In particular, when ρ is the maximally mixed state, then no coherence exists in the quantum system. This shows that the sum of the entropy of the quantum system and the amount of the coherence of quantum system is always smaller than a given fixed value: the larger S(ρ), the smaller C RE(ρ). For any quantum state ρ on the Hilbert space, the relative entropy of coherence 1 is defined as In the following, we only consider the measure of relative entropy of coherence. Recently, we also find that the measure of coherence induced by the fidelity does not satisfy condition (C2b) and an explicit example is presented 13. However, the measure of coherence induced by the squared Hilbert-Schmidt norm satisfies conditions (C1), (C2a), (C3), but not (C2b). It has been shown that the relative entropy and l 1-norm satisfy all conditions. We know that the condition (C2b) is important as it allows for sub-selection based on measurement outcomes, a process available in well controlled quantum experiments 1. Note that conditions (C2b) and (C3) automatically imply condition (C2a). (C2b) Monotonicity for average coherence under subselection based on measurements outcomes: C(ρ) ≥ ∑ n p n C(ρ n), where and for all. (C2a) Monotonicity under incoherent completely positive and trace preserving maps (ICPTP) Φ, i.e., C(ρ) ≥ C(Φ(ρ)). (C1) C(ρ) ≥ 0 for all quantum states ρ and C(ρ) = 0 if and only if. proposed that any proper measure of the coherence C must satisfy the following conditions: In this way, we can give a clear quantitative relation between the discord and the deficit. These two facts are the reasons that we study in detail the explicit expressions of the discord and the deficit in terms of the relative entropy of coherence in the bipartite quantum system. Since the incoherent states under two different bases are unitarily equivalent, then there are same matrix elements under the different bases for given quantum state. In the bipartite quantum system, based on the projective measurement in which the relative entropy of coherence is quantified, we obtain that the increased entropy produced by the local projective measurement is equal to the sum between the quantum correlation destroyed by this measurement and the relative entropy of coherence of the measured subsystem. Meanwhile, we find that the relative entropy of coherence satisfies the super-additivity. As an application, we discuss the relations between the entanglement and the coherence. Firstly, we derive an uncertainty-like expression which states that the sum of the coherence and the entropy in quantum system is bounded from the above by log 2 d, where d is the dimension of the quantum system. We only focus on particular the entropic form, also called relative entropy of coherence, which enjoys the properties of physical interpretation and being easily computable 1. In the present work, we will resolve the above questions via quantum coherence. In other word, is there a more clear quantitative relations between them? Curiously, up to now, no attempt for a transformed framework between them has been reported. If only one-way classical communication from one party to another is allowed, they showed that the one-way quantum deficit is an upper bound of quantum discord via the local von Neumman measurements on the party. 31 discussed the relationship between the discord and quantum deficit in the bipartite quantum system. There have been much interest in characterizing and interpreting their applications in quantum information processing 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30. Quantum discord 15, 16 and quantum deficit 17 have been viewed as two possible quantifiers for quantum correlations. On the other hand, it is well known that entanglement does not account for all nonclassical correlations (or quantum correlations) and that even correlation of separable state does not completely be classical. We know that quantum coherence and the entanglement are related to quantum superposition, but we are not sure of the exact relations between quantum coherence and the entanglement, is there a quantitative relation between the two of them? Quantum coherence has received a lot of attentions 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. Within such a framework for the coherence, one can define suitable measures, include the relative entropy and the l 1- norm of coherence 1 and a measure by the Wigner-Yanase-Dyson skew information 3. Recently, a rigorous framework to quantify coherence has been proposed 1 (or see early work 2). Quantum coherence is a common necessary condition for both entanglement and other types of quantum correlations and it is also an important physical resource in quantum computation and quantum information processing. Quantum coherence arising from quantum superposition plays a central role in quantum mechanics.
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