Algebraic and geometric properties of local transformations
- URL: http://arxiv.org/abs/2004.09405v1
- Date: Mon, 20 Apr 2020 16:01:31 GMT
- Title: Algebraic and geometric properties of local transformations
- Authors: Denis Rosset, \"Amin Baumeler, Jean-Daniel Bancal, Nicolas Gisin,
Anthony Martin, Marc-Olivier Renou, Elie Wolfe
- Abstract summary: Some properties of physical systems can be characterized from their correlations.
In this work, we single out the set of deterministic local maps as the one satisfying two equivalent constructions.
Surprisingly, the study of these fundamental properties has deep and practical applications.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- 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.
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