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The concept of the inertia tensor is employed to depict the mass distribution and rotational inertia of a solid or rigid object. This tensor is expressed through a three-by-three matrix. Each component within this matrix corresponds to varying moments of inertia about specific axes.
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The tensor as an informational resource.

Matthias Christandl1

  • 1Department of Mathematical Sciences, University of Copenhagen, Copenhagen 2100, Denmark.

PNAS Nexus
|October 11, 2024
PubMed
Summary
This summary is machine-generated.

We introduce new information-theoretic preorders to compare tensors, which are multidimensional arrays used in data, computation, and quantum information. These preorders help understand tensor transformations and generalize existing methods for analyzing computational complexity.

Keywords:
algebraic complexityquantum computationquantum informationtensortensor network

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Area of Science:

  • Mathematics
  • Computer Science
  • Quantum Information

Background:

  • Tensors are multidimensional arrays crucial for data storage, computational relations, and quantum entanglement.
  • Understanding tensor transformations is key to deciphering data structures, computational complexity, and quantum information.

Purpose of the Study:

  • To propose a novel family of information-theoretically constructed preorders on tensors.
  • To enable comparison and assessment of transformations between tensors.

Main Methods:

  • Constructing preorders by placing tensor copies on hypergraph edges and allowing vertex transformations.
  • Inducing preorders from possible transformations within a sequence of hypergraphs.
  • Generalizing Strassen's asymptotic restriction preorder.

Main Results:

  • Derived general properties of the proposed preorders.
  • Introduced associated asymptotic notions of tensor rank.
  • Unified recent results on tensor rank nonadditivity, tensor networks, and algebraic complexity.

Conclusions:

  • The new preorders offer a unifying framework for studying tensors.
  • This work provides a valuable perspective for applied mathematics, physics, and computer science, as well as pure mathematics.