Algebraic and geometric properties of local transformations

TL;DR

This paper explores the algebraic and geometric properties of local transformations in physical systems, revealing their role in decomposing correlations and classifying inequalities in quantum and causal scenarios.

Contribution

It characterizes deterministic local maps through operational and axiomatic definitions, and studies their algebraic properties with practical applications in quantum information theory.

Findings

Decomposition of correlation space into nonsignaling, signaling, and normalization components.

Invariant subspaces under local transformations aid in classifying Bell and causal inequalities.

Liftings of correlation boxes extend to causal and Bell inequalities, enhancing their analysis.

Abstract

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.