The distribution of coherence in multipartite systems is examined. We use a new coherence measure with entropic nature and metric properties, based on the quantum Jensen-Shannon divergence. The metric property allows for the coherence to be decomposed into various contributions, which arise from local and intrinsic coherences. We find that there are trade-off relations between the various contributions of coherence, as a function of parameters of the quantum state. In bipartite systems the coherence resides on individual sites or distributed among the sites, which contribute in a complementary way. In more complex systems, the characteristics of the coherence can display more subtle changes with respect to the parameters of the quantum state. In the case of the XXZ Heisenberg model, the coherence changes from a monogamous to a polygamous nature. This allows us to define the shareability of coherence, leading to monogamy relations for coherence. The concept of wave particle duality introduced the importance of quantum coherence in physical phenomena such as low temperature thermodynamics [1], quantum thermodynamics [2-4], nanoscale physics [5], biological systems [6,7], and is one of the most basic aspects of quantum information science [8]. For this reason, understanding quantum coherence has a long history and is of fundamental importance to many fields. In quantum optics [9,10], the approach has been typically to examine quantities such as phase space distributions and higher order correlation functions [11]. While this method distinguishes between quantum and classical coherence, it does not quantify coherence in a rigorous sense. More recently, a procedure to quantify coherence using methods of quantum information science was developed [12][13][14][15]. In the seminal work of Ref. [12], basic quantities such as incoherent states, incoherent operations, maximally coherent states were defined and the set of properties a functional should satisfy to be considered as a coherence measure were listed.One fundamental task that is desirable is to pinpoint what part of a quantum system is responsible for any coherence that is present. To understand the possibilities, let us consider a two qubit system as an example. Coherence is a basis-dependent quantity [15,16], and the reference incoherent states are chosen as |0 , |1 . We can consider then two types of states which possess coher- S is the set of separable states in a fixed basis b. ρ d is the solution of (2) and σ min S is the solution of (4).ence, (|0 − |1 )(|0 − |1 ) and |0 |0 − |1 |1 . In the former, the coherence lies on each qubit, while the latter has a kind of collective coherence, i.e. entanglement. An interesting aspect of this is that the types of coherence are complementary to each other -an increase in one type leads to a corresponding decrease in the other. In order to have maximum coherence on a particular qubit, it is optimal to create a superposition on each one, which excludes entanglement. On the other hand, for the Bell state, tracing out one o...

We study quantum coherence in a spin chain with both symmetric exchange and antisymmetric Dzyaloshinsky-Moriya couplings. Quantum coherence is quantified using the recently introduced quantum Jensen-Shannon divergence, which has the property that it is easily calculable and has several desirable mathematical properties. We calculate exactly the coherence for arbitrary number of spins at zero temperature in various limiting cases. The σ z σ z interaction tunes the amount of coherence in the system, and the antisymmetric coupling changes the nature of the coherence. We also investigate the effect of non-zero temperature by looking at a two-spin system and find similar behavior, with temperature dampening the coherence. The characteristic behavior of coherence resembles that of entanglement and is opposite to that of discord. The distribution of the coherence on the spins is investigated and found that it arises entirely due to the correlations between the spins.Quantum coherence is one of the central concepts in quantum physics, and hence its detection and quantification is a fundamental task. Traditionally, the distinction between quantum and classical coherence is made using phase space distributions 1,2 and higher order correlation functions 3 . While this gives some insight into the nature of coherence, these techniques do not quantify coherence in a rigorous sense. Recently a scheme for measuring coherence was developed by Baumgratz and co-workers in ref. 4 based on the framework of quantum information theory. Fundamental quantities such as incoherent states, maximally coherent states and incoherent operations which are needed for the development of the procedure were introduced and analyzed. Rapid developments have been made in understanding the theory of quantum coherence through numerous works 5-16 and in using it as a resource in quantum information theory [17][18][19][20][21][22][23] . Investigations on the role of quantum coherence in thermodynamic processes [24][25][26] , assisted subspace discrimination 27 , quantum state merging 28 and in the generation of gaussian entanglement 29 have been carried out. Currently there is a lot of interest in applying the procedure of quantifying coherence and relating them to experimental quantities in feasible systems like Bose-Einstein condensates 30,31 , cavity optomechanical system 32,33 and spin systems [34][35][36][37][38][39] .In ref. 4 the set of properties a functional should satisfy in order to be considered as a coherence measure were proposed. Based on these developments several functions were introduced to serve as measures of coherence 4,40-47 . All these measures of coherence can be broadly classified into either the entropic or the geometric class of measures. This depends on whether the measure is based on the entropy functional or has a metric nature which can give rise to a geometric structure. A mathematical functional is deemed to be a metric if it obeys the axioms of distance and satisfies the triangle inequality. In ref. 47 we introduced a new...

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A two-parameter generalization of Boltzmann-Gibbs-Shannon entropy based on natural logarithm is introduced. The generalization of the Shannon-Khinchin axioms corresponding to the two-parameter entropy is proposed and verified. We present the relative entropy, Jensen-Shannon divergence measure and check their properties. The Fisher information measure, the relative Fisher information, and the Jensen-Fisher information corresponding to this entropy are also derived. Also the Lesche stability and the thermodynamic stability conditions are verified. We propose a generalization of a complexity measure and apply it to a two-level system and a system obeying exponential distribution. Using different distance measures we define the statistical complexity and analyze it for two-level and five-level system.