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Noninvertible symmetries in quantum systems act as quantum operations, extending beyond traditional group-based symmetries. This framework, demonstrated with the quantum Ising chain, reveals new insights into quantum information theory.

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Area of Science:

  • Theoretical Physics
  • Quantum Field Theory
  • Many-Body Systems
  • Quantum Information Theory

Background:

  • Traditional symmetries in quantum field theories and many-body systems are described by invertible operations within groups.
  • Generalizations of symmetries are being explored to include noninvertible operations.

Purpose of the Study:

  • To demonstrate that noninvertible symmetries act on local operators via quantum operations.
  • To highlight the role of these operations in quantum information theory.
  • To illustrate this concept using a concrete physical example.

Main Methods:

  • Conceptual framework extending group theory to noninvertible operations.
  • Characterization of symmetry actions as completely positive maps (quantum operations).
  • Application to the Kramers-Wannier duality of the 1D quantum Ising chain.

Main Results:

  • Noninvertible symmetries induce quantum operations (completely positive maps) on local operators.
  • These operations encompass both unitary evolutions and measurements.
  • The Kramers-Wannier duality serves as a key example of these noninvertible symmetry operations.

Conclusions:

  • Noninvertible symmetries provide a richer description of quantum systems than traditional symmetries.
  • The framework of quantum operations offers a powerful tool for understanding these generalized symmetries.
  • This work bridges concepts from quantum field theory, many-body physics, and quantum information.