The role of coherence in quantum thermodynamics has been extensively studied in the recent years and it is now well-understood that coherence between different energy eigenstates is a resource independent of other thermodynamics resources, such as work. A fundamental remaining open question is whether the laws of quantum mechanics and thermodynamics allow the existence of a coherence distillation machine, i.e., a machine that, by possibly consuming work, obtains pure coherent states from mixed states, at a nonzero rate. This is related to another fundamental question: Starting from many copies of noisy quantum clocks which are (approximately) synchronized with a reference clock, can one distill synchronized clocks in pure states, at a non-zero rate? Surprisingly, we find that the answer to both questions is negative for generic (full-rank) mixed states. However, at the same time, it is possible to distill a sub-linear number of pure coherent states with a vanishing error.
Detecting quantum coherence Observing quantum coherence is a tricky business. Even before we fire up the femtosecond laser systems and start our search, we need to ask ourselves some basic questions: What is coherence? How does coherence manifest in measurements? What methods can we use to detect and measure coherence? Only then, can we begin to interpret data to search for coherence and to try to make sense of the observed signals. In this section, I will discuss detecting quantum coherence both in theory and in practice. The goal of this section is to present a unified view of experimental approaches in broad strokes and to provide useful references to guide further exploration. What is ‘quantum coherence’? Firstly, the precise meaning of ‘quantum coherence’ must be pinned down. I define quantum coherences to be off-diagonal elements of the density matrix representing the ensemble. (I am explicitly setting aside questions of ‘quantum’ versus ‘classical’ coherences; let us simply assume that the system in question is best described quantum mechanically and presume that the coherences in such a system are also quantum in nature. The simple fact is that we don't yet have the tools to answer this debate definitively.) This definition, however, is still insufficient. The density matrix and the magnitudes of its off-diagonal elements in particular depend on the basis set used to write down the matrix. For the purposes of this chapter, I will consider the density matrix only in the Hamiltonian eigenbasis.
We comment on recent numerical studies concerning coupled 1D electron liquids (F. Mila and D. Poilblanc, Phys. Rev. Lett. 76, 287 (1996)). In particular, we point out that the importance of integrability observed in the results of these authors for the quantity
P(t) is really an indication of the importance of integrability for questions of ergodicity, rather than questions of coherence. We discuss why this is so, and why the quantity
<δN(t)> is a more appropriate function for determining whether or not interliquid hopping is coherent.