Extending Noether's theorem by quantifying the asymmetry of quantum states

Understand

Noether's theorem is a fundamental result in physics stating that every symmetry of the dynamics implies a conservation law.

• It is, however, deficient in several respects: (i) it is not applicable to dynamics wherein the system interacts with an environment, and (ii) even in the case where the system is isolated, if the quantum state is mixed then the Noether conservation laws do not capture all of the consequences of the symmetries.
• To address these deficiencies, we introduce measures of the extent to which a quantum state breaks a symmetry.
• Such measures yield novel constraints on state transitions: for nonisolated systems, they cannot increase, while for isolated systems they are conserved.