodules over finite-dimensional algebras. We establish the equivalence between these approaches, following the work by M. Daws [Daw24]. Along the way, we explore the role of quantum graphs as operator systems within the context of quantum information theory, serving as an extension of confusability graphs in classical information theory. Furthermore, we explore the concept of quantum isomorphism between quantum graphs. We use the standard definition of quantum isomorphism, defined via the quantum adjacency matrix approach, to introduce an equivalent notion in the bimodules approach (employing the equivalence between the approaches). The goal of this work is to present and contribute to the growing knowledge of quantum graphs and isomorphisms between them.

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quant-pharXiv:2604.12690•Apr 15, 2026
In this thesis, we investigate two different notions of quantum graphs. The first approach is through quantum adjacency matrices, while, the second approach is through bimodules over finite-dimensional algebras. We establish the equivalence between these approaches, following the work by M. Daws [Daw24]. Along the way, we explore the role of quantum graphs as operator systems within the context of quantum information theory, serving as an extension of confusability graphs in classical information theory. Furthermore, we explore the concept of quantum isomorphism between quantum graphs. We use the standard definition of quantum isomorphism, defined via the quantum adjacency matrix approach, to introduce an equivalent notion in the bimodules approach (employing the equivalence between the approaches). The goal of this work is to present and contribute to the growing knowledge of quantum graphs and isomorphisms between them.
We introduce an operator-algebraic framework for Morita equivalence of quantum graphs based on ΔΔΔ-equivalence of operator systems introduced by Eleftherakis, Kakariadis and Todorov. Adopting the perspective of Weaver, we view quantum graphs as quantum relations, that is, operator systems endowed with a bimodule structure over the commutant of a von Neumann algebra. Within this framework, we show that two irreducibly acting quantum graphs are Morita equivalent if and only if they are both full pullbacks of a common quantum graph. This extends a result of Eleftherakis, Kakariadis and Todorov for graph operator systems to the quantum graph setting. In passing we construct a true-twin reduction analogue for an irreducibly acting quantum graph. We further characterise the case where we have simultaneous TRO-equivalence of the quantum graphs and their associated algebras, thus giving a second, stronger notion of Morita equivalence. In the special case of noncommutative graphs, corresponding to the zero-error quantum communication setting, the two notions coincide and we obtain a characterisation in terms of strong co-homomorphisms of noncommutative graphs. Finally, we show that connectivity, the independence number, Shannon capacity, quantum complexity and subcomplexity, Haemers bound, and the Lovász number are invariant under Morita equivalence.
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.
In the past few decades, the connections between non-local games arising in quantum information theory and the theory of operator algebras have undergone a phase of significant development. Operator algebras provide a particularly fruitful framework for approaching questions of non-locality in quantum systems, as the input-output behavior of measurements on bipartite quantum systems can be encoded through noncommutative operator algebras and their state spaces. This thesis is the compilation of a series of papers, written by the author in collaboration with Ivan G. Todorov, utilizing this framework by using the theory of operator algebras and completely positive maps to study questions involving non-local games. ☐ Using the simulation paradigm from information theory, we define a quantized version of homomorphisms and isomorphisms between classical hypergraphs, generalizing quantum homomorphisms and isomorphisms of graphs from the literature. We show that these quantum homomorphisms and isomorphisms constitute pre-orders and equivalence relations, respectively. We use quantum homomorphisms of hypergraphs to provide multiple examples of strict separation between correlation classes of varying types. Specializing to the case when our underlying hypergraphs arise from classical non-local games, we define quantum non-local game homomorphisms and isomorphisms. We also show how the existence of a homomorphism or isomorphism of some fixed type between games is reflected in a comparison of optimal and asymptotic game values with respect to this type. These non-local game homomorphisms are defined via a new class of non-signalling correlations which we introduce here. We develop the multivariate tensor product theory in the category of operator systems, which we then use to provide a characterization of game homomorphisms in terms of state spaces for various tensor products of canonically associated operator systems. We then apply these results to the study of synchronous non-local games, defining jointly synchronous correlations and showing how they correspond to tracial states of tensor products of C*-algebras canonically associated to each game party. ☐ We then move to a second level of quantization by proposing a "quantized" notion of hypergraphs (where "quantum" here is in the sense of a non-commutative analogue for a discrete combinatorial object), and introduce quantum homomorphisms between these quantum objects. We provide an initial foray into the properties quantum morphisms between quantum hypergraphs display, showing they satisfy analogous properties to quantum homomorphisms between classical hypergraphs. We also indicate initial connections these quantum homomorphisms between quantum hypergraphs have to the study of quantum input-output games. We end the thesis showing that homomorphisms of a local type between quantum hypergraphs is closely related, and in some cases identical, to the TRO equivalence of finite-dimensionally acting operator spaces canonically associated with each hypergraph. This suggests a quantum information theoretic approach to Morita equivalence in the category of operator spaces.
Quantum non-local games with quantum inputs or outputs have been formulated in several different languages, including rank-one and quantum XOR games, support maps between projection lattices, probabilistic quantum hypergraphs, and Frobenius-algebraic rule operators on finite quantum sets. We give a unified operator-algebraic comparison of these models by assigning to each rule its winning transformation space: the operator space of transformations accepted with certainty. We introduce R\mathcal RR-projection-test quantum games with finite-dimensional input and output von Neumann algebras and an arbitrary, possibly infinite-dimensional, referee von Neumann algebra R\mathcal RR. Their winning transformation spaces are precisely operator spaces with a natural bimodule structure, equivalently rectangular quantum relations. We compare this formalism with projection-lattice games, hypergraph quantum games, and the graphical rule-operator definition. Projection-lattice games capture exactly the reflexive winning bimodules. Hypergraph quantum games admit value-preserving projection-test realisations, and, after passing to perfect transformations, describe the same reflexive part as projection-lattice games. Using Daws' technique, we also translate rule operators to projections in tensor products of finite-dimensional von Neumann algebras. This identification places graphical rules in the same operator-bimodule framework and preserves both values and perfectness. Finally, we compare concurrency with the synchronicity conditions of Goldberg and of Bochniak--Kasprzak--Sołtan. The framework is illustrated by classical, rank-one, quantum XOR, colouring, and quantum graph homomorphism and isomorphism games.
The graph isomorphism (GI) problem is the computational problem of finding a permutation of vertices of a given graph G1G_1G1​ that transforms G1G_1G1​ to another given graph G2G_2G2​ and preserves the adjacency. In this work, we propose a quantum algorithm to determine whether there exists such a permutation. To find such a permutation, we introduce isomorphic equivalent graphs of the given graphs to be tested. We proof that the GI problem of the equivalent graphs is equivalent to the GI problem of the given graphs. The idea of the algorithm is to determine whether there exists a permutation can transform the eigenvectors of the adjacency matrix of the equivalent graphs each other. The cost time of the algorithm is polynomial.