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Journal of Mathematical Physics

Some remarks on assignment maps

Fabio Masillo, G. Scolarici, Luigi Solombrino

Published Jan 1, 2011• 15 citations
Publisher

Abstract

We study the properties of general linear assignment maps, showing that positivity axiom can be suitably relaxed, and propose a new class of dynamical maps (generalized dynamics). A puzzling result, arising in such a context in quantum information theory, is also discussed.
DOI
10.1063/1.3525832

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Assignment maps are mathematical operators that describe initial system-environment states for open quantum systems. We re-examine the notion of assignments that account for correlations between the system and the environment and show that these maps can be made linear at the expense of giving up positivity or consistency of the map. We study the role of positivity and consistency of the map and show the effects of relaxing these. Finally, we establish a connection between the violation of the positivity of linear assignments and the no-broadcasting theorem.
The constraints imposed by the initial system-environment correlation can lead to nonpositive dynamical maps. We find the conditions for positivity and complete positivity of such dynamical maps by using the concept of an assignment map. Any initial system-environment correlations make the assignment map nonpositive, while the positivity of the dynamical map depends on the interplay between the assignment map and the system-environment coupling. We show how this interplay can reveal or hide the nonpositivity of the assignment map. We discuss the role of this interplay in Markovian models.
This paper develops a theory-neutral structural result concerning relational descriptions in physics. Many modern frameworks assign physical meaning through interactions rather than observer-independent intrinsic properties. This raises a general composition problem: if facts are generated locally within interaction contexts, what determines their coherent comparison across later chains or networks of interactions? Within a minimal framework of interacting systems, locally assigned relational facts, and maps between fact contexts, the paper establishes an interface necessity theorem: local fact assignments together with bare accessibility structure are not, in general, sufficient to determine unique cross-context composition outcomes whenever later comparison, inheritance, or aggregation of records is permitted. Some additional structure governing admissibility, translation, filtering, compatibility, or equivalence between relational fact spaces is generically required. This additional layer is termed interface structure and is used operationally rather than ontologically. No new force, hidden variable, or replacement dynamics is proposed. Instead, the result identifies a structural requirement linking local relationality to broader coherence. The paper discusses implications for relational quantum mechanics, decoherence-based narratives of record formation, Wigner-type multi-observer scenarios, and broader network descriptions of physical information. This work is related to the broader Horizons-as-Dynamical-Interface Framework (HDIF) program.