The signaling dimension in generalized probabilistic theories

• URL: http://arxiv.org/abs/2311.13103v2
• Date: Mon, 1 Jul 2024 04:39:37 GMT
• Title: The signaling dimension in generalized probabilistic theories
• Authors: Michele Dall'Arno, Alessandro Tosini, Francesco Buscemi,
• Abstract summary: The signaling dimension of a given physical system quantifies the minimum dimension of a classical system required to reproduce all input/output correlations of the given system. We show that it suffices to consider extremal measurements with rayextremal effects, and we bound the number of elements of any such measurement in terms of the linear dimension. For systems with a finite number of extremal effects, we recast the problem of characterizing the extremal measurements with ray-extremal effects.
• Score: 48.99818550820575
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The signaling dimension of a given physical system quantifies the minimum dimension of a classical system required to reproduce all input/output correlations of the given system. Thus, unlike other dimension measures - such as the dimension of the linear space or the maximum number of (jointly or pairwise) perfectly discriminable states - which examine the correlation space only along a single direction, the signaling dimension does not depend on the arbitrary choice of a specific operational task. In this sense, the signaling dimension summarizes the structure of the entire set of input/output correlations consistent with a given system in a single scalar quantity. For quantum theory, it was recently proved by Frenkel and Weiner in a seminal result that the signaling dimension coincides with the Hilbert space dimension. Here, we derive analytical and algorithmic techniques to compute the signaling dimension for any given system of any given generalized probabilistic theory. We prove that it suffices to consider extremal measurements with ray-extremal effects, and we bound the number of elements of any such measurement in terms of the linear dimension. For systems with a finite number of extremal effects, we recast the problem of characterizing the extremal measurements with ray-extremal effects as the problem of deriving the vertex description of a polytope given its face description, which can be conveniently solved by standard techniques. For each such measurement, we recast the computation of the signaling dimension as a linear program, and we propose a combinatorial branch and bound algorithm to reduce its size. We apply our results to derive the extremal measurements with ray-extremal effects of a composition of two square bits (or squits) and prove that their signaling dimension is five, even though each squit has a signaling dimension equal to two.

Related papers

• The signaling dimension of two-dimensional and polytopic systems [2.4554686192257424]
  The signaling dimension of any given physical system represents its classical simulation cost. We propose a branch and bound division-free algorithm for the exact computation of the symmetries of any given polytope. We apply our algorithm to compute the exact value of the signaling dimension for all rational Platonic, Archimedean, and Catalan solids.
  arXiv  Detail & Related papers  (2024-07-25T02:54:21Z)
• Gaussian Entanglement Measure: Applications to Multipartite Entanglement of Graph States and Bosonic Field Theory [50.24983453990065]
  An entanglement measure based on the Fubini-Study metric has been recently introduced by Cocchiarella and co-workers. We present the Gaussian Entanglement Measure (GEM), a generalization of geometric entanglement measure for multimode Gaussian states. By providing a computable multipartite entanglement measure for systems with a large number of degrees of freedom, we show that our definition can be used to obtain insights into a free bosonic field theory.
  arXiv  Detail & Related papers  (2024-01-31T15:50:50Z)
• Normalizing flows for lattice gauge theory in arbitrary space-time dimension [135.04925500053622]
  Applications of normalizing flows to the sampling of field configurations in lattice gauge theory have so far been explored almost exclusively in two space-time dimensions. We discuss masked autoregressive with tractable and unbiased Jacobian determinants, a key ingredient for scalable and exact flow-based sampling algorithms. For concreteness, results from a proof-of-principle application to SU(3) gauge theory in four space-time dimensions are reported.
  arXiv  Detail & Related papers  (2023-05-03T19:54:04Z)
• The signaling dimension of physical systems [2.28438857884398]
  The signaling dimension of a physical system is the minimum dimension of a classical channel. In 2015, Frenkel and Weiner showed that the signaling dimension of any quantum system is equal to its Hilbert space dimension.
  arXiv  Detail & Related papers  (2022-10-27T06:46:52Z)
• Tight Cram\'{e}r-Rao type bounds for multiparameter quantum metrology through conic programming [61.98670278625053]
  It is paramount to have practical measurement strategies that can estimate incompatible parameters with best precisions possible. Here, we give a concrete way to find uncorrelated measurement strategies with optimal precisions. We show numerically that there is a strict gap between the previous efficiently computable bounds and the ultimate precision bound.
  arXiv  Detail & Related papers  (2022-09-12T13:06:48Z)
• Relating measurement disturbance, information and orthogonality [0.38073142980732994]
  In the general theory of quantum measurement, one associates a positive semidefinite operator on a $d$-dimensional Hilbert space to each of the $n$ possible outcomes of an arbitrary measurement. This restriction allows us to more precisely state the quantum adage: information gain of a system is always accompanied by unavoidable disturbance. We identify symmetric informationally complete quantum measurements as the unique quantum analogs of a perfectly informative and nondisturbing classical ideal measurement.
  arXiv  Detail & Related papers  (2021-05-05T14:19:40Z)
• A Topological Approach to Inferring the Intrinsic Dimension of Convex Sensing Data [0.0]
  We consider a common measurement paradigm, where an unknown subset of an affine space is measured by unknown quasi- filtration functions. In this paper, we develop a method for inferring the dimension of the data under natural assumptions.
  arXiv  Detail & Related papers  (2020-07-07T05:35:23Z)
• Convex Geometry and Duality of Over-parameterized Neural Networks [70.15611146583068]
  We develop a convex analytic approach to analyze finite width two-layer ReLU networks. We show that an optimal solution to the regularized training problem can be characterized as extreme points of a convex set. In higher dimensions, we show that the training problem can be cast as a finite dimensional convex problem with infinitely many constraints.
  arXiv  Detail & Related papers  (2020-02-25T23:05:33Z)
• Quantum probes for universal gravity corrections [62.997667081978825]
  We review the concept of minimum length and show how it induces a perturbative term appearing in the Hamiltonian of any quantum system. We evaluate the Quantum Fisher Information in order to find the ultimate bounds to the precision of any estimation procedure. Our results show that quantum probes are convenient resources, providing potential enhancement in precision.
  arXiv  Detail & Related papers  (2020-02-13T19:35:07Z)
• Compact convex structure of measurements and its applications to simulability, incompatibility, and convex resource theory of continuous-outcome measurements [0.0]
  We define the measurement space $mathfrakM(E)$ as the set of post-processing equivalence classes of continuous measurements on $E. We show that the robustness measures of unsimulability and incompatibility coincide with the optimal ratio of the state discrimination probability of measurement(s) relative to that of simulable or compatible measurements.
  arXiv  Detail & Related papers  (2020-02-10T02:42:11Z)

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.