# A no-broadcasting theorem for quantum asymmetry and coherence and a trade-off relation for approximate broadcasting

## TL;DR

This paper proves a quantum-specific no-broadcasting theorem for asymmetry and coherence, showing that symmetric dynamics impose fundamental constraints on quantum systems, with implications for quantum reference frames and thermodynamics.

## Contribution

It introduces a no-go theorem for broadcasting asymmetry in quantum systems and establishes a quantum information-disturbance tradeoff for symmetric operations.

## Key findings

- Impossibility of broadcasting asymmetry in bounded quantum systems
- Quantitative tradeoff relation for asymmetry distribution
- Asymmetry measures violate sub-additivity and super-additivity

## Abstract

Symmetries of both closed and open-system dynamics imply many significant constraints. These generally have instantiations in both classical and quantum dynamics (Noether's theorem, for instance, applies to both sorts of dynamics). We here provide an example of such a constraint which has no counterpart for a classical system, that is, a uniquely quantum consequence of symmetric dynamics. Specifically, we demonstrate the impossibility of broadcasting asymmetry (symmetry-breaking) relative to a continuous symmetry group, for bounded-size quantum systems. The no-go theorem states that if two initially uncorrelated systems interact by symmetric dynamics and asymmetry is created at one subsystem, then the asymmetry of the other subsystem must be reduced. We also find a quantitative relation describing the tradeoff between the subsystems. These results cannot be understood in terms of additivity of asymmetry, because, as we show here, any faithful measure of asymmetry violates both sub-additivity and super-additivity. Rather, they must be understood as a consequence of an (intrinsically quantum) information-disturbance principle. Our result also implies that if a bounded-size quantum reference frame for the symmetry group, or equivalently, a bounded-size reservoir of coherence (e.g., a clock with coherence between energy eigenstates in quantum thermodynamics) is used to implement any operation that is not symmetric, then the quantum state of the frame/reservoir is necessarily disturbed in an irreversible fashion, i.e., degraded.