arXiv subjects

Alexandros Chatzinikolaou

Publications and source records attributed to Alexandros Chatzinikolaou.

8 recordsLinked to original sources

arXiv2026

The hypergraph isomorphism game, Hopf algebras and Galois extensions

We develop an algebraic and operational framework for quantum isomorphisms of hypergraphs, using tools from compact quantum group theory. We introduce a new synchronous version of the hypergraph isomorphism game whose game algebra uniformly encodes multiple notions of quantum isomorphisms of hypergraphs. We show that there exist hypergraphs that are quantum isomorphic but not classically isomorphic. For graphs, we show that the $*$-algebra of the hypergraph isomorphism game is a quotient of the $*$-algebra of the graph isomorphism game. We further prove that the hypergraph game algebra forms a bi-Galois extension over the quantum automorphism groups of the underlying hypergraphs. This allows us to deduce that the algebraic notion of a quantum isomorphism of hypergraphs coincides with the operational one coming from the existence of perfect quantum strategies. Viewing games themselves as hypergraphs, we analyze isomorphisms and the transfer of strategies within this setting. Finally, we construct a $*$-algebra whose representation theory characterizes distinct classes of quantum isomorphisms between non-local games.

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arXiv2026

Quantum non-local games: Quantum relations, projection lattices and rule operators

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 $\mathcal R$-projection-test quantum games with finite-dimensional input and output von Neumann algebras and an arbitrary, possibly infinite-dimensional, referee von Neumann algebra $\mathcal R$. 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.

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arXiv2026

Absolutely dilatable bimodule maps

We characterise absolutely dilatable completely positive maps on the space of all bounded operators on a Hilbert space that are also bimodular over a given von Neumann algebra as rotations by a suitable unitary on a larger Hilbert space followed by slicing along the trace of an additional ancilla. We define the local, quantum and approximately quantum types of absolutely dilatable maps, according to the type of the admissible ancilla. We show that the local absolutely dilatable maps admit an exact factorisation through an abelian ancilla and show that they are limits in the point weak* topology of conjugations by unitaries in the commutant of the given von Neumann algebra. We show that the Connes Embedding Problem is equivalent to deciding if all absolutely dilatable maps are approximately quantum.

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arXiv2026

Morita equivalence for quantum graphs

We introduce an operator-algebraic framework for Morita equivalence of quantum graphs based on $Δ$-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.

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arXiv2026

On the embeddings of selfadjoint operator spaces

We investigate when a map on a selfadjoint operator space $E$ is an embedding, i.e., when its unitisation in the sense of Werner is completely isometric. Combining with results of Russell, of Ng, and of Dessi, the second and the last author, it is shown that this is equivalent to: (a) extending bounded positive functionals on each matrix level with the same norm; (b) extending quasistates to quasistates in each matrix level; (c) extending completely bounded completely positive maps with the same cb-norm; and (d) the map being a gauge maximal isometry in the sense of Russell. If $E$ is approximately positively generated and $\mathrm{C}^*(E)$ is unital, or if $E_{sa}$ is singly generated, then completely positive maps on $E \subseteq \mathcal{B}(H)$ have completely positive extensions on $\mathrm{C}^*(E)$, but possibly not with the same cb-norm; and this is not enough for the inclusion $E \subseteq \mathrm{C}^*(E)$ to be an embedding. We show that the inclusion $E \subseteq \mathrm{C}^*(E)$ is always an embedding when $E$ is completely approximately 1-generated, and we fully resolve the case when $E_{sa}$ is singly generated. Combining with the works of Salomon, Humeniuk--Kennedy--Manor, and previous work of the third author, we show that if the inclusion $E \subseteq \mathrm{C}^*(E)$ is an embedding, then rigidity at zero, in the sense of Salomon, coincides with $E$ being approximately positively generated. Consequently, we show that $E$ is approximately positively generated if and only if $M_n(E)$ is approximately positively generated for all $n\in \mathbb{N}$, thus extending a previous result of Humeniuk--Kennedy--Manor to the approximation setting. As an application we show that hyperrigidity of $E$ in $\mathrm{C}^*(E)$ allows to identify $\mathrm{C}^*(E)$ as the C*-envelope of $E$ in several (non-unital) contexts.

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arXiv2025

Cantor correlations I. Operator systems and Cantor games

We study no-signalling correlations over Cantor spaces, placing the product of infinitely many copies of a finite non-local game in a unified general setup. We define the subclasses of local, quantum spatial, approximately quantum and quantum commuting Cantor correlations and describe them in terms of states on tensor products of inductive limits of operator systems. We provide a correspondence between no-signalling (resp. approximately quantum, quantum commuting) Cantor correlations and sequences of correlations of the same type over the projections onto increasing number of finitely many coordinates. We introduce Cantor games, and associate canonically such a game to a sequence of finite input/output games, showing that the numerical sequence of the values of the games in the sequence converges to the corresponding value of the compound Cantor game.

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arXiv2025

Operator systems, contextuality and non-locality

We introduce an operator system, universal for the probabilistic models of a contextuality scenario, and identify its maximal C*-cover via the right C*-algebra of a canonical ternary ring of operators, arising from a hypergraph version of stochastic operator matrices. We study dilating contextuality scenarios, which have the property that each positive operator representation thereof admits a dilation to a projective representation on a larger Hilbert space, and characterise them via the equality of the aforementioned universal operator system and the operator system arising from the canonical generators of the respective hypergraph C*-algebra. We characterise the no-signalling probabilistic models over a pair of contextuality scenarios of different types, which arise from either the positive operator representations or from the projective representations of these scenarios, in terms of states on operator system tensor products. Generalising the notion of a synchronous no-signalling correlation to the hypergraph framework, we define coherent probabilistic models associated with a given contextuality scenario and characterise various classes thereof via different types of traces of the hypergraph C*-algebra, associated with the scenario. We establish several equivalent formulations of the Connes Embedding Problem in terms of no-signalling probabilistic models and hypergraph operator systems.

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arXiv2025

On coproducts of operator $\mathcal{A}$-systems

Given a unital $\boldsymbol{C}^{*}$-algebra $\mathcal{A}$, we prove the existence of the coproduct of two faithful operator $\mathcal{A}$-systems. We show that we can either consider it as a subsystem of an amalgamated free product of $\boldsymbol{C}^{*}$-algebras, or as a quotient by an operator system kernel. We introduce a universal $\boldsymbol{C}^{*}$-algebra for operator $\mathcal{A}$-systems and prove that in the case of the coproduct of two operator $\mathcal{A}$-systems, it is isomorphic to the amalgamated over $\mathcal{A}$, free product of their respective universal $\boldsymbol{C}^{*}$-algebras. Also, under the assumptions of hyperrigidity for operator systems, we can identify the $\boldsymbol{C}^{*}$-envelope of the coproduct with the amalgamated free product of the $\boldsymbol{C}^{*}$-envelopes. We consider graph operator systems as examples of operator $\mathcal{A}$-systems and prove that there exist graph operator systems whose coproduct is not a graph operator system, it is however a dual operator $\mathcal{A}$-system. More generally, the coproduct of dual operator $\mathcal{A}$-systems is always a dual operator $\mathcal{A}$-system. We show that the coproducts behave well with respect to inductive limits of operator systems.

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