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.

Related papers

• Gauge-Fixing Quantum Density Operators At Scale [0.0]
  We provide theory, algorithms, and simulations of non-equilibrium quantum systems. We analytically and numerically examine the virtual freedoms associated with the representation of quantum density operators.
  arXiv  Detail & Related papers  (2024-11-05T22:56:13Z)
• Unsupervised Representation Learning from Sparse Transformation Analysis [79.94858534887801]
  We propose to learn representations from sequence data by factorizing the transformations of the latent variables into sparse components. Input data are first encoded as distributions of latent activations and subsequently transformed using a probability flow model.
  arXiv  Detail & Related papers  (2024-10-07T23:53:25Z)
• A Relationship Between Nonphysical Quasi-probabilities and Nonlocality Objectivity [0.0]
  We show that breaking the positive-semidefinite requirement allows for the construction of states that are nonlocal under one component labelling but local when the labelling is interchanged. This is an observer-dependent nonlocality, showing the connection between nonlocal objectivism and negative quasi-probabilities.
  arXiv  Detail & Related papers  (2024-07-26T19:39:21Z)
• Interval Effect Algebras and Holevo Instruments [0.0]
  This article begins with a study of convex effect-state spaces. We point out that such spaces are equivalent to interval effect algebras that generate an ordered linear space and possess an order-determining set of states. We then discuss operations and instruments on interval effect algebras under the assumption of an unrestrictive condition.
  arXiv  Detail & Related papers  (2024-03-27T18:16:24Z)
• Nonparametric Partial Disentanglement via Mechanism Sparsity: Sparse Actions, Interventions and Sparse Temporal Dependencies [58.179981892921056]
  This work introduces a novel principle for disentanglement we call mechanism sparsity regularization. We propose a representation learning method that induces disentanglement by simultaneously learning the latent factors. We show that the latent factors can be recovered by regularizing the learned causal graph to be sparse.
  arXiv  Detail & Related papers  (2024-01-10T02:38:21Z)
• Showcasing a Barren Plateau Theory Beyond the Dynamical Lie Algebra [0.0]
  Barren plateaus have emerged as a pivotal challenge for variational quantum computing. We provide an exact formula for the loss function variance that is valid for arbitrary input states and measurements. While parameterized matchgate circuits are not efficiently simulable in general, our results suggest that the structure allowing for trainability may also lead to classical simulability.
  arXiv  Detail & Related papers  (2023-10-17T18:04:43Z)
• Bell inequalities with overlapping measurements [52.81011822909395]
  We study Bell inequalities where measurements of different parties can have overlap. This allows to accommodate problems in quantum information. The scenarios considered show an interesting behaviour with respect to Hilbert space dimension, overlap, and symmetry.
  arXiv  Detail & Related papers  (2023-03-03T18:11:05Z)
• Equivariant Disentangled Transformation for Domain Generalization under Combination Shift [91.38796390449504]
  Combinations of domains and labels are not observed during training but appear in the test environment. We provide a unique formulation of the combination shift problem based on the concepts of homomorphism, equivariance, and a refined definition of disentanglement.
  arXiv  Detail & Related papers  (2022-08-03T12:31:31Z)
• Projective characterization of higher-order quantum transformations [0.0]
  This work presents a framework for characterizing higher-order quantum transformations using superoperator projectors. The main novelty of this work is the introduction in the algebra of the 'prec' connector. This allows to assess the possible signaling structure of any maps characterized within the projective framework.
  arXiv  Detail & Related papers  (2022-06-13T14:36:28Z)
• Eigen Analysis of Self-Attention and its Reconstruction from Partial Computation [58.80806716024701]
  We study the global structure of attention scores computed using dot-product based self-attention. We find that most of the variation among attention scores lie in a low-dimensional eigenspace. We propose to compute scores only for a partial subset of token pairs, and use them to estimate scores for the remaining pairs.
  arXiv  Detail & Related papers  (2021-06-16T14:38:42Z)
• Exponentially Weighted l_2 Regularization Strategy in Constructing Reinforced Second-order Fuzzy Rule-based Model [72.57056258027336]
  In the conventional Takagi-Sugeno-Kang (TSK)-type fuzzy models, constant or linear functions are usually utilized as the consequent parts of the fuzzy rules. We introduce an exponential weight approach inspired by the weight function theory encountered in harmonic analysis.
  arXiv  Detail & Related papers  (2020-07-02T15:42:15Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.