Theoretical physicists from Kavli IPMU have demonstrated that any non-invertible symmetry operation in theoretical physics is a quantum operation. This establishes a close relationship between the two rapidly evolving fields of theoretical physics: quantum information theory and non-invertible symmetries in particle and condensed matter theories. Their findings were published in Physical Review Letters on November 6th, 2024.
A summary of this study showing that any operation of non-invertible symmetries is a quantum operation. Image Credit: Kavli IPMU
In physics, symmetry can provide vital information about a theory’s features. For example, if S-poles replace the N-poles in a magnetic field, and the S-poles by N-poles all at once, the forces on objects and energy stored in the magnetic field stay constant, despite the fact that the magnetic field's direction has changed. This is because the magnetic field equations are symmetric with regard to the process of swapping the N and S poles.
In recent years, the concept of symmetry has been generalized in numerous directions in the theoretical study of particle physics and condensed matter physics, becoming an important research topic. One such generalization is non-invertible symmetry. The operation of conventional symmetries is always invertible. There is a reverse operation to undo it. Non-invertible symmetry, on the other hand, permits some non-invertibility in certain symmetry procedures.
In the past few years, quantum information theory has also drawn increasing interest from physicists. Quantum computers are based on this hypothesis.
A core idea in quantum processing involves executing various operations on memory known as quantum bits or qubits. Any undoable operation is represented by a mathematical operation known as a unitary transformation.
Non-invertible operations with no reverse action, such as measuring quantum bits, are equally significant. These operations utilize a more generalized unitary transformation idea known as a quantum operation.
Several years ago, mathematician Marcel Bischoff and his coworkers proposed that the operation of non-invertible symmetries is quantum. However, their notion was presented in a framework only relevant to physical systems with specific attributes and is unfamiliar to most of the physics community.
Now, Masaki Okada, a graduate student at the University of Tokyo School of Science, and Yuji Tachikawa, a professor at the University of Tokyo Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), have demonstrated that this concept can be effectively proved in a framework commonly utilized in particle physics and condensed matter physics.
Until now, no one understood the universal property of non-invertible symmetry operations. Okada and Tachikawa's work supplied an answer: any operation involving non-invertible symmetries is a quantum operation.
Journal Reference:
Okada, M. and Tachikawa, Y. (2024) Noninvertible Symmetries Act Locally by Quantum Operations. Physical Review Letters. doi.org/10.1103/PhysRevLett.133.191602