2020/04/20 by Denis Rosset, Ämin Baumeler, Rosset, Denis +11
Computer Science · Mathematics · Physics and Astronomy · #Algebra over a field #Algebraic number #Computer science #Discrete mathematics #FOS: Physical sciences #Generalization #Invariant (physics) #Invertible matrix #Mathematical proof #Mathematics #Nondeterministic algorithm #Pure mathematics #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #Set (abstract data type) #Theoretical computer science #quant-ph
paper · pdf · doi:10.48550/arxiv.2004.09405
46 pages split in three independent parts (8+21+11). Numerous figures
arxiv created 2020/04/20 · openalex publication_date 2020/04/20 · arxiv updated 2020/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08
Some properties of physical systems can be characterized from their correlations. In that framework, subsystems are viewed as abstract devices that receive measurement settings as inputs and produce measurement outcomes as outputs. The labeling convention used to describe these inputs and outputs does not affect the physics; and relabelings are easily implemented by rewiring the input and output ports of the devices. However, a more general class of operations can be achieved by using correlated preprocessing and postprocessing of the inputs and outputs. In contrast to relabelings, some of these operations irreversibly lose information about the underlying device. Other operations are reversible, but modify the number of cardinality of inputs and/or outputs. In this work, we single out the set of deterministic local maps as the one satisfying two equivalent constructions: an operational definition from causality, and an axiomatic definition reminiscent of the definition of quantum completely positive trace-preserving maps. We then study the algebraic properties of that set. Surprisingly, the study of these fundamental properties has deep and practical applications. First, the invariant subspaces of these transformations directly decompose the space of correlations/Bell inequalities into nonsignaling, signaling and normalization components. This impacts the classification of Bell and causal inequalities, and the construction of assemblages/witnesses in steering scenarios. Second, the left and right invertible deterministic local operations provide an operational generalization of the liftings introduced by Pironio [J. Math. Phys., 46(6):062112 (2005)]. Not only Bell-local, but also causal inequalities can be lifted; liftings also apply to correlation boxes in a variety of scenarios.