We investigate the coherence measures induced by fidelity and trace norm, based on the coherence quantification recently proposed by Baumgratz et al. [T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014)]. We show that the fidelity of coherence does not in general satisfy the monotonicity requirement as a measure of coherence under the subselection of the measurement condition. We find that the trace norm of coherence can act as a measure of coherence for qubits and some special class of qutrits with some restrictions on the incoherent operators, while the general case needs to be explored further.

We study the quantification of coherence in infinite dimensional systems, especially the infinite dimensional bosonic systems in Fock space. We show that given the energy constraints, the relative entropy of coherence serves as a well-defined quantification of coherence in infinite dimensional systems. Via using the relative entropy of coherence, we also generalize the problem to multi-mode Fock space and special examples are considered. It is shown that with a finite average particle number, increasing the number of modes of light can enhance the relative entropy of coherence. With the mean energy constraint, our results can also be extended to other infinite-dimensional systems.Quantum coherence arising from quantum superposition principle is a fundamental aspect of quantum physics [1]. The laser [2] and superfluidity [3] are examples of quantum coherence, whose effects are evident at the macroscopic scale. However, the framework of quantification of coherence has only been methodically investigated recently. The first attempt to address the classification of quantum coherence as physical resources by T. Baumgratz et. al., who have established a rigorous framework for the quantification of coherence based on distance measures in finite dimensional setting [4]. With such a fundational framework for coherence, one can find the appropriate distance measures to quantify coherence in a fixed basis by measuring the distance between the quantum stateρ and its nearest incoherent state. After the framework for coherence has been proposed, it receives increasing attentions. Up to now, all the results for quantifying the quantum coherence are assumed the finite dimensional setting, which is neither necessary nor desirable. In consideration of the relevant physical situations such as quantum optics states of light, it must require further investigations on infinite dimensional systems.In this paper, we aim to investigate the quantification of coherence in infinite dimensional systems. Specificly, we focus on the infinite dimensional bosonic systems in Fock space [10] which are used to describe the most notable quantum optics states of light [11] and Gaussian states [12][13][14]. We show that when considering the energy constraints, the relative en- * Electronic address: liyongm@snnu.edu.cn † Electronic address: hfan@iphy.ac.cn tropy of coherence serves as a well-defined quantification of coherence in infinite dimensional systems and the l 1 norm of coherence fails. Via using the relative entropy of coherence, we also generalize the results to multi-mode Fock space and special examples are considered. It is shown that with a finite average particle number, increasing the number of modes of light can enhance the relative entropy of coherence. Our results can also be extended to other infinite-dimensional systems with energy constraints. Our work investigates special and experimentally relevant cases and the most general and easy to use quantifiers, which is significant and essential in quantum physics as well as quantum opt...