# Robustness of Noether's principle: Maximal disconnects between conservation laws and symmetries in quantum theory

## TL;DR

This paper explores the limits of Noether's principle in quantum channels, showing that symmetry constraints often do not imply strict conservation laws and revealing fundamental bounds on quantum transformations related to symmetries.

## Contribution

It introduces a quantitative framework for understanding how symmetry constraints in quantum channels relate to conservation laws and identifies fundamental limits on symmetric quantum operations.

## Key findings

- Bounds on deviation from conservation laws under symmetric channels
- Limits on spin polarization and inversion using symmetric operations
- Connections between unitarity, complementary channels, and purity

## Abstract

To what extent does Noether's principle apply to quantum channels? Here, we quantify the degree to which imposing a symmetry constraint on quantum channels implies a conservation law, and show that this relates to physically impossible transformations in quantum theory, such as time-reversal and spin-inversion. In this analysis, the convex structure and extremal points of the set of quantum channels symmetric under the action of a Lie group $G$ becomes essential. It allows us to derive bounds on the deviation from conservation laws under any symmetric quantum channel in terms of the deviation from closed dynamics as measured by the unitarity of the channel. In particular, we investigate in detail the $U(1)$ and $SU(2)$ symmetries related to energy and angular momentum conservation laws. In the latter case, we provide fundamental limits on how much a spin-$j_A$ system can be used to polarise a larger spin-$j_B$ system, and on how much one can invert spin polarisation using a rotationally-symmetric operation. Finally, we also establish novel links between unitarity, complementary channels and purity that are of independent interest.