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Article History

Received: 23 October 2020

Accepted: 15 March 2024

First Online: 28 May 2024

Declarations

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: After the paper was completed, we became aware of the work [], in which the authors define quantum-to-classical no-signalling correlations and study a version of the homomorphism game from a non-commutative to a classical graph. Although these can be viewed as dual to our classical-to-quantum correlations (Sect. ), there is no direct duplication of results in the current paper with those in []. <b>(ii) </b> The article [], published before the acceptance of the present paper, studies in detail quantum graph homomorphisms of different types, providing operator-algebraic characterisations thereof. In particular, in [, Theorem 5.5], we obtain a version of Propositionfor homomorphisms of quantum type, showing that the existence of such a homomorphism is equivalent to the existence of a local homomorphism from a quantum graph ampliated by a maximally entangled state. This implies that homomorphisms of quantum type constitute a special class of the entanglement-assisted non-commutative graph homomorphisms defined in [, Definition 15]. The identification of an operational characterisation of the latter notion remains an open problem.