Polytopic Quantum Resource Theories: Geometry and Structures

• URL: http://arxiv.org/abs/2606.00429v1
• Date: Fri, 29 May 2026 23:41:04 GMT
• Title: Polytopic Quantum Resource Theories: Geometry and Structures
• Abstract summary: Quantum resource theories provide a unifying framework to quantify, compare, manipulate quantum resources under well-defined operational constraints.<n>Here, we consider any resource theory where the set of free states can be expressed as a combination of a set of quantum states.<n>We formulate a novel representation of PQRTs that reveals the underlying geometry of these theories.
• Score: 0.09851812512860353
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Quantum resource theories provide a unifying framework to quantify, compare, and manipulate quantum resources under well-defined operational constraints. Here, we consider any resource theory where the set of free states can be expressed as a convex combination of a set of quantum states, referred to as extremal states and name them as polytopic quantum resource theories (PQRT). These include some of the most studied resource theories, such as coherence and magic. We formulate a novel tensorial representation of PQRTs that reveals the underlying geometry of these theories and provides insight into the origin of the resources. We further address a fundamental question in resource theories that when two theories should be regarded as physically equivalent, and to this purpose we introduce notions of homomorphism and isomorphism that compare both the structure of free states and the allowed transformations. Using the tools we develop, we find results revealing the geometrical and structural foundations of such theories. Interestingly, we find that all polytopic resource theories with a fixed number of pure extremal points are equivalent under a physical map, up to normalisation. Additionally, we introduce linearly independent polytopic resource theories (resource theory of ``basis-non-convexity''), where the set of extremal free states forms a basis of the quantum density operators. We further study the categorical structures of PQRTs beyond single systems.

Related papers

• Inverse Problem of Alchemical Resource Theory: Replication and Programming Single Out Imaginarity and Parity Asymmetry [0.0]
  We study the inverse problem of alchemical resource theories, which we define as resource theories admitting a resource that can both replicate itself exactly and universally program quantum instruments.<n>We show that for qubit single systems only two nontrivial universality theories survive, up to a common local unitary change of basis: parity asymmetry and imaginarity.
  arXiv  Detail & Related papers  (2026-09-11T15:52:01Z)
• Conceptual and Geometric Foundations for a Teleparallel Approach to Quantum Gravity [45.88028371034407]
  We revisit quantum field theory in curved spacetime as a semi-classical framework for quantum matter on classical geometries.<n>We outline a teleparallel framework based on coframe and spin-connection variables, where gravity is encoded in torsion rather than string curvature.
  arXiv  Detail & Related papers  (2026-06-08T15:02:26Z)
• Revisiting the Role of State Texture in Gate Identification and Fixed-Point Resource Theories [40.119993515374325]
  We revisit a protocol for identifying controlled-NOT gates versus single-qubit-only gates in quantum circuits.<n>We show that a more general fidelity-based formulation succeeds for nearly all laboratory bases.<n>We introduce a family of "fixed-point resource theories" that includes fixed-point instances of the theories of state texture, genuine coherence, purity, and athermality.
  arXiv  Detail & Related papers  (2026-02-26T00:22:58Z)
• Instrument-based quantum resources: quantification, hierarchies and towards constructing resource theories [6.2334188104823225]
  Quantum resource theories provide us with an elegant framework for studying these resources quantitatively and rigorously.<n>While numerous state-based quantum resource theories have already been investigated, instrument-based resource theories remain largely unexplored.<n>We study several instrument-based resource theories, namely (1) the resource theory of information preservability, (2) the resource theory of (strong) entanglement preservability, (3) the resource theory of (strong) incompatibility preservability, (4) the resource theory of traditional incompatibility, and (5) the resource theory of parallel incompatibility.
  arXiv  Detail & Related papers  (2025-08-12T17:58:30Z)
• Minimal operational theories: classical theories with quantum features [41.94295877935867]
  We introduce a class of probabilistic theories, where system dynamics are constrained to the minimal set of operations.<n>Specifically, the allowed instruments are limited to those derived from compositions of preparations, measurements, swap transformations, and conditional tests.<n>We demonstrate that minimal theories with conditioning and a spanning set of non-separable states satisfy two quantum no-go theorems.
  arXiv  Detail & Related papers  (2024-08-02T16:24:09Z)
• Is there a finite complete set of monotones in any quantum resource theory? [39.58317527488534]
  We show that there does not exist a finite set of resource monotones which completely determines all state transformations. We show that totally ordered theories allow for free transformations between all pure states.
  arXiv  Detail & Related papers  (2022-12-05T18:28:36Z)
• Investigating the Global Properties of a Resource Theory of Contextuality [0.0]
  We will investigate some properties of a resource theory for quantum contextuality. This characteristic ensures the impossibility of interpreting the results of quantum measurements as revealing properties that are independent of the set of measurements being made. We will use the so called cost and yield monotones, extending the results of reference Quantum 4, 280 ( 2020) to general contextuality scenarios.
  arXiv  Detail & Related papers  (2022-10-07T00:33:49Z)
• Stochastic approximate state conversion for entanglement and general quantum resource theories [41.94295877935867]
  An important problem in any quantum resource theory is to determine how quantum states can be converted into each other. Very few results have been presented on the intermediate regime between probabilistic and approximate transformations. We show that these bounds imply an upper bound on the rates for various classes of states under probabilistic transformations. We also show that the deterministic version of the single copy bounds can be applied for drawing limitations on the manipulation of quantum channels.
  arXiv  Detail & Related papers  (2021-11-24T17:29:43Z)
• An introductory review on resource theories of generalized nonclassical light [0.0]
  Quantum resource theory is perhaps the most revolutionary framework that quantum physics has ever experienced. Generalized quantum optical framework strives to bring in several prosperous contemporary ideas.
  arXiv  Detail & Related papers  (2021-03-23T05:10:44Z)
• Resource theory of imaginarity: Quantification and state conversion [48.7576911714538]
  Resource theory of imaginarity has been introduced, allowing for a systematic study of complex numbers in quantum mechanics and quantum information theory. We investigate imaginarity quantification, focusing on the geometric imaginarity and the robustness of imaginarity, and apply these tools to the state conversion problem in imaginarity theory. Our study reveals the significance of complex numbers in quantum physics, and proves that imaginarity is a resource in optical experiments.
  arXiv  Detail & Related papers  (2021-03-02T15:30:27Z)
• Quantum simulation of gauge theory via orbifold lattice [47.28069960496992]
  We propose a new framework for simulating $textU(k)$ Yang-Mills theory on a universal quantum computer. We discuss the application of our constructions to computing static properties and real-time dynamics of Yang-Mills theories.
  arXiv  Detail & Related papers  (2020-11-12T18:49:11Z)
• Topological Quantum Gravity of the Ricci Flow [62.997667081978825]
  We present a family of topological quantum gravity theories associated with the geometric theory of the Ricci flow. First, we use BRST quantization to construct a "primitive" topological Lifshitz-type theory for only the spatial metric. We extend the primitive theory by gauging foliation-preserving spacetime symmetries.
  arXiv  Detail & Related papers  (2020-10-29T06:15:30Z)
• Framework for resource quantification in infinite-dimensional general probabilistic theories [6.308539010172309]
  Resource theories provide a general framework for the characterization of properties of physical systems in quantum mechanics and beyond. We introduce methods for the quantification of resources in general probabilistic theories (GPTs) We show that a given resource state enables in channel discrimination tasks over all resourceless states. We demonstrate applications of the robustness to several resources of physical relevance: optical nonclassicality, entanglement, genuine non-Gaussianity, and coherence.
  arXiv  Detail & Related papers  (2020-09-23T18:00:20Z)

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.