  Finding the consequences of symmetry for open system quantum dynamics is a
problem with broad applications, including describing thermal relaxation,
deriving quantum limits on the performance of amplifiers, and exploring quantum
metrology in the presence of noise. The symmetry of the dynamics may reflect a
symmetry of the fundamental laws of nature, a symmetry of a low-energy
effective theory, or it may describe a practical restriction such as the lack
of a reference frame. In this paper, we apply some tools of harmonic analysis
together with ideas from quantum information theory to this problem. The
central idea is to study the decomposition of quantum operations---in
particular, states, measurements and channels---into different modes, which we
call modes of asymmetry. Under symmetric processing, a given mode of the input
is mapped to the corresponding mode of the output, implying that one can only
generate a given output if the input contains all of the necessary modes. By
defining monotones that quantify the asymmetry in a particular mode, we also
derive quantitative constraints on the resources of asymmetry that are required
to simulate a given asymmetric operation. We present applications of our
results for deriving bounds on the probability of success in nondeterministic
state transitions, such as quantum amplification, and a simplified formalism
for studying the degradation of quantum reference frames.
