Noninvertible Symmetries Act Locally by Quantum Operations

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.