Modes of asymmetry: The application of harmonic analysis to symmetric quantum dynamics and quantum reference frames

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 or 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.

Figure 1

Circuit diagrams depicting how an asymmetric state on one system can be used together with symmetric operations to simulate an asymmetric channel or an asymmetric measurement on another system.

Figure 2

Any linear time-invariant process transforms an input signal of frequency $\omega$ to an output signal of the same frequency. In other words, linearity together with time invariance implies that the process cannot change the frequency of the input. It follows that any linear time-invariant system can be uniquely specified by a complex function $f\left(\omega \right)$ specifying the change in amplitude and phase of the mode $\omega$. This explains why Fourier analysis is extremely useful for the study of these processes.