How to quantify coherence: Distinguishing speakable and unspeakable notions

Abstract

Quantum coherence is a critical resource for many operational tasks. Understanding how to quantify and manipulate it also promises to have applications for a diverse set of problems in theoretical physics. For certain applications, however, one requires coherence between the eigenspaces of specific physical observables, such as energy, angular momentum, or photon number, and it makes a difference which eigenspaces appear in the superposition. For others, there is a preferred set of subspaces relative to which coherence is deemed a resource, but it is irrelevant which of the subspaces appear in the superposition. We term these two types of coherence unspeakable and speakable, respectively. We argue that a useful approach to quantifying and characterizing unspeakable coherence is provided by the resource theory of asymmetry when the symmetry group is a group of translations, and we translate a number of prior results on asymmetry into the language of coherence. We also highlight some of the applications of this approach, for instance, in the context of quantum metrology, quantum speed limits, quantum thermodynamics, and nuclear magnetic resonance (NMR). The question of how best to treat speakable coherence as a resource is also considered. We review a popular approach in terms of operations that preserve the set of incoherent states, propose an alternative approach in terms of operations that are covariant under dephasing, and we outline the challenge of providing a physical justification for either approach. Finally, we note some mathematical connections that hold among the different approaches to quantifying coherence.

Preservation of quantum coherence under Lorentz boost for narrow uncertainty wave packets

“…Recently, resource theory of asymmetry or reference frame has emerged which treats individual formulations of coherence measures as special cases [18][19][20][21][22][23][24]. It has been noted that both operational and geometric perspectives are in general significant for resource theory [15]- [17], [28], [45].…”

“…In the frame of O Λ moving with velocity v = tanh αẑ, the state of the particle is given by 24) with the axis of rotation being along the directionẑ ×x = y. Substituting Eq.…”

“…Defined in a quantitative manner based on the framework of resource theory [16], [17], [18][19][20][21][22][23][24] quantum coherence may be exploited to perform quantum tasks. Several operational measures of quantum coherence have been proposed [25], [26], enabling it to be used for detection of genuine non-classicality in physical states.…”

Witnessing Multipartite Entanglement by Detecting Asymmetry

“…The lower bound O H is the first faithful experimental quantifier of asymmetry for finite-dimensional systems (a second one appeared in [63]). Thus, on the experimental side, we call for a demonstration of our study.…”

Coherence, asymmetry, and quantum macroscopicity

“…Recently, there have been attempts to interpret quantum properties as a resource. Resource theories based on the notion of free operations may provide useful tools to quantify nonclassical phenomena, covering a diverse range of topics that include entanglement [12,13], asymmetry [14][15][16][17][18][19][20], and quantum thermodynamics [21][22][23][24][25], all of which have been studied within this framework.…”