Quantum Coherence

More accurately, the resource theory of quantum coherence, aims to describe the phenomenon of quantum superposition, and clarify its connections with other sui generis quantum effects such as (in the case of several particles) entanglement. It is a particular (but possibly the most illustrative, along with entanglement) case of quantum resource theory:

A quantum resource theory is based on two main ingredients, the free states and the free operations. [...] The free states — states which do not possess any resource and are cheap to prepare — are the incoherent states, that is, quantum states which are diagonal in a fixed reference basis.

—Experimental Progress on Quantum Coherence: Detection, Quantification and Manipulation. K. Wu, A. Streltsov, B. Regula, G. Xiang, C. Li and G. Guo in Adv. Quantum Technol. 4:2100040 (2021).

coherence is basis dependent, so wherever we talk about coherence we must be clear which basis is presupposed.

—Quantifying coherence of Gaussian states. J. Xu in Phys. Rev. A 93:032111 (2016).

The most important paper is by Baumgratz et al. who study it in finite-dimensional Hilbert spaces, measuring coherence through the distance between the state and its closest incoherent state. The first work in that direction, however, is an unpublished arXiv preprint by Johan Aberg with a better terminology than "coherence" (namely, "superpositions"). Streltsov et al. used entanglement

A crime in terminology

From Streltsov et al.

From Wu et al.

From Killoran et al.

Interesting papers

• Quantum resource theories. E. Chitambar and G. Gour in Rev. Mod. Phys. 91:025001 (2019). (review)
• Quantifying Coherence. T. Baumgratz, M. Cramer and M. Plenio in Phys. Rev. Lett. 113:140401 (2014).
• How to quantify coherence: Distinguishing speakable and unspeakable notions. I. Marvian and R. W. Spekkens in Phys. Rev. A 94:052324 (2016).
• Quantum coherence as a resource. A. Streltsov, G. Adesso and M. B. Plenio in Rev. Mod. Phys. 89:041003 (2017). (review)
• Maximal coherence and the resource theory of purity. A. Streltsov, H. Kampermann, S. Wölk, M. Gessner and D. Bruß in New J. Phys. 20:053058 (2018). links quantum coherence to purity.
• Converting Nonclassicality into Entanglement. N. Killoran, F. Steinh and n. M. Plenio in Phys. Rev. Lett. 116:080402 (2016).

Continuous variables

The original theory was developped for finite-dimensional Hilbert spaces. There has been several attempts at extending it to infinite spaces:

• Quantifying coherence of Gaussian states. J. Xu in Phys. Rev. A 93:032111 (2016).
• Quantifying coherence in infinite-dimensional systems. Y. Zhang, L. Shao, Y. Li and H. Fan in Phys. Rev. A 93:012334 (2016).
• Quantifying the Coherence between Coherent States. K. C. Tan, T. Volkoff, H. Kwon and H. Jeong in Phys. Rev. Lett. 119:190405 (2017).
• Hierarchy of continuous-variable quantum resource theories. G. Gianfelici, H. Kampermann and D. Bruß in New J. Phys. 23:113008 (2021).