Quantifying coherence in infinite-dimensional systems

We study the quantification of coherence in infinite dimensio nal 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 infini te 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 nu mber, 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 co herence in a fixed basis by measuring the distance between the quantum state ˆ � and its nearest incoherent state. After the framework for coherence has been proposed, it receives increasing attentions. A. Streltsov et. al. have used entanglement to quantify quantum coherence, which provides the operational quantification of coherence [8]. S. Du et. al. focused on the interconversion of coherent states by means of incoherent operations using the concept of majorization relations [7]. Z. Xi et. al. have given a clear quantitative analysis and operational connections between relative entropy of coherence, quantum discord and one-way quantum deficit in the bipartite quantum system [6]. T. Bromley et. al. have found freezing conditions in which coherence remains unchanged during the nonunitary dynamics [5]. 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 q uantum optics states of light, it must require further investig ations on infinite dimensional systems. In this paper, we aim to investigate the quantification of coherence in infinite dimensional systems. Specificly, we fo cus 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 [12‐14]. We show that when considering the energy constraints, the relative en